Lesson 11

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

Provide students with access to scientific calculators.

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.

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.

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.

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

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.

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?”

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

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.)