In this activity, students consider the relationship between length and width for different rectangles with the same given area and are asked to compare strategies for finding various lengths and widths. Students make sense of how the graph shows how the width changes when the length changes and what the plotted points on the graph mean in the context of the problem.

The discussion should focus on the connection between the situation, the equation (or another strategy) for finding combinations that make the area 36 square feet, and the graph that represents the relationship between length and width in the different rectangles.

Invite students to share their responses and reasoning to the last two questions. Discuss  questions such as:

• “What does the point in the graph mean? In general, what does each point represent?” (The point represents a rectangle with length 2 feet and height 18 feet. Each point represents a rectangle with an area of 36 square feet.)
• “Why does it make sense that the graph falls as you move to the right?” (The length and width are factors of the same product, so if one increases the other has to decrease.)
• “Where do you see the area 36 square feet in each equation or strategy for finding combinations, and the graph that shows those combinations?” (In the equations, 36 is the product of the length and width. In the graph, the coordinates of each point multiply to 36.)

If time permits, consider asking students:

• “What would the graph look like if it were to extend more to the right? Can you name some points on the graph and describe their coordinates?” (Points could be , , . would be a fraction less than 1 as gets larger than 36 because and have to multiply to 36. The points will keep getting closer to the horizontal axis as gets smaller.)

This activity presents a situation with a similar structure to the area situation in the activity about making a banner. Students consider different combinations of base areas and heights that keep the volume of a rectangular box at 225 cubic inches. They complete a table for given values of area and height, write an equation relating the area and height, and graph the relationship.

If students have completed the activity about making a banner, prompt them to think about similarities and differences they noticed between the activities when synthesizing the lesson. In comparing the two situations and then using that insight to solve the problems in this activity, students practice looking for and making use of structure (MP7).

Invite students to share how they went about finding the missing values in the table, writing an equation that represents the relationship of the two quantities, and deciding on which variable depends on the other. Then discuss questions such as:

• “Is one equation more helpful than the other when figuring out the design for the cereal box? If so, which one, and why is it more helpful? If not, why not?”
• “What might be some benefits of using a table to represent the relationship between the two quantities in this relationship?”
• “What might be some benefits of writing an equation or creating a graph?” 

Consider asking the following questions:

• "How does this situation compare to the one in the ‘Making a Banner’ activity?" (It involved measurements. We knew the product, but one or both of the two factors were unknown. The structure of the equations was similar: they both had the dependent variable equal to a number divided by the independent variable.)
• In each situation, are there any values of the variables that do not make sense? Explain. (For the banner, having a very small width and very long length does not make sense, and vice versa. For example, a banner that foot wide and 288 feet long is not useful. For the cereal box, a prism that is very short with a very large base area does not make sense, and vice versa. For example, a cereal box that is 225 inches tall and has a base area of 1 in is not useful.)

In this activity, students consider a relationship in which one quantity doubles each time the other increases by 1. the exponent is a variable. Students interpret a given equation with an exponent that is a variable and explain what it means in the context of the situation.

Monitor for students who connect this activity to the lessons on exponents, or who recognize that the quantities in this relationship are changing with respect to each other in a different manner than previous examples they have seen (MP7). Ask these students to  share during the discussion. 

Focus the discussion on the connections between the table, the graph, and the equation that describe this situation. Invite students to share their responses. Consider asking questions such as:

• “Which gives you a better idea of how the mosquito population changes with the day in the study—the data table, the equation, or the graph? How is that so?”
• “How is the relationship between the quantities in this situation like the ones in an earlier activity?” (When the quantities are graphed as points, they form a curve. There is one operation that relates one quantity to the other.) “How is it different?” (In this case, we can double the number of mosquitos each day to get the number for the next day. In earlier activities, it was necessary to divide a number, either 36 or 225, by one of the values each time.)
• “Think back to the activities about a dot pattern and about rice grains on a chessboard. How is this situation about mosquitos similar to those situations?” (They are all about multiplying by the same factor repeatedly.) 
• “How would tables of data, graphs, and equations compare?” (The values for one quantity in the table increase much faster than the values of the other. The equations would all involve an exponent. The vertical values on the graph all increase very quickly.)