Lesson 11

This warm-up introduces students to a scale without units and invites them to interpret it using what they have learned about scales so far.

As students work and discuss, notice those who interpret the unitless scale as numbers having the same units, as well as those who see “1 to 100” as comparable to using a scale factor of 100. Invite them to share their thinking later.

Remind students that, until now, we have worked with scales that each specify two units—one for the drawing and one for the object it represents. Tell students that sometimes scales are given without units.

Arrange students in groups of 2. Give students 2 minutes of quiet think time and another minute to discuss their thinking with a partner.

Students might think that when no units are given, we can choose our own units, using different units for the 1 and the 100. This is a natural interpretation given students’ work so far. Make note of this misconception, but address it only if it persists beyond the lesson.

Solicit students’ ideas about what the scale means and ask for a few examples of how it could tell us about measurements in the park. If not already mentioned by students, point out that a scale written without units simply tells us how many times larger or smaller an actual measurement is compared to what is on the drawing. In this example, a distance in the park would be 100 times the corresponding distance on the map, so a distance of 12 cm on the map would mean 1,200 cm or 12 m in the park.

Explain that the distances could be in any unit, but because one is expressed as a number times the other, the unit is the same for both. 

Tell students that we will explore this kind of scale in this lesson.

In this activity, students use a scale drawing and a scale expressed without units to calculate actual lengths. Students will need to make a choice about which units to use, and some choices make the work easier than others.

Monitor for several paths students may take to determine actual heights of the objects in the drawing. Their choice of units could influence the number of conversions needed and the efficiency of their paths (as shown in the sample student responses). Select students with the following approaches, sequenced in this order, to share during the discussion.

• Measure in cm, find cm for actual spacecraft, then convert to m
• Measure in cm, convert to m for scale drawing, then find spacecraft measurement in m

One other approach students may use is to measure the scale drawing using an inch ruler. This leads to an extra conversion from inches to centimeters or meters. Ask them to consider the unit of interest. Discuss and highlight strategic choices of units during whole-class debriefing.

You will need the Apollo Lunar Module blackline master for this activity.

Tell students that Neil Armstrong and Buzz Aldrin were the first people to walk on the surface of the Moon. The Apollo Lunar Module was the spacecraft used by the astronauts when they landed on the Moon in 1969. Consider displaying a picture of the landing module such as this one. Tell students that the landing module was one part of a larger spacecraft that was launched from Earth.

Solicit some guesses about the size of the spacecraft and about how the height of a person might compare to it. Explain to students that they will use a scale drawing of the Apollo Lunar Module to find out.

Arrange students in groups of 2. Give each student a scale drawing of the Apollo Lunar Module (from the blackline master). Provide access to centimeter and inch rulers. Give students 3–4 minutes to complete the first two questions. Ask them to pause briefly and discuss their responses with their partner before completing the rest of the questions.

Students are asked to find heights of people if they are drawn “to scale.” Explain that the phrase means “at the same scale” or “at the specified scale.”

If students are unsure how to begin finding the actual length of the landing gear or actual height of the spacecraft, suggest that they first find out the length on the drawing.

Students may measure the height of the spacecraft in centimeters and then simply convert it to meters without using the scale. Ask students to consider the reasonableness of their answer (which is likely around 0.14 m) and remind them to take the scale into account.

Invite selected students who measured using a centimeter ruler to share their strategies and solutions for the first two questions. Consider recording their reasoning for all to see. Highlight the multiplication of scaled measurements by 50 to find actual measurements. For example, the height of each leg is about 350 cm because \(50 \boldcdot 7 = 350\). 

Discuss whether or how units matter in problems involving unitless scales:

• Does it matter what unit we use to measure the drawing? Why or why not?
• Which unit is more efficient for measuring the height of the lunar module on the drawing—inches or centimeters? (Since the question asks for a height in meters, centimeters would be more efficient since it means fewer conversions. If the question asks for actual height in feet, inches would be a more strategic unit to use.)

Ask a few other students to share their responses to the last two questions. Select those who gave their heights in different units to share their solutions to the last problem. Highlight that, regardless of the starting unit, finding the length on the scale drawing involves dividing the actual measurement by 50. In other words, actual measurements can be translated to scaled measurements with a scale factor of \(\frac{1}{50}\).

If time permits, consider displaying a photograph of one of the astronauts next to the Lunar Module, such as shown here, as a way to visually check the reasonableness of students’ solutions.

In this activity, students explore the connection between a scale with units and one without units. Students are given two equivalent scales (one with units and the other without) and are asked to make sense of how the two could yield the same scaled measurements of an actual object. They also learn to rewrite a scale with units as a scale without units.

Students will need to attend to precision (MP6) as they work simultaneously with scales with units and without units. A scale of 1 inch to 16 feet is very different than a scale of 1 to 16, and students have multiple opportunities to address this subtlety in the activity.

As students work, identify groups that are able to reason clearly about why the two scales produce the same scale drawing. Two different types of reasoning to expect are:

• Using the two scales and the given dimensions of the parking lot to calculate and verify the student calculations.
• Thinking about the meaning of the scales, that is, in each case, the actual measurements are 180 times the measurements on the scale drawing.

Ask students: “Is it possible to express the 1 to 50 scale of the Lunar Module as a scale with units? If so, what units would we use?” Solicit some ideas. Students are likely to say “1 inch to 50 inches,” and “1 cm to 50 cm.” Other units might also come up. Without resolving the questions, explain to students that their next task is to explore how a scale without units and one with units could express the same relationship between scaled lengths and actual lengths.

Keep students in the same groups. Provide access to rulers. Give partners 3–4 minutes to complete the first question and another 3–4 minutes of quiet work time for the last two questions.

Some students may have trouble getting started. Suggest that they begin by treating each scale separately and find out, for instance, how a scale of 1 inch to 15 feet produces a drawing that is 8 inches by 5 inches.

Select a couple of previously identified groups to share their responses to the first question and a couple of other groups for the other questions.

Highlight how scaled lengths and actual lengths are related by a factor of 180 in both scales, and that this factor is shown explicitly in one scale but not in the other.

• In the case of 1 to 180, we know that actual lengths are 180 times as long as scaled lengths (or scaled lengths are \(\frac {1}{180}\) of actual lengths).
  If the scaled lengths are given in inches, we can use scaled lengths to find actual lengths in inches and, if desired, convert to feet afterward, and vice versa.
• In the case of 1 in to 15 ft, though we know that actual measurements are not 15 times longer than their corresponding measurements on a drawing (because 15 feet is not 15 times larger than 1 inch), it is not immediately apparent what factor relates the two measurements. Converting the units helps us see the scale factor. Since 1 foot equals 12 inches and \(15 \boldcdot 12=180\), the scale of 1 in to 15 feet is equivalent to the scale of 1 in to 180 in, or 1 to 180.