tag:blogger.com,1999:blog-16285396224021451322024-03-13T09:09:24.728-07:00OnlineMathProMath Help & Silly SongsMrWAGNEEZYhttp://www.blogger.com/profile/14636358345686928129noreply@blogger.comBlogger15125tag:blogger.com,1999:blog-1628539622402145132.post-28350343291564161092013-08-19T11:53:00.000-07:002013-08-19T11:53:57.759-07:00Math Study<div class="separator" style="clear: both; text-align: center;">
<a href="http://youtu.be/-T-0N7X8nMU" target="_blank"><img border="0" height="96" src="http://3.bp.blogspot.com/--HWrJwA59e4/UhJoqNxr2jI/AAAAAAAAAHg/ex_aqBfSj-s/s320/MathStudy.JPG" width="320" /></a></div>
<br />
Summer vacation is waning fast, and it's time to get back to school.<br />
<br />
For my first installment of the new school year, I'd like to talk a little bit about an aspect of test preparation that many people do not take seriously enough: Sleep.<br />
<br />
I know that some students take a very last-minute approach to preparing for tests (or finishing projects, or any number of other school-related activities). I have been in this position myself, and it is quite stressful.<br />
<br />
As {insert name of deadline here} approaches, people discover that they can't fit all of the things they should be doing into the amount of time that is available to them. All too often, the thing that gets discarded is sleep.<br />
<br />
I'm not going to go into the health issues associated with sleep deprivation... needless to say, you'll be healthier if you get enough sleep. Also, there are the obvious safety concerns... sleep-deprived drivers can be just as dangerous as drunk drivers.<br />
<br />
My goal here is to draw attention to the academic ramifications of missing out on sleep.<br />
<br />
Personally, I think that students need to consider the type of academic activity they are preparing for when deciding how much sleep to cut out of their schedule. Students need to understand that skipping sleep <em>will</em> affect their performance. The goal is to find an acceptable trade-off between needed preparation and lower performance quality.<br />
<br />
Some activities do not require much performance. For example, if a big project is due then the student only needs to be able to stumble into the classroom and turn the project in. Students can afford to miss more sleep in a situation like this.<br />
<br />
However, if the student will be required to present his/her project to the class (and will be graded on the presentation) then more sleep is advised. It doesn't make much sense to stay up all night to enhance the quality of a project if the presentation ends up being terrible.<br />
<br />
When preparing for a test, there is typically a point at which the loss of brain power due to sleep deprivation outweighs the benefits of more cramming. Continuing to stay awake beyond this point does more damage than good.<br />
<br />
This gets a bit tricky because different people need different amounts of sleep. Therefore, I can't possibly prescribe a last-minute cramming strategy that will work for everyone...<br />
<br />
...and I guess that's the real point of this post: If you plan ahead and don't procrastinate, then you won't have to worry about staying up and losing a significant amount of sleep the night before a big presentation or exam.<br />
<br />
My latest Silly Math Song was inspired by this concept:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://youtu.be/-T-0N7X8nMU" target="_blank"><img border="0" height="96" src="http://1.bp.blogspot.com/--HWrJwA59e4/UhJoqNxr2jI/AAAAAAAAAHc/ZPhNwrLZqu0/s320/MathStudy.JPG" width="320" /></a></div>
<br />
If you are a student, parent, or teacher, I hope you have a wonderful school year.<br />
<br />
-------------------------------<br />
For math tutorials and silly math songs, visit <a href="http://www.onlinemathpro.com/">www.onlinemathpro.com</a><br />
MrWAGNEEZYhttp://www.blogger.com/profile/14636358345686928129noreply@blogger.com1tag:blogger.com,1999:blog-1628539622402145132.post-59863720382343466402013-05-10T12:25:00.000-07:002013-05-10T12:25:45.289-07:00Deconstructing Chocolate Math
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Every
year, mass emails go out challenging people to calculate their age using “Chocolate
Math”.<span style="mso-spacerun: yes;"> </span>In a minute, I’ll deconstruct the
2013 version of this fun math activity – but first, try out this (simpler)
activity:<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">I’m
going to read your mind, with the help of a little mathematical trickery.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpFirst" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -0.25in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">1.<span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";"> </span></span></span><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Think
of a number – any number will work, but it’s easiest if you choose a smaller number
(between 1 and 10).<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -0.25in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">2.<span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";"> </span></span></span><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Add
7.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -0.25in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">3.<span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";"> </span></span></span><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Take
the result and add your original number.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -0.25in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">4.<span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";"> </span></span></span><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Take
the result and add 9.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -0.25in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">5.<span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";"> </span></span></span><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Take
the result and divide it by 2.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -0.25in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">6.<span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";"> </span></span></span><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Take
the result and subtract your <i style="mso-bidi-font-style: normal;">original</i>
number.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpLast" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">The number you are now thinking of is… … … <b style="mso-bidi-font-weight: normal;">8.</b><o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">This
result always holds (assuming you do the math correctly), regardless of your
choice of starting number.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Do
you know why?<span style="mso-spacerun: yes;"> </span>(If not, don’t worry.<span style="mso-spacerun: yes;"> </span>I’ll give the explanation a little later.)<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Now,
let’s take a look at Calculating your Age by using Chocolate Math – 2013 Version.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Follow
these steps to use chocolate math to calculate your age:<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpFirst" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l1 level1 lfo2; text-indent: -0.25in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">1.<span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";"> </span></span></span><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Pick
the number of times a week that you would like to have chocolate.<span style="mso-spacerun: yes;"> </span>(More than once, but less than 10 times.)<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l1 level1 lfo2; text-indent: -0.25in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">2.<span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";"> </span></span></span><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Multiply
this number by 2<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l1 level1 lfo2; text-indent: -0.25in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">3.<span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";"> </span></span></span><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Add
5<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l1 level1 lfo2; text-indent: -0.25in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">4.<span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";"> </span></span></span><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Multiply
the result by 50 (followed by some joke about getting a calculator)<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l1 level1 lfo2; text-indent: -0.25in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">5.<span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";"> </span></span></span><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">If
you have already had your birthday this year, add 1763.<span style="mso-spacerun: yes;"> </span>If not, add 1762.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l1 level1 lfo2; text-indent: -0.25in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">6.<span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";"> </span></span></span><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Subtract
the four-digit year that you were born.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpLast" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">You now have a three-digit number.<span style="mso-spacerun: yes;"> </span>The first digit is your original number (i.e.
how many times you want to have chocolate each week).<span style="mso-spacerun: yes;"> </span>The next two numbers equal… YOUR AGE!<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">To
discover why this works (and why this version will only work in the year 2013),
we’ll need to make use of some basic Algebra skills.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Every
step of this activity is dictated to you, EXCEPT the first step… there is no
way to know how many times you will decide you want chocolate each week.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">So
– let’s define a variable.<span style="mso-spacerun: yes;"> </span>Let<span style="mso-spacerun: yes;"> </span><i style="mso-bidi-font-style: normal;">x</i><span style="mso-spacerun: yes;"> </span>represent the number of times you want
chocolate each week.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">By
writing algebraic expressions for each of the following steps, we can discover
what is happening.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Pick
a number:<span style="mso-tab-count: 2;"> </span><span style="mso-tab-count: 1;"> </span><i style="mso-bidi-font-style: normal;">x</i><o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Multiply
by 2<span style="mso-tab-count: 3;"> </span><span style="mso-tab-count: 1;"> </span>2<i style="mso-bidi-font-style: normal;">x<o:p></o:p></i></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Add
5<span style="mso-tab-count: 4;"> </span><span style="mso-tab-count: 1;"> </span>2<i style="mso-bidi-font-style: normal;">x</i>
+ 5<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Multiply
by 50<span style="mso-tab-count: 2;"> </span><span style="mso-tab-count: 1;"> </span>50(2<i style="mso-bidi-font-style: normal;">x</i> + 5) = 100<i style="mso-bidi-font-style: normal;">x</i> + 250<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">At
this point, you have a choice based on whether or not you’ve had your birthday
this year.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">If
yes, add 1763<span style="mso-tab-count: 2;"> </span><span style="mso-tab-count: 1;"> </span>100<i style="mso-bidi-font-style: normal;">x</i> + 250 + 1763 = 100<i style="mso-bidi-font-style: normal;">x</i> +
2013<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">If
no, add 1762<span style="mso-tab-count: 2;"> </span><span style="mso-tab-count: 1;"> </span>100<i style="mso-bidi-font-style: normal;">x</i> + 250 + 1762 = 100<i style="mso-bidi-font-style: normal;">x</i> +
2012<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">The
expression now contains 100<i style="mso-bidi-font-style: normal;">x</i> plus the
last year you had a birthday.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">By
subtracting your birth year, you will end up with 100<i style="mso-bidi-font-style: normal;">x</i> + your age.<span style="mso-spacerun: yes;"> </span>Therefore, the
first digit of your three-digit number will be the number of times you want
chocolate each week, and the last two digits will be your age.<o:p></o:p></span></div>
<br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">(By
the way, this activity doesn’t work for people who are more than 99 years old…
how discriminatory!!!)<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Here’s
an interesting thought – there are many ways to force an algebraic expression
of either 100<i style="mso-bidi-font-style: normal;">x</i> + 2013<span style="mso-spacerun: yes;"> </span>or<span style="mso-spacerun: yes;"> </span>100<i style="mso-bidi-font-style: normal;">x</i> + 2012.<span style="mso-spacerun: yes;"> </span>This means that, if you were so inclined, you
could make up your own set of steps that would calculate someone’s age using
Chocolate Math.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Now
– if you understand the math behind the Chocolate Math activity, you can
probably follow the steps in the initial activity to discover why those steps
will <i style="mso-bidi-font-style: normal;">always</i> yield a result of <b style="mso-bidi-font-weight: normal;">8</b>.<span style="mso-spacerun: yes;">
</span>If you don’t want to work through the process on your own, here it is:<o:p></o:p></span></div>
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">.<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">.<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">.<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">.<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">.<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">.<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">.<o:p></o:p></span></b><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Think
of a number<span style="mso-tab-count: 2;"> </span><span style="mso-tab-count: 1;"> </span><i style="mso-bidi-font-style: normal;">x<o:p></o:p></i></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Add
7<span style="mso-tab-count: 4;"> </span><span style="mso-tab-count: 1;"> </span><i style="mso-bidi-font-style: normal;">x</i>
+ 7<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Add
your original number<span style="mso-tab-count: 1;"> </span><span style="mso-tab-count: 1;"> </span><i style="mso-bidi-font-style: normal;">x</i>
+ 7 + <i style="mso-bidi-font-style: normal;">x</i> = 2<i style="mso-bidi-font-style: normal;">x</i> + 7<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Add
9<span style="mso-tab-count: 4;"> </span><span style="mso-tab-count: 1;"> </span>2<i style="mso-bidi-font-style: normal;">x</i>
+ 7 + 9 = 2<i style="mso-bidi-font-style: normal;">x</i> + 16<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Divide
