Lesson 7

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 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.

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, the same paths are likely, except that students will multiply by 2,000 and reverse the unit conversion. Identify students who use different approaches so they can share later.

Have students close their books or devices. Display an image of Tunisia’s flag. Explain that Tunisia holds the world record for the largest version of a country flag. The record-breaking flag is nearly four soccer fields in length. Solicit from students a few guesses for a scale that would be appropriate to create a scale drawing of the flag on a sheet of paper. If asked, provide the length of the flag (396 m) and the size of the paper (letter size: \(8\frac12\) inches by 11 inches, or about 21.5 cm by 28 cm).

After hearing some guesses, explain to students that they will now solve problems about the scale and scale drawing of the giant Tunisian flag.

Arrange students in groups of 3–4. Provide access to a metric unit conversion chart. Give students 4–5 minutes of quiet work time, and then another 5 minutes to collaborate and discuss their work in groups.

During work time, assign one sub-problem from the second question for each group to present.

Students may be confused about whether to multiply or divide by 2,000 (or to multiply by 2,000 or by \(\frac {1}{2,000}\)) when finding the missing lengths. Encourage students to articulate what a scale of 1 to 2,000 means, or remind them that it is a shorthand for saying “1 unit on a scale drawing represents 2,000 of the same units in the object it represents.” Ask them to now think about which of the two—actual or scaled lengths—is 2,000 times the other and which is \(\frac{1}{2,000}\) of the other.

For the third question relating the area of the real flag to the scale model, if students are stuck, encourage them to work out the dimensions of each explicitly and to use this to calculate the scale factor between the areas.

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 \(\frac{1}{2,000}\), or multiply by \(\frac{1}{2,000}\) first, and then convert the quotient into centimeters.

Invite previously identified students to display and share their responses for the sub-problems in the second question. After each person shares, solicit questions or comments from the class. 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.

Make sure students understand why the scale factor for the area of the two flags is 4,000,000. (Both the length and the height of the large flag are 2,000 times the length and height of the small flag. So the area of the large flag is \(2,\!000 \boldcdot 2,\!000\) times the area of the small flag. Alternatively, there are 10,000 square centimeters in a square meter, so in square centimeters, the area of the large flag is 1,053,360,000. Dividing this by the area of the small flag in square centimeters, 261.36, also gives 4,000,000.)