Unit 5, Lesson 11
Filling Containers
Grade 8
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This Warm-up prompts students to compare four different objects and think about how they are similar and different from objects they have encountered in previous activities and grade levels. 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.
Arrange students in groups of 2–4. Display the objects for all to see. If time allows, after 30 seconds of quiet think time, invite 2–3 students to briefly share what they notice all of the figures have in common (for example, they all look like three-dimensional objects). The purpose of this initial share out is to support all students in naming common attributes before they work to identify more specific features that a group of only three out of the four have.
Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three shapes that go together and can explain why. Next, tell each student to share their response with their group and then together find as many sets of three as they can.
Which three go together? Why do they go together?
Invite each group to share one reason why a particular set of three go together. 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 three go together, attend to students’ explanations, and ensure the reasons given are correct.
During the discussion, prompt students to explain the meaning of any geometric terminology they use, such as “side,” “edge,” “radius,” or “surface area,” and to clarify their reasoning as needed. Consider asking:
MLR8: Discussion Supports
To Gather
Graduated cylinders
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).
In the digital version of the activity, students use an applet to do the experiment. The applet allows students to fill a digital cylinder that displays the volume of water added and the height of the water. The applet also lets students change the radius of the cylinder. The digital version may reduce barriers for students who need support with fine-motor skills and students who benefit from extra processing time.
The goal of this discussion is for students to understand how the experiment can be modeled with a linear function.
Invite 2–3 groups to share their graphs, and display these for all to see. Consider asking students the following questions:
Conclude the discussion by asking students to predict how the graph would change if their cylinder had double the diameter. Alternatively, if it was possible to work with graduated cylinders with different diameters, display those graphs now. 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. Ask students to consider which line must represent which cylinder. Ask, “How did the slope of each graph change as the radius increased?” (The larger the radius, the less steep the slope. 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.)
MLR1: Stronger and Clearer Each Time
None
Previously, 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 (MP2). The goal of this activity is for students to practice connecting words to graphs of functions.
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 asking half of the class to work on the first question and the other half to work on the second question, then complete the last question together as part of the Activity Synthesis.
The graph shows the height vs. volume function of an unknown container.
What shape could this container have? Explain how you know, and draw a possible container.
The graph shows the height vs. volume function of a different unknown container.
What shape could this container have? Explain how you know, and draw a possible container.
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.
When filling a shape like a cylinder with water, we can see how the dimensions of the cylinder affect things like the changing height of the water. For example, let's say we have two cylinders, D and E, with the same height, but D has a radius of 3 cm, and E has a radius of 6 cm.
If we pour water into both cylinders at the same rate, the height of water in D will increase faster than the height of water in E because of its smaller radius. This means that if we made graphs of the height of water as a function of the volume of water for each cylinder, we would have two lines, and the slope of the line for Cylinder D would be greater than the slope of the line for Cylinder E.
Help us improve by sharing suggestions or reporting issues.
Arrange students in groups of 3–4. Be sure students know how to measure using a graduated cylinder. If needed, display a graduated cylinder filled to a specific measurement for all to see, and demonstrate to students how to read the measurement. Give each group access to a graduated cylinder and water.
If possible, consider assigning different groups graduated cylinders with different radii to compare and contrast their graphs during the Activity Synthesis.
Give groups 8–10 minutes to work on the task, and follow with a whole-class discussion.
Your teacher will give you a graduated cylinder, water, and some other supplies. Your group will use these supplies to investigate the height of water in the cylinder as a function of the water volume.
| volume (ml) | ||||||
|---|---|---|---|---|---|---|
| height (cm) |
Building Toward