## Fill 'Er Up

- Lesson

In this lesson, students use an interactive applet to investigate the formula for the volume of a rectangular prism. Students will construct two origami boxes and use centimeter cubes to measure and compare the volume of the boxes. Students will also analyze how changing the dimensions of the prism affects its volume.

Prior to teaching this lesson, familiarize yourself with the Cubes applet.

Introduce the lesson by holding up a familiar household prism such
as a clear plastic storage container. Use the object to review
vocabulary including: *rectangular prism*, *length*, *width*, *height*, *surface
area*, and *volume*. Ask students to identify the three linear
measurements: length, width, and height. Start by labeling the bottom
face and its length and width. The height is the perpendicular distance
between the top and bottom faces. Explain to students that length is
sometimes called depth. The applet that students will explore uses the
terms width, depth, and height.

To illustrate the concept of volume, drop some inch or centimeter cubes into the box. Let students know that in today's lesson they will explore volume by interacting with virtual boxes and by making origami boxes. Project the Cubes applet, and show students how to navigate it.

Distribute the Fill 'Er Up Activity Sheet to each student. Have students work with partners and explore prisms. Then, have them complete Questions 1–6.

Distribute two sheets of paper and some centimeter cubes to each student. Be sure to place enough centimeter cubes that students can measure the length, width, and height of their prisms, but not so many that they can completely fill their prisms. Have students work in larger groups of 3–4, providing peer support during the origami. Each student will use the Folding Directions to create a physical model of a rectangular prism. Some students will experience difficulty following the written and pictorial directions. Try to assign a student with strong spatial reasoning to guide their group members through the folding. As an instructional accommodation, consider providing completed models for students who have difficulty.

Discuss with students how the volume can be determined without completely filling the prism with cubes. Some students may need to fill their prism completely.

Summarize key concepts of the lesson at the end of class. Ask volunteers to explain how they approximated the volume of the box using centimeter cubes. The first two rows from the table in Question 2 will reinforce the concept that doubling all three dimensions will result in a box with a volume that is eight times as large. Show symbolically that when the dimensions of a prism are doubled, the volume of the new box will be eight times as large as the original box.

*V*=_{original}*lwh**V*= (2_{new}*l*)(2*w*)(2*h*) = 8*lwh*

Reinforce this idea by showing a box that was made using paper with double the dimensions. Verify using visual estimation that the volume of this box is 8 times as large as the original box. You might also want to go over the last row from the table in Question 2 because it involves algebraic reasoning.

Showcase a well-made origami model. This is a good opportunity to highlight the accomplishment of a student who does not participate in class often or applaud the efforts of a low achiever.

### Reference

Higginson, William and Lynda Colgan. "Algebraic Thinking Through Origami." Mathematics Teaching in Middle School 6: 343.

- Computers or tablets with internet connection
- Computer projector
- Fill'er Up Activity Sheet
- 8.5 x 11 inch paper, colored if possible: 2 sheets per student
- Folding Instructions
- Centimeter cubes

**Assessment Options**

- Students complete a 3-2-1 Journal Entry: "3 mathematical terms I reviewed today; 2 things I did in class; 1 question I still have."
- Require students to complete a ticket-to-leave. Students should respond to a prompt like 'Give the dimensions of two non-congruent prisms that have the same volume.' They should justify their answers by showing their reasoning.

**Extensions**

- Students who show an interest in origami could construct a modular origami box.
- Students research another mathematical origami model that is not a rectangular prism and share it with the class.
- Explore the surface area of rectangular prisms, take special notice of how surface area changes as the dimensions of a prism change.

**Questions for Students**

1. Which dimensions can be measured in a prism?

[Length (depth), width, and height are the three dimensions of a prism.]

2. Define volume in your own words.

[Volume is the amount of cubes it takes to fill an object.]

3. How do you calculate the volume of a prism?

[Multiply the area of the prism's base by the area of its height.]

4. If you know the volume of a prism, and you know the width and depth, how can you determine the height?

[Divide the volume of the prism by the product of the width and depth.]

5. If the width and depth of a prism double, but the height stays the same, how will the volume change?

[The volume will quadruple.]

6. How will halving all three dimensions affect the volume of a prism?

[The new volume will be 1/8 the volume of the original prism.]

**Teacher Reflection**

- How much background on volume, if any, did your students have prior to this lesson?
- Which questions on the activity sheet were confusing for your students?
- Which folding instructions challenged your students?
- What difficulties did students experience that you had not anticipated?
- What student accommodations can you make next time?
- What did the summary reveal about the students' understanding?
- How can you modify the lesson to help students with learning differences succeed?
- What extensions would you suggest for more capable students?
- In what ways did you meet the needs of students who learn best by: hearing, seeing, using technology, doing, and interacting with others?

### Learning Objectives

- Understand and apply the formula for the volume of a rectangular prism.
- Use origami to create a three-dimensional prism from a two-dimensional sheet of paper.

### NCTM Standards and Expectations

- Understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.

- Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.

- Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders.