Completing the Square (Part 1)

5 min

Narrative

This activity reinforces the meaning of perfect squares and the fact that a perfect square can appear in different forms. To recognize a perfect square, students need to look for and make use of structure (MP7).

Student Task

Select all expressions that are perfect squares. Explain how you know.

  1. (x+5)(5+x)(x+5)(5+x)
  2. (x+5)(x−5)(x+5)(x-5)
  3. (x−3)2(x-3)^2
  4. x−32x-3^2
  5. x2+8x+16x^2+8x+16
  6. x2+10x+20x^2+10x+20

Sample Response

1, 3, and 5. Sample reasoning: Each of them is the product of an expression and itself. x2+8x+16x^2+8x+16 is equivalent to (x+4)(x+4)(x+4)(x+4).

Synthesis

Display the expressions for all to see. Invite students to share their responses, and record them for all to see. For each expression that they consider a perfect square, ask them to explain how they know. For expressions that students believe aren’t perfect squares, ask them to explain why not.

For the last expression, x2+10x+20x^2+10x+20, students may reason that it is not a perfect square because:

  • The constant term 20 is not a perfect square. (Most students are likely to reason this way.)
  • If 20 is seen as (20)2(\sqrt {20})^2, then the coefficient of the linear term would have to be 2202\sqrt {20}, not 10.

Though students have been dealing mostly with rational numbers, the second line of reasoning is also valid and acceptable.

Standards
Addressing
  • A-SSE.2·Use the structure of an expression to identify ways to rewrite it. <em>For example, see x<sup>4</sup> — y<sup>4</sup> as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).</em>
  • A-SSE.2·Use the structure of an expression to identify ways to rewrite it. <em>For example, see x<sup>4</sup> — y<sup>4</sup> as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).</em>
  • A-SSE.2·Use the structure of an expression to identify ways to rewrite it. <em>For example, see x<sup>4</sup> — y<sup>4</sup> as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).</em>
  • A-SSE.2·Use the structure of an expression to identify ways to rewrite it. <em>For example, see x<sup>4</sup> — y<sup>4</sup> as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).</em>
  • A-SSE.2·Use the structure of an expression to identify ways to rewrite it. <em>For example, see x<sup>4</sup> — y<sup>4</sup> as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).</em>
  • A-SSE.A·Interpret the structure of expressions
  • HSA-SSE.A·Interpret the structure of expressions.
  • HSA-SSE.A.2·Use the structure of an expression to identify ways to rewrite it. <span>For example, see <span class="math">\(x^4 - y^4\)</span> as <span class="math">\((x^2)^2 - (y^2)^2\)</span>, thus recognizing it as a difference of squares that can be factored as <span class="math">\((x^2 - y^2)(x^2 + y^2)\)</span>.</span>

10 min

20 min