Saturday, December 26, 2009

Math visualization: (x + 1)2

You may remember this from high-school algebra (or perhaps earlier for some). Expand (x + 1)2 using the FOIL method.

(x + 1)2 = (x + 1)(x + 1)
= x2 + x + x + 1
= x2 + 2x + 1

A neat way to visualize this equality, and hopefully help remember the factorization of the resulting polynomial, is to look at how the pieces fit together.


When they're arranged in this way, it's easy to see that the pieces squeeze together to form a larger square that has a length of (x + 1) on a side, proving the equality.


Related posts:

Six Visual Proofs
Multiplying Two Binomials

16 comments:

Tim Kington said...

Visualizing math like this is something I've always done. It always used to surprise me when I would find out that other students were memorizing stuff like this instead of really understanding it. I think most of math should be taught in a visual way like this. You get a visceral understanding of FOIL, the difference of two squares, why multiplication as it's taught in 3rd grade works, etc.

Bill the Lizard said...

Tim,
I was a memorizer when I first took Algebra in high school. That changed the following year when I took Geometry and a light-switch went off for me. Things suddenly started to make sense. It was like that moment that many of us have when we take Physics, learn Bohr's model of the atom, and realize that this is something we should have been taught before we were ever allowed to take Chemistry.

I'll be tutoring my son in Algebra next semester, so I'll be sure to come across a lot more examples like this one. I'll post them when I do.

Jeff Moser said...

Great picture! It reminds me of the intuitive approach that betterexplained.com often takes.

Bill the Lizard said...

Jeff,
Thanks for the encouragement! I'll definitely be trying to add more graphics to my posts, since I find that I understand things better when they're presented visually.

You also reminded me, I need to add Better Explained to my blog roll.

Mark said...

I love it... My dad taught me that trick many many years ago and I ended up using it primarily for the reverse approach (factorisation). It's a bit sluggish being a brute force but I usually found it easier than polynomial division for the sort of problems we were given in high school.

Another favourite trick was the 'magic' triangle for the A = BC type equations. Draw a triangle like this (http://imagebin.ca/view/v05xNO.html) and to find the missing unknown just cover it and the equation is displayed.

Ted said...

Just found this blog a couple weeks ago, and it is already one of my favorites.

This diagram can easily be relabeled to demonstrate (x+y)^2, or extended to demonstrate (x+y)^3.

Bill the Lizard said...

Mark,
Yes, this is definitely easier than polynomial division if you recognize that the polynomial fits the right pattern. Of course, my teachers in high school and college always made sure at least one problem on every test required that the division be done long hand. :(

I can't believe I've never seen that magic triangle before. It's so simple! Thanks for sharing it.

Bill the Lizard said...

Ted,
Thanks for reading. I should have more stuff like this pretty soon, and I plan on increasing my SICP rate. My goal is to finish the book in the first half of the new year.

I had also realized that this diagram could be extended to cover (x+y)^2 shortly after I posted it. I hadn't thought to try (x+y)^3, though. Thanks pointing it out.

DiMaN said...

In Montessori school there are a lot of such visual representations for math.

Check out this post in my blog about it.

Ashley Jennings said...

Bill, love it! So simple and understated--yet so undeniably correct. You can see it and touch it now. Goodbye rote memorization--hello Mr. Lightbulb. Have you considered teaming up with the Jason Project? A dynamic duo you two would be. Like Batman and Robin.

Bill the Lizard said...

DiMaN,
I only recently heard of the Montessori school when it was mentioned on Hacker News. Thanks for sharing the link to your article about it.

Bill the Lizard said...

Ashley,
I'd never heard of the Jason Project before, so thanks for mentioning it. I'll have to look into it further to see if there's something I can contribute.

Kartik said...

This reminds me of the "proof" that Gauss came up with for 1+2+...+n=n*(n+1)/2

Just write the numbers in ascending and descending order in two rows. Each column adds up to (n+1). The formula is obtained naturally.

Anonymous said...

((x+1)x2)+1 = 9 squares

Anonymous said...

Please how can I solve (x + 1)12

Thanks

Anonymous said...

What about 2- (x + 1)?
The answer given is -x +1