Lesson 18: Computing Actual Lengths from a Scale Drawing

NYS COMMON CORE MATHEMATICS CURRICULUM 7•1 Lesson 18

Lesson 18: Computing Actual Lengths from a Scale Drawing Date: 7/26/15

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Lesson 18: Computing Actual Lengths from a Scale Drawing

Student Outcomes

§ Given a scale drawing, students compute the lengths in the actual picture using the scale. Students identify the scale factor in order to make intuitive comparisons of size, and then devise a strategy for efficiently finding actual lengths using the scale.

Classwork

Example 1 (14 minutes): Basketball at Recess?

The first example has students building upon the previous lesson by applying the scale factor to find missing dimensions. This leads into a discussion of whether this method is the most efficient and whether they could find another approach that would be simpler, as demonstrated in Example 2. Guide students to record responses and additional work in their student materials.

§ How can we use the scale factor to write an equation relating the scale drawing lengths to the actual lengths?

ú The scale factor is the constant of proportionality, or the 𝑘 in the equation 𝑦 = 𝑘𝑥 or 𝑥 = !! or even 𝑘 =

!! where 𝑥 is the actual length, 𝑦 is the scale drawing length, and 𝑘 is the value of the ratio of the

drawing length to the corresponding actual length.

§ How can we use the scale factor to determine the actual measurements?

ú Divide each drawing length, 𝑦, by the scale factor, 𝑘, to find the actual measurement, x. This is illustrated by the equation 𝑥 = !

!.

§ How can we reconsider finding an actual length without dividing?

ú We can let the scale drawing be the first image and the actual picture be the second image. We can calculate the scale factor that relates the given scale drawing length, 𝑥, to the actual length,𝑦. If the actual picture is an enlargement from the scale drawing, then the scale factor is greater than one or 𝑘 > 1. If the actual picture is a reduction from the scale drawing, then the scale factor is less than one or 𝑘 < 1.

Scaffolding: A reduction has a scale factor less than 1, and an enlargement has a scale factor greater than 1.

MP.2



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Example 1: Basketball at Recess?

Vincent proposes an idea to the Student Government to install a basketball hoop along with a court marked with all the shooting lines and boundary lines at his school for students to use at recess. He presents a plan to install a half-‐court design as shown below. After checking with school administration, he is told it will be approved if it will fit on the empty lot that measures 𝟐𝟓 feet by 𝟕𝟓 feet on the school property. Will the lot be big enough for the court he planned? Explain.

Scale Drawing Lengths 𝟏 in. 𝟐 in. 𝟏 𝟐𝟑 in.

Actual Court Lengths 𝟏𝟓 ft. 𝟑𝟎 ft. 𝟐𝟓 ft.

Scale Factor: The value of k will be greater than 1 because the actual court is larger than the scale drawing. Calculating the value of the ratio of 𝟏𝟓 feet to 𝟏 inch, which corresponds to (𝟏𝟓 ∙ 𝟏𝟐) inches to 1 inch, will give a the scale factor of 𝟏𝟖𝟎.

Let 𝒌 = 𝟏𝟖𝟎, let 𝒙 represent the scale drawing lengths in inches, and let 𝒚 represent the actual court lengths in inches. Calculate the actual lengths, and convert the result from inches to feet as a final step.

To find actual length: 𝒚 = 𝟏𝟖𝟎𝒙

𝒚 = 𝟏𝟖𝟎(𝟐)

𝒚 = 𝟑𝟔𝟎 inches, or 𝟑𝟎 feet

To find actual width: 𝒚 = 𝟏𝟖𝟎𝒙

𝒚 = 𝟏𝟖𝟎 𝟏𝟐𝟑

𝒚 = 𝟏𝟖𝟎𝟏 ∙ 𝟓𝟑

𝒚 = 𝟑𝟎𝟎 inches, or 𝟐𝟓 feet

The actual court measures 𝟐𝟓 feet by 𝟑𝟎 feet. Yes, the lot will be big enough for the court Vincent planned. The court will take up the entire width of the lot.

Example 2 (5 minutes): Garden Dimensions

Guide the whole class through the completion of the examples below while encouraging student participation through questioning. Students should record the information in their student materials.

Hold a discussion with students regarding the use of the word scale.

Scale Drawing: 𝟏 inch on the drawing corresponds to 𝟏𝟓 feet of actual length.



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§ Where have you seen this term used?

ú Bottom of a map, blueprint, etc.

§ The word scale refers to a type of ratio. 1 cm represents 20 m is an example of a ratio relationship, and the ratio 1: 20 is sometimes called a scale ratio or a scale. Why isn’t it called the scale factor?

ú The scale factor in a scaled drawing is always a scalar between distances measured in the same units.

§ Do we always need to use the scale factor in order to find actual measurements from a scale drawing, or could we just use the given scale ratio (or scale)? (Example 2 will show this.)

§ Take a few minutes to try to find the actual length of the garden. Give your answer in meters. Be prepared to explain you how you found your answer.

