## Scaling Away

• Lesson
6-8
1

Students will measure the dimensions of a common object, multiply each dimension by a scale factor, and examine a model using the multiplied dimensions. Students will then compare the surface area and volume of the original object and the enlarged model.

In this lesson, students will examine a rectangular prism or cylinder. Before this lesson, have students bring in common objects that are either cylinders or rectangular prisms. Have a collection of additional items ready to distribute if needed.

As a prerequisite to this lesson, students must be able to compute the surface area and volume. The applets listed below can be used to provide a visual demonstration of surface area and volume of prisms in class, or students can use these applets to explore on their own.

To engage students in this lesson, show a scale model, such as a Hot Wheels® toy car or an HO scale train replica.

The Hot Wheels® Monster Jam® car on the Scaling Overhead is built on a 1:24 scale; HO scale train replicas are built on a 1:87 scale. That is, the scale factor is the number by which each dimension of the original object is multiplied to find the corresponding dimension of the model. (Note that in these examples, the scale factor is less than 1, which means that the model is smaller than the original object. In this lesson, students will consider a model for which the scale factor is greater than 1, which means that the model will be larger than the original.)

Ask students when they have seen a scale model. They may mention models of buildings, neighborhoods, or monuments. Most students have seen miniature replicas of famous places, such as the Eiffel Tower or Empire State Building.

Say to students, "Suppose you wanted to build an exact replica of your object, enlarging it by using a scale factor of 5. Picture in your mind how it would look. How many times larger do you think the surface area will become? How many times larger do you think the volume will become?"

Distribute the Scaling Away Activity Sheet. Explain to students that they will record their hypotheses, measure the dimensions of their object, and compute its surface area and volume. Then, they will choose a scale factor, determine the dimensions of a model using that scale factor, and compute the surface area and volume of the model. Finally, they will examine what happens to the surface area and volume. After working through the questions on the activity sheet, the students will discuss their findings as a class.

For Question 1, students are to predict what they think will happen to the surface area and volume of an object when each dimension is increased. Allow students to write a conjecture, and then discuss as a class. Explain to students that they will test their conjectures during this lesson.

After the discussion, distribute rulers or tape measures and have students work independently on Questions 2-4. As students work, check for accuracy in their measurements and calculations. Discuss the advantage of computing with decimals in the metric system instead of fractions in the customary system.

When computing surface area and volume, a useful mnemonic for students is the FSC2 method: Formula, Substitute, Compute, Correct units.

To help students compute the surface area of their object, you may want to use the Surface Area Overhead. This overhead helps students to organize the information when finding the area of each of the parts of a three-dimensional object. (A similar template could be used to find the volume.)

Note that questions about surface area appear before questions about volume on the worksheet because of the transition from one to two to three dimensions—that is, from length to area to volume. However, students may be more successful computing volume before computing surface area.

Students next determine the dimensions of the model using their scale factor from Question 5, as well as the surface area and volume in Questions 6-7. A common misconception is to simply take the previous calculations and multiply by the scale factor to find the surface area and volume of the model. This is not correct! Make sure that students use the same procedure that was used previously, which involves identifying the correct formula, substituting, computing, and labeling the solution with the correct units.

If students compute accurately, they will discover how much larger the model is than the original. By answering Questions 8-9, students should notice the following:

That is, the ratio between the length of the object and the length of the model is not equal to the ratio of surface areas, and neither of these is equal to the ratio of volumes.

The Scaling Ratios Overhead may be helpful for finding the ratios.

Earlier in the lesson, students chose a scale factor, n. Students should now find that the numerator for the ratio of surface areas is n2, and the numerator for the ratio of volumes is n3. Allow students to compare their answers. Students who chose the same scale factors should verify that they got the same ratios in Questions 8-9.

The following algebraic explanation may help students understand why the surface increased by a factor of n2 and why the volume increased by a factor of n3.

If the original object was a prism, it had dimensions l, w, and h.

The dimensions of the model are equal to the dimensions of the object multiplied by the scale factor n, which are nl, nw, and nh.

