Even though a student could use this post as a “how to” guide for multiplying binomials, this is intended to be a “look for” post for parents who are working with algebra students at home.
Let’s look at a typical example…
Multiply (3x + 2)(x – 5)
Traditionally, there is one method taught – FOIL.
Multiply the First terms, then the Outer terms, then the Inner terms, and finally the Last terms.
There is nothing magical about FOIL. It’s just one way to organize work. The big idea is that each term in one binomial needs to be multiplied by each term in the other one. (Think “sharing is caring!”)
But there are different ways to organize work that may make more sense to the person doing the work. Many times, a student thinks they need permission to do it a different way.
(If your teacher isn’t flexible about the approach, show it to them privately first. Most will give full credit if the mathematics is correct. What may seem like “inflexibility” is really just an attempt to avoid confusion.)
It’s more important to understand the mathematics under the algebra than to follow a set of steps without understanding what is going on.
If a student can write out their process and can correctly justify their thinking when asked “why did you do that?”, let them find a way to organize the work however it makes sense to them.
Teaching “rules” about how to do things implies that there is ONE way to do a thing, which…
1) isn’t true, and
2) actually prevents learning mathematics.
Most students stall out when they need to apply the FOIL algorithm to multiplying polynomials with more than 2 terms.
A student who doesn’t understand what FOIL is really doing won’t attempt a problem like (x+5)(3x 2 + 2x + 4) without someone else showing them how to do it.
A better outcome comes when they were encouraged to organize multiplying binomials differently, asked to explain why they think it will work, and then praise the creativity and grit for figuring out a “new” way to solve the problem. An empowered student will trust her intuition and attempt a new problem on her own.
Here are some approaches I’ve seen done when students forgot FOIL and had to figure out another way…
A distributive property approach…
Think of (3x + 2) as a single number and “distribute” it to the x and to the -5.
(This one gets a bit tricky when they need to distribute “backwards” on the second line. If your student is using this approach, watch for errors here.)
The vertical method of organizing work is familiar to students because it looks like the way they learned how to multiply 2-digit numbers in elementary school.
The old skill from earlier grades:
Same technique applied to (3x + 2)(x – 5):
Let’s “level up” now…
Multiply (x + 5)(3x 2 + 2x + 4)
What happens when a student who memorized FOIL is now asked to move up to the next level…
However, students who have been empowered to organize their work in the way that makes sense to them, usually attack this problem on their own.
They may still make some mistakes and that’s ok. Mistakes are attempts and that means there will be a successful solution at some point soon.
What we want to teach kids to avoid is the habit of up before starting. ANY attack strategy is better than giving up before trying something.
I’m sure there are more than two ways to multiply a binomial and a trinomial, but let’s compare the two methods we’ve been looking at here.
First, a distributive property approach and then a vertical approach…
Both approaches arrived at the same solution.
There is a bigger benefit to not forcing kids to do their work one way – their approach may shine light on their misconceptions. For example, when forced to use FOIL on a problem like
(3x + 2)(x – 5),
they might multiply the first terms, 3x and x, the last terms, 2 and -5, and arrive at their solution of 3x 2 – 10. When I see that, I suggest the vertical approach and that will help them see why there is a middle term.
Thoughts? I’d love to hear from you!
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Thanks for reading!