Lesson 12: Completing the Square (Part 1)

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.

$(x+5)(5+x)$
$(x+5)(x-5)$
$(x-3)^2$
$x-3^2$
$x^2+8x+16$
$x^2+10x+20$

Sample Response

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

Activity Synthesis (Teacher Notes)

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, $x^2+10x+20$ , 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 $(\sqrt {20})^2$ , then the coefficient of the linear term would have to be $2\sqrt {20}$ , 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. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
A-SSE.2·Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
A-SSE.2·Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
A-SSE.2·Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
A-SSE.2·Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
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. For example, see $x^4 - y^4$ as $(x^2)^2 - (y^2)^2$, thus recognizing it as a difference of squares that can be factored as $(x^2 - y^2)(x^2 + y^2)$.

10 min

20 min

Knowledge Components

All skills for this lesson

No KCs tagged for this lesson

Completing the Square (Part 1)

5 min

Narrative

Student Task

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

$(x+5)(5+x)$
$(x+5)(x-5)$
$(x-3)^2$
$x-3^2$
$x^2+8x+16$
$x^2+10x+20$

Sample Response

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

Activity Synthesis (Teacher Notes)

For the last expression, $x^2+10x+20$ , 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 $(\sqrt {20})^2$ , then the coefficient of the linear term would have to be $2\sqrt {20}$ , 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. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
A-SSE.2·Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
A-SSE.2·Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
A-SSE.2·Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
A-SSE.2·Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
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. For example, see $x^4 - y^4$ as $(x^2)^2 - (y^2)^2$, thus recognizing it as a difference of squares that can be factored as $(x^2 - y^2)(x^2 + y^2)$.

Completing the Square (Part 1)

Warm-upPerfect or Imperfect?5 minKCs

Narrative

Student Task

Sample Response

ActivityBuilding Perfect Squares10 minKCs

ActivityDipping Our Toes in Completing the Square20 minKCs

Knowledge Components

Completing the Square (Part 1)

Warm-upPerfect or Imperfect?5 minKCs

Narrative

Student Task

Sample Response

ActivityBuilding Perfect Squares10 minKCs

ActivityDipping Our Toes in Completing the Square20 minKCs

5 min

10 min

20 min

5 min

10 min

20 min