This Warm-up prompts students to attend to precision in measurements, which will be important in upcoming work.
Launch
Arrange students in groups of 2. Give students 1 minute of quiet think time to estimate the length of their own foot in centimeters or inches, and a moment to share their estimate with a partner. Then, ask them to complete the task.
Activity
None
If a student uses a ruler like this to measure the length of their foot, which choices would be appropriate measurements? Select all that apply. Be prepared to explain your reasoning.
inches
inches
23.47659 centimeters
23.5 centimeters
23.48 centimeters
Here is a scale drawing of an average seventh-grade student’s foot next to a scale drawing of a foot belonging to the person with the largest feet in the world. Estimate the length of the larger foot.
Scale drawing of an average seventh-grade student's foot next to a scale drawing of a foot belonging to the person with the largest feet in the world.
Student Response
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Building on Student Thinking
Some students may say the large foot is about inches or about 9 centimeters long, because they assume the ruler shown in the first question is at the same scale as the feet shown in the second question. Explain that the images are drawn at different scales.
Activity Synthesis
Select a few students to share the measurements they think would be appropriate based on the given ruler. Consider displaying the picture of the ruler for all to see and recording students' responses on it. After each response, poll the class on whether they agree or disagree.
If students consider B, C, or E to be an appropriate measurement, ask them to share how to get such a level of precision on the ruler. Make sure students understand that reporting measurements to the nearest of an inch or to the hundred-thousandths of a centimeter would not be appropriate (i.e., show that the ruler does not allow for these levels of precision).
Choice E of 23.48 cm may merit specific attention. With the ruler, it is possible to guess that the hundredths place is an 8. This may even be correct. The problem with reporting the measurement in this way is that someone who sees this might misinterpret it and imagine that an extremely accurate measuring device was used to measure the foot, rather than this ruler. The way a measurement is reported reflects how the measurement was taken.
Next, invite students to share their estimates for the length of the large foot. Since it is difficult to measure the length of these feet very precisely, these measurements should not be reported with a high level of precision; the nearest centimeter would be appropriate.
Math Community
After the Warm-up, display the revised Math Community Chart created from student responses in Exercise 3. Tell students that today they are going to monitor for two things:
“Doing Math” actions from the chart that they see or hear happening.
“Doing Math” actions that they see or hear that they think should be added to the chart.
Provide sticky notes for students to record what they see and hear during the lesson.
10.2
Activity
Standards Alignment
Building On
6.G.A.1
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
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.
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
In this activity, students create a scale drawing and then compare multiple scale drawings of the same thing at different scales. Each group member uses a different scale to calculate scaled lengths of the same plot of land, draw a scale drawing, and calculate its scaled area.
Next, each group member uses a different scale to calculate scaled lengths of the same plot of land, draw a scale drawing, and calculate its scaled area. The group then orders the different drawings and analyzes them. They think about how many square meters of actual area are represented by one square centimeter on each drawing. Students are likely to determine this value in two ways:
By visualizing what a centimeter square represents at a given scale (for example, at a scale of 1 cm to 5 m, each 1 cm2 represents , or 25 m2).
By dividing the actual area represented by the scale drawing by the area of their scale drawing.
As students examine the relationships between scale, lengths in scale drawings, and area in scale drawings, they practice looking for and making use of mathematical structure (MP7).
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Launch
Display this map of a neighborhood in Philadelphia for all to see. Tell students that they are going to create a new map of the triangular piece of land at a different scale.
Tell students that the actual base of the triangle is 120 meters and its actual height is 90 meters. Ask, “What is the area of the plot of land?” (5,400 square meters—one half the base times the height of the triangle.)
Arrange students in groups of 5–6 and provide access to centimeter graph paper. Assign each student in a group a different scale (from the blackline master) to use to create a scale drawing. Remind students to include the units in their measurements. Give students 4–5 minutes of quiet work time to answer the first 3 questions.
Use Collect and Display to direct attention to words collected and displayed from an earlier lesson. Invite students to borrow language from the display as needed, and update it throughout the lesson. Give students 3–4 minutes to work on the last question in their 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
None
Here is a map showing a plot of land in the shape of a right triangle.
Your teacher will assign you a scale to use. On centimeter graph paper, make a scale drawing of the plot of land. Make sure to write your scale on your drawing.
What is the area of the triangle you drew? Explain or show your reasoning.
How many square meters are represented by 1 square centimeter in your drawing?
After everyone in your group is finished, order the scale drawings from largest to smallest. What do you notice about the scales when your drawings are placed in this order?
Activity Synthesis
The goal of this discussion is to compare the structure of larger and smaller scale drawings of the same thing, both in terms of length and area. Direct students’ attention to the reference created using Collect and Display. Ask students to share what they noticed about the scale drawings placed in order. Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond.
To highlight some of the patterns students may have noticed, consider asking questions such as:
“How does a change in the scale influence the size of the drawings?” (As the length being represented by 1 cm gets larger, the size of the drawing decreases.)
“Look at the drawings with the scales ‘1 cm to 5 m’ and ‘1 cm to 15 m.’ How do the lengths in these two drawings compare? How do the areas compare?” (The lengths are three times as long in the drawing with the scale ‘1 cm to 5 m.’ The area is 9 times as great.)
