Lesson 4

Scaling and Area

4.1: Squares and Roots (5 minutes)

Warm-up

In grade 8, students used evaluated square roots of small perfect squares and determined that $$\sqrt{2}$$ is irrational. Here, these concepts are revisited in preparation for working with the effects of scaling on area.

Student Facing

1. What number times itself equals 25?
2. What number times itself equals 81?
3. What number times itself equals 10?

Student Response

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Anticipated Misconceptions

Students may initially believe there is no number that multiplies by itself to give 10. Prompt them to think outside of the whole numbers.

Activity Synthesis

Remind students that another word for multiplying a number by itself is squaring the number. We represent this with an exponent of 2. For example, we write 52 = 25.

We can take this process backwards, too. If we start with a number $$x$$ and look for the value that squares to give $$x$$, that value is said to be the square root of $$x$$. For example, we write $$\sqrt{25}=5$$. The number 25 is called a perfect square because its square root is a whole number. The number 10 is not a perfect square. We can simply write $$\sqrt{10}$$ for its square root, or we can find an approximation of about 3.2, either with the square root function on the calculator or by doing mental estimation.

4.2: Scaling Up a Rectangle (20 minutes)

Activity

The purpose of this activity is for students to calculate areas of rectangles on a grid, organize that information in a table, and look for structure in the table and in an algebraic expression to create a rule that describes the relationship between scale factor and area.

As students determine the pattern that emerges over several scale factors, they are expressing regularity in repeated reasoning (MP8).

Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention

Student Facing

Here is a rectangle with length 5 units and width 2 units.

1. What is the area of the rectangle?
2. Dilate rectangle $$ABCD$$ from point $$A$$ by a scale factor of 2. Calculate the area of the image.
3. Dilate rectangle $$ABCD$$ from point $$A$$ by a scale factor of 3. Calculate the area of the image.
4. Complete the table.
scale factor area of image in square units factor by which the area changed
0.5
1
2
2.5
3
4
5. Write an expression for the area of a rectangle with length $$\ell$$ and width $$w$$.
6. Imagine dilating the rectangle with length $$\ell$$ and width $$w$$ by a factor of $$k$$. Write expressions for the dimensions of the dilated rectangle.
7. Write an expression for the area of the dilated rectangle.
8. Use your work to draw a conclusion about what happens to the area of a rectangle when it’s dilated by a scale factor of $$k$$.

Student Response

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Student Facing

The image shows a dilation of an equilateral triangle by scale factors of 1, 2, 3, and 4. At each stage, the dilated shape is partitioned into triangles congruent to the original.

1. Why is the difference in the number of original triangles that fit into the image between 2 successive scale factors always odd?
2. What does this image tell us about the sum of the first $$n$$ odd numbers?

Student Response

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Activity Synthesis

The goal of the discussion is to establish that scaling a rectangle by a factor of $$k$$ has the result of multiplying the area by $$k^2$$. Here are some questions for discussion:

• “How do the algebraic expression and the numbers in the table relate to each other?” (The algebra says that the area of a figure scaled by $$k$$ should change by $$k^2$$. We saw that, for example, when $$k=3$$, the area changed by a factor of $$3^2=9$$.)
• “Did the rule hold for scale factors that weren’t whole numbers?” (Yes. For example, when $$k=0.5$$, the area changed by $$0.5^2=0.25$$.)
• “Suppose we scale the area of the original 2 by 5 rectangle by a factor of $$k$$. What expression would give its area?” ( $$10k^2$$ )
• “What would be different if the original rectangle had an area of $$A$$ square units instead of 10?” (The area of the scaled rectangle would be $$Ak^2$$. Regardless of what the original length and width are, they each get multiplied by $$k$$, so the area is multiplied by $$k^2$$.)
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share correct values from the table, present an incorrect answer and explanation. For example, “The original rectangle $$ABCD$$ has an area of 10 square units. If the scale factor is 3, then the area of the larger rectangle is 30 square units, because 3 times 10 is 30.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who clarify the definition of scale factor and how it affects the side lengths of the rectangle. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to justify why the area of the larger rectangle is 9 times greater than the original rectangle. This will help students evaluate and improve on the written mathematical arguments of others.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

4.3: What About Other Shapes? (10 minutes)

Activity

The purpose of this activity is to generalize scaling properties of rectangles. The area of any shape can be approximated with rectangles. So, the property that the area of a rectangle is multiplied by $$k^2$$ when it’s dilated using a scale factor of $$k$$ applies to all two-dimensional shapes.

Launch

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for the second question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How do you know that the area of each rectangle is multiplied by $$k^2$$?”, and “How do you know that the area of the dilated blob is also multiplied by $$k^2$$?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify their reasoning for why the area of any shape is multiplied by $$k^2$$ when dilated by scale factor $$k$$.
Design Principle(s): Optimize output (for explanation); Cultivate conversation

Student Facing

Andre says, “We know that if a rectangle is scaled by a factor of $$k$$, the area scales by a factor of $$k^2$$. Does this apply to other shapes?”

Jada says, “Here’s a shape that’s not a rectangle. Say its area is $$A$$ square units. Let’s draw some rectangles on it that get smaller and smaller to fit the remaining empty space. With enough rectangles we can come close to covering the whole blob.”

