## Warm-Up: Cavalieri's Principle - Section 1: Warm-Up

# Real World Volume Context Problems

Lesson 12 of 14

## Objective: Students will be able to identify 2-D cross sections of 3-D figures while applying Cavalieri's Principle.

#### Warm-Up

*20 min*

In this Warm-Up, I give students a right triangular prism and an oblique parallelogram prism with equal length. I give the dimensions of the base of the right triangular prism and ask students to determine the base and height of the parallelogram base to target students’ understanding of Cavalieri’s Principle.

Like all warm-ups in my class, I ask students to work individually first. I circulate the room, looking for evidence in my students’ work—calculations for the triangle base area, factors of 36 for the parallelogram base—that indicate they are ready to discuss their ideas (**MP1**). I ask students to share out with their small group, and after a few minutes, facilitate a debrief discussion with the whole class. While it is not typically my style, I cold call students to share out different combinations of bases and heights for the parallelogram—this allows me to call on at least one student from each group to share out an answer, which is helpful for increasing participation across my classroom. I then choose at least two Recorder/Reporters to read out a high-quality written summary of the warm-up work from their group to highlight the academic and geometric vocabulary I want to see in their writing.

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#### Homework Review

*20 min*

At this point in the unit, I know students need time to check their work, look over their notes, ask questions, and practice. The last lesson’s Homework targeted students to calculate volumes of a range of prisms, cylinders, pyramids, and cones.

In Problem #7, students must convert units, which many often struggle with; this problem provides yet another opportunity to reinforce scale factor and its impact on volume since 12 inches in 1 foot mean there are 12^{3} inches in 1 foot^{3}.

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I want students to be able to solve volume problems that require them to consider the world around them. Homework: Real World Contexts requires students to determine the volume of a tree trunk, pipes, sausages in a can, spheres that have been packed into a box, and Earth and Mercury.

Problem #4 can pose some challenges to students, requiring them to visualize 2D cross-sections of 3-D shapes as they consider the arrangement of the sphere balls in the box.

Problem #6 can also pose some challenges to students, particularly because they need to show how they know in two different ways, requiring them to calculate and compare the volume of Earth and Mercury, as well as to apply the notion of ratios of similarity.

We will spend most of today's lesson working on last night's and tonight's problem sets. It is okay for the students (and I) to have a breather once in a while.

#### Resources

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- UNIT 1: Creating Classroom Culture to Develop the Math Practices
- UNIT 2: Introducing Geometry
- UNIT 3: Transformations
- UNIT 4: Discovering and Proving Angle Relationships
- UNIT 5: Constructions
- UNIT 6: Midterm Exam Review
- UNIT 7: Discovering and Proving Triangle Properties
- UNIT 8: Discovering and Proving Polygon Properties
- UNIT 9: Discovering and Proving Circles Properties
- UNIT 10: Geometric Measurement and Dimension
- UNIT 11: The Pythagorean Theorem
- UNIT 12: Triangle Similarity and Trigonometric Ratios
- UNIT 13: Final Exam Review

- LESSON 1: Sectors of Circles
- LESSON 2: Making Sense of Area Formulas for Triangles, Parallelograms, Trapezoids, and Kites
- LESSON 3: Making Sense of Area Formulas for Regular Polygons and Circles
- LESSON 4: Strategies for Decomposing 2-D Figures
- LESSON 5: Sector Area Application: The Grazing Goat
- LESSON 6: Surface Area and Area Differentiation
- LESSON 7: Extreme Couponing: Pizza Edition
- LESSON 8: Area "Quest"
- LESSON 9: Introduction to Volume: Origami Boxes
- LESSON 10: Origami Boxes Gallery Walk
- LESSON 11: Volume Formulas, Cavalieri's Principle, and 2-D Cross-Sections
- LESSON 12: Real World Volume Context Problems
- LESSON 13: Ratios of Similarity and 3D Solids Generated by Revolving 2D Figures
- LESSON 14: Volume "Quest"