Lesson 12

The purpose of this warm-up is for students to compare different objects that may not be familiar and think about how they are similar and different from objects they have encountered in previous activities and grade levels. To allow all students to access the activity, each object has one obvious reason it does not belong. Encourage students to move past the obvious reasons (e.g., Figure A has a point on top) and find reasons based on geometrical properties (e.g., Figure D when looked at from every side is a rectangle). During the discussion, listen for important ideas and terminology that will be helpful in upcoming work of the unit.

Arrange students in groups of 2. Display the image for all to see. Ask students to indicate when they have noticed one object that does not belong and can explain why. Give students 2 minutes of quiet think time and then time to share their thinking with their partner.

Ask students to share one reason why a particular object might not belong. Record and display the responses for all to see. After each response, ask the rest of 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, prompt students to explain the meaning of any terminology they use, such as diameter, radius, vertex, edge, face, or specific names of the figures: sphere, cylinder, cone, rectangular prism.

In this activity, students investigate how the height of water in a graduated cylinder is a function of the volume of water in the graduated cylinder. Students make predictions about how the graph will look and then test their prediction by filling the graduated cylinder with different amounts of water, gathering and graphing the data (MP4).

Select groups to share the graph for the third question and display it for all to see. Consider asking students the following questions:

• “What do you notice about the shape of your graph?”
• “What is the independent variable of your graph? Dependent variable?”
• “How does this graph differ from what you predicted the shape would be?”
• “For the last question, what point did you choose, and what does that point mean in the context of this activity?”
• “What would the endpoint of the graph be?” (There is a maximum possible volume for the cylinder. Once it’s filled, any extra water will spill out and not raise the water height.)

Ask students to predict how the graph would change if their cylinder had double the diameter. After a few responses, display this graph for all to see:Explain that each line represents the graph of a cylinder with a different radius. One cylinder has a radius of 1 cm, another has a radius of 2 cm, and another has a radius of 3 cm. Have students consider which line must represent which cylinder. Ask, “how did the slope of each graph change as the radius increased?” (As the radius is larger, the slope is less steep. This is because for a cylinder with a larger base, the same volume of water will not fill as high up the side of the cylinder.)

In the previous activity, students were given a container and asked to draw the graph of the height as a function of the volume. In this activity, students are given the graph and asked to draw a sketch of the container that could have generated that height function. Since students have worked on the two previous activities, they have an idea of what the data for a graduated cylinder and a graduated cylinder with twice the diameter looks like and can use that information to compare to while working on this task.

Arrange students in groups of 2. Give students 3–5 minutes of quiet work time and then time to share their drawings with their partner. Follow with a whole-class discussion.

If time is short, consider having half of the class work on the first question and the other half work on the second question and then complete the last question as part of the Activity Synthesis.

Select students to share the different containers they drew. Display their drawings and the graph for all to see. Ask students to explain how they came up with their drawing and refer to parts in the graph that determined the shape of their container.