Lesson 13

The purpose of this warm-up is for students to recognize prisms and their bases. This concept reinforces what was discussed in the previous lesson where students found the volume of different prisms and non-prisms. Students first determine if a given figure is a prism or not and then shade and describe the base of the prism. As students work on the task, monitor for students who are using precise language to describe the reason a figure is a prism.

Arrange students in groups of 2. Give students 1 minute of quiet work time followed by time to discuss their answers with a partner. Follow with a whole-class discussion.

If students struggle to see why figure B is a prism, ask them to point out the base of the figure. It might be helpful to remind them that the base of the figure might not always be on the bottom.

Select previously identified students to share their reasoning. Invite students to share the bases they shaded and the drawings of the cross sections. If not mentioned by students, remind them that a figure is considered a prism if the cross section, when cut parallel to the base, has the same shape as the base of the figure.

In this activity, students practice mentally dissecting a prism with a non-rectangular base into simpler prisms. The dissection corresponds to a dissection of the base into simpler figures. This expands on students’ ability to calculate the area of a base of a figure that has a rectangular base because here the base is not a rectangle. This prepares students to calculate the volume of this figure and other figures in future lessons.

As students work in their groups, monitor for the different ways students are decomposing or constructing the base of the figure into more familiar shapes.

Arrange students in groups of 3. Display the image for all to see throughout the activity. Tell students this is what the heart-shaped box mentioned in the task statement looks like.

Give students 2–3 minutes of quiet work time followed by a whole-class discussion.

Select students to share whose method they decided to use and why. Ask students: 

• “Whose method could not be used? Why not?” (Lin’s, because we don’t know the base and  height of the trapezoids.)
• “How did you find the areas of the base?”
• “What was different about the base of this figure in comparison to other bases we have worked with?” (This base needs to be decomposed to calculate its area.)
• “What was the first thing you did to find the volume?” (Use the area of the base and multiply it by the height of the figure.)
• “Why would a chocolatier want to know the volume of a heart shaped box like this?” (He may want to know how many candies can fit inside of a box.)

Explain to students that they might encounter figures that have non-rectangular bases in future activities or lessons. It will be important for them to think about different strategies to calculate the area of the base.

In this activity, students practice finding the volume of another prism with a non-rectangular base by applying the formula \(\text{Volume} = (\text{Area of the base}) \boldcdot (\text{Height of the prism})\).

As students work on the task, monitor for students who decompose or compose the base of the figure into more familiar shapes.

Arrange students in groups of 2. Give students 1–2 minutes of quiet work time followed by time to discuss their work with a partner. Follow with a whole-class discussion.

If students mistake the rectangle for the base of the figure, ask students how we know that this figure is a prism and what the base of this figure needs to be in order to consider it a prism.

Select previously identified students to share the different methods for calculating the area of base. If not brought up by students, explain to students that the base of this figure can be either decomposed into rectangles and triangles or composed into a larger rectangle by adding two additional triangles. Ask students how they used the area of the base to calculate the volume of the figure (area of the base multiplied by the height).