The Pythagorean Theorem Animation October 31, 2012

I’m embarrassed to admit this, but I had never realized that the Pythagorean Theorem’s a^2 + b^2 = c^2 actually meant “the area of the square made by the side length C is the same as the sum of the areas made by the squares made of the other two sides’ side lengths” until I was in graduate school.  When I saw it for the first time in Professor Oliver Knill’s class, aside from his animation being totally bad ass, I was totally blown away by the realization of what I had never realized.

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So here are some animations, starting with Oliver’s, that show how awesomely dynamic Pythagoras’s theorem is:

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Wanna be a Math Hero? Answer these questions! September 17, 2012

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These Math students need YOUR help.

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If you’ve checked all recent posts on Facebook, refreshed your Twitter page until it can be refreshed no more, all of your Pinterest friends seem to be on vacation and your email is all read, why not answer some Math questions?

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http://www.algebra.com/Answer/

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I’ve been addicted to this site since last night, which in 2012 terms is an eternity.  All I can imagine are kids all over the US toiling away at their Math homework, one hand on head, one wrapped around a pencil, foregoing food, sleep, showering, just to get tomorrow’s math work complete in time for their teachers to put a small check in the corner.   Hey, maybe a few teachers are stickerers, I don’t know.  Personally, I’m a grape-flavored stamper.  So here I come to the rescue!  The THANK YOU! emails are cool to get; I do feel a bit like a hero today.

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Questions range from “Plz help me graph y = 45x + 40” to “What is the square root of 1 – i?“  So try it out!  It’s a great way to put that advanced degree to good use!

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The Coolest Math Class Ever. Seriously! June 6, 2011

I was lucky to get into Oliver Knill’s class Teaching Math With a Historical Perspective, one of the choices within the Harvard Extension School’s ALM in Mathematics for Teaching program.  It changed the way I think and the way I teach.  He explained complex topics, such as code breaking and non-verbal proofs, with such ease.  He was inspiring and made me look at math in a way I had never before – from a historical perspective!  His site is worth checking out, especially if you can not get into his class!  (Click on the red circle to go to Oliver’s site)

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My Harvard Math for Teaching Thesis: Complete! And ready to share… March 20, 2011

After many many years of jumping through many many hoops, I am finally graduating with my MA in Mathematics for Teaching in May.  My thesis, Negative Number Misconceptions in High School: An Intervention Using the ZeroSum Ruler is right now at the printers being printed and bound.  I don’t know about you, but that instantaneous feeling of relief after taking a final exam or passing in a final paper stopped hitting me sometime in college.  So now, I’m just feeling a bit burnt out.  OK, completely burnt out.  But I’m sure it will hit me soon since it kind of needs to; I need to now get in a post-Bach program to get my Initial teaching license.  I like to do things backwards.

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So here it is for download!  For all to read!  Or maybe to just glance.  In my study, the ZeroSum ruler proved effective in reducing eleventh grade error on integer addition and subtraction problems (especially with negative integers).  If I wasn’t so burnt out, I’d want to test it with younger kids.  Imagine how our world would be if my eleventh graders actually mastered integers when they learned them in, and only in, 7th grade.  But that’s in my thesis.]

 

Why we add fractions the way we do… a visual tour January 15, 2011

Why do we add fractions the way we do- by getting the common denominator?  A legitimate question!  The following is a visual explanation of why we need to do so…

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When we add two fractions with different denominators, we have been taught to “find the common denominator”, then add the numerators only.  But why not add the denominators too? 

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Let’s try to add 2/7 + 3/5.  We know from our standard algorithm that we would change both denominators to 35, then change the numerators by making sure to keep the ratio between the numerator and the denominator in tact.  We’d end up with 10/35 + 21/35 = 31/35. 

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But how can we see why this works?  Let’s first look at both as pictures: 

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The picture on the left represents “2 out of 7”, and the picture on the right, “3 out of 5”.

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To add means to combine, and fractions- with the exception of improper fractions- represent amounts less than one.  So, we want to combine these two shaded regions into one, or, if we can make more than one, we want to see how many “ones” we can make.  But the shaded bars aren’t the same size.  And how big is “one”?

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In the algorithm we’d “find the common denominator”, but what does this mean and look like?  It means we have to change the look of these two fractions so that their numerators represent portions of whole broken up into the same amount of pieces.  To do this, we break the picture on the left up into fifths and the picture on the right up in to sevenths. 

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Now our two fraction pictures are broken up into the same amount of pieces, and each piece is the same size. 

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To justify this, let’s look at the dimensions of each area…

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The fraction picture on the left has an area of (7×5).  The fraction picture on the right has an area of (5×7).  Because of the commutative property, we know that 7×5 = 5×7, so both fraction pictures have an equal total area (denominator), and that area is 35 spaces.  For this same reason, the shaded spaces in both pictures are also all the same size.   

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But now we have 21 (out of 35) shaded pieces in the fraction picture on the left and 10 (out of 35) shaded pieces in the fraction picture on the right.  Can we do this?  Is 21/35 the same as 3/5?  Is 10/35 the same as 2/7?

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Because we can break a “whole” up into as many pieces as we want, these fractions are equal.  For example, if two people both had a liter of soda each, one could give small cups of soda to 21 friends, and the other friend could give larger cups of soda to 10 friends.  If both empty their bottles, both gave out the same amount of soda despite giving it out to different numbers of friends. 

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It’s the same with these two fractions.  Once we have set shaded regions (3/5 and 2/7), we can break these regions up into an infinite number of pieces and still have 3/5 and 2/7.

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Now we can begin adding one to the other.

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A total of 31 spaces are filled in when we take the shaded spaces from the fraction picture on the left and add them to the picture on the right.  So, 31 out of 35 spaces are now filled in, or “31/35”.

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If we had used the standard algorithm, we would have added the numerators of the fractions (after we found the common denominators) to get this 31. 

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2/7 + 3/5 = 10/35 + 21/35 = 31/35, or 4/35 less than a whole.   

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For why we need to first find the common denominator, see two or three posts down…

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When does -22 + 5 = -27? December 6, 2010

My graduate thesis is a study of the long-term effects the ZeroSum ruler has on eleventh grade student understanding of negative integers.  By eleventh grade, students should easily be able to answer “-22 + 5 =”, but on a diagnostic test given to 57 students, 40.35% of the students answered this problem incorrectly.  Why does this matter?  It matters because it shows that students did not learn the relationship between negative and positive numbers in elementary or middle school.  By the time they get to me in eleventh grade and need to be fluent in equation manipulation, answering “-22 + 5 = -27″ is a real problem. 

 

My thesis was set up the following way:

1: Diagnostic test: eight simple sums and differences of integers  (ie: ’22 + 5=”) without a ZeroSum ruler or calculator

2: Introduction to the ZeroSum ruler with examples

3: Three activities, spaced out over 2 weeks,  using the ZeroSum ruler

4: A post test within days of the last activity (no ZeroSum ruler or calculator)

5: A delayed retention test one month after the last activity (no ZeroSum ruler or calculator)

 

Because the attendance rates of students in Boston Public Schools is not the best, especially by the 11th and 12th grades,  a subgroup of 31 students was identified who took the diagnostic test, participated in at least 2 of the 3 activities with the ZeroSum ruler, took the post test, and took the delayed retention test.  The data shows a 62% decrease in student error from the diagnostic test to the delayed retention test because of the ZeroSum Ruler!  These results indicate that the ZeroSum ruler works to improve student comprehension long-term even without the ruler.

 

Pretty exciting stuff.

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