by 2<span style="mso-tab-count: 3;"> </span><span style="mso-tab-count: 1;"> </span>(2<i style="mso-bidi-font-style: normal;">x</i> + 16) ÷ 2 = <i style="mso-bidi-font-style: normal;">x</i> + 8<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Subtract
your original number<span style="mso-tab-count: 1;"> </span><i style="mso-bidi-font-style: normal;">x</i> + 8 – <i style="mso-bidi-font-style: normal;">x</i> = <b style="mso-bidi-font-weight: normal;">8</b><o:p></o:p></span></div>
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><o:p> </o:p></span><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">With
some basic algebra skills, you too can read minds!</span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">----------------------------------------------</span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">For math tutorials and Silly Math Songs, visit: <a href="http://www.onlinemathpro.com/">www.onlinemathpro.com</a></span></div>
MrWAGNEEZYhttp://www.blogger.com/profile/14636358345686928129noreply@blogger.com1tag:blogger.com,1999:blog-1628539622402145132.post-70918237236692979622013-03-28T07:15:00.000-07:002013-03-28T07:15:13.773-07:00Factoring SongI've got an NBA-themed article in the works, but decided to postpone finishing it in order to work on my latest Silly Math Song.<br />
<br />
<strong>FACTORING SONG: FACTOR GANGNAM STYLE </strong>teaches the basics of factoring, as well as reducing fractions that contain polynomials.<br />
<br />
The information is pretty rapid-fire as the song covers all of these types of factoring:<br />
<br />
Factoring out the GCF<br />
Factoring the Difference of Squares<br />
Factoring Quadratic Trinomials with a Leading Coefficient of 1<br />
Factoring by Grouping<br />
Factoring the Sum and Difference of Cubes<br />
Factoring General Quadratic Trinomials (ax^2 + bx + c)<br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/pX57IW5p-YI?feature=player_embedded' frameborder='0'></iframe></div>
<br />
<br />
It's interesting to note that the vast majority of quadratic expressions are <em>not</em> factorable - but factoring is such a great exercise in mathematical logic, and has been part of the core Algebra curriculum for years.<br />
<br />
Hope you enjoy the song.<br />
<br />
--------------------------------------------------<br />
<br />
For Math Tutorials and more Silly Math Songs, visit <a href="http://www.onlinemathpro.com/">www.onlinemathpro.com</a>MrWAGNEEZYhttp://www.blogger.com/profile/14636358345686928129noreply@blogger.com0tag:blogger.com,1999:blog-1628539622402145132.post-1781074716684935292013-02-28T16:28:00.002-08:002013-02-28T16:30:29.843-08:00The Road Not Taken (Logic Puzzle)<em><span style="font-family: Calibri;"></span></em><br />
<iframe border="0" frameborder="0" height="60" marginwidth="0" scrolling="no" src="http://rcm.amazon.com/e/cm?t=onlmatpro-20&o=1&p=26&l=ur1&category=computers_accesories&banner=1AT7SAZES6GMFS62C5R2&f=ifr" style="border: currentColor;" width="468"></iframe>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;"></span><br />
<span style="font-family: Calibri;">In Robert
Frost’s famous poem, <u>The Road Not Taken</u> (sometimes mistakenly referred
to as “The Road Less Travelled”), a traveler is faced with the difficult choice
of which road he should follow.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">This brings
to mind a classic logic puzzle, with mathematical implications.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Here is my
poetic interpretation of the puzzle.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<u><span style="font-family: Calibri;">THE ROAD
NOT TAKEN – Mr. Wagneezy Version<o:p></o:p></span></u></div>
<span style="font-family: Calibri;">(Line 1 by
Robert Frost)<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">Two roads diverge in a yellow wood<o:p></o:p></span></b></div>
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">One leads to certain death<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">The other leads to riches untold<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">I stop to catch my breath<o:p></o:p></span></b><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">I soon discover I have no clue<o:p></o:p></span></b></div>
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">Exactly which road is which<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">I look to the right, and then to the left<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">My eyes begin to twitch<o:p></o:p></span></b><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">Suddenly two gnomes appear<o:p></o:p></span></b></div>
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">From out of nearby briars<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">One of them is a truthful gnome<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">The other one is a liar<o:p></o:p></span></b><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">In looking at these gnomes, alas<o:p></o:p></span></b></div>
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">I cannot tell the difference<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">Which one speaks truth?<span style="mso-spacerun: yes;"> </span>Which one speaks lies?<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">I fight the urge to wince<o:p></o:p></span></b><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">These seemingly identical gnomes<o:p></o:p></span></b></div>
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">Both know which road to take<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">But instead of making it clear to me<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">They make it nearly opaque<o:p></o:p></span></b><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">The gnomes agree that one of them<o:p></o:p></span></b></div>
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">Will answer a single question<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">Once I get the answer<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">They will end the conversation<o:p></o:p></span></b><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">I still can’t tell which one speaks truth<o:p></o:p></span></b></div>
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">And which one is the liar<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">They smirk at me, these pesky gnomes<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">That came out of the briars<o:p></o:p></span></b><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">I must determine what to say<o:p></o:p></span></b></div>
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">It is a daunting task<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">To get the gnomes to reveal the way<o:p></o:p></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span style="font-family: Calibri;">What question should I ask?<o:p></o:p></span></b><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">-----------------------------------------------------------------<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Successfully
getting these gnomes to show us which road to choose will take careful
planning.<span style="mso-spacerun: yes;"> </span>Since we have no way of
knowing which gnome tells the truth and which gnome lies, we must be able to
come up with a question that both gnomes would answer the same way.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Obviously,
the direct approach (e.g. “Which road leads to untold riches?”) will not work,
because the truth-teller would point to one road while the liar would point to
the other road.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Therefore,
we must ask an indirect question – one that incorporates both the truth and the
lie.<span style="mso-spacerun: yes;"> </span>We can accomplish this by asking
one gnome to tell us which road <i style="mso-bidi-font-style: normal;">the other
gnome</i> would point us towards.<span style="mso-spacerun: yes;"> </span>(E.g.
“If I asked the other gnome which road leads to untold riches, which road would
he point to?”)<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">The logic
here is that the truth about a lie gives the same result as a lie about the
truth.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">More
specifically, if we happen to talk to the truth-teller, he will point to the
wrong road because that is the road the liar would have pointed to.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">On the other
hand, if we happen to talk to the liar, he will also point to the wrong road
because that is <i style="mso-bidi-font-style: normal;">not </i>the road the
truth-teller would have pointed to.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">In either
case, the wrong road will be indicated, and we can choose the other road to
travel on.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">-----------------------------------------------------------------<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">For movie
buffs – a version of this puzzle appeared in the movie <a href="http://www.amazon.com/gp/product/B00000K3D4/ref=as_li_qf_sp_asin_tl?ie=UTF8&camp=1789&creative=9325&creativeASIN=B00000K3D4&linkCode=as2&tag=onlmatpro-20" target="_blank">Labyrinth</a><img alt="" border="0" height="1" src="http://www.assoc-amazon.com/e/ir?t=onlmatpro-20&l=as2&o=1&a=B00000K3D4" style="border: none !important; margin: 0px !important;" width="1" />
:<o:p></o:p></span></div>
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/NWw9y8UHs9c?feature=player_embedded' frameborder='0'></iframe></div>
<br />
<span style="font-family: Calibri;"></span><br />
<span style="font-family: Calibri;"></span><br />
<span style="font-family: Calibri;"></span><br />
<span style="font-family: Calibri;"></span><br />
<span style="font-family: Calibri;"></span><br />
<span style="font-family: Calibri;"><div class="MsoNormal" style="margin: 0in 0in 0pt;">
<o:p><span style="font-family: Calibri;"> </span></o:p></div>
<span style="font-family: Calibri;">-----------------------------------------------------------------<o:p></o:p></span></span><br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Earlier, I
mentioned that this puzzle has mathematical implications.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Let’s
deconstruct this puzzle into a similar mathematical question.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">I hope you
agree with me that telling the truth is <i style="mso-bidi-font-style: normal;">positive</i>,
and lying is <i style="mso-bidi-font-style: normal;">negative</i>.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Consider
mathematical operations that can be performed on two numbers.<span style="mso-spacerun: yes;"> </span>Suppose one of the numbers is positive, and
the other number is negative, but we DON’T KNOW WHICH IS WHICH.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Which
operations are guaranteed to give us results with the same sign, regardless of
which number is positive?<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">If we
arbitrarily choose ±2 and ±3 for our numbers and use them to explore each
operation, we get the following:</span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;"><o:p></o:p></span> </div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-86f1pRD4GOg/US_xsbqRr3I/AAAAAAAAAGU/xw1W4c6keDc/s1600/Ops&Negatives.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="480" src="http://3.bp.blogspot.com/-86f1pRD4GOg/US_xsbqRr3I/AAAAAAAAAGU/xw1W4c6keDc/s640/Ops&Negatives.jpg" width="640" /></a></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<o:p><span style="font-family: Calibri;"></span></o:p> </div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">As we can
see, <i style="mso-bidi-font-style: normal;">multiplication</i> and <i style="mso-bidi-font-style: normal;">division</i> are the only operations that
fit the bill.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">(Not
coincidentally, multiplication and division have the same priority in the <a href="http://www.onlinemathpro.com/tutorials/order-of-operations-song" target="_blank">Order of Operations</a>.)<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Therefore,
the logical argument<o:p></o:p></span></div>
<br />
<div align="center" class="MsoNormal" style="margin: 0in 0in 0pt; text-align: center;">
<span style="font-family: Calibri;">“The truth about a lie is equivalent to a lie about the
truth”<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">seems to
match up with the mathematical concepts<o:p></o:p></span></div>
<br />
<div align="center" class="MsoNormal" style="margin: 0in 0in 0pt; text-align: center;">
<span style="font-family: Calibri;">“A negative times a positive is equivalent to a positive
times a negative”<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">and<o:p></o:p></span></div>
<br />
<div align="center" class="MsoNormal" style="margin: 0in 0in 0pt; text-align: center;">
<span style="font-family: Calibri;">“A negative divided by a positive is equivalent to a
positive divided by a negative”.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">By the way –
multiplication and division both exhibit the desired property because<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpFirst" style="margin: 0in 0in 0pt 1.25in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">1)</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";">
</span></span></span><span style="font-family: Calibri;">Multiplying is the same thing as dividing by the
reciprocal.</span></div>
<div class="MsoListParagraphCxSpLast" style="margin: 0in 0in 0pt 1.25in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">2)</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";">
</span></span></span><span style="font-family: Calibri;">Dividing is the same thing as multiplying by the
reciprocal.</span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt 2in;">
<span style="font-family: Calibri;">and</span></div>
<div class="MsoListParagraph" style="margin: 0in 0in 0pt 1.25in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">3)</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";"> </span></span></span><span style="font-family: Calibri;">A
number’s sign (positive or negative) is the same as the sign of the number’s
reciprocal.<o:p></o:p></span></div>
<o:p><span style="font-family: Calibri;"> </span></o:p><br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">May all your
roads be wisely chosen.</span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">-----------------------------------------------------------------</span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">For Math Tutorials & Silly Math Songs, visit <a href="http://www.onlinemathpro.com/">www.onlinemathpro.com</a>.</span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;"></span><br />
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MrWAGNEEZYhttp://www.blogger.com/profile/14636358345686928129noreply@blogger.com0tag:blogger.com,1999:blog-1628539622402145132.post-42454041115573343172013-02-08T16:38:00.000-08:002013-02-08T16:38:53.002-08:00What’s the difference between 90% and 100%? (Traits of HIGHLY successful math students)<br />
<iframe border="0" frameborder="0" height="90" marginwidth="0" scrolling="no" src="http://rcm.amazon.com/e/cm?t=onlmatpro-20&o=1&p=48&l=ur1&category=electronicsrot&f=ifr" style="border: currentColor;" width="728"></iframe>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;"></span><br />
