The Distributive Property (“FOIL”) Through Pictures December 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.

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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

(a)  x (d) = ad

(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 !

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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!

 

 

 

 

 

Freshmen Everywhere Dream…. of FOIL? – Video June 17, 2010

All too common math mistakes! June 16, 2010

 

One of my grad school professors called the mistake (a+b)^2 = a^2 + b^2 the “freshman dream”.  I guess he meant freshmen in college, but freshmen in high school make the same mistake.  Come to think of it, a lot of people make the same mistake.  What other mistakes are common in Algebra?

 

Here are just a few I can think of:

 

x = 1.     NO.  Well, sometimes, but definitely not always.

  

-7 + 5 = -12.  NO.  This one’s a never and the reason I developed the ZeroSum ruler.

  

(a+b)^2 = a^2 + b^2.  NOPE.  There’s a middle term in there.  Write it out.  Don’t be lazy.  Find it. 

  

1/2 = 2.  NO.  “1 out of 2″ is not 2.

  

(x + 2) + (x - 5) = x^2 – 3x - 10.  NO.  This is an addition problem, not a multiplication problem.

  

(x+2) – (x - 5) = -3.  NOPE again.  That little – sign is a -1 in disguise and needs to be distributed.

  

(2/5)(7/3) = ….<blank>….  NO.  Multiply the numerators together and the denominators together to get 14/15.  Easy!

  

“Thirty percent of 140 = 140/.3″.  NO.  You’re going to get a huge number.  “OF” means to multiply, not divide.

  

|-5| = 5, so |5| = -5.  NO.  Absolute value is always a positive number.  It represents a distance.  Even if you walk backwards, you’re still moving some distance.

  

“find x^2 if x = -5″.  Answer: -25.  NO.  Jut because the TI-83 says -25 doesn’t mean it’s right.  Calculators use PEMDAS, which states multiplication comes after exponents.  Remember a few mistakes back that the little – sign is a -1 in disguise.  It’s the same here.  If you put “-5^2″ into the calculator, the calculator will square 5 and then multiply by -1.  To square the entire -5, use ( ).  Or just remember that a negative number raised to any even power will always be positive. 

 

(x^4)(x^5) = x^20.  NO!  This is a common mistake kids make.  But if they start by thinking about what “x^4″and “x^5″ mean, it’s easy to see that (x^4)(x^5) expands to (xxxx)(xxxxx).  Parenthesis right next to each other tell us to multiply, so you end up with (xxxxxxxxx) or x^9 or x^(4+5).

 

(x^4)^5 = x^9.  No, too.  x^4 = (xxxx) and the “^5″ means you have 5 (xxxx)’s.  So (x^4)^5 expands to (xxxx)(xxxx)(xxxx)(xxxx)(xxxx).  Parenthesis right next to each other still say “multiply” so we have 20 x’s in a row, or x^20.

 

I know there are more.  What did I miss?