Equivalent Quadratic Expressions

5 min

Narrative

In this activity, students recall that an area diagram can be used to illustrate multiplication of a number and a sum. This work prepares them to use diagrams to reason about the product of two sums that are variable expressions.

Launch

Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.

Student Task

<p><strong>Rectangle divided into 2 smaller rectangles. Top labeled 6. Left side labeled 3 and 4.</strong></p>

  1. Explain why the diagram shows that 6(3+4)=6⋅3+6⋅46(3+4) = 6 \boldcdot 3 + 6 \boldcdot 4.
  2. Draw a diagram to show that 5(x+2)=5x+105(x+2) = 5x + 10.

Sample Response

  1. The entire diagram represents the product 6(3+4)6(3+4). The upper half represents 6⋅36 \boldcdot 3 while the lower half represents 6⋅46 \boldcdot 4.
  2. Diagrams should show a large rectangle with two sub-rectangles, with one sub-rectangle representing 5⋅x5 \boldcdot x, and the other representing 5⋅25 \boldcdot 2.
Activity Synthesis (Teacher Notes)

Make sure students understand that the expressions 6⋅3+6⋅46 \boldcdot 3 + 6 \boldcdot 4 and 6(3+4)6(3+4) are two ways of representing the area of the same rectangle.

  • One expression treats the area of the largest rectangle as a sum of the areas of the two smaller rectangles that are 6 by 3 and 6 by 4.
  • The other expression describes the area of the largest rectangle as a product of its two side lengths, which are 6 and the sum of 3 and 4.

We can reason about 5(x+2)5(x+2) and 5x+105x + 10 the same way. 5(x+2)5(x+2) can represent the area of a large rectangle that is 5 by x+2x+2, and 5x+105x + 10 can represent the area of a large rectangle composed of two smaller ones whose areas are 5x5x and 5⋅25 \boldcdot 2 (or 10). When we express 5(x+2)5(x+2) as 5x+105x + 10, we are applying the distributive property.

Standards
Building On
  • 6.EE.3·Apply the properties of operations to generate equivalent expressions. <em>For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.</em>
  • 6.EE.A.3·Apply the properties of operations to generate equivalent expressions. <span>For example, apply the distributive property to the expression <span class="math">\(3 (2 + x)\)</span> to produce the equivalent expression <span class="math">\(6 + 3x\)</span>; apply the distributive property to the expression <span class="math">\(24x + 18y\)</span> to produce the equivalent expression <span class="math">\(6 (4x + 3y)\)</span>; apply properties of operations to <span class="math">\(y + y + y\)</span> to produce the equivalent expression <span class="math">\(3y\)</span>.</span>
Building Toward
  • A-SSE.3·Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
  • A-SSE.3·Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
  • A-SSE.3·Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
  • HSA-SSE.B.3·Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

10 min

20 min