<span style="font-family: Calibri;">The
differences between the students who fail in math and the students who get
decent grades – C and above – are generally clear cut, and easily identifiable
(unless there is a learning disability involved).<span style="mso-spacerun: yes;"> </span>Most of these differences aren’t even math
specific… failing students in any subject can generally improve their grades by
improving one or more of the following areas:<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpFirst" style="margin: 0in 0in 0pt 0.75in; mso-add-space: auto; mso-list: l1 level1 lfo1; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">1)</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";">
</span></span></span><span style="font-family: Calibri;">Get organized</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.75in; mso-add-space: auto; mso-list: l1 level1 lfo1; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">2)</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";">
</span></span></span><span style="font-family: Calibri;">Do (and turn in) assignments on time</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.75in; mso-add-space: auto; mso-list: l1 level1 lfo1; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">3)</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";">
</span></span></span><span style="font-family: Calibri;">Be willing/able to seek help when needed</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.75in; mso-add-space: auto; mso-list: l1 level1 lfo1; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">4)</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";">
</span></span></span><span style="font-family: Calibri;">Be willing to put in the time/effort necessary
to be successful</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.75in; mso-add-space: auto; mso-list: l1 level1 lfo1; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">5)</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";">
</span></span></span><span style="font-family: Calibri;">Practice/Prepare for tests</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.75in; mso-add-space: auto; mso-list: l1 level1 lfo1; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">6)</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";">
</span></span></span><span style="font-family: Calibri;">If the people you hang out with aren’t committed
to success, hang out with different people</span></div>
<div class="MsoListParagraphCxSpLast" style="margin: 0in 0in 0pt 0.75in; mso-add-space: auto; mso-list: l1 level1 lfo1; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">7)</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";">
</span></span></span><span style="font-family: Calibri;">Don’t accept defeat easily (“If at first you
don’t succeed, try, try again.”)<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">However, I
recently found myself wondering about what sets the truly excellent math
students apart from the merely-great math students.<span style="mso-spacerun: yes;"> </span>I have a number of students earning a grade
of A or A- (at or above 90%).<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Why do a
select few of these students <i style="mso-bidi-font-style: normal;">consistently</i>
earn grades near 100% (or above 100%, if extra credit is offered) in math
class?<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">In analyzing
my students, I have noticed a few traits that differentiate merely-great math
students from highly successful math students.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpFirst" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l0 level1 lfo2; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">1.</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";">
</span></span></span><span style="font-family: Calibri;">Merely-great math students tend to believe that
knowing a lot and being good at math is the most important component of good
test taking.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto;">
<span style="font-family: Calibri;">Highly successful math students understand that
knowing a lot and being good at math is actually the <i style="mso-bidi-font-style: normal;">second</i> most important component of test taking.<span style="mso-spacerun: yes;"> </span>The most important component is being able to
successfully <u>communicate your knowledge and math excellence to your
teacher/professor</u>.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto;">
<span style="font-family: Calibri;">There are many merely-great math students who turn
in tests containing problems with ambiguity in the solutions.<span style="mso-spacerun: yes;"> </span>Their brains might have been doing all the
right steps, but their work is suspect.<span style="mso-spacerun: yes;">
</span>Highly successful math students provide complete solutions that are <i style="mso-bidi-font-style: normal;">clearly</i> mathematically sound.<span style="mso-spacerun: yes;"> </span>As a result, they tend to receive higher
scores on their tests.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto;">
<span style="font-family: Calibri;">Laziness also plays a role in this.<span style="mso-spacerun: yes;"> </span>Many merely-great math students seem to
follow the philosophy, “When in doubt, show less work.”<span style="mso-spacerun: yes;"> </span>Most highly successful math students follow the
philosophy, “When in doubt, show more work.”<o:p></o:p></span></div>
<br />
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l0 level1 lfo2; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">2.</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";">
</span></span></span><span style="font-family: Calibri;">Merely-great math students don’t tend to place a
high value on neatness.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto;">
<span style="font-family: Calibri;">Highly successful math students tend to make
neatness a priority.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto;">
<span style="font-family: Calibri;">I can’t count how many times I’ve been grading work
and discovered that a student got the wrong answer due to a mistake caused by
poor handwriting.<span style="mso-spacerun: yes;"> </span>I’ve seen “4” turn
into “9”, “z” turn into “2”, “7” turn into “1” and a myriad of other sloppy
mistakes.<span style="mso-spacerun: yes;"> </span>These mistakes often prove to
be the difference between a 95% test and a 100% test.<o:p></o:p></span></div>
<br />
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l0 level1 lfo2; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">3.</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";">
</span></span></span><span style="font-family: Calibri;">Merely-great math students use lectures and
class time to learn what they need to know in order to succeed.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto;">
<span style="font-family: Calibri;">Highly successful math students also use lectures
and class time to learn what they need to know, but they tend to use this as a <i style="mso-bidi-font-style: normal;">starting point</i> in their learning
process.<span style="mso-spacerun: yes;"> </span>They are adept at looking in
textbooks for additional examples in order to enhance the material covered in a
lecture.<span style="mso-spacerun: yes;"> </span>They are also eager to explore
alternative methods and they have a desire to know WHY a particular method
works or doesn’t work.<o:p></o:p></span></div>
<br />
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l0 level1 lfo2; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">4.</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";">
</span></span></span><span style="font-family: Calibri;">Merely-great math students do their best to find
out what material will be covered on a test.<span style="mso-spacerun: yes;">
</span>(Some are even quite assertive in trying to get specific review topics
from their instructors.)<span style="mso-spacerun: yes;"> </span>They then review
this material diligently to make sure they can complete the test successfully.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto;">
<span style="font-family: Calibri;">Highly successful math students emphasize material
they know will be covered on a test when studying.<span style="mso-spacerun: yes;"> </span>However, they also attempt to review/practice
all of the other concepts from a chapter.<span style="mso-spacerun: yes;">
</span>This is helpful in (at least) two ways:<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 1.75in; mso-add-space: auto; mso-list: l2 level1 lfo3; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">1.</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";">
</span></span></span><span style="font-family: Calibri;">Since concepts are often interrelated, highly
successful students gain a deeper understanding of the “key concepts” by
broadening their review.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpLast" style="margin: 0in 0in 0pt 1.75in; mso-add-space: auto; mso-list: l2 level1 lfo3; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">2.</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";">
</span></span></span><span style="font-family: Calibri;">Teachers who put extra credit items on their
tests will often draw from these “other” concepts.<span style="mso-spacerun: yes;"> </span>When these extra credit test items show up,
highly successful students are ready for them.<o:p></o:p></span></div>
<br />
<br />
<div class="MsoListParagraphCxSpFirst" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto; mso-list: l0 level1 lfo2; text-indent: -0.25in;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font-family: Calibri;">5.</span><span style="font-size-adjust: none; font-stretch: normal; font: 7pt/normal "Times New Roman";">
</span></span></span><span style="font-family: Calibri;"><span style="mso-spacerun: yes;"> </span>Merely-great
math students sometimes put too much emphasis on being right at the expense of
focusing on what went wrong.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto;">
<span style="font-family: Calibri;">Highly successful math students attempt to learn as
much as possible from their mistakes so that they can avoid making the same
kind of mistakes in the future.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpMiddle" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto;">
<span style="font-family: Calibri;"><span style="mso-spacerun: yes;"> </span>I can think
of a number of students who are experts at “nickel-and-diming” teachers out of
extra points.<span style="mso-spacerun: yes;"> </span>They submit work that is
good (but not great) and then have to verbally defend their work and try to
convince the teacher that they deserve 100% credit. They rejoice when their
arguments are fruitful and they receive a small increase in score… their
mission has been accomplished.<span style="mso-spacerun: yes;"> </span>Sadly,
these students tend to get in the habit of submitting less-than-stellar work
and find themselves arguing with their teachers a lot.<o:p></o:p></span></div>
<br />
<div class="MsoListParagraphCxSpLast" style="margin: 0in 0in 0pt 0.5in; mso-add-space: auto;">
<span style="font-family: Calibri;">Highly successful math students, on the other hand, are
willing to admit when their work is less than ideal.<span style="mso-spacerun: yes;"> </span>They may argue the merits of their work with
their teacher in an attempt to get more points, but their chief concern is
learning how to produce stellar work in the future that will be above reproach.<o:p></o:p></span></div>
<br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">The above
list of traits is not exhaustive – there are probably more traits of highly
successful math students that I’ve missed.<span style="mso-spacerun: yes;">
</span>(If you’d like to add to my list, leave a comment below.)<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Also, the
above list does not come from a scientific study.<span style="mso-spacerun: yes;"> </span>It is simply an anecdotal summary of my own
personal observations.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">I hope it
will help you as you strive for true excellence.<o:p></o:p></span></div>
<br />
---------------------------------------------<br />
<br />
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<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"></span><br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"></span><br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Here’s
an interesting math puzzle involving probability…<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 14pt; line-height: 115%;"><span style="color: #351c75;"><span style="font-size: large;">1
out of every 10,000 citizens of a country has a deadly virus.<span style="mso-spacerun: yes;"> </span>A blood test can be used to determine whether
or not somebody has the virus.<span style="mso-spacerun: yes;"> </span>The test
is accurate 98% of the time.<span style="mso-spacerun: yes;"> </span>There are 1
million people in the country, and each person is tested.<span style="mso-spacerun: yes;"> </span>A man who tested positive for the virus talks
to his doctor, who informs the man that there is a 98% chance that he does, in
fact, have the virus.<span style="mso-spacerun: yes;"> </span>The doctor is not
correct in making this statement… what is the actual probability that the man
has the deadly virus?<o:p></o:p></span></span></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">To
see the answer, scroll down to the bottom.<span style="mso-spacerun: yes;">
</span>If you would like help in understanding the logic behind the solution,
keep reading.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">First
of all, we have to understand what probability is.<o:p></o:p></span></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-lHUIIEWyStc/UPt5dSLB7QI/AAAAAAAAAGE/2q4WaqMkkhQ/s1600/Probability01.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-lHUIIEWyStc/UPt5dSLB7QI/AAAAAAAAAGE/2q4WaqMkkhQ/s1600/Probability01.jpg" /></a></div>
<br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">We
know that 1 out of every 10,000 people actually have the virus.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Therefore,
out of 1 million people, 100 people have the virus.<span style="mso-spacerun: yes;"> </span>(1,000,000 ÷ 10,000 = 100)<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">The
probability that a citizen of the country has the virus is 100 ÷ 1,000,000 =
0.0001 = 0.01%.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">This
probability is <i style="mso-bidi-font-style: normal;">independent</i> of the
blood test.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Once
the blood test is administered to every citizen in the country, the question
becomes a bit more difficult.<span style="mso-spacerun: yes;"> </span>We need to
determine the probability that a person who tested positive for the virus does,
in fact, have the virus.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">So,
how many people tested positive for the virus?<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">98%
of people who have the virus tested positive.<span style="mso-spacerun: yes;">
</span>98% of 100 is 98.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">If
100 people have the virus, then 999,900 people do not have the virus.
