Factoring Polynomials with Cheese and Crackers

: Modeling Bejeweled scores with Base10 blocks.

Math wasn’t my strongest subject when I was a grade-school student. It was full of mysterious and unintelligble machinations with numbers that made no sense to me. Why do I have to find a common demonimator? Why does that number change signs when we move it to the other side of the equal sign? How come I keep adding zeros in long division? Why does counting digits behind a decimal places in a multiplication equation give me the number of decimals in the product?

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It was all so much magic to me inasmuch as I couldn’t understand what was happening even though I could see it there in front of me. If nothing else, it has made me a much stronger Math teacher today because I understand what kids are thinking, I know what questions they have about processes and my goal is to remove the magic and mystery of math while retaining the marvel at what math can do for us. Math is the language with which we describe patterns.

As a parent, I love pointing out and creating patterns with the boy. When you strip away the magic shortcuts we were taught in solving equations and focus on what it means in nature, we can expose our children to very advanced topics from a very young age.

Take the humble Cheerios,

how many cheerios are there in this group? (counting, subatizing)
if I add two more, how many are there now? (addition)
ooh… you ate three of them, how many are left (subtraction)
let’s make a rectangle (arrays)
how many around the outside? (perimeter)
how many all together (area)
let’s count them all by row: 3, 6, 9 (multiples)
you have 3 rows of 5 that’s 15! and 5 columns of 3 – that’s 15 too (commutative properties of multiplication)
let’s share those with mommy – one for you, one for me, one for mom… how many each? (division)
can you make a full square with those 15? no? what if you have one more? yes! 16 can make a square (square numbers)
how many are on each side of that square? 4 is the root of that square! (square roots)
how many do you have to eat to make the next smallest square?

It isn’t hard to see that simply playing, talking, exploring patterns, while using the vocabulary of math will lead to significant understanding. The boy had a hundred grid poster at the foot of his bed since he was one year old – at bed time we’d count across, read columns, and find all the numbers that met certain criteria (where are all the numbers that end in 5? What about all the ones that start with 3?) Now he has a multiplication grid. (Proud dad moment) A couple weeks ago, at age 8, the boy pointed out to me in his own language, that the sum of sequential odd numbers starting with 1 always results in the square of the number of odd numbers in the equation. For example:

1 + 3 = 4 there are two odd numbers, 2 squared is 4

1 + 3 + 5 + 7 + 9 = 25 there are five odd numbers, 5 squared is 25

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 = 169 there are 13 odd numbers, 13 squared is 169

Cool, eh?

Elsewhere I posted about the cumulative effects of this kind of interaction. Over time, as you help your child recognize patterns, and he start to recognize them on his own, the understanding will come. In school, when the teacher introduces new concepts, he will already have some experience with that language, manipulating objects following those patterns.

How far can you take this? How about factoring polynomials with cheese, crackers, and pretzels!

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This entry was posted on Sunday, December 18th, 2011 at 10:52 am and is filed under Learning Ideas, Numeracy. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

2 Comments

Bon Crowder says:

May 15, 2012 at 3:38 am

Great article, Miles!

And I’m inspired to get a 100 grid poster. (How is it that we don’t!? Daughter is almost 3 and I’m a hard core math mom. #badmom)

Although I have to say that I really don’t understand the factoring picture. I love that it can be done, but so far I’ve yet to figure out how people factor with pictures. Heading to think on it now.

Thanks bunches!
Milesmac says:

May 19, 2012 at 10:33 am

Thanks for the comment, Bon! I’m working on an explanation for the factoring process. The Boy, at this point, doesn’t quite get what it all means, but he can can play with the manipulatives, create the array and determine the factors. Connecting math processes with with real objects is my goal. The rest will come cheers!