Lesson 11

Prisms Practice

11.1: New Heights (10 minutes)

Warm-up

This warm-up helps students recall the use of the Pythagorean Theorem and right triangle trigonometry. They’ll use these concepts in subsequent activities.

Launch

Provide students with access to scientific calculators.

Student Facing

Calculate the height of each solid. Round your answers to the nearest tenth if needed.

A

Right cone. Base radius = 6 point 5 inches. Cone height = 17 point 3 inches.

B

Oblique cylinder. The side of the cylinder has length 12 millimeters and is at a 50 degree angle to the base.

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

The height is not drawn in on the cylinder, so students may not be able to visualize the right triangle involved in the problem. Encourage them to draw in the height and think back to other problems from prior units involving angles and degree measurements.

Students may be uncertain how to round the first answer. Remind them that 16.0 is considered a “rounded” answer because the additional decimal digits to the right have been dropped.

Activity Synthesis

Ask students what strategies they used, and how they chose those strategies. Invite them to explain why they can use the Pythagorean Theorem for the cone problem but not the cylinder problem. Remind students that they can use their reference chart to recall the trigonometric ratio definitions.

11.2: The Choice is Yours (15 minutes)

Activity

The purpose of this activity is to give students the opportunity to reflect on their own understanding and apply that understanding to solve problems involving volumes, the Pythagorean Theorem, and trigonometry.

Look for students who choose a right prism with which they can straightforwardly compute \(Bh\), students who choose the oblique cylinder that doesn’t involve trigonometry, and students who choose a prism that involves trigonometry.

Launch

Arrange students in groups of 2. Tell students to discuss their thinking with their partner. If they disagree, they should work to reach agreement.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for the first question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “Why do you think this solid is the easiest to calculate its volume?”, “Why do you think this solid is the most difficult to calculate its volume?”, and “How do you know that the volume calculation will require trigonometry?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify their reasoning for the level of difficulty of volume calculations.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Representation: Internalize Comprehension. Activate background knowledge about the Pythagorean theorem and trigonometric ratios.
Supports accessibility for: Memory; Conceptual processing

Student Facing

Here are several solids.

A

Right Triangular prism. Triangular base, Base =11 centimeters, height = 5 centimeters. prism height = 8 centimeters 

B

Oblique cylinder. Radius =5. Slant length = 10. Angle formed with surface =40 degrees 

C

Rectangular prism. Base side lengths =7 and 5 feet. Prism height = 10 feet 

D

Right Triangular prism. Triangular base, base= 7 meters, height = 8 meters. Prism height = 11 meters 

E

Cylinder. Radius = 20 inches. Height = 15 inches 

F

Right Triangular prism. Triangular bases, base = 4 centimeters. Adjacent angle = 30 degrees. Prism height = 6 point 5 centimeters.

G

Oblique cylinder. Base, capital C = 4 meters. Perpendicular height = 2 meters.

  1. Without doing any calculating, identify 2 solids you think would have the least difficult volume calculations and 2 solids that would have the most difficult volume calculations. Be prepared to explain your reasoning.
  2. Choose 3 of the solids. At least 1 should be from your “least difficult” list and 1 should be from your “most difficult” list. Calculate the volumes of the solids you chose. Round your answers to the nearest tenth if needed.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

The images show a solid and the two-dimensional figure that was rotated to generate it.

Three cylinders sitting one on top of another. The middle cylinder is larger, in diameter, and slightly taller than the other two cylinders.
An eight sided figure. 

Find 4 different positive integers for the values of \(A,B,C,\) and \(D\) so that the total volume of the solid is \(297\pi\) cubic units.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Students may not be sure how to visualize the height of the oblique cylinder with the 40-degree angle. Remind them that they can add lines to the diagram.

Activity Synthesis

The purpose of the discussion is for students to share strategies for finding volumes of prisms and cylinders. Ask students:

  • “For which solids did you think the volume would be easy to calculate? Why?”
  • “For which solids did you think the volume would be hard to calculate? Why?”
  • “For the solids with missing measurements, how did you decide which dimensions you needed to calculate?”
  • “What were some of dimensions that were missing from the figures?”

11.3: The Cayan Tower (10 minutes)

Activity

The purpose of this activity is for students to apply what they have learned about prisms and cross sections to a context.