(1,000,000 – 100 = 999,900)<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">98%
of people who don’t have the virus tested negative.<span style="mso-spacerun: yes;"> </span>This means that 2% of people who don’t have
the virus tested positive.<span style="mso-spacerun: yes;"> </span>2% of 999,900
is 19,998.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">The
total number of people who tested positive is 98 + 19,998 = 20,096.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<i style="mso-bidi-font-style: normal;"><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Now</span></i><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">
we can calculate the probability that the man who tested positive does, in
fact, have the virus.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">20,096
people tested positive for the virus.<span style="mso-spacerun: yes;"> </span>Of
these, 98 people actually have the virus.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Therefore,
the probability that the man has the virus is 98 ÷ 20,096 = 0.0049 = </span><b style="mso-bidi-font-weight: normal;"><span style="font-family: "Times New Roman","serif"; font-size: 14pt; line-height: 115%;">0.49%</span></b><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">That’s
less than half of one percent!<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">When
considering this fact, it is tempting to think that the test is practically
worthless – but this is not the case.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Before
the test was administered, all we knew was that 0.01% (one 100<sup>th</sup> of
a percent) of the population had the virus.<span style="mso-spacerun: yes;">
</span>Now a group has been isolated in which about half of a percent has the
virus.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">If
this group is re-tested, the probability that a person testing positive does,
in fact, have the virus will be about 24%.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">If
the group that tests positive a 2<sup>nd</sup> time is re-tested, the
probability that a person testing positive does, in fact, have the virus will
be about 92%.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">It’s
amazing to see how quickly accuracy is gained through the process of
re-testing.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">And
now you know a little bit of the mathematics that goes into being a good
doctor.<o:p></o:p></span></div>
<br />
------------------------------<br />
<br />
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<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">How do you cut a pizza into 10 pieces that are the same size? Skip to the bottom to find out. However, if you want to know how NOT to cut a pizza into 10 equally-sized pieces – and maybe learn a little math in the process – keep reading.</span><br />
<br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">In
addition to teaching math, I also sponsor a high school class.<span style="mso-spacerun: yes;"> </span>Recently, the class I sponsor held a
fundraiser at which pizza was sold.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Normally,
when pizzas are delivered for our fundraisers, each pizza is cut into 8 pieces
that are roughly the same size.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">However,
the pizzas for this fundraiser had all been cut into 10 pieces.<span style="mso-spacerun: yes;"> </span>I was somewhat annoyed – and somewhat amused
– to see that the person who had cut the pizzas had done a very poor job of
making equally-sized pieces.<span style="mso-spacerun: yes;"> </span>Some pieces
appeared to be oversized, while others were quite obviously undersized.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Naturally,
everybody wanted to buy the oversized pieces, while nobody wanted to buy the
undersized pieces.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">I
began to think about the skills that a person needs to have in order to be a
good pizza cutter:<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><span style="mso-tab-count: 1;"> </span>1.<span style="mso-spacerun: yes;">
</span>Good hand-eye coordination<o:p></o:p></span></div>
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><span style="mso-tab-count: 1;"> </span>2.<span style="mso-spacerun: yes;">
</span>Good motor skills<o:p></o:p></span><br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><span style="mso-tab-count: 1;"> </span>3.<span style="mso-spacerun: yes;">
</span>Good math skills<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">…Huh?<span style="mso-spacerun: yes;"> </span>Good math skills?!<o:p></o:p></span></div>
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><o:p></o:p></span><br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">That’s
right.<span style="mso-spacerun: yes;"> </span>The difference between mediocrity
and greatness in the field of pizza cutting boils down to math.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">In
order to be a mediocre pizza cutter, one only needs to be able to cut a pizza
(or pieces of pizza) in half.<span style="mso-spacerun: yes;"> </span>Most
people can do this well enough.<span style="mso-spacerun: yes;">
</span>Unfortunately, if this is the extent of one’s pizza cutting skills, then
it will only be possible to cut pizzas into a number of equally-sized pieces
that is a power of 2.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Powers
of 2 are:<span style="mso-spacerun: yes;"> </span>2<sup>1</sup> = <b style="mso-bidi-font-weight: normal;">2</b>,<span style="mso-spacerun: yes;">
</span>2<sup>2</sup> = <b style="mso-bidi-font-weight: normal;">4</b>,<span style="mso-spacerun: yes;"> </span>2<sup>3</sup> = <b style="mso-bidi-font-weight: normal;">8</b>,<span style="mso-spacerun: yes;"> </span>2<sup>4</sup> = <b style="mso-bidi-font-weight: normal;">16</b>,<span style="mso-spacerun: yes;">
</span>etc…<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">If
you are considering how much of the pizza each piece makes up, it’s better to
consider powers of one-half<span style="font-family: Calibri;">.</span></span></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-gCzDiPcfQyU/UOyvG1kf04I/AAAAAAAAAFc/YkYToMq-zxU/s1600/PowersOfHalf.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-gCzDiPcfQyU/UOyvG1kf04I/AAAAAAAAAFc/YkYToMq-zxU/s1600/PowersOfHalf.jpg" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-jgKv1A7ke_o/UOyvGHucmaI/AAAAAAAAAFU/rEboZAmLCPY/s1600/Pizza01.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="299" src="http://2.bp.blogspot.com/-jgKv1A7ke_o/UOyvGHucmaI/AAAAAAAAAFU/rEboZAmLCPY/s640/Pizza01.jpg" width="640" /></a></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Since
8 is a power of two, even a mediocre pizza cutter can cut a pizza into 8 pieces
that are roughly the same size.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">On
the other hand, in order to cut a pizza into 10 equally-sized pieces one must
know that</span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-sq0m6SmLtMM/UOyu3abRAlI/AAAAAAAAAFI/eAGLcz9gh-U/s1600/equation01.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-sq0m6SmLtMM/UOyu3abRAlI/AAAAAAAAAFI/eAGLcz9gh-U/s1600/equation01.jpg" /></a></div>
</div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">{If
you hate fractions, consider the fact that the prime factorization of 10 is
2·5.}<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Therefore,
after the initial cut dividing the pizza into halves, the pizza cutter must
divide each half into <b style="mso-bidi-font-weight: normal;">five equally-sized
pieces</b>.<span style="mso-spacerun: yes;"> </span>If the person cutting the
pizza makes the mistake of cutting each half into two equally-sized pieces,
then there is no chance of ending up with 10 equally-sized pieces because <b style="mso-bidi-font-weight: normal;">five is not divisible by two</b>. (This is how NOT to cut a pizza into 10 equally-sized pieces.)</span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-2dEbUqdOuX0/UOyvGE2lA-I/AAAAAAAAAFg/sXnacVoDTWU/s1600/Pizza02.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="http://1.bp.blogspot.com/-2dEbUqdOuX0/UOyvGE2lA-I/AAAAAAAAAFg/sXnacVoDTWU/s640/Pizza02.jpg" width="640" /></a></div>
</div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">This
appears to be where our pizza cutter made his/her fatal mistake.<span style="mso-spacerun: yes;"> </span>(Mathematically fatal, that is.)<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">A
good, mathematically sound pizza cutter would have made the first cut dividing
the pizza into halves.<span style="mso-spacerun: yes;"> </span>Then, he/she
would have begun cutting the halves into fifths.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-zxhdsTCyB28/UOyvGL5B_TI/AAAAAAAAAFQ/XbxRlWLZhgg/s1600/Pizza03.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="http://2.bp.blogspot.com/-zxhdsTCyB28/UOyvGL5B_TI/AAAAAAAAAFQ/XbxRlWLZhgg/s640/Pizza03.jpg" width="640" /></a></div>
</div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">If
you really want to delve into the mathematical implications of this series of
cuts, consider how each successive cut takes off a bigger fraction of the big
piece of pizza remaining:<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-e44NSj6uS90/UOyvGt40V7I/AAAAAAAAAFY/dmduDSpAL04/s1600/Pizza04.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="http://2.bp.blogspot.com/-e44NSj6uS90/UOyvGt40V7I/AAAAAAAAAFY/dmduDSpAL04/s400/Pizza04.jpg" width="400" /></a></div>
</div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">One
last thing to consider:<span style="mso-spacerun: yes;"> </span>I believe that
people who cut pizzas for the big pizza chains (Domino’s, Pizza Hut, etc.) use
a cutting instrument that reaches all of the way across a pizza – a <i style="mso-bidi-font-style: normal;">diameter</i>, assuming that the cut passes
through the center.<span style="mso-spacerun: yes;"> </span>If this is the case,
then it is impossible to cut a pizza into an odd number of pieces by making cuts through the center.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">If
you’re feeling devious, try ordering a pizza and requesting that it be cut into
nine pieces.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">On
the other hand, maybe you shouldn’t… the person cutting the pizza might
actually try to do it.<o:p></o:p></span></div>
<br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">***Disclaimer:<span style="mso-spacerun: yes;"> </span>I love pizza.<span style="mso-spacerun: yes;">
</span>I think that pizza should be one of the main food groups.<span style="mso-spacerun: yes;"> </span>Therefore, by extension, I love the people
who prepare pizza for me – even if they don’t cut the pizza into equally-sized
pieces.<span style="mso-spacerun: yes;"> </span>If you are a professional pizza
cutter, I hope that you are not offended by my mathematical musings.<span style="mso-spacerun: yes;"> </span>On the contrary, I salute you and thank you
for all that you do for pizza lovers like me.***</span><br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"></span><br />
<br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"></span><br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"></span><br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"></span><br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">---------------------------------------------------<o:p></o:p></span></span></div>
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">
</span>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">For
math tutorials and silly math songs, visit <a href="http://www.onlinemathpro.com/">www.onlinemathpro.com<o:p></o:p></a></span></span></div>
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">
<o:p></o:p></span><br />
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<br />
<br />
Happy New Year! <span style="mso-spacerun: yes;"> </span>I hope your year is
off to a great start.<o:p></o:p><br />
<br />
If you like to contemplate numbers like I do, you may have wondered whether
or not 2013 is a prime number. After a quick review of basic divisibility rules
(yeah, <a href="http://www.onlinemathpro.com/tutorials/divisibility-song" target="_blank"><span style="color: blue;">there's a song for that</span></a>) we can see that 2013 is divisible by 3.<o:p></o:p><br />
<br />
(2 + 0 + 1 + 3 = 6, and 3 goes into 6.)<o:p></o:p><br />
<br />
Factoring out 3, we have:<span style="mso-spacerun: yes;"> </span>2013 =
3(671)<o:p></o:p><br />
<br />
So, is 671 a prime number?<span style="mso-spacerun: yes;"> </span>It doesn’t
fit any of the divisibility rules for 2 through 10.<span style="mso-spacerun: yes;"> </span>However, as it turns out, 671 is divisible by
11.<o:p></o:p><br />
<br />
Factoring out 11, we have:<span style="mso-spacerun: yes;"> </span>671 =
11(61)<o:p></o:p><br />
<br />
Therefore, the prime factorization of 2013 is 3·11·61.<o:p></o:p><br />
<br />
In general, how can we tell whether or not a number is divisible by 11?<o:p></o:p><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Here's a fairly easy <strong>Divisibility Rule for 11:<o:p></o:p></strong></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">1)
Find the sum of every other digit in the number (1<sup>st</sup> digit + 3<sup>rd</sup>
digit + 5<sup>th</sup> digit + …)<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">2)
Find the sum of the digits not used in step 1 (2<sup>nd</sup> digit + 4<sup>th</sup>
digit + 6<sup>th</sup> digit + …)<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">If
these sums differ by 0, or a multiple of 11, then the original number is
divisible by 11.<o:p></o:p></span></div>
<br />
{If you want to know why this rule works, continue reading. If you don't care why it works, but would like to see another neat divisibility rule for 11, skip to the bottom.}<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
</div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<u><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Why
this rule Works<o:p></o:p></span></u></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Think
back to elementary school, when you used pencil and paper to multiply.<span style="mso-spacerun: yes;"> </span>What does it look like when a number is
multiplied by 11?<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Here
is an example:<span style="mso-tab-count: 2;"> </span>12345<o:p></o:p></span></div>
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><span style="mso-tab-count: 3;"> </span><span style="mso-spacerun: yes;"> </span><u>× <span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>11<o:p></o:p></u></span><br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><span style="mso-tab-count: 4;"> </span>12345<o:p></o:p></span><br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><span style="mso-tab-count: 3;"> </span><span style="mso-spacerun: yes;"> </span><u>+</u></span><u><span style="font-family: "Times New Roman","serif"; font-size: 6pt; line-height: 115%;"> </span></u><u><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span>123450<span style="color: white; mso-themecolor: background1;"><o:p></o:p></span></span></u><br />
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><span style="mso-tab-count: 3;"> </span><span style="mso-spacerun: yes;"> </span></span><span style="color: black; font-family: "Times New Roman","serif"; font-size: 6pt; line-height: 115%; mso-themecolor: text1;"><span style="mso-spacerun: yes;"> </span></span><span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">135795<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><o:p> </o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">Let’s take a look at a general case, in which
a 5-digit number is multiplied by 11.<span style="mso-spacerun: yes;">
</span>The conclusions we draw can be generalized for multiplying any length of
number by 11.<o:p></o:p></span></div>
<br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><span style="mso-tab-count: 4;"> </span>ABCDE<o:p></o:p></span></div>
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><span style="mso-tab-count: 3;"> </span><span style="mso-spacerun: yes;"> </span><u>×<span style="mso-spacerun: yes;">
