##
* *Reflection: Advanced Students
Volume Formulas, Cavalieri's Principle, and 2-D Cross-Sections - Section 3: You Choose: Practice Thinking About Volume or Take it Further

I wanted to design a lesson that would allow all my students to see how area and volume relate to one another—specifically how the area of a 2D cross-section relates to the volume of the 3D solid from which it was taken. However, since making sense of Cavalieri’s Principle is a “plus” standard in the Common Core, I knew I had to teach this lesson at a level appropriate for each student.

Student engagement was very high in the group that wanted to make sense of the volume formula for a sphere. By differentiating this segment of the lesson, I gave my students high cognitive tasks that targeted them at the “right” level. In particular, the group that took on the sphere volume task grappled with really important mathematical ideas. They had to decipher what exactly each variable represented in the cross-sections taken from two different 3D solids, figure out how the variables related to one another, and consider how they would write a proof about the areas of these cross-sections to then derive the volume formula for a sphere. Overall, I was really proud of my students who engaged in this work because they used the kind of thinking that is often used in Calculus BC and later math classes.

The groups that worked on the prism volume task were able to see that they could quadruple a prism’s volume by quadrupling either the prism’s height or base area. Because one of the groups had actually built four rectangular prisms, they physically saw that doubling (not quadrupling) the dimensions of the base would quadruple the prism’s volume, an important concept I wanted students to take away from the task.

*Why Differentiate?*

*Advanced Students: Why Differentiate?*

# Volume Formulas, Cavalieri's Principle, and 2-D Cross-Sections

Lesson 11 of 14

## Objective: Students will be able to identify the shapes of 2D cross-sections of 3D objects and use geometric shapes, their measures, and properties, to describe objects.

## Big Idea: In a differentiated lesson based on student choice, students will be able to practice finding the volume of prisms or apply Cavalieri's Principle to determine the volume formula for a sphere.

*80 minutes*

#### Homework Review

*20 min*

To begin class today I ask my students to compare their work on the Origami Box Homework with others, making corrections and changes in a colorful pen. Since this homework requires students to exercise algebraic manipulation, it is really important for them to have a chance to explain how they performed their procedures and made sense of algebraic expressions to each other (**MP3**).

When I debrief this homework assignment with the whole class, I make sure to call attention to one or two of the Origami Box posters to help students recall how similar solids’ volumes and surface areas increase by the scale factor^{3} or scale factor^{2}.

#### Resources

*expand content*

#### Launch Volume

*25 min*

I use this powerpoint to formally launch this part of my Measurement and Dimensionality unit. Since my students have had previous experiences dealing with volume, I want to activate their prior knowledge and create intrigue by introducing them to **Cavalieri's Principle** right from the beginning, in a motivating candy bar context.

*expand content*

Since students have seen volume before in their elementary and middle school experiences, they exhibit a range of comfort levels with solving volume problems. For this reason, it is important to differentiate students' work to engage students at appropriate levels. In this lesson, I give students the choice between practicing calculating the volume of prisms (triangular, rectangular, pentagonal, and hexagonal), and "taking it further," where we will apply Cavalieri's Principle to determine the volume formula for a sphere (**MP8**).

Since I facilitate the "taking it further" small group discussion while other students work on “Thinking About Volume,” I make sure to physically position myself so I can easily monitor students as they work. After I ensure students’ understand the comparison of the cross-sections of the hemisphere with the cross-sections of the cylinder containing the cone, I leave the group to work out the algebra while I check in with the students who are calculating the volumes of the prisms they have built.

*expand content*

I want to assess students’ understanding of Cavalieri’s Principle and their facility with determining the 3-D solids made by 2-D cross-sections. I ask students to work on this check for understanding individually to give me an accurate picture of students’ understanding. I collect these exit tickets as students leave the classroom, selecting model student writing samples to share out with the class in the next lesson.

*expand content*

This homework assignment gives students the opportunity to consider cross-sections of 3D figures and to solve volume problems.

*expand content*

Hi Rebecca! Thanks for posting this comment. Â I uploaded the files now so you can see the nets I used for the Thinking About Volume task.

| 2 years ago | Reply

In the Thinking about Volume task, from what objects do the students take measurements and what does the instruction "Cut out and build prisms from your nets." mean? Â Thanks

| 2 years ago | Reply##### Similar Lessons

###### Old and New Knowledge of Circles

*Favorites(0)*

*Resources(13)*

Environment: Urban

###### The Pyramid Stack (Part 1)

*Favorites(0)*

*Resources(22)*

Environment: Rural

###### Exploring Circumference, Area and Cavalieri's principle

*Favorites(8)*

*Resources(19)*

Environment: Suburban

- 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"