Launch

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to increase awareness of the language used to talk about the cross sections and volume of the Cayan Tower. Before revealing the questions in this activity, display the image of the Cayan Tower along with the statement: “The building on the left side of the picture is called the Cayan Tower,” and ask students to write down possible mathematical questions that could be asked about the tower. Invite students to compare their questions before revealing the actual questions. Listen for and amplify any questions about the cross sections and volume of the Cayan Tower. For example, “What is the area of the base of the Cayan Tower?” and “What is the volume of the Cayan Tower?”
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

The building on the left side of the picture is called the Cayan Tower. It’s in the city of Dubai. The tower is about 306 meters tall. It’s made up of identical floors that are each rotated slightly compared to the one underneath it.

Photograph of Cayan Tower 

floor plan

Chevron composed of 2 equalivalent parallelograms, top and bottom sides of each = 35 meters. Circle in middle between parallelograms, diameter= 17 meters

Each floor is the same chevron shape that is approximately 2 parallelograms put together, with the dimensions shown in the image. The circle in the floor plan shows the cross section of the core, which is used to circulate air and carry pipes and wiring throughout the building.

  1. The area of the Cayan Tower’s base is \(57 \boldcdot 35\) or 1,995 square meters. Why is it possible to find the area of the chevron shape by just multiplying its width and height?
  2. Describe how the total volume of the building (including the core) can be calculated.
  3. What shape is the core of the building, whose cross section is shown in the floor plan as a circle?
  4. Describe how the volume of the building’s core can be calculated, including describing the measurements that would be used.
  5. What percentage of the building’s volume is taken up by its core?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The goal of the discussion is to connect Cavalieri’s Principle to the tower volume calculation. Ask students to imagine a building that is identical to the Cayan Tower, except that the floors are rotated so they line up with each other, forming a right prism. Ask students:

  • “How are the floors of the building like cross sections? How are they different?” (They are horizontal slices like cross sections, but cross sections have no thickness, whereas the floors do have thickness.)
  • “Even though the floors aren’t exactly cross sections, we can use reasoning involving cross sections to compare the volumes of the original building and our imaginary rotated building. How do the volumes compare and why?” (The original building and the one with rotations have equal-area “cross sections” (or rather, equal-volume floors) at all heights, so their volumes are the same.)
  • “How does this reasoning help us find the volume of the tower?” (The expression \(Bh\) gives the volume of the imaginary rotated building because it is a prism. This volume is equal to the volume of the tower.)
Engagement: Develop Effort and Persistence. Break the class into small discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Language; Social-emotional skills; Attention

Lesson Synthesis

Lesson Synthesis

Ask students what measurements are needed to calculate volumes of various solids: rectangular prisms, prisms with triangular bases, and cylinders. Invite students to create a list of formulas and strategies they might use when finding volumes of these solids. The list may include the Pythagorean Theorem, trigonometry, area formulas for triangles and circles, and the relationships between a circle’s radius, its diameter, and its circumference. Ask students under what conditions the Pythagorean Theorem and trigonometry can be used. Point out that right triangles don’t always come pre-drawn—sometimes students may need to add lines to a figure.

11.4: Cool-down - Cylinder with a Hole (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

The formula \(V=Bh\), where \(V\) stands for volume, \(B\) is the area of the base, and \(h\) is the height, applies to right and oblique cylinders and all prisms. Sometimes, though, additional calculations are needed to find missing measurements before the formula can be applied.

To calculate the volume of this cylinder, first find the area of the base, which is a circle of radius 5 cm. Its area is \(25\pi\) square centimeters. The cylinder’s height isn’t given. To find the value of the height, notice that a right triangle is formed by the 15-cm diagonal line, the 10-cm diameter of the circle, and the height of the cylinder. The diagonal line is the triangle’s hypotenuse. By the Pythagorean Theorem, \(10^2+h^2=15^2\). That means \(100+h^2=225\). Subtracting 100 from each side gives \(h^2=125\), so \(h=\sqrt{125}\) centimeters.

Cylinder. Radius of base = 5 centimeters. Slant length of cylinder = 15 centimeters 

Now, the volume of the cylinder is the area of the base multiplied by the height. This is \(25\pi\boldcdot\sqrt{125}\) cubic centimeters, or approximately 878.1 cubic centimeters.