</span><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>11<o:p></o:p></u></span><br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><span style="mso-tab-count: 4;"> </span>ABCDE<o:p></o:p></span><br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><span style="mso-tab-count: 3;"> </span><span style="mso-spacerun: yes;"> </span><u>+<span style="mso-spacerun: yes;">
</span>ABCDE0<o:p></o:p></u></span><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><o:p> </o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">The answer will have the following values for
its digits:<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><span style="mso-tab-count: 1;"> </span>1<sup>st</sup>
digit:<span style="mso-tab-count: 1;"> </span>A<o:p></o:p></span></div>
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><span style="mso-tab-count: 1;"> </span>2<sup>nd</sup>
digit:<span style="mso-tab-count: 1;"> </span>(A + B)<o:p></o:p></span><br />
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><span style="mso-tab-count: 1;"> </span>3<sup>rd</sup>
digit:<span style="mso-tab-count: 1;"> </span>(B + C)<o:p></o:p></span><br />
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"> 4<sup>th</sup> digit:<span style="mso-tab-count: 1;"> </span>(C + D)<o:p></o:p></span><br />
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">5<sup>th</sup> digit:<span style="mso-tab-count: 1;"> </span>(D + E)<o:p></o:p></span><br />
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">6<sup>th</sup> digit:<span style="mso-tab-count: 1;"> </span>E<o:p></o:p></span><br />
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><o:p> </o:p></span><br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">The sum of digits 1, 3 and 5 is A + (B + C) +
(D + E)<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">The sum of digits 2, 5, and 6 is (A + B) + (C
+ D) + E<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">Clearly these sums have a difference of 0.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">But… what if we need to carry when adding
ABCDE + ABCDE0?<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">Exactly what happens when we carry?<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">If a digit’s value exceeds 9, then we
subtract 10 from that value, and add 1 to the value of the digit to the left.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">For example:<span style="mso-tab-count: 2;"> </span>39<o:p></o:p></span></div>
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><span style="mso-tab-count: 2;"> </span><span style="mso-spacerun: yes;"> </span></span><span style="color: black; font-family: "Times New Roman","serif"; font-size: 6pt; line-height: 115%; mso-themecolor: text1;"><span style="mso-spacerun: yes;"> </span></span><span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><span style="mso-spacerun: yes;"> </span><u>+<span style="mso-spacerun: yes;"> </span>7<o:p></o:p></u></span><br />
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><span style="mso-tab-count: 2;"> </span><span style="mso-spacerun: yes;"> </span></span><span style="color: black; font-family: "Times New Roman","serif"; font-size: 6pt; line-height: 115%; mso-themecolor: text1;"><span style="mso-spacerun: yes;"> </span></span><span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">46<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><span style="mso-spacerun: yes;"> (</span>9 + 7
= 16.<span style="mso-spacerun: yes;"> </span>16 – 10 = 6.<span style="mso-spacerun: yes;"> </span>We write 6 in the one’s place, and add 1 to
the digit in the ten’s place)<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"></span><br /></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">Subtracting 10 from a digit’s value and
adding 1 to the value of the digit to the left causes a net change of 11
between the sum of digits 1, 3 & 5 and the sum of digits 2, 4 & 6.<o:p></o:p></span></div>
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><o:p> </o:p></span><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">Let’s assume that in our generalized example,
(D + E) exceeds 9.<span style="mso-spacerun: yes;"> </span>In this case, since
we have to carry, the answer will have the following values for its digits:<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><span style="mso-tab-count: 1;"> </span>1<sup>st</sup>
digit:<span style="mso-tab-count: 1;"> </span>A<o:p></o:p></span></div>
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><span style="mso-tab-count: 1;"> </span>2<sup>nd</sup>
digit:<span style="mso-tab-count: 1;"> </span>(A + B)<o:p></o:p></span><br />
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><span style="mso-tab-count: 1;"> </span>3<sup>rd</sup>
digit:<span style="mso-tab-count: 1;"> </span>(B + C)<o:p></o:p></span><br />
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">4<sup>th</sup> digit:<span style="mso-tab-count: 1;"> </span>(C + D) + 1<o:p></o:p></span><br />
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">5<sup>th</sup> digit:<span style="mso-tab-count: 1;"> </span>(D + E) – 10 <o:p></o:p></span><br />
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">6<sup>th</sup> digit:<span style="mso-tab-count: 1;"> </span>E<o:p></o:p></span><br />
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><o:p> </o:p></span><br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">Now the sum of digits 1, 3 and 5 is<span style="mso-spacerun: yes;"> </span>A + B + C + D + E – 10<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">And the sum of digits 2, 4 and 6 is<span style="mso-spacerun: yes;"> </span>A + B + C + D + E + 1<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">These sums differ by 11.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">Every time we carry (subtract 10 from a
digit’s value and add 1 to the value of the digit to the left) we are causing a
net change of 11 between the sum of digits 1, 3 & 5 and the sum of digits
2, 4 & 6.<span style="mso-spacerun: yes;"> </span>These changes may
accumulate, or cancel each other out.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">If we carry twice, the sums could differ by
22 (if the net changes of 11 accumulate) or they could differ by 0 (if the net
changes cancel each other out).</span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">By generalizing this explanation, we can show
that for <i style="mso-bidi-font-style: normal;">any</i> multiple of 11, the sum
of digits 1, 3, 5, etc. will differ from the sum of digits 2, 4, 6, etc. by 0,
or a multiple of 11.<o:p></o:p></span></div>
<br />
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">Here’s an even easier <strong>divisibility rule for
11</strong>:<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">Subtract the last digit from the number
formed by the remaining digits.<span style="mso-spacerun: yes;"> </span>(You can
repeat this step as often as necessary.)<span style="mso-spacerun: yes;">
</span>If the resulting number is divisible by 11, then the original number is
divisible by 11.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">(Example:<span style="mso-spacerun: yes;">
</span>671 is divisible by 11 because 67 – 1 = 66, and 66 is divisible by 11.)<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">Why does this work?<span style="mso-spacerun: yes;"> </span>I bet you can figure it out…<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">*Hint: It might be helpful to once again
consider this problem:<o:p></o:p></span></div>
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><o:p> </o:p></span><br />
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><span style="mso-tab-count: 4;"> </span>ABCDE<o:p></o:p></span><br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><span style="mso-tab-count: 3;"> </span><span style="mso-spacerun: yes;"> </span><u>×<span style="mso-spacerun: yes;">
</span><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>11<o:p></o:p></u></span><br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><span style="mso-tab-count: 4;"> </span>ABCDE<o:p></o:p></span><br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><span style="mso-tab-count: 3;"> </span><span style="mso-spacerun: yes;"> </span><u>+<span style="mso-spacerun: yes;">
</span>ABCDE0<o:p></o:p></u></span><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><o:p> </o:p></span></div>
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">If you want to try out an explanation, feel
free to post a comment below.</span><br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">Or, you can share
on Facebook:<span style="mso-spacerun: yes;"> </span><a href="http://www.facebook.com/pages/OnlineMathProcom/134423926682041"><span style="color: blue;">www.facebook.com/pages/OnlineMathProcom/134423926682041</span></a></span></div>
<br />
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">Here’s to a great (not to mention,
mathtastic) year in 2013.<o:p></o:p></span><br />
<br />
--<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">-------------------------------------<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;">For math tutorials and silly math songs,
visit<span style="mso-spacerun: yes;"> </span><a href="http://www.onlinemathpro.com/"><span style="color: blue;">www.onlinemathpro.com</span></a></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-themecolor: text1;"><o:p></o:p></span> </div>
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<br />MrWAGNEEZYhttp://www.blogger.com/profile/14636358345686928129noreply@blogger.com0tag:blogger.com,1999:blog-1628539622402145132.post-33804999411524766662012-12-27T17:32:00.000-08:002012-12-27T17:32:12.843-08:00Watermelons and New Years' Parties (Puzzle)<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Here’s a fun
math puzzle involving percents.<span style="mso-spacerun: yes;"> </span>The solution
is not terribly difficult – but it is surprising how many people end up getting
the answer wrong.<span style="mso-spacerun: yes;"> </span>Why not give it a try:<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">The owner of a farm stand puts a
bunch of watermelons on display at the beginning of the day.<span style="mso-spacerun: yes;"> </span>The watermelons have a mass of 200 kg.<span style="mso-spacerun: yes;"> </span>99% of the watermelons’ mass is water.<span style="mso-spacerun: yes;"> </span>It is hot, and during the day water evaporates
from the melons.<span style="mso-spacerun: yes;"> </span>At the end of the day, 98%
of the watermelons’ mass is water.<span style="mso-spacerun: yes;">
</span>Assuming that no watermelons were sold during the day, what is the mass
of the watermelons at the end of the day?<o:p></o:p></span></b></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">If you’d
like to check your answer, the correct solution is below.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<o:p><span style="font-family: Calibri;"> </span></o:p></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">If you’re
having a hard time, consider this puzzle. <span style="mso-spacerun: yes;"> </span>The math involved the same as the watermelon
problem, but the way in which the puzzle is presented tends to make the
solution easier to find.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Some guys planned a New Years’
party and invited only women.<span style="mso-spacerun: yes;"> </span>At the
party, there were a total of 200 people, 99% of whom were women.<span style="mso-spacerun: yes;"> </span>A lot of the women thought that the party was
lame, so they left.<span style="mso-spacerun: yes;"> </span>After one hour, 98%
of the people left at the party were women.<span style="mso-spacerun: yes;">
</span>How many total people were at the party after one hour?<o:p></o:p></span></b></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">This puzzle
is slightly easier to grasp because it explicitly states that there are men and
women.<span style="mso-spacerun: yes;"> </span>(And, it states that only women
leave the party.)</span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">See below for the solutions.</span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;"></span> </div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://www.onlinemathpro.com/tutorials/repeat-it" target="_blank"><img border="0" src="http://2.bp.blogspot.com/-r91-Aun9DYs/UNz0wuWIHmI/AAAAAAAAAEs/-ETDy_ASX6A/s1600/RepeatIt-Ad-001.jpg" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
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</div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<u><span style="font-family: Calibri;">Solution:
New Years’ party puzzle<o:p></o:p></span></u></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">It’s not
very hard to determine how many men and women there were at the beginning of
the party:<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Women: <span style="mso-spacerun: yes;"> </span>99% of 200 = 198<span style="mso-tab-count: 1;"> </span>Men:<span style="mso-spacerun: yes;"> </span>1% of 200 = 2<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Since only women
left the party, there are still two men after one hour.<span style="mso-spacerun: yes;"> </span>However, these two men now comprise 2% of the
total party crowd.<span style="mso-spacerun: yes;"> </span>2 is 2% of 100…
therefore, there are 100 people at the party after one hour.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<o:p><span style="font-family: Calibri;"> </span></o:p></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<u><span style="font-family: Calibri;">Solution:
Watermelon puzzle<o:p></o:p></span></u></div>
<br />
<span style="font-family: Calibri;">The
watermelon puzzle is a little trickier simply because the problem doesn’t
explicitly state that the watermelon is made up of water and solids.<span style="mso-spacerun: yes;"> </span>Therefore, many people get hung up focusing
on the water.<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">If 99% of
the watermelon mass is water, then 1% is made up of solids.<span style="mso-spacerun: yes;"> </span>1% of 200kg is 2 kg – so there are 2 kg of
solids.<span style="mso-spacerun: yes;"> </span>Solids do not evaporate, so at
the end of the day, there are still 2 kg of solids.<span style="mso-spacerun: yes;"> </span>This now comprises 2% of the total watermelon
mass.<span style="mso-spacerun: yes;"> </span>2 kg is 2% of 100 kg.<span style="mso-spacerun: yes;"> </span>Therefore, there are 100 kg of watermelons
left at the end of the day.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Happy New
Years’, Everyone!</span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">------------------------------</span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">For Math Tutorials and Silly Math Songs, visit <a href="http://www.onlinemathpro.com/">www.onlinemathpro.com</a></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;"></span> </div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;"></span> </div>
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<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
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MrWAGNEEZYhttp://www.blogger.com/profile/14636358345686928129noreply@blogger.com0tag:blogger.com,1999:blog-1628539622402145132.post-40263260906207859872012-12-13T16:24:00.000-08:002012-12-14T16:04:19.119-08:00Ready, Go, SET!<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Riddles
and puzzles are a good way to challenge yourself and increase your logic
skills.<span style="mso-spacerun: yes;"> </span>They also give valuable practice
in solving the story problems that seem to haunt so many math students.<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Here’s
a brain teaser that can be solved with simple math.<span style="mso-spacerun: yes;"> </span>(As is the case with many math puzzles, there
is more than one way to solve the problem.)<span style="mso-spacerun: yes;">
</span>If you think you’re up to it, try to answer the question before reading
the solutions and explanations at the bottom of this page.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<u><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">There
are 340 players in a <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">SET*</i></b> tournament.<span style="mso-spacerun: yes;"> </span>Each game in the tournament is played with 4
people.<span style="mso-spacerun: yes;"> </span>For each game, there is one
winner who moves on to the next round.<span style="mso-spacerun: yes;">
