Lesson 1

Accessing Areas and Pondering Perimeters

These materials, when encountered before Algebra 1, Unit 6, Lesson 1 support success in that lesson.

1.1: Which One Doesn’t Belong: Quadrilaterals (5 minutes)

Warm-up

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.

Launch

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.

Student Facing

Which one doesn’t belong? 

A

Trapezoid. Top left and bottom left angles = 90 degrees. Bases = 18 and 9 units. Perpendicular Height = 6 units. 

B

Rectangle. Length = 10 units. Width = 8 units.

C

A square with side length 9.

D

Parallelogram. Base 9 units, height 9 units.

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.

1.2: Inspect Some Rectangles (20 minutes)

Activity

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.

Launch

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.

Student Facing

Here are some rectangles.

3 rectangles A, B, C. Rectangle A, side lengths 2, 8. Rectangle B, side lengths 4, 6. Rectangle C, side lengths 1, 9.
  1. Which rectangle has the greatest perimeter?
  2. Which rectangle has the greatest area?
  3. Find a rectangle with the same perimeter as rectangle C that has an even greater area.
  4. For the remaining questions, tables are provided to organize your work. Rectangle D has a perimeter of 32 units.
    1. Find the side lengths of three different possible rectangles that have this perimeter.
    2. Find a pair of side lengths for rectangle D that give the greatest area in square units.
    3. Find a pair of side lengths for rectangle D that give the smallest area in square units.
      length (units) width (units) perimeter (units) area (square units)
             
             
             
             
             
  5. Rectangle E has an area of 36 square units.
    1. Find 3 pairs of side lengths that give this area.
    2. Find a pair of side lengths for rectangle E that give the greatest perimeter in whole-number units.
    3. Find a pair of side lengths for rectangle E that give the smallest perimeter in whole-number units.
      length (units) width (units) perimeter (units) area (square units)
             
             
             
             
             

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.)

1.3: Inspect Some Tables (20 minutes)

Activity

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).

Student Facing

Here are two tables. The first shows some measurements for Rectangle A, with a side length of 5 cm. The second shows some measurements of Rectangle B, which is a square.

  1. Complete the table for Rectangle A and be prepared to explain your reasoning.

    length (cm) width (cm) perimeter (cm) area (sq cm)
    5 1    
    5 2    
    5 4    
    5   20  
    5     40
    5   28  
    5     50
    5 \(x\)    
  2. Complete the table for Rectangle B and be prepared to explain your reasoning.

    length (cm) width (cm) perimeter (cm) area (sq cm)
    1 1    
    2 2    
    3 3    
    4   16  
      8    
          100
      \(x\)    
  3. Sketch the graph of each pair of quantities, where the width is plotted along the \(x\)-axis.

    1. \(x\) and the perimeter of Rectangle A

    2. \(x\) and the area of Rectangle A

    3. \(x\) and the perimeter of Rectangle B

    4. \(x\) and the area of Rectangle B

Blank coordinate grid.
Blank coordinate grid.
Blank coordinate grid.
Blank coordinate grid.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.)