The formal statement of the distributive law should read as follows:

If a, b, c and d are numbers, or algebraic expressions (same thing really) and b = c + d then ab = ac + ad

It is a by-product of the law that it tells you how to expand an expression with a bracketed factor.

In any case, what’s the big deal ?

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Filed under abstract, algebra, arithmetic, language in math, teaching

Tagged as abstract, addition, arithmetic, cynical, distributive, law, multiplication

It’s a big deal to K-12 teachers and students who have never thought of multiplication being represented by an area. It helps those first learning how to multiply numbers with 2 or more digits. It reinforces place value. It helps the young algebra students understand how to multiply (x + 3) (x – 6) without using the FOIL (first, outer, inner, last) algorithm, which makes no sense to them. It’s not that the FOIL method is bad, it’s that students memorize it with no understanding of the fact that it is the distributive property used twice. It helps the algebra students factor trinomials that can be factored. It helps algebra students complete the square physically, which I’m sure historically came well before the algebraic algorithm for completing the square (since geometry predates algebra). Most students are helped tremendously by visual representations of mathematical ideas.

My goal is to make math understandable to those who see it as a foreign language. I’m a teacher first, and a mathematician second, if I can be considered a mathematician at all. I majored in mathematics in college because it came easy to me. I never aspired to become a mathematician. I really never aspired to be anything. I fell into teaching through a series of serendipitous accidents. I have grown to be passionate about math education for all, not for the ones who “get it”, like I did. As a result, I think I “get it” now better than I ever did before.

“It’s a big deal to K-12 teachers and students who have never thought of multiplication being represented by an area.”

It really amazes me that teachers of math at any level cannot visualize multiplication in terms of area, or even in terms of rows and columns of dots. It is SO basic ! Also it ties in with the (not always approved) definition of multiplication as repeated addition.

Check this out:

http://mathwithbaddrawings.com/2014/12/03/the-sixth-sense-for-multiplication/

The point of the gif was to show that the distributive property is “obviously” applicable to numbers, when you see things as areas.

It is even possible to show that (-1)x(-1) = 1 using (a – 1)x(b – 1) when viewed as a chopped up area.

Regarding FOIL, I had never heard of it until I got into math blogging here in the Caribbean USA. I am pretty sure that I learned about expanding brackets (parentheses) with the area model, and that was a LONG time ago.