“Look at the drawings with the scales ‘1 cm to 5 m’ and ‘1 cm to 50 m.’ How do the lengths in these two drawings compare? How do the areas compare?” (The lengths are ten times as long in the drawing with the scale ‘1 cm to 5 m.’ The area is 100 times as great.)
Help students to observe and formulate these patterns:
As the number of meters represented by one centimeter increases, the lengths in the scale drawing decrease.
As the number of meters represented by one centimeter increases, the area of the scale drawing also decreases, but it decreases by the square of the factor for the lengths (because finding the area means multiplying the length and width, both of which decrease by the same factor).
10.3
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 are given a scale drawing to reproduce at a different scale. This builds on previous work they did creating scale drawings given the actual dimensions and different scales.
There are two different types of reasoning students may apply. Monitor for students who:
Use the given scale drawing to find the dimensions of the actual school playground and then use those measurements to find the dimensions of the new scale drawing.
Calculate the measurements in the new drawing directly from the original drawing, by multiplying each length by .
Plan to have students present in this order, from more concrete to more efficient. Some students may initially follow the first approach and then figure out the second approach after noticing regularity in their repeated calculations (MP8).
Launch
Tell students that they are going to reproduce a scale drawing using a different scale. The scale for the given drawing is 1 cm to 30 meters, and they are going to make a new scale drawing at a scale of 1 cm to 20 meters. Ask them if they think the new drawing will be larger or smaller than the given one.
Arrange students in groups of 3. Make sure students have access to their geometry toolkits. Give students 5 minutes of quiet work time, followed by 3–4 minutes of group discussion.
Select students who used each strategy described in the Activity Narrative to share later. Aim to elicit both key mathematical ideas and a variety of student voices, especially from students who haven't shared recently.
Activity
None
Here is a scale drawing of a playground.
The scale is 1 centimeter to 30 meters.
Make another scale drawing of the same playground at a scale of 1 centimeter to 20 meters.
How do the two scale drawings compare?
Student Response
Activity Synthesis
The purpose of this discussion is to make connections between scale drawings that are reproduced at a different scale and scaled copies. Invite previously selected students to share how they created the new scale drawing. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display their work for all to see.
Connect the different responses to the learning goals by asking questions such as:
“What is the same about these strategies? What is different?”
“Why do the different approaches lead to the same outcome?”
“Did anyone solve the problem the same way, but would explain it differently?”
The key takeaways are:
The new scale drawing is a scaled copy of the original scale drawing. The shapes are the same (both represent the same playground), but the sizes are different.
The scale factor from the original drawing to the new drawing is (or 1.5). Multiplying by 30 and then dividing by 20 gets the same result as just multiplying by .
The drawing with a scale of 1 cm to 20 m is larger than the drawing with a scale of 1 cm to 30 m. When 1 centimeter on the scale drawing represents a shorter actual distance, it takes more of those centimeters to describe the object.
If time permits, include area in the discussion of these scale drawings. Consider asking questions like:
“On the original map with the scale of 1 cm to 30 m, how much area does 1 square centimeter represent?” (900 cm2)
“On the new map with the scale of 1 cm to 20 m, how much area does one square centimeter represent?” (400 cm2)
“How many times as large as the original map is the new map?” (The side lengths are 1.5 times as long. The area is , or 2.25, times as large.)
MLR8 Discussion Supports. For each observation that is shared, invite students to turn to a partner and restate what they heard using precise mathematical language. Ask, “Who can restate what shared using mathematical language?” Advances: Listening, Speaking
Lesson Synthesis
Share with students “Today we reproduced a scale drawing at a different scale.”
If desired, use this example to review these concepts: “You have a map that uses the scale 1 cm to 200 m. You draw a new map of the same place using the scale 1 cm to 20 m.”
Consider asking students:
“How does your new map compare to your original map?” (It is larger. The lengths are 10 times as long, and the area is 100 times as large.)
“How much actual area does 1 cm2 on your new map represent?” (400 m2)
“How much actual area did 1 cm2 on your original map represent?” (40,000 m2)
Math Community
Invite 2–3 students to share what “Doing Math” actions they noticed. Record and display their responses for all to see, such as by adding check marks to any already listed items or adding new items near the chart for the class to consider adding. Next, give students 1–2 minutes with a partner to discuss any changes or revisions they think the chart needs. Tell students they can suggest revisions during the Cool-down.
Student Lesson Summary
Sometimes we have a scale drawing of something, and we want to create another scale drawing of it that uses a different scale. We can use the original scale drawing to find the size of the actual object. Then we can use the size of the actual object to figure out the size of our new scale drawing.
For example, here is a scale drawing of a park where the scale is 1 cm to 90 m.
The rectangle is 10 cm by 4 cm, so the actual dimensions of the park are 900 m by 360 m, because and .
Suppose we want to make another scale drawing of the park where the scale is 1 cm to 30 meters. This new scale drawing should be 30 cm by 12 cm, because and .
Another way to find this answer is to think about how the two different scales are related to each other. In the first scale drawing, 1 cm represented 90 m. In the new drawing, we would need 3 cm to represent 90 m. That means each length in the new scale drawing should be 3 times as long as it was in the original drawing. The new scale drawing should be 30 cm by 12 cm, because and .
Since the length and width are 3 times as long, the area of the new scale drawing will be 9 times as large as the area of the original scale drawing because .
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.
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
