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.
Launch
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.
Activity
None
A map of a park says its scale is 1 to 100.
What do you think that means?
Give an example of how this scale could tell us about measurements in the park.
Student Response
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Building on Student Thinking
Students might think that when no units are given, they can choose their 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.
Activity Synthesis
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.
7.2
Activity
Standards Alignment
Building On
Addressing
7.G.A.1
Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
In this activity, students use a scale drawing and a scale expressed without units to calculate actual lengths. They also use actual heights to calculate corresponding scaled heights. Students make choices about which units to use. Their choice of units could influence the number of conversions needed and the efficiency of their paths (as shown in the sample student responses). As students choose units and calculate scaled heights, they are reasoning quantitatively and abstractly (MP2).
For the problem about the height of the spacecraft, monitor for students who:
Measure the drawing in inches, calculate the actual height in inches, and then convert to meters.
Measure the drawing in centimeters, calculate the actual height in centimeters, and then convert to meters.
Measure the drawing in centimeters, convert to meters, and then calculate the actual height.
This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.
Launch
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. (The landing module was one part of a larger spacecraft that was launched from Earth.) Consider displaying a picture of the landing module such as this one.
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 copy of the blackline master. Provide access to centimeter and inch rulers.
Make sure that students understand what is meant by a “leg” of the spacecraft in the first question. The “legs” of the spacecraft are its landing gear. Students should measure one of the legs on the side of the spacecraft in the drawing because the one in the middle appears shorter due to foreshortening.
Give students 6–7 minutes of partner work time. Select work from students with different strategies, such as those described in the activity narrative, to share later.
Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, directing students to first find the length on the drawing. Then provide students with the scale of the drawing and prompt them to use that to find the actual length. Supports accessibility for: Organization, Attention
Activity
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Activity Synthesis
The goal of this discussion is to highlight how units matter in problems involving a scale without units. First, poll the class on their answers to the first question. Highlight the multiplication of scaled measurements by 50 to find actual measurements.
Next, display 2–3 approaches to the second question from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:
“How is the scale ‘1 to 50’ used in each method?”
“Why do the different approaches lead to the same outcome?”
“Are there any benefits or drawbacks to one approach compared to another?”
The key takeaways are that when we have a scale without units:
We can use any unit we want to measure the scale drawing.
We will get an actual measurement in that same unit.
We may choose what unit we use to measure the drawing based on how we want to express the final answer.
(Since the question asks for a height in meters, using centimeters as the unit would be more efficient than using inches because fewer conversions would be required. If the question asked for actual height in feet, then inches would be a more strategic unit to use.)
Next, poll the class on their answers to the third question. Highlight that finding the length on the scale drawing involves dividing the actual measurement by 50 (or multiplying by ).
If time permits, display the image from the blackline master. Sketch a stick figure that is 1.4 inches tall to represent Neil Armstrong standing next to the Apollo Lunar Module. Select students who gave their heights in different units to share their solutions to the last problem. Sketch a stick figure with its height to scale for each student who shares an answer. 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.
7.3
Activity
Standards Alignment
Building On
Addressing
7.G.A.1
Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
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.
This activity is the first time students consider equivalent scales in this course. During the Activity Synthesis, make sure to highlight the second question in the task so students are familiar with the concept. If necessary, consider asking “Are the scales of ‘1 inch to 1,000 inches’ and ‘1 centimeter to 10 meters’ equivalent? 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.
Launch
Tell students to 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: 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 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.
Engagement: Develop Effort and Persistence. Provide tools to facilitate information processing or computation, enabling students to focus on key mathematical ideas. For example, allow students to use calculators to support their reasoning. Supports accessibility for: Memory, Conceptual Processing
Activity Synthesis
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.
MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to the first question (explaining how they completed the table). Invite listeners to ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback they receive. Advances: Writing, Speaking, Listening
Lesson Synthesis
To help students generalize about scales with and without units, consider asking:
“What does it mean when the scale on a scale drawing does not include units?” (When a scale does not show units, the same unit is used for both the scaled distance and the actual distance.)
“How is a scale without units the same as a scale with units? How are they different?” (Both types of scale show the relationship between scaled and actual distances. When the scale doesn’t specify a unit, you can use any unit to measure the scale drawing and then calculate the actual distance in that same unit.)
“How can we express the scale ‘1 inch to 5 miles’ without units?” (Since there are 12 inches in a foot and 5,280 feet in a mile, this is the same as 1 inch to 63,360 inches, or 1 to 63,360.)
“If a map uses the scale ‘1 inch to 5 miles,’ how can we find the actual area of a region represented on the map?” (Find the area on the map in square inches and multiply by 25, because 1 square inch represents 25 square miles.)
Student Lesson Summary
Sometimes scales come with units, and sometimes they don’t. For example, a map of Nebraska may have a scale of 1 mm to 1 km. This means that each millimeter of distance on the map represents 1 kilometer of distance in Nebraska. Notice that there are 1,000 millimeters in 1 meter and 1,000 meters in 1 kilometer. This means there are or 1,000,000 millimeters in 1 kilometer. So, the same scale without units is 1 to 1,000,000, which means that each unit of distance on the map represents 1,000,000 units of distance in Nebraska. This is true for any choice of unit to express the scale of this map.
Here is some information about equal lengths that you may find useful.
1 centimeter (cm) = 10 millimeters (mm)
1 meter (m) = 1,000 millimeters (mm)
1 meter = 100 centimeters
1 kilometer (km) = 1,000 meters
Equal Lengths in Different Systems
1 inch = 2.54 centimeters
1 foot 0.30 meter
1 mile 1.61 kilometers
1 centimeter 0.39 inch
1 meter 39.37 inches
1 kilometer 0.62 mile
Standards Alignment
Building On
Addressing
7.G.A.1
Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
Your teacher will give you a scale drawing of the Apollo Lunar Module. It is drawn at a scale of 1 to 50.
Estimate the actual length of one leg of the spacecraft to the nearest 10 centimeters. Explain or show your reasoning.
Estimate the actual height of the spacecraft to the nearest meter. Explain or show your reasoning.
Neil Armstrong was 71 inches tall when he went to the Moon. How tall would he be in this scale drawing? Show your reasoning.
Sketch a stick figure to represent yourself standing next to the Apollo Lunar Module. Make sure the height of your stick figure is to scale. Show how you determined your height on the drawing.
Student Response
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Building on Student Thinking
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.
Activity
None
As of 2016, Tunisia holds the world record for the largest version of a national flag. It was almost as long as four soccer fields. The flag has a circle in the center, a crescent moon inside the circle, and a star inside the crescent moon.
Complete the table. Explain or show your reasoning.
flag length
flag height
height of
crescent moon
actual
396 m
99 m
at 1 to 2,000 scale
13.2 cm
Complete each scale with the value that makes it equivalent to the scale of 1 to 2,000. Explain or show your reasoning.
1 cm to ____________ cm
1 cm to ____________ m
1 cm to ____________ km
2 m to _____________ m
5 cm to ___________ m
____________ cm to 1,000 m
____________ mm to 20 m
What is the area of the large flag?
What is the area of the smaller flag?
The area of the large flag is how many times the area of the smaller flag?
Student Response
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Building on Student Thinking
Students may be confused about whether to multiply or divide by 2,000 (or to multiply by 2,000 or by ) 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 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.
