Rewriting Quadratic Expressions in Factored Form (Part 2)

5 min

Narrative

This Warm-up serves two purposes. The first is to recall that if the product of two numbers is negative, then the two numbers must have opposite signs. The second is to review how to add two numbers with opposite signs.

Student Task

  1. The product of the integers 2 and -6 is -12. List all the other pairs of integers whose product is -12.
  2. Of the pairs of factors you found, list all pairs that have a positive sum. Explain why they all have a positive sum.
  3. Of the pairs of factors you found, list all pairs that have a negative sum. Explain why they all have a negative sum.

Sample Response

  1. 1 and -12, -1 and 12, 2 and -6, -2 and 6, 3 and -4, -3 and 4
  2. -1 and 12, 6 and -2, 4 and -3. Sample reasoning: The positive number has a larger absolute value than the negative number (or equivalent).
  3. 1 and -12, 2 and -6, 3 and -4. Sample reasoning: The negative number has a larger absolute value than the positive number (or equivalent).
Activity Synthesis (Teacher Notes)

Ask students to share their list of factors. Once all the pairs are listed, highlight that each pair has a positive number and a negative number because the product we are after is a negative number, and the product of a positive and a negative number is negative.

Then, invite students to share their responses for the next two questions. Consider displaying a number line for all to see and using arrows to visualize the additions of factors. Make sure students understand that when adding a positive number and a negative number, the result is the difference of the absolute values of the numbers, and that sum takes the sign of the number that is farther from zero.

Standards
Building On
  • 7.NS.1·Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
  • 7.NS.2·Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
  • 7.NS.A.1·Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
  • 7.NS.A.2·Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
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