# ZeroSum Ruler (home)

## Blogging on math education and other related things

### (a+b)^2 = a^2 + b^2, right? (video)February 11, 2012

Disclaimer: Not “solved”. Simplified!

(click to view video)

### The Distributive Property (“FOIL”) Through PicturesDecember 15, 2011

The transitive property was always my favorite as it could be applied to so many situations.  I like chocolate, there is chocolate in those cookies, so I like those cookies.  Totally useful.

But a close runner up to this cookie property has got to be the Distributive Property.  With strange rules of “first, outer, inner, last”, I liked its mystery.  I could multiply two things together with no mention of a multiplication sign and somehow it meant something.  Something big.  I was doing real Algebra now.

It wasn’t until I became a teacher that I really had to think about what was being done.  My students would make mistakes when “F.O.I.L.ing” (I do not like this acronym.  What if one piece is a trinomial?) and I would attempt to explain what was happening.  It’s difficult to explain something that has been taken for granted for 15 years.  But as I made my way through my graduate program where being able to explain math was seen as the most important, I began to rethink this important property.

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The Example:

I always like to start with a concrete example.  Let’s take the problem “14 x 7”

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“14 x 7” is no easy problem for most of us as neither of these numbers is easy to work with.  To begin, let’s look at “14 x 7” as a geometric area in a picture:We can easily count up the small rectangles to find how many there are, though that would take time and leaves a lot of room for error.  Or, we could break the picture down into smaller pictures to make it easier to work with:

Here, we’ve broken “14 x 7” down into (10 + 4) x (5 + 2), or simply (10+4)(5+2).  Is this form familiar?

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Now we can see that “14 x 7” = (10 + 4)(5 + 2).  And now we can simply use multiplication to find the areas of the different colored pieces and add them up:

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10 x 5 = 50

10 x 2 = 20

4 x 5 = 20

4 x 2 = 8

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50 + 20 + 20 + 8 = 98!  And in fact, 14 x 7 = 98.

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The Generalization:

Now let’s make a generalization that we can apply to other similar problems:

Here, we’ve replaced all of the numbers with letters and we can rewrite the problem as:

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(a + b)(c + d)

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Using the method we used before, we multiply each colored piece to find its area and then add up all the areas to find the total:

(a)  x (c) = ac

(b)  x (c) = bc

(b)   x (d) = bd

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The area is: ac + ad + bc + bd  !

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Not the prettiest of answers, but done correctly.  Using this model, can you multiply (3x + 4)(5x + 2)?

We’ll use the same picture because “x” can stand for any number at all.

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We have:

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(3x + 4)(5x + 2)

(3x)(5x) = 15x2

(3x)(2) = 6x

(4)(5x) = 20x

(4)(2) = 8

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Putting the pieces together, we have the trinomial:

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15x2 + 26x + 8 !

The Error:

The biggest error I have seen with the Distribute Property is forgetting to multiply a piece or two.  Students sometimes will answer:

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(3x + 4)(5x + 2) = 15x2 + 20x + 8

Can you see what they forgot?  Can you imagine what other mistakes could be made?

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If you always remember the area of each piece, you will be The Best Distributor and Master of the Distributive Property!