Andre says, “These rectangles start to make a nice approximation of the blob. If we wanted to get closer, we could add even more rectangles. The sum of the areas of all the rectangles would add up to the area of the blob. I think we’re almost there!”

1. Suppose the blob is dilated by a factor of $$k$$. In doing this, the rectangles covering the blob also get dilated by a factor of $$k$$. How does the area of each dilated rectangle compare to the area of each original rectangle?
2. What does this tell you about the area of the dilated image? Explain your reasoning.
3. Suppose a circle has area 20 square inches and it’s dilated using a scale factor of 6. What is the area of the image? Explain or show your reasoning.

Student Response

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Anticipated Misconceptions

If students give an answer of 120 square units for the answer to the last part, ask them what happened when the rectangle in the previous activity was scaled by a factor of 3. Did the area also increase by a factor of 3?

Activity Synthesis

Ask students to share their thinking about what happens to the area of the blob when it is dilated by a scale factor of $$k$$. Display this image for all to see.

Label several of the rectangles in the original blob $$a_1, a_2, \ldots$$ . If students are not familiar with subscripts, explain that they are a way to label variables to make it easier to keep track of them. That is, $$a_1$$ is the area of the first rectangle in square units, $$a_2$$ is the area of the second rectangle, and so on.

Ask students how many rectangles there are in total (9) and how they can write an expression for the area of the original blob: $$a_1 + a_2 + \ldots + a_{9}$$.

Now ask students to write expressions for the areas of the corresponding rectangles in the dilated blob: $$k^2 \boldcdot a_1, k^2 \boldcdot a_2$$, and so on. Ask students how they could write the area of the dilated blob: $$k^2 \boldcdot a_1 + k^2 \boldcdot a_2 + \ldots + k^2 \boldcdot a_{9}$$.

We can use the distributive property to rewrite the expression for the blob’s area as $$k^2 (a_1 + a_2 + \ldots + a_{9})$$, which is $$k^2$$ times the sum of all the original rectangles. That means the area of the dilated figure is $$k^2$$ times the area of the original figure. The takeaway is that the area of any shape, no matter how irregular, gets multiplied by $$k^2$$ when the shape is dilated by a scale factor of $$k$$.

Most importantly, invite students to share strategies for how they solved the last problem. Ask them how the number 10 in the previous activity relates to this problem (the 10 was the area of the original rectangle, but for the circle, that number is 20).

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use color coding and annotations to highlight the connections between each rectangle inside the dilated blob and each term in the equation for the area of the dilated blob. More specifically, use the same color for the first rectangle inside the dilated blob and the expression for its area $$k^2 \boldcdot a_1$$.
Supports accessibility for: Visual-spatial processing

Lesson Synthesis

Lesson Synthesis

The goal of the discussion is to reinforce the idea that when a figure is dilated by a factor of $$k$$, its area changes by a factor of $$k^2$$.

Draw any non-rectangular shape and display it for all to see. Tell students the shape has an area of 40 square units. Ask students:

• “Suppose we dilate this shape by a factor of 2. Will the area of the dilated figure be twice the original?” (No, the area will be 4 times the original, or 160 square units.)
• “Suppose we dilate the shape by a factor of $$\frac12$$ or 0.5. Will the area of the dilated figure be half the original?” (No, it will be a quarter of the original, or 10 square units, because $$\left(\frac12 \right)^2=\frac14$$.)
• “Suppose we dilate the shape by a factor of 3.5. How would you find the area of the dilated figure?” (First, square the scale factor to get 3.52 = 12.25. Then multiply 12.25 by the area 40 square units to get 490 square units.)

4.4: Cool-down - Finding an Area (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

Say you have a 5 inch by 4 inch rectangle. Its area is 20 square inches. If you dilate the rectangle by a scale factor of 3, what do you think the area of the new rectangle will be?

The dilated rectangle’s dimensions can be written as $$3\boldcdot 5$$ and $$3\boldcdot 4$$. Substitute these values into the area expression $$\ell \boldcdot w$$ to get $$(3\boldcdot 5)(3\boldcdot 4)$$. Rearrange the numbers to get $$(3\boldcdot 3)(5\boldcdot 4)$$. This can be rewritten $$(3^2)(5\boldcdot 4)$$ or $$9 \boldcdot 20$$. So the area of the dilated rectangle is 180 square inches, or 9 times the original. The diagram confirms that 9 of the original rectangles fit into the dilated rectangle.

In general, when you scale a rectangle by a factor of $$k$$, the length and the width are both multiplied by $$k$$, so the area is multiplied by $$k^2$$.

What if you dilate a shape that is not a rectangle by a scale factor of $$k$$? Consider the rounded shape called an ellipse in the image. You can approximate the area of an ellipse by filling it with many rectangles. The sum of the areas of the rectangles will be a little less than the area of the ellipse because they don’t fill it entirely. If you want to get closer to the area of the ellipse, draw in more rectangles. If you continued the process infinitely, you could find the exact area of the ellipse this way.

If you dilated this ellipse using a scale factor of 4, each rectangle would become 16 times larger. This means that the area of the ellipse will increase by a factor of 16 as well. Any closed shape can be filled with rectangles that approximate its area. Because of this, when you scale any shape by a factor of $$k$$, its area is multiplied by a factor of $$k^2$$.