Lesson 18

The purpose of this warm-up is to prompt students to reason about what a set of organized points in a coordinate plane might mean. This activity invites students to explain their reasoning and hold mathematical conversations, and allows you to hear how they use terminology and talk about points in a coordinate plane. To allow all students to access the activity, there is not one correct answer so students are able to choose any figure as long as they can support their reasoning. As students share their responses, listen for important ideas and terminology that will be helpful in upcoming work.

Arrange students in groups of 2-4. Display the image of the four figures for all to see. Ask students to indicate when they have noticed one figure that does not belong and can explain why. Give students 1 minute of quiet think time and then time to share their thinking with their group. After everyone has conferred in groups, ask the group to offer at least one reason each figure doesn’t belong.

After students have conferred in groups, invite each group to share one reason why a particular figure might not belong. Record and display the responses for all to see. After each response, poll the rest of the class if they agree or disagree. Since there is no single correct answer to the question of which shape does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, prompt students with the following questions:

• "How would you describe the relationship between the two quantities represented by the two axes?"
• "Do you have any ideas about what quantities or relationships any of these graphs might represent?" (If students do have ideas, based on relationships they have explored or others they are thinking about, have them explain why the graph represents the relationship.)

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.  They make sense of how the graph shows what happens to the width 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 strategy for finding combinations that make the area 36, and the graph that represents the relationship between length and width in the different rectangles.

Some guiding questions:

• “What does the point \((2,18)\) (for example) in the graph mean? In general, what does each point represent?” (each point represents a rectangle with area 36. \((2,18)\) represents a rectangle with length 2 and height 18)
• “Why does it make sense that the graph falls as you move to the right?” (The length and width are factors of a fixed product, so if one increases the other has to decrease)
• “Where do you see the area 36 in each of the situation, the equation or strategy for finding combinations, and the graph that shows those combinations?” (in the graph, the coordinates of each point multiply to 36)
• “Describe what the graph would look like if it were to extend more to the right. Name some points on the graph and describe their coordinates.” (Points could be \((\frac{9}{10}, 40)\), \((\frac{18}{25},50)\), \((\frac12,72)\). \(w\) would be a fraction less than 1 as \(\ell\) gets larger than 36 because \(w\) and \(\ell\) have to multiply to 36. The points will keep getting closer to the \(x\)-axis as \(w\) gets smaller.

This activity presents a situation with a similar structure to the area situation in the Making a Banner activity. 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 Making a Banner activity, prompt them to think about similarities and differences they noticed between the activities. Invite students to share these thoughts during the discussion. Students have an opportunity to engage in MP7 as they notice the similar structures of the two situations and use that knowledge to solve the problems in this activity.

The purpose of the discussion is to connect this activity to the previous one and point out similarities and differences between them.

Consider asking the following questions:

• "How does this situation compare to the one in the Making a Banner activity?" (The structure of the equations was similar: the two quantities multiply to a constant.)
• "How did you decide on an equation that represents the relationship?"
• "How would the graph be different if height was on the vertical axis and area on the horizontal axis?" (It would look the same since the two numbers multiply to a constant.)

In this activity, students consider a doubling relationship where the exponent is a variable. 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. Have these students share during the discussion. 

The purpose of this discussion is to help students make connections between the table, graph, and equation that describe this situation. Ask students to share who connected 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. 

Consider asking some of these questions:

• “Which of the data, equation, or graph gives you more of a feel for how the mosquito population changes with the day in the study?”
• “Think back to the lesson on exponents with the dot pattern and the Genie activity. How do those situations compare with this mosquito situation? How would tables of data, graphs, and equations compare?”
• “What are some similarities and differences you noticed between all the relationships you've seen in the last few lessons?”