In this activity, students investigate polygons on a grid to determine whether or not they are scaled copies. The sides of the polygons are diagonal, so the side lengths cannot be easily determined using the grid.

First, students measure the angles using a protractor to see if corresponding angles have the same measure. Next, they find other corresponding distances that are not side lengths and see that these distances are related by a consistent scale factor. Students learn that in scaled copies, any corresponding distances—not limited to lengths of sides or segments—are related by the same scale factor.

Students must take care when they identify corresponding parts. As students work, urge them to attend to the order in which points or segments are listed (MP6).

If students are not sure what to make of the values in the table, encourage them to consider the corresponding distances of two figures at a time. For example, ask: “What do you notice about the corresponding vertical distances in and ? What about the corresponding horizontal distances in those two figures?”  

Display the completed tables for all to see. To highlight how all distances in a scaled copy (not just the side lengths of the figure) are related by the same scale factor, ask questions such as:

• “How do the distances in compare to the distances in ? Do the corresponding distances share the same scale factor?” (The distances in are each times the corresponding distance in .)
• “How do distances in compare to the distances in ? Do the corresponding distances share the same scale factor?” (The distances in are each times the corresponding distance in .)

In this activity students examine how the size of the scale factor is related to the original figure and the scaled copy. The activity serves several purposes:

1. To reinforce students’ awareness of scale factors.
2. To draw attention to how scaled copies behave when the scale factor is greater than 1, exactly 1, or less than 1.
3. To help students notice that reciprocal scale factors reverse the scaling.

As students notice that groups of scale factors have similar effects on the scaled copies, they are looking for regularity in repeated reasoning (MP8).

This is the first Card Sort activity of the course. An important aspect of this routine is to allow students time at the start to sort the cards into categories of their choosing. This step gives students the opportunity to familiarize themselves with the content of the cards without the additional pressure of organizing them in a specific fashion. It also provides insight into the aspects of each card that students attend to and the language they have to describe their observations.

Monitor for different ways students choose to categorize the cards, especially for students who group the cards in terms of: 

• Specific scale factors (such as 2, 3, , and so on).
• ranges of scale factors producing certain effects (for example, factors producing larger, unchanged, or smaller copies).
• Reciprocal scale factors (that is, one factor scales Figure A to B, and its reciprocal reverses the scaling).
   

Select groups to share their categories and how they sorted the scaled copies. Discuss as many different types of categories as time allows, but ensure that at least one set of categories discussed distinguishes between ranges of scale factors (greater than 1, exactly 1, and less than 1). 

If no groups sorted in terms of ranges of scale factors, consider asking:

• “What can we say about the scale factors that produced larger copies? Smaller copies? Copies that are the same size as the original?”

If no groups sorted in terms of reciprocal scale factors, consider asking:

• “Some cards had the same pair of figures on them, just in a reversed order (cards 1 and 7, cards 10 and 13). What do you notice about their scale factors?”

Highlight the two main ideas of the lesson:

• The effects of scale factors that are greater than 1, exactly 1, and less than 1.
• The reversibility of scaling—that is, if Figure B is a scaled copy of Figure A, then A is also a scaled copy of B. In other words, A and B are scaled copies of one another, and their scale factors are reciprocals.

This activity gives students a chance to apply what they know about scale factors, lengths, and angles and to create scaled copies without the support of a grid. Students work in groups of 6 to complete a jigsaw puzzle, with each group member scaling one piece of the puzzle with a scale factor of . The group then assembles the scaled pieces and examines the accuracy of their scaled puzzle.

As students work, notice how they measure distances and whether they consider angles. Depending on how students determine scaled distances, they may not need to transfer angles. Look out for students who measure only the lengths of drawn segments rather than distances, e.g., between the corner of a square and where a segment begins. Suggest that they consider other measurements that might help them locate the beginning and end of a segment.

Much of the conversations about creating accurate scaled copies will have taken place among partners, but consider coming together as a class to reflect on the different ways students worked. Ask questions such as:

• “How is this task more challenging than creating scaled copies of polygons on a grid?”
• “Besides distances or lengths, what helped you create an accurate copy?”
• “How did you know or decide which distances to measure?”
• “Before your drawings were assembled, how did you check if they were correct?”

Student responses to these questions may differ. For example, for Piece 6, the two lines can be drawn by measuring distances on the border of the puzzle piece; the angles work out correctly automatically. For Piece 1, however, to get the three lines that meet in a point in the middle of the piece just right, students can either measure angles, or extend those line segments until they meet the border of the piece and then measure distances.