Lesson 1

This warm-up prompts students to compare four figures. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

Arrange students in groups of 2–4. Display the figures for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as right angle, trapezoid, rectangle, square, parallelogram, area, or perimeter. Also, press students on unsubstantiated claims.

The purpose of this activity is to revisit and internalize the meanings of perimeter and area. To accomplish this, students play around with locking in one attribute (like area), coming up with lengths and widths that produce that area, and considering the resulting perimeter of each option.

Use one of the rectangles in the task, and ask students to find its perimeter and area. Ensure students have a chance to process what these terms mean before they proceed with the rest of the activity.

If students struggle to come up with side lengths of rectangles that meet the constraints, encourage them to sketch quick diagrams of rectangles. Consider providing graph paper for this purpose.

If calculators are needed to assist with numerical computations to access the task, provide them. If not, this activity is a good opportunity to practice some mental math.

Focus discussion on any consistencies that students noticed when perimeter was held constant and they considered side lengths that resulted in different areas, and vice versa. Some questions for discussion:

• “What do you notice about the shape of the rectangles that have larger or smaller perimeters when the area is 36? (Long skinny rectangles have the larger perimeters, and the more square-shaped rectangles have the smaller perimeters.)
• “What do you notice about the shape of the rectangles that have larger or smaller areas when the perimeter was 32?” (The more square-shaped the rectangles had larger areas.)

The purpose of this activity is for students to, by the end, plot points that represent a discrete graph of \(y=x^2\). When students make repeated numerical calculations and then generalize by expressing perimeter or area in terms of a variable, they are expressing regularity in repeated reasoning (MP8).

Possible questions for discussion:

• “Describe how the perimeter of the square changes as the width increases by 1.” (It’s a linear relationship, and every time \(x\) increases by 1, the perimeter increases by 4.)
• “Describe how the area of the square changes as the width increases by 1.” (Every time \(x\) increases by 1, the area increases by a greater and greater amount.)
• “Is the relationship between the width, \(x\), and the area of the square exponential? How do you know?” (The relationship is not exponential, because each time \(x\) increases by 1, the area doesn’t increase by a common factor.)