# Calculating Volumes of Compound Objects

## Mathematical goals

This lesson unit is intended to help you assess how well students solve problems involving measurement and in particular, to identify and help students who have difficulty:

- Computing measurements using formulas.
- Decomposing compound shapes into simpler ones.
- Using right triangles and their properties to solve real-world problems.

## Introduction

The unit is structured in the following way:

- Before the lesson, students attempt the task individually. You then review their work and formulate questions for them to answer, in order to improve their work.
- At the start of the lesson, students work individually to answer your questions. Next, they work in small groups on the same task, to produce better collective solutions than those they produced individually. Throughout their work they justify and explain their decisions to each other.
- In the same small groups, students critique some student responses to the
*Glasses*task, provided in the*Sample Responses to Discuss.*They correct errors in those responses. - In a whole-class discussion, students discuss their own solutions and explain how to correct the common misconceptions seen in the
*Sample Responses to Discuss*. - Finally, students work on their own again to improve their individual solutions.

## Materials required

- Each student will need two copies of the
*Glasses*task, the*Formula Sheet*, a mini-whiteboard, pen, and eraser. - Each small group of students will need a new copy of the
*Glasses*task and the*Sample Responses to Discuss.* - Some students may require the
*Glasses: Extension Questions.*Have copies of the*Hint Sheet*ready in case any students need it.

## Time needed

15 minutes before the lesson, a 1-hour lesson, and 15 minutes in a follow-up lesson. All timings are approximate and will depend on the needs of the students.

## Lesson Type

## Mathematical Practices

This lesson involves a range of *mathematical practices* from the standards, with emphasis on:

- MP1: Make sense of problems and persevere in solving them
- MP2: Reason abstractly and quantitatively
- MP3: Construct viable arguments and critique the reasoning of others
- MP4: Model with mathematics
- MP5: Use appropriate tools strategically
- MP6: Attend to precision
- MP7: Look for and make use of structure

## Mathematical Content Standards

This lesson asks students to select and apply mathematical content from across the grades, including the *content standards:*

- H.G-GMD: Explain volume formulas and use them to solve problems
- H.G-GMD: Visualize relationships between two-dimensional and three- dimensional objects
- H.G-SRT: Define trigonometric ratios and solve problems involving right triangles
- H.N-Q: Reason quantitatively and use units to solve problems