Lesson 6

By now, students understand that lengths in a scaled copy are related to the original lengths by the scale factor. Here they see that the area of a scaled copy is related to the original area by the square of the scale factor.

Students build scaled copies of a single pattern block, using blocks of the same shape to do so. They determine how many blocks are needed to create a copy at each specified scale factor. Each pattern block serves as an informal unit of area. Because each original shape has an area of 1 block, the \(\text{(scale factor)}^2\) pattern for the area of a scaled copy is easier to recognize.

Students use the same set of scale factors to build copies of three different shapes (a rhombus, a triangle, and a hexagon). They notice regularity in their repeated reasoning and use their observations to predict the number of blocks needed to build other scaled copies (MP8).

Some students may come up with one of these arrangements for the first question, because they assume the answer will take 2 blocks to build:

You could use one pattern block to demonstrate measuring the lengths of the sides of their shape, to show them which side they have not doubled.

Students may also come up with:

for tripling the trapezoid, because they triple the height of the scaled copy but they do not triple the length. You could use the process described above to show that not all side lengths have tripled.

Display a table with only the column headings filled in. For the first four rows, ask different students to share how many blocks it took them to build each shape and record their answers in the table.

To help student notice, extend, and generalize the pattern in the table, guide a discussion using questions such as these: 

• In the table, how is the number of blocks related to the scale factor? Is there a pattern?
• How many blocks are needed to build scaled copies using scale factors of 5 or 10? How do you know?
• How many blocks are needed to build a scaled copy using any scale factor \(s\)? 
• If we want a scaled copy where each side is half as long, how much of a block would it take? How do you know? Does the same rule still apply?

If not brought up by students, highlight the fact that the number of blocks it took to build each scaled shape equals the scale factor times itself, regardless of the shape (look at the table row for \(s\)). This rule applies to any factor, including those that are less than 1.

This activity extends the conceptual work of the previous one by adding a layer of complexity. Here, the original shapes are comprised of more than 1 block, so the number of blocks needed to build their scaled copies is not simply \(\text{(scale factor)}^2\), but rather \(n \times \text{(scale factor)}^2\), where \(n\) is the number of blocks in the original shape. Students begin to think about how the scaled area relates to the original area, which is no longer 1 area unit. They notice that the pattern \(\text{(scale factor)}^2\) presents itself in the factor by which the original number of blocks has changed, rather than in the total number of blocks in the copy.

As in the previous task, students observe regularity in repeated reasoning (MP8), noticing that regardless of the shapes, starting with \(n\) pattern blocks and scaling by \(s\) uses \(ns^2\) pattern blocks.

Also as in the previous task, the shape composed of trapezoids might be more challenging to scale than those composed of rhombuses and triangles. Prepare to support students scaling the red shape by offering some direction or additional time, if feasible. 

As students work, monitor for groups who notice that the pattern of squared scale factors still occurs here, and that it is apparent if the original number of blocks is taken into account. Select them to share during class discussion.  

Students may forget to check that the lengths of all sides of their shape have been scaled and end with an inaccurate count of the pattern blocks. Remind them that all segments must be scaled by the same factor.

The goal of this discussion is to ensure that students understand that the pattern for the number of blocks in the scaled copies depends both on the scale factor and on the number of blocks in the pattern.

Display a table with only the column headings filled in. Poll the class on how many blocks it took them to build each scaled copy using the factors of 2 and 3. Record their answers in the table.

Consider displaying the built shapes or pictures of them for all to see. 

Invite selected students to share the pattern that their groups noticed and used to predict the number of blocks needed for copies with scale factors 4, 5, and 6. Record their predictions in the table. Discuss:

• How does the pattern for the number of blocks in this activity compare to the pattern in the previous activity? Are they related? How?
• For each figure, how many blocks does it take to build a copy using any scale factor \(s\)?

In this activity, students transfer what they learned with the pattern blocks to calculate the area of other scaled shapes (MP8). In groups of 2, students draw scaled copies of either a parallelogram or a triangle and calculate the areas. Then, each group compares their results with those of a group that worked on the other shape. They find that the scaled areas of two shapes are the same (even though the starting shapes are different and have different measurements) and attribute this to the fact that the two shapes had the same original area and were scaled using the same scale factors.

While students are not asked to reason about scaled areas by tiling (as they had done in the previous activities), each scaled copy can be tiled to illustrate how length measurements have scaled and how the original area has changed. Some students may choose to draw scaled copies and think about scaled areas this way.

As students find the areas of copies with scale factors 5 and \(\frac35\) without drawing (for the last question), monitor for these methods, depending on their understanding of or comfort with the \(\text{(scale factor)}^2\) pattern:

• Scaling the original base and height and then multiplying to find the area
• Multiplying the original area by the square of the scale factor

Select students using each approach. Invite them to share their reasoning, sequenced in this order, during the discussion.

You will need the Area of Scaled Parallelograms and Triangles blackline master for this activity.

Arrange students in groups of 2. Provide access to geometry toolkits.

Distribute slips showing the parallelogram to half the groups and the triangle to the others. Give students 1 minute of quiet work time for the first question, and then time to complete the rest of the task with their partner.

Students may not remember how to calculate the area of parallelograms and triangles. Make sure that they have the correct area of 10 square units for their original shape before they calculate the area of their scaled copies.

When drawing their scaled copies, some students might not focus on making corresponding angles equal. As long as they scale the base and height of their polygon correctly, this will not impact their area calculations. If time permits, however, prompt them to check their angles using tracing paper or a protractor.

Some students might focus unnecessarily on measuring other side lengths of their polygon, instead of attending only to base and height. If time is limited, encourage them to scale the base and height carefully and check or measure the angles instead. 

Invite selected students to share their solutions. Then focus class discussion on two themes: how the values in the tables for the two shapes compare, and how students determined the scaled areas for the scale factors 5 and \(\frac35\). Ask questions such as: 

• What did you notice when you compared your answerswithanother group that worked with the other figure? (When the scale factors are the same, the scaled areas are the same, though the bases and heights are different.)
• How did you find the scaled areas for scale factors of 5 and \(\frac35\)? (By scaling the original base and height and multiplying the scaled measurements; by multiplying the original area by \(\text{(scale factor)}^2\).)
• How is the process for finding scaled area here the same as and different than that in the previous activities with pattern blocks? (The area units are different; the pattern of squaring the scale factor is the same.)

Highlight the connection between the two ways of finding scaled areas. Point out that when we multiply the base and height each by the scale factor and then multiply the results, we are essentially multiplying the original lengths by the scale factor two times. The effect of this process is the same as multiplying the original area by \(\text{(scale factor)}^2\).