Math Mondays: Some Murderous Fun with Pascal's Triangle

I am ready for a vacation.  I don't mind saying it.  I am exhausted, and my creativity is pushed to its limit.  With very little sleep and even less time, I am sharing a fun little trick I recently used just to add a little variety.

My students love the Murderous Maths series.  The books are full of a variety of math concepts told through engaging stories and many fun math tricks and observations.  One of my favorites appears in The Phantom X, a book primarily focused on Algebra topics.  This little gem uses Pascal's triangle (which I love for probability) to show patterns coefficients (p. 100-107 in my book).  Let me explain.

Let's say we want to find (a-b)^2.  That's pretty easy.  We just expand it with our favorite method and we get 

a^2 -2ab + b^2.  

Now, if we try (a-b)^3, we get 

a^3 - 3a^2b +3ab^2 - b^3.  

Do (a-b)^4 and soon students see the pattern in the powers of a and b in each term.  The "a exponent"  goes down by 1 with each term and the "b exponent" goes up by 1.  

The coefficient pattern is less clear. But as author Kjartan Poskitt points out in The Phantom X, Pascal's Triangle can give us the coefficients. Look at Pascal's triangle and the sums side by side.  See it?

Now, I can give my students (a-b)^8 or even (a-b)^12, and they can expand it without even multiplying.  

While this certainly doesn't make my students' math lives easier on a day-to-day basis, it's a fun little trick.  Plus, I love to share patterns in math.  There are so many!  And so much of math is in finding these patterns.  For an added challenge ask students to prove why this always works.  

Math Mondays is a bi-weekly blog post (2nd and 4th Monday of each month) sharing tips, ideas, resources, and products for teaching math.  If you have questions or think there is something I should include, you can leave me a message in the comments section below or at the store in the question and answer section.



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