When I left off yesterday, I pointed out a certain kind of algebra expression that tends to lead students to make mistakes. It was an expression like this:

8 – 2(3x – 7)

I pointed out that students often mistakenly combine the 8 and the 2, to get this:

= 6(3x – 7)

= 18x – 42

The theme of these posts is: how to help students avoid mistakes in algebra.

The solution to this problem, I have found, is the following:

**Use a mark that show students what gets combined and what stays separate.**

Of course, we as teachers are trained to tell students to use the order of operations, which mandates that students multiply before adding. And while that “should” work in this problem, it often does not, largely because students don’t see the “implied” multiplication between the 2 and what follows in parentheses. So I improvise, adding a notation that nudges students in the right direction.

I use a “double-slash” notation like this: // to show which terms stay separate. I like the double-slash because it looks definitive, appears significantly different from the single-slash fraction mark, and it’s quick and easy to make — just two quick flicks of a pencil.

Using the double-slash, here’s how you’d proceed:

8 – 2(3x – 7)

= + 8 // – 2(3x – 7) [notice also that jotting the + in front of the 8 enhances clarity]

Then students continue:

+ 8 // – 2(3x – 7)

= + 8 // – 6x + 14

Then use the double-slash again, this time to group like terms, like this:

+ 8 // – 6x + 14

= – 6x // + 8 + 14 [variable terms on the left; number terms on the right, in preparation for writing answer in descending order]

Finish like this:

– 6x // + 8 + 14

= – 6x + 22

Voila. All done, with no confusion and with clarity of thought.

Tomorrow: why this approach works, and how it could be used in other situations.

Hmmm…why not simply place a bracket, or another set of parentheses, around the expression as follows?

= 8 – [2(3x – 7)]

= 8 – [6x-14]

= 8 – 6x + 14

= 22 – 6x or -6x+22

Students seem to understand the distributive property even with the mistake you showed since they multiplied 6 timers the expression 3x-7.

Have you tried using additional parentheses or brackets?

Hi Dave. You are right that brackets would also work, in this instance. The problem, though, is that brackets do not work in a wide variety of situations in which many students need to separate certain terms, as you will see as I develop the idea of the double-slash further. Thanks for your response, though. Always interesting to hear from other math teachers.