MathChat

This is Part 6 in my series for helping students make fewer mistakes in algebra.

In this post I show how — by using the double-slash notation — students can avoid mistakes when factoring by grouping.

Let's Reduce Mistakes in Algebra!

Factoring by grouping confuses many students at first, but the double-slash helps cut through the confusion.

Take the problem:   6x² + 23x + 20

The first step, of course, is to find factors of 120 (6 x 20) that add up to 23. Once you find that those factors are 15 and 8, you rewrite the expression like this:

        6x² + 23x + 20

=              6x² + 15x + 8x + 20

But notice how much clearer it would be — and how much easier to proceed to the next step — if you use the double-slash between the second and third terms, like this:

              6x² + 23x + 20

=             6x² + 15x   //   + 8x + 20

Just by using the double-slash, the expression splits into two sub-expressions, each of which you factor individually. So the double-slash not only gives students bits of information that are more “bite-sized,” it also clues them in to what to do next. In this way, the notation itself prompts students to perform the next step.

The rest of the work looks like this:

=             6x² + 15x  //  + 8x + 20
=            3x(2x + 5)   //  + 4(2x+5)

At this point the double-slash, by pointing out the symmetry in the expression, highlights the two parenthetical terms  (2x + 5), which students must factor, like this:

3x(2x + 5)  //  + 4(2x+5)

=               (2x + 5)(3x + 4)

And the expression is now factored.

Help students get used to this process by giving them the following practice problems:

PRACTICE:

a)   2x² – 5x – 12

b)  3x² – 7x + 2

c)  4x² + 13x + 3

d)  5x² + 14x – 3

e)  6x² – 7x + 1

ANSWERS:

a)   2x² – 5x – 12  =  (2x + 3) (x – 4)

b)  3x² – 7x + 2  =   (3x – 1) (x – 2)

c)  4x² + 13x + 3  =  (4x + 1) (x + 3)

d)  5x² + 14x – 3  =  (5x – 1) (x – 3)

e)  6x² – 7x + 1  =  (6x – 1) (x – 1)