Illustrative Mathematics Grade 7 Unit 3 Lesson 7 Answer Key

There are two main goals for this discussion: for students to notice ways to be more efficient when comparing the areas of the regions and to be introduced to expressing answers in terms of pi.

Display Figures A, B, and C for all to see. Ask selected students to share their reasoning. Sequence the strategies from most calculations to most efficient.

If there were selected students who determined the areas were equal before calculating, ask them to share how they could tell. If there were no selected students, ask the class how we could determine that the area of the shaded regions in Figures A, B, and C were equal before calculating the answer of 30.96.

Next, focus the discussion on leaving answers in terms of \(\pi\) for each figure. Explain to students that in Figure A, the radius of the circle is 6, so the area of the circular region is \(\pi \boldcdot 6^2\). Instead of multiplying by an approximation of \(\pi\), we can express this answer as \(36\pi\). This is called answering in terms of \(\pi\) . Consider writing "\(36\pi\)" inside the large circle of Figure A.

Discuss:

In terms of \(\pi\), what is the area of one of the circular regions in Figure B? (\(9\pi\))
What is the combined area of all four circles in Figure B? (\(4 \boldcdot 9\pi\), or \(36\pi\))
What is the area of one of the circular regions in Figure C? (\(4\pi\))
What is the combined area of all nine circles in Figure C? (\(9 \boldcdot 4\pi\), or \(36\pi\))

Consider writing "\(9\pi\)" and "\(4\pi\)" inside some of the circles in Figures B and C. Explain that the area of the shaded region for each of these figures is \(144 - 36\pi\).

3 different images to compare areas made of circles.

Discuss how students' strategies differed between the first problem (about Figures A, B, and C) and the second problem (about Figures D and E) and why.

Ask students to express the area of Figures D and E in terms of \(\pi\). Record and display their answers of\(2 + 2\pi\) and \(4 + 1.5\pi\) for all to see. Ask students to discuss how they can tell Figure E's area is larger than Figure D's area when they are both written in terms of \(\pi\).

Writing, Speaking, Listening: MLR1 Stronger and Clearer Each Time. After students have determined which figure has the largest shaded region, ask students to show their work and provide a brief explanation of their reasoning. Ask each student to meet with 2–3 other partners for feedback. Provide students with prompts for feedback that will help them strengthen their ideas and clarify their language (e.g., "How did you find the radius of each circle in the figure?", "Why did you . . . ?", etc.). Students can borrow ideas and language from each partner to refine and clarify their original explanation. This will help students refine their own explanation and learn about different strategies to find area.
Design Principles(s): Optimize output (for explanation); Maximize meta-awareness