Completing the Square (Part 3)

5 min

Narrative

In this Warm-up, students make a conjecture about the relationship between values from a quadratic expression in factored form to values of the same expression in standard form by using a specific example. Previously, students have seen a pattern for expressions in the form (x+m)2(x+m)^2, and here they begin to generalize to a pattern for expressions in the form (kx+m)2(kx + m)^2 (MP8).

At this stage, it is not important for them to notice all of the patterns or express their conjecture in precise mathematical language. These patterns will be more closely explored later in the lesson.

Launch

Arrange students in groups of 2. Give students 3 minutes to expand the expression and notice a pattern they might be able to express as a conjecture. Students do not need to write out their conjecture, but should be able to express any patterns they notice during the Activity Synthesis.

Student Task

Previously, we saw that (x+3)2(x+3)^2 can be expanded to standard form as x2+2⋅3x+32x^2+2\boldcdot 3x+3^2.

  1. Expand (5x+3)2(5x+3)^2 into standard form.
  2. Be prepared to share a conjecture about the relationship between the coefficients 5 and 3 in the factored form and the values in standard form.

Sample Response

  1. 25x2+30x+925x^2+30x+9
  2. Sample response: The coefficient of the quadratic term in standard form seems to be the square of coefficient of xx in factored form. The constant term in standard form might be the square of the constant in factored form. The coefficient of the linear term in standard form might be double the product of the two numbers in the factored form.

Synthesis

Display the solution for the first question, then invite 1–2 students to share their conjectures. Encourage students to share even if their ideas are only partial.

It is okay if the conjectures are not complete or correct at this point. The conjectures will be tested in the following activities.

Standards
Building Toward
  • 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>
  • 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>

15 min

15 min

30 min