</span>Rounds will continue until there is just one winner left – the champion.<span style="mso-spacerun: yes;"> </span>In any round, if the number of players is not
a multiple of 4, then some players will be chosen randomly to advance
automatically so that the number of people playing in that round is divisible
by 4.<span style="mso-spacerun: yes;"> </span>(This will ensure that each game
that is played will have 4 players.) <span style="mso-spacerun: yes;"> </span>How
many total games will be played in the tournament?</span></u><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><o:p>{Scroll down for solutions}</o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
</div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">--------------------------------------------------<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">*I
love card games.<span style="mso-spacerun: yes;"> </span>I also love activities
that help people learn and/or increase their brain power.<span style="mso-spacerun: yes;"> </span><b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">SET</i></b> is a card game that is simple to
learn, fun for a wide range of players (recommended for players 6 years old and
above), and helps build brain power.<span style="mso-spacerun: yes;"> </span>As
a teacher and card game enthusiast, I give <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">SET</i></b> my full recommendation.<span style="mso-spacerun: yes;"> </span>(And now <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">SET-Junior</i></b> is available for younger
players.)<o:p></o:p></span></div>
<br />
<br />
<iframe frameborder="0" marginheight="0" marginwidth="0" scrolling="no" src="http://rcm.amazon.com/e/cm?t=onlmatpro-20&o=1&p=8&l=as1&asins=B00000IV34&ref=tf_til&fc1=000000&IS2=1&lt1=_blank&m=amazon&lc1=0000FF&bc1=000000&bg1=FFFFFF&f=ifr" style="height: 240px; width: 120px;"></iframe>
<iframe border="0" frameborder="0" height="250" marginwidth="0" scrolling="no" src="http://rcm.amazon.com/e/cm?t=onlmatpro-20&o=1&p=12&l=ur1&category=amazonhomepage&f=ifr" style="border: currentColor;" width="300"></iframe>
<iframe frameborder="0" marginheight="0" marginwidth="0" scrolling="no" src="http://rcm.amazon.com/e/cm?t=onlmatpro-20&o=1&p=8&l=as1&asins=B007XOY0UW&ref=tf_til&fc1=000000&IS2=1&lt1=_blank&m=amazon&lc1=0000FF&bc1=000000&bg1=FFFFFF&f=ifr" style="height: 240px; width: 120px;"></iframe>
<br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">--------------------------------------------------<o:p></o:p></span></div>
<br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<b style="mso-bidi-font-weight: normal;"><u><span style="font-family: "Times New Roman","serif"; font-size: 14pt; line-height: 115%;">Solutions/Explanations<o:p></o:p></span></u></b></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
</div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><u>Method
1 (slow)<o:p></o:p></u></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">To
solve the problem, many people get out pencil and paper and begin calculating
the results, round by round.<span style="mso-spacerun: yes;"> </span>For each
round, they keep track of the number of games, and then add these up at the end
to find the answer to the question.<span style="mso-spacerun: yes;"> </span>This
method can be quite time consuming.<span style="mso-spacerun: yes;"> </span>One
such solution is shown:<o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-rBkiXhfQ25I/UMptDTxpL_I/AAAAAAAAAEY/kQreOmuKHnk/s1600/SET_problem-001.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="480" src="http://1.bp.blogspot.com/-rBkiXhfQ25I/UMptDTxpL_I/AAAAAAAAAEY/kQreOmuKHnk/s640/SET_problem-001.jpg" width="640" /></a></div>
</div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
</div>
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><u>Method
2 (fast)</u></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">Of
course, there is another way to solve this problem – a way that takes much less
time.<span style="mso-spacerun: yes;"> </span>In the previous solution, the
focus was on the WINNERS from each round.<span style="mso-spacerun: yes;">
</span>(The number of players in rounds 2 – 5 was determined by figuring out
how many winners there were in the previous round and adding this amount to the
number of people who were lucky enough to advance automatically.)<span style="mso-spacerun: yes;"> </span>A much quicker solution involves focusing on
LOSERS:</span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">We
know <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">each
game that is played will have 4 players</i></b>.<span style="mso-spacerun: yes;"> </span>This means that each game is guaranteed to
have 3 losers.<span style="mso-spacerun: yes;"> </span>The tournament begins
with 340 players, and 339 of those players will be a loser at some point.<span style="mso-spacerun: yes;"> </span>All we need to do is divide 339 by 3 to find
our answer: There will be 113 games.<o:p></o:p></span></div>
<br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">When
looking at a tough problem, we often hurt ourselves by focusing on the wrong
part of the problem.<span style="mso-spacerun: yes;"> </span>Sometimes it is
helpful to take a step back, reexamine the problem, and figure out if our focus
is in the right place.</span><br />
<br />
<br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"></span>For math tutorials and silly math songs, visit <a href="http://www.onlinemathpro.com/">www.onlinemathpro.com</a><br />
<br />
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MrWAGNEEZYhttp://www.blogger.com/profile/14636358345686928129noreply@blogger.com0tag:blogger.com,1999:blog-1628539622402145132.post-6997032604016499442012-12-05T14:14:00.000-08:002012-12-05T14:18:52.157-08:00Percentiles?!<span style="font-family: Calibri;"></span><br />
<iframe border="0" frameborder="0" height="60" marginwidth="0" scrolling="no" src="http://rcm.amazon.com/e/cm?t=onlmatpro-20&o=1&p=42&l=ur1&category=holiday12&banner=1NRWBFX10ZVFX29GK202&f=ifr" style="border: currentColor;" width="234"></iframe>
<span style="font-family: Calibri;"></span><br />
<span style="font-family: Calibri;"></span><br />
<span style="font-family: Calibri;">We take
tests throughout our school years and most of us understand how the scoring
works.<span style="mso-spacerun: yes;"> </span>Take the number of points earned,
divide by the number of points possible, move the decimal a couple of spaces
right and, voila, we have a <i style="mso-bidi-font-style: normal;">percent</i>
that represents our score. (If you want to learn more about percents, there's a <a href="http://www.onlinemathpro.com/tutorials/percents---finding-percent-of-change" target="_blank"><span style="color: blue;">song for that</span></a>.)</span><br />
<span style="font-family: Calibri;"><o:p></o:p></span><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-8cX47L9UNa8/UL_DY1qc1cI/AAAAAAAAADg/B5m2YHislxI/s1600/TestScore.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-8cX47L9UNa8/UL_DY1qc1cI/AAAAAAAAADg/B5m2YHislxI/s1600/TestScore.jpg" /></a></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">But then,
once a year, we take a standardized test and when the score is reported, we are
given a <i style="mso-bidi-font-style: normal;">percentile</i>.<span style="mso-spacerun: yes;"> </span>What in the world is a percentile?<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Unfortunately,
we don’t learn much about percentiles in school.<span style="mso-spacerun: yes;"> </span>However, we do learn about the median, which
I discussed in my <u><a href="http://onlinemathpro.blogspot.com/2012/11/nfl-salaries-part-2-is-average-really.html" target="_blank"><span style="color: blue;">last blog entry</span></a></u>, and also about quartiles.<span style="mso-spacerun: yes;"> </span>Let’s review briefly:<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">If all of
the values in a data set are arranged in order, then the middle number (or, if
there are two middle numbers, their average) is the <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">median</i></b>.<span style="mso-spacerun: yes;"> </span>This number divides the data set into two
equal groups:<span style="mso-spacerun: yes;"> </span>Half of the values in the
set are above the median, and half of the values are below the median.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">If all of
the values in a data set are arranged in order, then the <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">quartiles</i> </b>(there are three
of them) are the values that split the data set into four equal groups:<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">One-fourth
(25%) of the values in the set are less than or equal to the 1<span style="font-size: small;"><sup>st</sup>
quartile, and three-fourths (75%) of the values are greater than or equal to
the 1<sup>st</sup> quartile.<o:p></o:p></span></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Half (50%)
of the values in the set are less than or equal to the 2<span style="font-size: small;"><sup>nd</sup> quartile,
and half (50%) of the values are greater than or equal to the 2<sup>nd</sup>
quartile.<span style="mso-spacerun: yes;"> </span>(The 2<sup>nd</sup> quartile
is the same as the median.)<o:p></o:p></span></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Three-fourths
(75%) of the values in the set are less than or equal to the 3<span style="font-size: small;"><sup>rd</sup>
quartile, and one-fourth (25%) of the values are greater than or equal to the 3<sup>rd</sup>
quartile.</span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-4_r_rV6V6FQ/UL_EFGpnylI/AAAAAAAAAD4/TMUD-QCCOd4/s1600/Quartiles001.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="96" src="http://1.bp.blogspot.com/-4_r_rV6V6FQ/UL_EFGpnylI/AAAAAAAAAD4/TMUD-QCCOd4/s400/Quartiles001.jpg" width="400" /></a></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">If you
understand how quartiles work, then it’s not much of a leap to understand
percentiles.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">If all of
the values in a data set are arranged in order, then the <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">percentiles</i></b> (there are 99
of them) are the values that split the data into 100 equal groups.<span style="mso-spacerun: yes;"> </span>(Note: some people call these <i style="mso-bidi-font-style: normal;">centiles.</i>)<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">1% of the
values in the set are less than or equal to the 1<span style="font-size: small;"><sup>st</sup> percentile, and
99% of the values are greater than or equal to the 1<sup>st</sup> percentile.<o:p></o:p></span></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">2% of the
values in the set are less than or equal to the 2<span style="font-size: small;"><sup>nd</sup> percentile, and
98% of the values are greater than or equal to the 2<sup>nd</sup> percentile.<o:p></o:p></span></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">And so on…<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">Because they
split the data set into so many groups, percentiles are only useful in
analyzing data sets that are very large – like the number of students who take
standardized tests.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">If we
understand how percentiles work, then interpreting standardized test scores
becomes much easier.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">If a score
is reported as 20<span style="font-size: small;"><sup>th</sup> percentile, then 20% of test-takers scored at or
below that level.<o:p></o:p></span></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">If a score
is reported as 90<span style="font-size: small;"><sup>th</sup> percentile, then 90% of test-takers scored at or
below that level.<o:p></o:p></span></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">That wasn’t
so hard was it?<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">So, why are
there so many people who struggle with percentiles?<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: Calibri;">I blame
politicians.<o:p></o:p></span></div>
<o:p><span style="font-family: Calibri;"> </span></o:p><br />
<span style="font-family: "Calibri","sans-serif"; line-height: 115%; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin;">In the recent years, politicians arguing over
tax rates have consistently discussed “Americans in the top two percent of
earners”.<span style="mso-spacerun: yes;"> </span>They would do us all a great
service if they would instead argue about “Americans at or above the 98<span style="font-size: small;"><sup>th</sup>
percentile of earnings”.<span style="mso-spacerun: yes;"> </span>It wouldn’t
make politicians any more likely to work well together, but at least people
would understand standardized test scores better!</span></span><br />
<span style="font-family: Calibri;"></span><br />
<span style="font-family: Calibri;"></span><br />
<iframe border="0" frameborder="0" height="90" marginwidth="0" scrolling="no" src="http://rcm.amazon.com/e/cm?t=onlmatpro-20&o=1&p=48&l=ur1&category=holiday12&banner=1Y1KXYKEJW933MBW8NR2&f=ifr" style="border: currentColor;" width="728"></iframe><a href="http://www.onlinemathpro.com/products-and-product-reviews/affiliate-disclosures" target="_blank">
<span style="font-family: Calibri;"></span></a><br />
<span style="font-family: "Calibri","sans-serif"; font-size: xx-small; line-height: 115%; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin;"><a href="http://www.onlinemathpro.com/products-and-product-reviews/affiliate-disclosures" target="_blank">Affiliate Disclosure</a></span>MrWAGNEEZYhttp://www.blogger.com/profile/14636358345686928129noreply@blogger.com0tag:blogger.com,1999:blog-1628539622402145132.post-44060570743624697632012-11-30T10:31:00.000-08:002012-11-30T16:24:08.486-08:00NFL Salaries (part 2) - Is Average really Average?<iframe border="0" frameborder="0" height="60" marginwidth="0" scrolling="no" src="http://rcm.amazon.com/e/cm?t=onlmatpro-20&o=1&p=26&l=ur1&category=giftcardsseasonal&banner=11BJK66JEY1YVQRT8302&f=ifr" style="border: currentColor;" width="468"></iframe>
<a href="http://4.bp.blogspot.com/-zPaFR-YPEn0/ULj4QFO15NI/AAAAAAAAADA/Gglq42Benlc/s1600/AverageMedian.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://4.bp.blogspot.com/-zPaFR-YPEn0/ULj4QFO15NI/AAAAAAAAADA/Gglq42Benlc/s1600/AverageMedian.jpg" /></a><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"></span><br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"></span><br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">In
my <a href="http://onlinemathpro.blogspot.com/2012/11/thanksgiving-nfl-salary-analysis.html" target="_blank"><span class="Apple-style-span" style="color: blue;">last blog entry</span></a>, I discussed average NFL salaries.<span style="mso-spacerun: yes;"> </span>I had initially hoped to talk about <i style="mso-bidi-font-style: normal;">median</i> salaries but, with my limited
internet searching skills, was not able to find enough information about the
history of median NFL salaries.<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
</div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">When
analyzing data, why do we often focus on the median instead of the average
(mean)?<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
</div>
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">The
fact is that the average (mean) is not always so average (typical).<span style="mso-spacerun: yes;"> </span>If we want to get a good picture of what a <i style="mso-bidi-font-style: normal;">typical</i> NFL player makes, we should
analyze median salaries.</span><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><o:p> </o:p></span><br />
<br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"></span><br />
<div align="right">
<table align="right" border="0" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="margin: auto 6.75pt; mso-cellspacing: 0in; mso-padding-alt: 0in 0in 0in 0in; mso-table-anchor-horizontal: column; mso-table-anchor-vertical: paragraph; mso-table-left: right; mso-table-lspace: 9.0pt; mso-table-overlap: never; mso-table-rspace: 9.0pt; mso-table-top: .05pt; mso-yfti-tbllook: 1184;">
<tbody>
