This activity extends the conceptual work of the previous one by adding a layer of complexity. Students continue building scaled copies with pattern blocks, but now the original shapes are comprised of more than 1 block. Therefore, the number of blocks needed to build their scaled copies is not simply , but rather , where 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 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 pattern blocks and scaling by uses 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.

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.

This is the first time Math Language Routine 7: Compare and Connect is suggested in this course. In this routine, students are given a problem that can be approached using multiple strategies or representations, and they record their method for all to see. They then compare and identify correspondences across strategies by means of a teacher-led gallery walk with commentary or teacher think-aloud (such as “I notice . . . I wonder . . .”). A typical discussion prompt is “What is the same and what is different?”, comparing their own strategy to the others. The purpose of this routine is to allow students to make sense of mathematical strategies and, through constructive conversations, develop awareness of the language used as they compare, contrast, and connect other ways of thinking to their own.

If pattern blocks are not available, consider using the digital version of the activity. In the digital version, students use an applet to fill in outlines of scaled copies with pattern blocks. The applet allows students to move and rotate the pattern blocks. The digital version may help students build the shapes quickly and accurately so they can focus more on the number of pattern blocks it takes to make the scaled copy rather than on how the blocks are arranged.

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

Display a table with only the column headings filled in. Ask students to share 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 pictures of the built shapes to help students see where the numbers in the table came from.

Next, focus the discussion on the question about predicting the number of blocks needed to make copies with scale factors 4, 5, and 6. Display the approaches from previously selected students for all to see. Record their predictions in the table. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do the solutions have in common? How are they different?”
• “How does the scale factor show up in each method?”
• “How does the pattern for the number of blocks in this activity compare to the pattern in the previous activity?”
• “For each figure, how many blocks does it take to build a copy using any scale factor ?” ( for Figure D, for Figure E, and for Figure F. In each case, it’s the number of blocks in the original figure times the .)

Once students can explain 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, move on to the next activity.
 

In this activity, students transfer what they learned with the pattern blocks about the structure of scaled copies to calculate the area of other scaled shapes (MP7). 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. They attribute this to the fact that the two shapes had the same original area and were scaled using the same scale factors.

Two blue shapes. One parallelogram base 5, height 2. One triangle base 4, height 5.

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.

Monitor for students who use these strategies to find the areas of copies with scale factors 5 and :

• Scale the original base and height and then calculate the area of the new shape.
• Multiply the original area by the square of the scale factor.

Plan to have students present in this order, from more concrete to more abstract.
 

The purpose of this discussion is to make clear that with scaled copies, the area changes by the square of the scale factor by which the sides change.

To establish that the areas of the scaled shapes are the same for both the parallelogram and the triangle, ask questions such as:

• “What did you notice when you compared your answers to the first part with another group that worked with the other figure?” (The areas for each scale factor were the same, even though the bases and heights were different.)
• “Why do you think that is?” (Because the areas of the original shapes were equal.)

Invite previously selected students to share how they determined the scaled areas for the scale factors 5 and . Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students' work for all to see. To make it easier for students to compare the approaches, show both strategies for the same shape. If time permits, you can also show strategies for the other shape.

Connect the different responses to the learning goals by asking questions such as:

• “Did anyone solve the problem the same way, but would explain it differently?”
• “Why do the different approaches lead to the same outcome?”
• “How is the process for finding scaled area here the same as finding the number of pattern blocks in the previous activity? How is it different?” (The pattern of squaring the scale factor is the same. The area units are different.)

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 .