Lesson 6Scaling and Area

Learning Goal

Let’s build scaled shapes and investigate their areas.

Learning Targets

I can describe how the area of a scaled copy is related to the area of the original figure and the scale factor that was used.

Lesson Terms

area
corresponding
reciprocal
scale factor
scaled copy

Warm Up: Scaling a Pattern Block

Your teacher will give you some pattern blocks. Work with your group to build the scaled copies described in each question.

Pictures of a blue parallelogram, a green triangle, and a red trapezoid.

Problem 1

Use the applets to explore the pattern blocks. Work with your group to build the scaled copies described in each question.

How many blue rhombus blocks does it take to build a scaled copy of Figure A:

open applet in presentation mode

Where each side is twice as long?
Where each side is 3 times as long?
Where each side is 4 times as long?

Print Version

How many blue rhombus blocks does it take to build a scaled copy of Figure A:

Where each side is twice as long?
Where each side is 3 times as long?
Where each side is 4 times as long?

Problem 2

How many green triangle blocks does it take to build a scaled copy of Figure B:

open applet in presentation mode

Where each side is twice as long?
Where each side is 3 times as long?
Using a scale factor of 4?

Print Version

How many green triangle blocks does it take to build a scaled copy of Figure B:

Where each side is twice as long?
Where each side is 3 times as long?
Using a scale factor of 4?

Problem 3

How many red trapezoid blocks does it take to build a scaled copy of Figure C:

open applet in presentation mode

Using a scale factor of 2?
Using a scale factor of 3?
Using a scale factor of 4?

Print Version

How many red trapezoid blocks does it take to build a scaled copy of Figure C:

Using a scale factor of 2?
Using a scale factor of 3?
Using a scale factor of 4?

Problem 4

Make a prediction: How many blocks would it take to build scaled copies of these shapes using a scale factor of 5? Using a scale factor of 6? Be prepared to explain your reasoning.

Activity 1: Scaling More Pattern Blocks

Your teacher will assign your group one of these figures, each made with original-size blocks.

Pictures of shapes made with pattern blocks. The first one is made up of blue parallelograms, next is red trapezoids, and then green triangles.

Problem 1

In the applet, move the slider to see a scaled copy of your assigned shape, using a scale factor of 2. Use the original-size blocks to build a figure to match it. How many blocks did it take?

open applet in presentation mode

Print Version

Build a scaled copy of your assigned shape using a scale factor of 2. Use the same shape blocks as in the original figure. How many blocks did it take?

Problem 2

Your classmate thinks that the scaled copies in the previous problem will each take 4 blocks to build. Do you agree or disagree? Explain you reasoning.

Problem 3

Start building a scaled copy of your assigned figure using a scale factor of 3. Stop when you can tell for sure how many blocks it would take. Record your answer.

Problem 4

How many blocks would it take to build scaled copies of your figure using scale factors 4, 5, and 6? Explain or show your reasoning.

Problem 5

How is the pattern in this activity the same as the pattern you saw in the previous activity? How is it different?

Are you ready for more?

How many blocks do you think it would take to build a scaled copy of one yellow hexagon where each side is twice as long? Three times as long?
Figure out a way to build these scaled copies.
Do you see a pattern for the number of blocks used to build these scaled copies? Explain your reasoning.

Activity 2: Area of Scaled Parallelograms and Triangles

Problem 1

Your teacher will give you a figure with measurements in centimeters. What is the area of your figure? How do you know?

Problem 2

Work with your partner to draw scaled copies of your figure, using each scale factor in the table. Complete the table with the measurements of your scaled copies.

scale factor	base (cm)	height (cm)	area (cm²)
$1$
$2$
$3$
$\frac{1}{2}$
$\frac{1}{3}$

Problem 3

Compare your results with a group that worked with a different figure. What is the same about your answers? What is different?

Problem 4

If you drew scaled copies of your figure with the following scale factors, what would their areas be? Discuss your thinking. If you disagree, work to reach an agreement. Be prepared to explain your reasoning.

scale factor	area (cm²)
$5$
$\frac{3}{5}$

Lesson Summary

Scaling affects lengths and areas differently. When we make a scaled copy, all original lengths are multiplied by the scale factor. If we make a copy of a rectangle with side lengths 2 units and 4 units using a scale factor of 3, the side lengths of the copy will be 6 units and 12 units, because $2 \cdot 3 = 6$ and $4 \cdot 3 = 12$ .

Two rectangles: The first rectangle has a horizontal length labeled 4 and vertical width labeled 2. The second rectangle has a horizontal length labeled 12 and vertical width labeled 6.

The area of the copy, however, changes by a factor of (scale factor)². If each side length of the copy is 3 times longer than the original side length, then the area of the copy will be 9 times the area of the original, because $3 \cdot 3$ , or $3^{2}$ , equals 9.

Two rectangles. The first rectangle has the vertical side labeled 2 and the horizontal side labeled 4. The second rectangle has the vertical side labeled 6 and the horizontal side labeled 12. Two horizontal dashed lines and 2 vertical dashed lines are drawn in the second rectangle dividing it into 9 identical smaller rectangles.

In this example, the area of the original rectangle is 8 units² and the area of the scaled copy is 72 units², because $9 \cdot 8 = 72$ . We can see that the large rectangle is covered by 9 copies of the small rectangle, without gaps or overlaps. We can also verify this by multiplying the side lengths of the large rectangle: $6 \cdot 12 = 72$ .

Lengths are one-dimensional, so in a scaled copy, they change by the scale factor. Area is two-dimensional, so it changes by the square of the scale factor. We can see this is true for a rectangle with length $l$ and width $w$ . If we scale the rectangle by a scale factor of $s$ , we get a rectangle with length $s \cdot l$ and width $s \cdot w$ . The area of the scaled rectangle is $A = (s \cdot l) \cdot (s \cdot w)$ , so $A = (s^{2}) \cdot (l \cdot w)$ . The fact that the area is multiplied by the square of the scale factor is true for scaled copies of other two-dimensional figures too, not just for rectangles.