### Purpose

To teach students that most things have built-in limits in scale, but that knowing something’s extreme limits of scale can be revealing.

### Context

This lesson is designed to build upon students' ability to build structures and to learn about mathematical and engineering relationships like length, area, and volume. Students will find that they can measure very large or very small objects if they understand and implement certain principles.

Research suggests that, when approaching models, some students may demonstrate a qualitative knowledge or reasoning that is directly connected with prior experience with models. For other students, learning and applying new laws too rapidly will impede their ability to reason qualitatively. Therefore, it is important to take some time to get a feel for the limits of your students’ understanding of models. (*Benchmarks for Science Literacy*, p. 357.)

Ideas for this lesson are taken from the Exploratorium’s Structures Around the World website. In this lesson, students will learn to see scale patterns in larger and smaller shapes. The lesson will help students discover how volume, length, and area can proportionately change in cylinders that are scaled up and down in size. Before approaching this lesson, be sure that students have a basic understanding of geometric principles like measurement of height, width, area, and diameter (or circumference).

### Planning Ahead

Read the "Context" and "What It Is" section of the Cylinders and Scale online lesson for information about conducting this activity in your classroom.

### Motivation

Review what a cylinder is and how a cylinder is measured. Then invite students to think about how cylinders are used in the world around them. Ask them to name as many cylinders as they can. (Examples might include a soft drink can, potato chip canister, paper towel center, grain silo, pipeline, etc.)

Ask students why they think cylinders are different sizes. For example, what do they think the main difference is between a pipeline the circumference of a soda can and a pipeline as large in circumference as a car tire? (The larger pipeline can accommodate a higher capacity, or volume of liquid.)

Now ask students to imagine a canister of potato chips. Ask them to imagine that the manufacturer would like to make a new line of “super chips” that are 40 percent larger. Since the original canisters are too small, what does his company have to do to make the "super chip" canister larger?

Introduce the different-sized canisters filled with water and an empty measuring cup. Introduce them as cylinders of different shapes and volumes.

Ask students which container they think holds the most water (volume). Write their votes on the board and then pour the water from each into the measuring cup, making notes of the actual volumes. Were students surprised? What did they learn about cylinders?

### Development

Have students go to Making Cylinders: Step One on the Exploratorium website.

Before following the steps in the lesson, demonstrate how students will make cylinders. If your students have had some experience scaling cubes or other shapes, ask the questions listed on the bottom of the first page of the online lesson:

- What kind of pattern of growth do you expect as you scale up?
- How much like or unlike the patterns of cube growth do you expect the cylinder's pattern to be?

Now lead students through Step One, being careful that each step is understood and performed accurately. In the beginning of Step Two, students learn about the measurements of height, circumference, area, and volume.

Before moving on to the end of Step Two (on the third page), discuss these questions:

- What part of the rectangle is the height of the cylinder? (The short length.)
- What part of the rectangle will equal the circumference? (The longer length.)
- How is the area measured? (The area of the rectangle equals the area of the side of the cylinder.)
- How is the volume measured? (It is measured by the amount of sand it will hold.)

Continuing on page three, ask students to make and measure double, triple, and quadruple sized cylinders. Have them trace the outlines on a sheet of chart paper before taping them into cylinders. Explain that doing so will help them measure, compare, and visualize their results more concretely.

Ask students to follow instructions for measuring and filling out their charts as directed, but focus on the second means of measurement to help them compare and find growth patterns. Have them record these patterns on the chart.

Ask students:

- How is a cylinder scaled up by double? (By doubling the height [h] and circumference [c].)
- When measuring growth patterns, what are the two ways of measuring linear and area measurements? (Method one: measure lengths in centimeters and areas in the number of square centimeters covered; method two [the more effective way for this lesson]: measure the height (h), circumference (c), and diameter (d) of the smallest cylinder and determine how many base units are required in larger cylinders [questions listed mid-page on page four].)

Now, as outlined in the online lesson, use graph paper to measure the area of the circles by the number of graph cubes required. Measure volume by the number of film canisters required to fill each cylinder. Ask students to experiment with changes in volume by widening and closing the cylinder circumference and pouring in sand.

Ask students:

- Why is measuring the area of a cylinder not easy? (Because its base is a circle.)
- How is volume measured? (By filling the cylinders (film canisters) with units of sand.)
- If the shape weren't round, how would it change the volume? (That can vary with shape; extreme changes in shape can equal more extreme changes in volume.)

### Assessment

In the Discussing Results section, have students tape their cylinders to the side of the chart.

Ask students:

- What did you notice about the relationship between linear dimensions vs. surface and volume dimensions? (Linear dimensions grow in smaller increments compared to surface and volume, which grow more rapidly.)
- What two things do you need to do to make larger cylinders? (Change the height and the circumference.)
- What happens to the volume when you double only a cylinder's height? (The volume doubles.)
- What happens when you double just the circumference? (It doubles the length and width and increases the area of the base by four times.)
- Do you think that you can use what you've learned to measure almost any size cylinder and learn about its basic dimensions? Why or why not?

After the discussion, ask students to experiment by changing the height and/or circumference in various ways (by triple, quadruple, and so on).

In closing, lead students in a discussion of questions such as the following:

- What might be the limits on how big or small a cylinder can be?
- How can what you learned about units and measurement be applied to understanding any object, regardless of size?

### Extensions

For a related Science NetLinks lesson, see Shapes at Work. In this lesson, students look for shapes in both the natural and designed world and begin to learn more about how to build and design structures.

Check out What’s In a Shape?, a Science NetLinks lesson that uses tangrams to explore characteristics of shapes.

For high quality mathematics investigations, see NCTM’s Illuminations site.

Go to NyeLabs for an interesting project (Barometer in a Bottle) in which students learn about volume and atmospheric pressure with a jar and other materials. (NOTE: Flash-enabled browsers make the site more exciting!). When at the site, click on the helix, then on Home Demos, Planetary Science, Earth Science, and finally Barometer in a Bottle.

Invite students to measure cylinders in their everyday worlds! Encourage them to apply what they’ve learned to measure cylinders of different scales at home, school, and in the neighborhood.