<tr style="height: 0.15in; mso-row-margin-right: 5.6%; mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td colspan="3" style="background: rgb(23, 21, 143); border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 94.4%;" width="94%"><h2 class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: center;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><span style="color: white;">Indianapolis Colts – 2012 Salaries<o:p></o:p></span></span></h2>
</td>
<td colspan="2" style="background-color: transparent; border: 0px rgb(0, 0, 0); mso-cell-special: placeholder; padding: 0in;" width="5%"><div class="MsoNormal" style="margin: 0in 0in 10pt;">
</div>
</td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 1;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">Rk<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">Player<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">Salary<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 2;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">1<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Dwight
Freeney</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$19,035,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 3;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">2<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Antoine
Bethea</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$6,225,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 4;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">3<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Robert
Mathis</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$5,750,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 5;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">4<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Andrew
Luck</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$4,015,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 6;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">5<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Reggie
Wayne</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$3,500,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 7;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">6<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Adam
Vinatieri</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$3,200,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 8;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">7<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Antonio
Johnson</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$2,000,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 9;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">8<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Samson
Satele</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$1,866,667<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 10;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">9<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Anthony
Castonzo</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$1,818,431<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 11;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">10<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Cory
Redding</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$1,716,667<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 12;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">11<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Jerry
Hughes</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$1,691,250<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 13;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">12<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Winston
Justice</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$1,500,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 14;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">13<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Austin
Collie</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$1,417,388<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 15;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">14<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Donald
Brown</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$1,385,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 16;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">15<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Pat
McAfee</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$1,322,500<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 17;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">16<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Drew
Stanton</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$1,225,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 18;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">17<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Pat
Angerer</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$1,132,500<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 19;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">18<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Tom
Zbikowski</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$1,033,333<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 20;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">19<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Coby
Fleener</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$971,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 21;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">20<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Vontae
Davis</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$957,500<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 22;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">21<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Fili
Moala</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$873,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 23;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">22<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="mso-field-code: " HYPERLINK ";"><span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">Trai Essex</span></span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$825,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 24;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">23<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="mso-field-code: " HYPERLINK ";"><span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">Mewelde Moore</span></span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$825,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 25;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">24<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Jerraud
Powers</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$784,625<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 26;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">25<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Darius
Butler</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$770,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 27;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">26<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Mike
McGlynn</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$700,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 28;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">27<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Moise
Fokou</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$627,438<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 29;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">28<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Drake
Nevis</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$622,278<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 30;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">29<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Donnie
Avery</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$615,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 31;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">30<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="mso-field-code: " HYPERLINK ";"><span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">Justin King</span></span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$615,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 32;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">31<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Delone
Carter</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$571,301<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 33;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">32<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Dwayne
Allen</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$565,826<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 34;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">33<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Ricardo
Mathews</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$551,628<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 35;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">34<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Kavell
Conner</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$551,309<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 36;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">35<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Josh
Gordy</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$540,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 37;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">36<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Jeff
Linkenbach</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$540,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 38;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">37<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Seth
Olsen</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$540,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 39;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">38<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Matt
Overton</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$540,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 40;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">39<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">T.Y.
Hilton</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$521,750<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 41;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">40<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Sergio
Brown</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$490,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 42;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">41<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Cassius
Vaughn</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$490,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 43;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">42<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Joe
Lefeged</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$467,667<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 44;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">43<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="mso-field-code: " HYPERLINK ";"><span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">Kris Adams</span></span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$465,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 45;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">44<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Mario
Addison</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$465,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 46;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">45<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Martin
Tevaseu</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$465,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 47;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">46<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Joe
Reitz</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$455,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 48;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">47<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Vick
Ballard</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$426,140<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 49;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">48<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">LaVon
Brazill</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$409,670<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 50;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">49<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="mso-field-code: " HYPERLINK ";"><span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">Chandler Harnish</span></span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$401,474<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 51;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">50<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Mario
Harvey</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$390,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 52;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">51<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Justin
Hickman</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$390,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 53;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">52<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Dominique
Jones</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$390,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 54;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">53<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Nathan
Palmer</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$390,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 55;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">54<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Bradley
Sowell</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$390,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 56;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">55<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="color: blue; font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman"; text-decoration: none; text-underline: none;">Jerrell
Freeman</span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div align="right" class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">$375,000<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"></td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"></td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 57;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
</div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
</div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
</div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
</div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
</div>
</td>
</tr>
<tr style="height: 0.15in; mso-yfti-irow: 58; mso-yfti-lastrow: yes;">
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in 0in 0in 6pt; width: 10.98%;" width="10%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
</div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 50.22%;" width="50%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; mso-fareast-font-family: "Times New Roman";">Source: Fox Sports*<o:p></o:p></span></div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 33.22%;" width="33%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
</div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 3.42%;" width="3%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
</div>
</td>
<td style="background-color: transparent; border: rgb(0, 0, 0); height: 0.15in; padding: 0in; width: 2.18%;" width="2%"><div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: column; mso-element-anchor-vertical: paragraph; mso-element-frame-hspace: 9.0pt; mso-element-left: right; mso-element-top: .05pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly;">
</div>
</td>
</tr>
</tbody></table>
</div>
<div style="text-align: left;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">As an example, let’s look at the salaries for the 2012 Indianapolis Colts.</span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
</div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
</div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">This
team is somewhat of an anomaly because they have the highest-paid player in the
league (Dwight Freeney) <i style="mso-bidi-font-style: normal;">and</i> the
lowest total team payroll in the league.<span style="mso-spacerun: yes;">
</span>This makes the Colts an exaggerated microcosm of the NFL – a league in
which most players earn a high salary, but a few players earn an <i style="mso-bidi-font-style: normal;">insanely</i> high salary.<o:p></o:p></span></div>
<br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">So,
how much money does the typical<i style="mso-bidi-font-style: normal;"> </i>player
on the Colts earn? <o:p></o:p></span></div>
<br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">If
we use average salary (rounded to the nearest dollar) the answer is $1,469,115
– but is this really <i style="mso-bidi-font-style: normal;">typical</i>?<span style="mso-spacerun: yes;"> </span>43 players, representing78 percent of the
team, make less than this amount.<o:p></o:p></span></div>
<br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">The
average salary is artificially inflated by the top-earning players on the
list.<span style="mso-spacerun: yes;"> </span>If we take the top three salaries
(Freeney, Bathea, and Mathis) off of the list and recalculate the average, it
becomes $957,526 (again, rounded to the nearest dollar).<span style="mso-spacerun: yes;"> </span>Removing three players caused the average
salary to decrease by about 35%.<o:p></o:p></span></div>
<br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">It
seems more reasonable to use median salary and state that the typical player on
this team earns $622,278.<span style="mso-spacerun: yes;"> </span>This amount is
not artificially inflated by the top earners on the list.<span style="mso-spacerun: yes;"> </span>If we remove the top three salaries, the
median becomes $615,000.<span style="mso-spacerun: yes;"> </span>This number is
about 1% lower than the original median.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"></span><br /></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">If
the example of the Colts does not convince you that median is the better way to
go, consider this example:<o:p></o:p></span></div>
<br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">A
group of one hundred people decide to look in their wallets to see how much
money each person is carrying.<span style="mso-spacerun: yes;">
</span>Ninety-nine of the people are carrying $1 and one person is
carrying $1,000,000.<span style="mso-spacerun: yes;"> </span>The average amount
of money per person is $10,000.99 ($1,000,099 ÷ 100).<span style="mso-spacerun: yes;"> </span>Does this mean that the typical person in
this group is carrying over ten thousand dollars in his/her wallet?<span style="mso-spacerun: yes;"> </span>Of course not.<o:p></o:p></span></div>
<br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">In
situations like these, <i style="mso-bidi-font-style: normal;">median</i> gives
the best approximation of the typical case – the average is not really very
average.<o:p></o:p></span></div>
<br />
----------------------------------------------------------------------------<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
</div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><o:p>For Math Tutorials and Silly Math Songs, visit</o:p></span></div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
</div>
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; line-height: 115%;"><o:p><a href="http://www.onlinemathpro.com/"><span style="color: blue; font-size: x-large;">www.onlinemathpro.com</span></a></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 10pt; line-height: 115%;">(*Source:
</span><span style="font-family: "Times New Roman","serif"; font-size: 9pt; line-height: 115%;"><span style="mso-spacerun: yes;"> </span></span><span style="color: #0070c0; font-family: "Arial","sans-serif"; font-size: 9pt; line-height: 115%;"><a href="http://msn.foxsports.com/nfl/team/indianapolis-colts/salary/67048?q=indianapolis-colts">http://msn.foxsports.com/nfl/team/indianapolis-colts/salary/67048?q=indianapolis-colts</a></span><span style="font-family: "Times New Roman","serif"; font-size: 10pt; line-height: 115%;">)</span><br />
<br />
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MrWAGNEEZYhttp://www.blogger.com/profile/14636358345686928129noreply@blogger.com0tag:blogger.com,1999:blog-1628539622402145132.post-82611718476688271392012-11-20T17:01:00.000-08:002012-11-24T10:27:05.418-08:00Thanksgiving: NFL Salary Analysis<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;">As we
celebrate Thanksgiving and its many wonderful traditions, my mind is drawn to
three of the most commonly enjoyed institutions of the holiday.<span style="mso-spacerun: yes;"> </span>I like to refer to them as the 3 F’s:<o:p></o:p></span></span><br />
<br />
<div align="center" class="MsoNormal" style="margin: 0in 0in 10pt; text-align: center;">
<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;"><strong><em>Family, Food, and Football<o:p></o:p></em></strong></span></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;">The NFL fan in
me loves the fact that our Thanksgiving celebrations can include what I
consider to be the best of all professional sports.<span style="mso-spacerun: yes;"> </span>The math teacher in me is thankful for the
fact that sports give us so many opportunities to observe math being modeled in
real life.<o:p></o:p></span></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;">Since we’re
on the topic of being thankful, I think it would be interesting to analyze
something that most professional athletes can be <i style="mso-bidi-font-style: normal;">very </i>thankful for… salaries.<o:p></o:p></span></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;">In 1967, the
average NFL salary was about $25,000 and most players made less money than the
average working American.<span style="mso-spacerun: yes;"> </span>In contrast, last
year (2011) the average NFL salary was about $1,900,000 and every player made
quite a bit more money than the average American.<o:p></o:p></span></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;">So, exactly how
have NFL salaries grown over the last 45 years?<span style="mso-spacerun: yes;">
</span>Plotting average NFL salaries can help us analyze the trends:<o:p></o:p></span></span></div>
<a href="http://3.bp.blogspot.com/-wNK-jR-H20Y/UKwmwUJOf-I/AAAAAAAAACg/wegOMb7iKHw/s1600/Slide1.GIF" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="480" src="http://3.bp.blogspot.com/-wNK-jR-H20Y/UKwmwUJOf-I/AAAAAAAAACg/wegOMb7iKHw/s640/Slide1.GIF" width="640" /></a><br />
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;">It appears
that NFL salary growth was fairly slow at first.<span style="mso-spacerun: yes;"> </span>However, through the years, the growth rate
has increased dramatically.<o:p></o:p></span></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;">So, what
type of mathematical model best fits this data?<span style="mso-spacerun: yes;">
</span>Let’s begin by looking at a linear model.<span style="mso-spacerun: yes;"> </span>Using the first and last years in this study,
and letting the year 1967 correspond with <i style="mso-bidi-font-style: normal;">x</i>
= 0, we get the equation:<o:p></o:p></span></span></div>
<br />
<div align="center" class="MsoNormal" style="margin: 0in 0in 10pt; text-align: center;">
<span style="font-family: Calibri;"><i style="mso-bidi-font-style: normal;"><span style="font-size: 12pt; line-height: 115%;">y</span></i><span style="font-size: 12pt; line-height: 115%;"> = 42,613.64<i style="mso-bidi-font-style: normal;">x</i> + 25,000<o:p></o:p></span></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;">This equation
indicates that salaries started at $25,000 in 1967, and then increased steadily
at a rate of $42,613.64 per year after that.<span style="mso-spacerun: yes;">
</span>Here is a graph of the linear model:<o:p></o:p></span></span></div>
<a href="http://3.bp.blogspot.com/-uESTmnJjFyo/UKwnDXNcf5I/AAAAAAAAACo/j7QYoxbHQHw/s1600/Slide2.GIF" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="480" src="http://3.bp.blogspot.com/-uESTmnJjFyo/UKwnDXNcf5I/AAAAAAAAACo/j7QYoxbHQHw/s640/Slide2.GIF" width="640" /></a><br />
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;">In analyzing
the graph, it is clear that this model doesn’t fit the data points very well.<span style="mso-spacerun: yes;"> </span>Let’s see how an exponential model fits the points.<span style="mso-spacerun: yes;"> </span>Using a regression program to approximate an equation
matching this data (again letting the year 1967 correspond with <i style="mso-bidi-font-style: normal;">x</i> = 0) I came up with:<o:p></o:p></span></span></div>
<br />
<div align="center" class="MsoNormal" style="margin: 0in 0in 10pt; text-align: center;">
<span style="font-family: Calibri;"><i style="mso-bidi-font-style: normal;"><span style="font-size: 12pt; line-height: 115%;">y</span></i><span style="font-size: 12pt; line-height: 115%;"> = 38,128.42<i style="mso-bidi-font-style: normal;">e</i><sup>0.09461<i style="mso-bidi-font-style: normal;">x</i><o:p></o:p></sup></span></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;">This
equation is a little more difficult to interpret.<span style="mso-spacerun: yes;"> </span>It indicates that salaries started at
$38,128.42 in 1967 and then grew continuously at a rate of 9.461% per year. <span style="mso-spacerun: yes;"> </span>We may be tempted to think that this model is
not very accurate, since it indicates a starting salary that is much different
than what we know to be the case.<span style="mso-spacerun: yes;">
</span>However, when we analyze the graph we see that this model fits the data
quite well.<o:p></o:p></span></span></div>
<a href="http://2.bp.blogspot.com/-72gY-yVRsFk/UKwnXGtLw1I/AAAAAAAAACw/wMuyT7sUuCA/s1600/Slide3.GIF" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="480" src="http://2.bp.blogspot.com/-72gY-yVRsFk/UKwnXGtLw1I/AAAAAAAAACw/wMuyT7sUuCA/s640/Slide3.GIF" width="640" /></a><br />
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;">So, what
does it all mean?<o:p></o:p></span></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;">Nobody can
predict the future with absolute certainty.<span style="mso-spacerun: yes;">
</span>However, if we are going to make predictions about the salaries of NFL
players in the future, it would be best to use the model that most closely fits
the data that we have – the exponential model.<span style="mso-spacerun: yes;">
</span>(By the way, according to this model, the average NFL player will make
$9,212,715.57 in the year 2025.)<o:p></o:p></span></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;">Happy
Thanksgiving!</span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;">-----------------------------------------------</span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;">For Online Tutorials and Silly Math Songs, visit <a href="http://www.onlinemathpro.com/">www.onlinemathpro.com</a></span></span></div>
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-size: 12pt; line-height: 115%;"><span style="font-family: Calibri;"></span></span> </div>
MrWAGNEEZYhttp://www.blogger.com/profile/14636358345686928129noreply@blogger.com0tag:blogger.com,1999:blog-1628539622402145132.post-36209630269744679122012-11-15T17:19:00.000-08:002012-11-24T10:26:17.170-08:00Studying for Algebra TestsI often hear students and/or parents who want to know the best way to study for an Algebra test.<br />
<br />
Unlike tests in other subjects, success on Algebra tests generally doesn't hinge solely on one's ability to memorize facts. This explains why some students can experience success in other subject areas by cramming at the last minute, but fail miserably in Algebra.<br />
<br />
The fact is, Algebra tests require students to develop problem-solving skills and then use those skills to solve a variety of problems. On Algebra tests, students are given problems that are similar to problems done in class or assigned as homework, but that have been modified slightly.<br />
<br />
This doesn't happen as much in other subject areas. (In a U.S. History unit, for example, students may learn that the Declaration of Independence was adopted on July 4, 1776. When it comes time for the test, this date hasn't been modified... It's still July 4, 1776.)<br />
<br />
In preparing for Algebra tests, students need to work to ensure that their problem-solving skills are strong. They should feel confident that they can solve problems without assistance from a parent, teacher, or another student.<br />
<br />
For most students, this means... practice. I think it's great when students help each other work through homework problems. I also encourage students to seek help from teachers and/or parents. However, students who do not practice doing problems <em>on their own</em> often end up struggling on tests.<br />
<br />
(This explains why many students get good homework scores, but poor test scores.)<br />
<br />
When students ask me how to prepare for an Algebra test, I tell them to do these three things:<br />
<br />
1) Memorize vocabulary/math facts emphasized in the chapter.<br />
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2) Work through a review assignment. Get help from the teacher, a parent, or other students if you need to - but make sure your work and answers are correct.<br />
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3) Choose problems from the review assignment that you completed with assistance. Re-work these problems <em>on your own</em>, using the completed assignment as an answer key. You may need to do this multiple times for a problem, but eventually you will strengthen the skills (and confidence) that you will need to succeed on the test.<br />
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I realize that some students have legitimate issues with test anxiety. (By the way - text anxiety due to the fact that you failed to study/prepare is <em>not</em> a legitimate issue.) These students need to take additional non-mathematical measures to try to minimize their anxiety.<br />
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However, for most students, I find that following the above steps generally leads to success on tests.<br />
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One last note: "Success" is a relative term. Mathematics is one of the most unfair subjects I know of. Some students can ace tests with minimal preparation time, while other students put in hours of prep work and struggle to get a B or a C. I can't guarantee that my study tips will help you ace your Algebra tests, but they should help you increase your scores.<br />
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For Math Tutorials and Silly Math Songs, visit <a href="http://onlinemathpro.com/">OnlineMathPro.com</a>MrWAGNEEZYhttp://www.blogger.com/profile/14636358345686928129noreply@blogger.com0tag:blogger.com,1999:blog-1628539622402145132.post-38625241342174985172012-05-17T15:08:00.003-07:002012-05-17T15:25:10.938-07:00Solving Quadratic EquationsFirst blog... all right!<br />
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Did you ever wonder why a 2nd-degree polynomial is called "QUADRATIC"? Doesn't "quad" mean "4"?</div>
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And, are you having trouble memorizing the QUADRATIC FORMULA?</div>
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I did a quick intro & song on FotoFlo M (awesome iPhone app) for my students... yes - we do sing it in class.</div>
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The tune is "pop goes the weasel" ...here's the link:</div>
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<u><a href="http://www.thefotoflo.com/v/BByjh9S" target="_blank">Quadratic Formula Song</a></u></div>
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Caution: If a quadratic equation doesn't contain a 1st-degree term ("x" to the 1st power) then b = 0 in the quadratic formula.</div>MrWAGNEEZYhttp://www.blogger.com/profile/14636358345686928129noreply@blogger.com0