How to Decrease Algebraic Mistakes – Part 6

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.

No Mistakes

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)

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2 Responses to “How to Decrease Algebraic Mistakes – Part 6”

  1. zainou says:

    I find the formula, brilliant and applicable! Its fun because i have not tried a mathematical solution for so long and I could still remember!

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  2. Josh Rappaport says:

    Hi Zainou, Glad that you liked the post and the procedure for factoring by grouping. Does the double-slash notation make it easier to simplify the expression for you? It sounds like you’re in touch with teachers in your country. Feel free to share these posts with teachers in general, math teachers in particular. Thanks for reading!

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