The Distributive Property, Part 3

10 min

Narrative

In this activity, students are prompted to explain why two expressions with a variable represent the area of a shaded rectangle that is part of a larger rectangle of unknown width.

In this partner activity, students take turns sharing their initial ideas and first drafts. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). As students revise their writing, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6).

This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.

Launch

Give students 2 minutes of quiet work time, followed by a whole-class discussion.

Student Task

A rectangle with side lengths 6 cm and ww cm is partitioned into two smaller rectangles, as shown in the diagram.

A rectangle with height labeled 6 and total width labeled w. A portion of the width "w" is labeled 4.
A rectangle with height labeled 6 and total width labeled w. Rectangle is partition into two smaller rectangles.  First rectangle shares height of 6 and width of 4. Second smaller rectangle has an area shaded blue.

Explain why each of these expressions represents the area, in cm2, of the shaded region.

  • 6w−246w-24
  • 6(w−4)6(w-4)

Sample Response

Sample responses:

  • 6w−246w-24: The area, in cm2, of the entire rectangle is 6w6w. The area of the unshaded rectangle is 6⋅46 \boldcdot 4 or 24 cm2. Subtracting the area of the unshaded rectangle from the area of the entire rectangle, 6w−246w-24, gives the area of the shaded rectangle.
  • 6(w−4)6(w-4): The length of the shaded rectangle is w−4w-4 cm. Its width is 6 cm, so its area, in cm2, is 6(w−4)6(w-4).

Synthesis

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to the Warm-up. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

If time allows, display these prompts for feedback:

  • “‾\underline{\hspace{.5in}} makes sense, but what do you mean when you say. . . ?”
  • “Can you describe that another way?”
  • “How do you know . . . ? What else do you know is true?”

Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.

After Stronger and Clearer Each Time, invite students to share their second draft explanation for 6(w−4)6(w - 4). Highlight the ways students connect the terms 6 and w−4w-4 to the length and width of the shaded rectangle.

Standards
Addressing
  • 6.EE.2·Write, read, and evaluate expressions in which letters stand for numbers.
  • 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.2·Write, read, and evaluate expressions in which letters stand for numbers.
  • 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>

15 min

15 min