Original surface area:

2lw + 2lh + 2wh

New surface area:

 2(nl)(nw) + 2(nl)(nh) + 2(nw)(nh) = 2n2lw + 2n2lh + 2n2wh = n2 (2lw + 2lh + 2wh)

Original volume:

lwh

New volume:

(nl)(nw)(nh) = n3(lwh)

A similar argument could be used to show why the surface area and volume of a cylinder increase by the same factors.

As students discover this relationship, they will understand the effects of scale factors on volume and surface area. More importantly, they will begin to develop an understanding of why square units are used for area and why cube units are used for volume.

It is most important for students to discover this relationship on their own. If they cannot write their own conclusion at first, be patient. Exploring the results for other scale factors, hearing about the results of their classmates, and investigating other objects may help students to grasp this important mathematical concept.

Assessment Options

1. The Scaling Away Activity sheet may be collected and evaluated.
2. Have students present their work to the class. During the presentations, every student should keep a chart showing object, scale factor, ratio of surface areas, and ratio of volumes. By the end of the presentations, students will have a large amount of data, and they should be able to see the pattern of n, n2, and n3.
3. The students may build their scale model at home to provide additional experience with measurement. Students may then see hands on how many of their original objects will fit on the surface of their model (if the scale factor is n, then n2 of their objects will fit on the surface). They will also be able to see that with a scale factor of n, then n3 of their objects will fit inside.
4. Students may write a description of the mathematics used in their project. They should include a description of the conclusions they reached based on this lesson as well as a statement of the important mathematics they learned.
5. Students may apply what they have learned by solving the following problem:
A school is being remodeled. Using a scale of 1:20, the architect built a scale model so the design could be shown to parents, teachers, students, and the school board. If the school is 20 feet tall, and if the architect used a scale of 1:20, how tall would the model be? If the school is 100 feet long, how long would the model be? If the width of the model is 14 inches, what is the width of the school?
6. Have students react to this statement: "I made a model using a scale factor of 5. My model is five times the size of the original." Explain why you agree or disagree with this statement.

Extensions

1. Have students examine cubes with side lengths of 1 cm, 2 cm, 3 cm, 4 cm, and 5 cm. (Snapping centimeter cubes may be used to build these larger cubes.) Compute the surface area of each, and create a graph showing the relationship between side length and surface area. As the side length increases, what happens to the surface area? Does it increase at a constant rate? Describe the shape of the graph. If the side length is n, what is the surface area? Students may also make a similar graph comparing the side length to the volume. If the side length is n, what is the volume?
2. Have students interview an architect or engineer. They can share their model with her and examine the models that she builds. Find out how ratios and scale factors are used in their careers.

Questions for Students

1. What happens to the volume when an object is enlarged by a given scale factor? What happens to the surface area?

[Both the surface area and volume increase when an object is enlarged.]

2. What is the ratio of the surface area of the original object to the surface area of the model? What is the ratio of the volumes? How does this compare to the ratio of the side lengths?

[When an object is enlarged (or, for that matter, shrunk) by a scale factor n, the resulting surface area is n2 times the original surface area, and the resulting volume is n3 times the original volume. Consequently, the ratio of side lengths is 1:n, the ratio of surface areas is 1:n2, and the ratio of volume is 1:n3.]

Teacher Reflection

• Were concepts were presented too abstractly? Too concretely? How would you change them?
• Was students’ level of enthusiasm high or low? Explain why.
• What background knowledge was needed for this lesson? How did you verify that students had the prerequisite skills they needed to be successful with this lesson?
• What adjustments will you make when you teach this lesson again?

### Learning Objectives

Students will:

• Measure the dimensions of a rectangular prism or cylinder, and determine the dimensions of an enlarged model using a scale factor of 3, 4, 5, 6, 7, or 8.
• Compute surface area and volume and determine the ratio of surface areas and volumes between the object and the model.
• Learn that if the dimensions of an object are multiplied by a scale factor n, then the surface area is multiplied by n2 and the volume is multiplied by n3.

### NCTM Standards and Expectations

• Understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.
• Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders.
• Solve problems involving scale factors, using ratio and proportion.