Lesson 4

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.

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

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.

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).

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 \(k^2\).)

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.

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?

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).