In this partner activity, students take turns identifying equivalent scales, including some expressed without units and some with units. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

A key insight to uncover here is that when comparing scales, it can be helpful to convert them into equivalent scales in a particular format (for example, without units, or using the same units).

If groups have trouble getting started, encourage them to think about different ways to express a scale, both with units and without units.

Students may sort the cards by the types (metric or customary; with units or without units) rather than by common scale factors. Remind students that scales that are equivalent have the same factor relating their scaled lengths to actual lengths.

Students may think that scales in metric units and those in customary units cannot be equivalent. For example, they may think that “1 inch to 1,000 inches” belongs in one group and “1 cm to 10 m” belongs in another. If this misconception arises and is not resolved in group discussions, address it during the Activity Synthesis.

Much of the discussion takes place between partners, so a whole-class debrief may be necessary only to tie up any loose ends. Invite a few students to share how their group reasoned about a couple of the scales (for example, cm to 500 m, 1 mm to 1 m).

• “What were some ways you handled . . .”
• “Describe any difficulties you experienced and how you resolved them.”
• “Which matches were tricky? Explain why.”

Address any questions that arose during sorting, common misconceptions, or unsettled disagreements between groups. For example, students may still be unclear about whether scales in customary and metric units can be equivalent. Consider asking “Can ‘1 inch to 1,000 inches’ and ‘1 centimeter to 10 meters’ both go in the same group? Why or why not?” Help students see that as long as the two scales represent the same scale factor, they are equivalent and will produce the same scale drawing.

If time permits, consider asking students to order their groups of equivalent scales, starting with the ones that would produce the smallest drawing of the same actual thing to the ones that would produce the largest drawing. Invite students to explain their reasoning.

In this activity, students use a scale without units to find actual and scaled distances that involve a wider range of numbers, from 0.02 to 2,000. They also return to thinking about how the area of a scale drawing relates to the area of the actual thing. As students make sure to include the correct units on their answers and while explaining their reasoning, they are attending to precision (MP6).

Students are likely to find scaled lengths in one of two ways: 1) by first converting the measurement in meters to centimeters and then dividing by 2,000, or 2) by dividing the measurement by 2,000 and then converting the result to centimeters. To find actual lengths, similar paths are likely, except that students will multiply by 2,000 and reverse the unit conversion.

Select a few students with differing solution paths to share their responses to the first question. Record and display their reasoning for all to see. Highlight two different ways for dealing with unit conversions. For example, in finding scaled lengths, one can either first convert the actual length in meters to centimeters and then multiply by , or multiply by first, and then convert the quotient to centimeters.

Poll the class on their responses for the second question. If there is disagreement about any of the answers, invite a few students to share their reasoning. Emphasize that all of the scales are equivalent because in each scale, a factor of 2,000 relates scaled distances to actual distances. Reiterate the fact that a scale does not have to be expressed in terms of 1 scaled unit, as is shown in the last three sub-questions, but that 1 is often chosen because it makes the scale factor easier to see and can make calculations more efficient.

If time permits, help students recognize that the areas of the two flags are related by a factor of 4,000,000. Both the length and the height of the large flag are 2,000 times the corresponding side of the small flag, so the area of the large flag is times the area of the small flag. Another way to look at it is that there are 10,000 square centimeters in a square meter, so in square centimeters, the area of the large flag is 1,045,440,000. Dividing this by the area of the small flag in square centimeters, 261.36, also gives 4,000,000.

In this activity, students use the floor plan of a recreation center to calculate the area of two swimming pools. This helps reinforce the relationship that the area of an actual region is related to its area on a scale drawing by the .

Previously, whenever students were asked to use a scale drawing to calculate the area of an actual region, they were able to find the dimensions of the actual region as an intermediate step. Each time, students were prompted to notice the pattern. Some students may have already become comfortable using this relationship to calculate the actual area directly from the scaled area, without needing to calculate the actual dimensions as an intermediate step.

The purpose of this activity is to help all students internalize this more efficient method. The question about the rectangular pool can be solved either way, but for the question about the kidney-shaped pool, students must rely on the relationship between scaled area and actual area. As students apply the  pattern they have seen in other activities to determine the scaled area, they are making use of structure (MP7).

As students work, monitor for the different ways that students find the area of the large rectangular pool, such as:

• By first finding the actual side lengths of the pool in meters and then multiplying them.
• By calculating the scaled area in square centimeters and multiplying that area by 25 (or ​​).

The goal of this discussion is to help students appreciate the efficiency of using the relationship to calculate the area of an actual region.

Select a few students with differing solution paths to share their responses to the question about the large rectangular pool. Record and display their reasoning for all to see. Highlight the two different strategies described in the activity narrative. Consider asking questions such as:

• “Why do the different approaches lead to the same outcome?”
• “How is the scale used in each approach?”
• “What might be some benefits of using one method over another for finding the actual area?”

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to the last question (about the kidney-shaped pool) by correcting errors, clarifying meaning, and adding details. 

• Display this first draft:
  “Since the scale is 1 to 500, I multiplied 3.2 times 500. This equals 1,600 so the kidney-sized pool has an actual area of 1,600.”
  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft. 
• Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.