The learning goals for this lesson are:

Describe (orally and in writing) how scale factors of 1, less than 1, and greater than 1 affect the size of scaled copies.
Explain and show (orally and in writing) how to recreate the original figure given a scaled copy and its scale factor.
Recognize (orally and in writing) the relationship between a scale factor of a scaled copy to its original figure is the “reciprocal” of the scale factor of the original figure to its scaled copy.

Students have used the definition of scale factor as a multiplier, but may still not be convinced that we can make a smaller scaled copy of a figure by multiplying. They may still describe scaling down by a scale factor of 1/2 as "divide by 2", and name 2 as the scale factor. In this lesson, we confront that head on!

We also introduce the idea that 1 can be a scale factor. If the copy looks the same, it may seem like we haven't done anything to it. Thinking about 1 as a possible scale factor gives us some completeness. We can describe what will happen when we use any positive number as a scale factor: We either make a smaller or larger copy, or we make something that looks exactly the same as what we started with. Those are the only three options, and we can achieve them all with multiplying.

Let's take a closer look at Activity 2: Scaled Copies Card Sort. Print and cut out the set of cards (found in "Additional Materials"). Notice in the directions that we're just asked to sort based on the scale factor, and we're not even asked to sort into a particular number of categories. Sort the cards a few different ways and describe your sort.

Let's take a closer look at Card 10 and Card 13. Students might put these cards in the same category because there is a "3-ness" about them. They might say we multiply the side lengths in A to get to B on Card 10, but we divide by 3 on Card 13. Remember that scale factors are used to multiply, so we need to describe dividing by 3 in a way that uses multiplication. That means we should multiply by 1/3. Related to our learning goal, a scale factor greater than 1 produces a copy larger than the original, and a scale factor smaller than 1 produces a copy that is smaller than the original.

We should also notice that 3 and 1/3 are reciprocals of each other. In the Lesson Summary, we will see another pair of scale factors that are not a whole number and a unit fraction. Will students expect those to work the same way?

In Activity 4: Missing Figure, Factor, or Copy, we get a chance to utilize this reciprocal idea in question 3, shown below.

Given our discussion in the card sort, do we expect the original figure to be smaller or larger than the copy?

We can think about what the original might look like in two ways. Let's focus on the right side of the figure. It has a length of 6. Some number was multiplied by 3232 to 6. I might reason that 6 is 112112 times 4, so the right side of the original figure must be 4. Each segment in the copy is 112112 times the size of the original.

Or, I can work from the copy to the original by using the reciprocal of the scale factor that got us from the original to the copy. I can scale the copy by 2323 to get a right side length of 4. How might it benefit students to check the reciprocals in each question in this activity?

Lesson 5: The Size of the Scale Factor