Difference of Two Squares (Part 1)

In Algebra I, we drill into student’s heads the formula for the difference of two squares:

x^2 - y^2 = (x-y)(x+y)

While this formula can be confirmed by just multiplying out the right-hand side, innovative teachers can try to get students to do some exploration to guess the formula for themselves. For example, teachers can use some cleverly chosen multiplication problems:

9 \times 11 = 99

19 \times 21 = 399

29 \times 31 = 899

39 \times 41 = 1599

Students should be able to recognize the pattern (perhaps with a little prompting):

9 \times 11 = 99 = 100 - 1

19 \times 21 = 399 = 400 - 1

29 \times 31 = 899 = 900 - 1

39 \times 41 = 1599 = 1600 - 1

Students should hopefully recognize the perfect squares:

9 \times 11 = 99 = 10^2 - 1

19 \times 21 = 399 = 20^2 - 1

29 \times 31 = 899 = 30^2 - 1

39 \times 41 = 1599 = 40^2 - 1,

so that they can guess the answer to something like 59 \times 61 without pulling out their calculators.

green lineContinuing the exploration, students can use a calculator to find

8 \times 12 = 96

18 \times 22 = 396

28 \times 32 = 896

38 \times 42 = 1596

Students should be able to recognize the pattern:

8 \times 12 = 10^2 - 4

18 \times 22 = 20^2 - 4

28 \times 32 = 30^2 - 4

38 \times 42 = 40^2 -4,

and perhaps they can even see the next step:

8 \times 12 = 10^2 - 2^2

18 \times 22 = 20^2 - 2^2

28 \times 32 = 30^2 - 2^2

38 \times 42 = 40^2 -2^2.

From this point, it’s a straightforward jump to

(10-2) \times (10+2) = 10^2 - 2^2

(20-2) \times (20+2) = 20^2 - 2^2

(30-2) \times (30+2) = 30^2 - 2^2

(40-2) \times (40+2) = 40^2 -2^2,

leading students to guess that (x-y)(x+y) = x^2 -y^2.

 

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1 Comment

  1. Difference of Two Powers (Index) | Mean Green Math

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