The purpose of this activity is for students to plot points that represent the length and width of a rectangle with a given perimeter. Since the perimeter is twice the length plus twice the width, decreasing the length by a certain amount will mean that the width has to increase by the same amount for the perimeter to stay the same.

Monitor for and select students with the following approaches for Jada’s rectangle if it is 2.5 cm long to share in the Activity Synthesis:

• Use the perimeter formula to find unknown values. 
• Use the relationship that the length and width together are half the perimeter to find unknowns.
• Reason that if the length increases by 1, the width decreases by 1 (or any other value such as 0.5), using the values in the table or graph to justify.

The approaches are sequenced from more concrete to more abstract to help students understand the relationship between the length and width of rectangles with a fixed perimeter. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven’t shared recently. For an example for each approach, look at the Student Responses.

• “How did you find the width of Jada’s rectangle if it is 2.5 cm long?”
• Invite previously selected students to share in the given order. Record or display their work for all to see.
• Connect students’ approaches by asking:
  • “How are ways of solving the problem the same?” (All the ways to solve the problem involved adding and subtracting or an increase and a decrease. The perimeter formula involved adding and subtracting. If length increased by 0.5, the width decreased by 0.5.)
  • “How are they different?” (Some approaches used equations with the given length and perimeter. Other approaches used two pairs of values for length and width to observe patterns of increasing and decreasing, then applied it to the new length value.)
• If no one notices the increasing and decreasing relationship, display student work that shows lengths increasing by 1 cm in each row and a graph of the points, or display the table and graph from the Student Responses.
• Connect students’ approaches to the Learning Goal by asking:
  • “What happens to the width when the length increases by 1? Why?” (The width decreases by one. This makes sense because the sum needs to say the same or else the perimeter changes.)
  • “How does the table show this relationship?” (If you compare values in the table such as  and , the length goes up 1 and the width goes down 1. It makes sense because the sum needs to stay the same or else the perimeter changes.)
  • “How does the graph show this?” (For each point I plotted, I can go right 1 and down 1 and find another possible length and width.)

The purpose of this activity is to investigate the possible lengths and widths of a rectangle with a given area. Since the area is the product of length and width, this means that the main operation being used is multiplication or division, contrasting with the previous activity where students investigated the perimeter which is the sum of the side lengths of a rectangle. This means that the calculations are more complex and some of the coordinates of the points that students plot will either be decimals or fractions depending on how students express them. There are some important common characteristics between the lengths and widths for a given area and for a given perimeter which will be examined in the Activity Synthesis (MP7, MP8):

• When the length increases, the width decreases
• The length and width can be switched to get another possible length and width pair

• Invite students to share their responses for the width of a rectangle that is 5 cm long.
• “How did you calculate the value?” (I knew that 5 times the width was 16 so the width is or  cm.)
• “How did you know where to plot that length and width pair?” (I looked for 5 on the horizontal axis and then I had to estimate where was on the vertical axis. I put it a little above 3 but closer to 3 than to 4.)
• “How was determining the possible lengths and widths for a given area the same as determining the possible lengths and widths for a given perimeter?” (When the length increases, the width decreases. When the width decreases, the length increases. I can flip the order of the length and width and get another rectangle.)
• “How are the length and width pairs for rectangles with area 16 different from the length and width pairs for rectangles with perimeter 12?” (I was looking for a total of 16 instead of a total of 12. I have to multiply the side lengths rather than add them. When the length decreases by 1 for the perimeter, the width increases by 1. For area, when the length decreases, the width increases but the relationship is more complex.)
• Consider drawing some rectangles with an of area 16 on the coordinate grid with the lower left corner of each rectangle at . Ask students what they notice about the coordinates of the upper right corners of each rectangle. (They represent the length and width of the corresponding rectangle.)