A geometric look at x^3 – y^3

Yesterday my older son started the section about factoring sums and differences of cubes in Art of Problem Solving’s Introduction to Algebra book. After he did some problems we talked about a geometric way to understand how to factor x^3 - 1.

 

In the evening we looked at x^3 - y^3:

 

It is neat to see these algebraic identities appear geometrically. The next thing we are going to look at is a geometric way to understand how x^3 + 1 factors.

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Comments

2 Comments so far. Leave a comment below.
  1. When I saw that you made these videos about making geometric representations of equations and their factors I knew that as soon as I got time I would sit down and really study them. Just about a year ago, @hedge (I think) wrote a post about visualizing what equations LOOK like: reading her post was the first time that I understood that these kinds of equations could be represented with blocks. (…just to give you some context about this, it has only been a few years since I realized that all equations can be graphed, a fact which amazed me).

    So I am sitting here studying your videos and really enjoying them.

    I had a hard time sorting out how to transition from the representation of the block arrangement that you started out with to the blocks that represent the factored version of the equation. So (naturally) I started drawing pictures…

    Finally, the best way for me to make sense of it was rearrange the blocks to represent the factored equation. I justified this by thinking that I would draw a 3 x 4 array differently than a 6 x 2 array, though recognizing these are equivalent quantities, factored differently.

    After you factored your x^3 +1 equation, then said that x=3, your equation was now 4(3^2 – 3+1) . This factored version of the equation now seemed to tell me to draw 4 stacks of 3^2 from which point it seemed easier to me to subtract 3 from one of the top squares, then add back 1.

    Wondering what your thoughts are on creating a different arrangement for the factored equations. I am going to keep playing around with this… having a great time sorting this out.

  2. oops. should have said, “this factored version of the equation now seemed to tell me to draw 4 stacks of 3^2 from which point it seemed easier to me to subtract 4 x 3 ( the whole top of the stack) and then add bacck 4 x 1.

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