Allow for students to share approaches with the class. Students could calculate the scale factor and follow the steps from Example 1, or they may realize that it is not necessary to find the scale factor. They may apply the scale ratio and work the problem using the ratio 1: 20, perhaps setting up the proportional relationship 𝑦 = 20𝑥, where 𝑥 represents the drawing measurement and 𝑦 represents the actual length.

§ So then, what two quantities does the constant of proportionality, 𝑘, relate? ú The constant of proportionality relates the drawing length to the actual length, when converted to the

same units if a scale factor is being used. If just the scale ratio is used, then the quantities do not need to be converted to the same units.

§ What method was more efficient? Explain why.

ú Allow for students to respond. If we apply the scale ratio, it requires fewer steps.

§ Then why would we consider the scale factor?

ú The scale factor gives us a sense of the comparison. In this example, the scale factor is 2,000, so the scale drawing lengths are !

!,!!! of the actual lengths. It is not always easy to see that comparison when

you are basing your calculations on the scale. The scale factor helps us reason through the problem and make sense of our results.

§ Now, go back and find the actual width of the garden using the scale ratio.

Elicit responses from students, including an explanation of how they arrived at their answers. Record results on the board for students to see, and be sure students have recorded correct responses in their student materials.

Example 2: Garden Dimensions

The diagram shown represents a garden. The scale is 𝟏 centimeter for every 𝟐𝟎 meters. Each square in the drawing measures 𝟏 cm by 𝟏 cm. Find the actual length and width of the garden based upon the given drawing.



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Scale Length Width Drawing, 𝒙 𝟏 cm 𝟖 cm 𝟒 cm Actual, 𝒚 𝟐𝟎 m (or 𝟐,𝟎𝟎𝟎 cm) 𝟏𝟔𝟎 m (or 𝟏𝟔,𝟎𝟎𝟎 cm) 𝟖𝟎 m (or 𝟖,𝟎𝟎𝟎 cm)

Method 1:

Using the given scale: 𝟏 cm of scale drawing length corresponds to 𝟐𝟎 m of actual length.

𝒌 = 𝟐𝟎 the value of the ratio of the drawing length to the actual length (in different units)

To find actual length: 𝒚 = 𝟐𝟎𝒙 where 𝒙 represents the the scale drawing measurements in centimeters and 𝒚 represents the actual measurement in meters

𝒚 = 𝟐𝟎(𝟖) substitute the scale drawing length in place of 𝒙

𝒚 = 𝟏𝟔𝟎 The actual length is 𝟏𝟔𝟎 m.

To find actual width: Divide the actual length by 𝟐 since its drawing width is half the length. The actual width is 𝟖𝟎 m.

Method 2:

Using the scale factor: 𝟏 cm of scale drawing length corresponds to 𝟐,𝟎𝟎𝟎 cm of actual length.

𝒌 = 𝟐,𝟎𝟎𝟎 the value of the ratio of the drawing length to the actual length (in the same units)

To find actual length: 𝒚 = 𝟐,𝟎𝟎𝟎𝒙 where 𝒙 represents the drawing measurement in centimeters and 𝒚 represents the actual measurement in centimeters.

𝒚 = 𝟐,𝟎𝟎𝟎(𝟖) substitute the scale drawing length in place of 𝒙 𝒚 = 𝟏𝟔,𝟎𝟎𝟎 The actual length is 𝟏𝟔,𝟎𝟎𝟎 cm, or 𝟏𝟔𝟎 m.

To find actual width: 𝒚 = 𝟐,𝟎𝟎𝟎𝒙

𝒚 = 𝟐,𝟎𝟎𝟎(𝟒) substitute the scale drawing width in place of 𝒙

𝒚 = 𝟖,𝟎𝟎𝟎

The actual width is 𝟖,𝟎𝟎𝟎 cm, or 𝟖𝟎 m.



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Example 3 (10 minutes)

Example 3

A graphic designer is creating an advertisement for a tablet. She needs to enlarge the picture given here so that 𝟎.𝟐𝟓 inches on the scale picture will correspond to 𝟏 inch on the actual advertisement. What will be the length and width of the tablet on the advertisement?

Using a Table:

Using an Equation:

To find the constant of proportionality: 𝟏

𝟎.𝟐𝟓= 𝟒

𝟏= 𝟒

𝒌 = 𝟒 (scale factor since units of measure are the same; it is an enlargement)

To find actual length: 𝒚 = 𝟒𝒙 where 𝒙 represents the picture measurement and 𝒚 represents the

actual advertisement measurement

𝒚 = 𝟒 𝟏 𝟏𝟒 Substitute the picture length in place of 𝒙.

𝒚 = 𝟓 in.

To find actual width: 𝒚 = 𝟒 𝒙

𝒚 = 𝟒 𝟏 𝟏𝟖 Substitute the picture width in place of 𝒚.

𝒚 = 𝟒 𝟏𝟐

The tablet will be 𝟓 inches by 𝟒 𝟏𝟐 inches on the actual advertisement.

§ Is it always necessary to write and solve an equation 𝑦 = 𝑘𝑥 to find actual measurements?

ú Guide students to conclude that the actual measurement can be found by applying any of the three relationships: 𝑦 = 𝑘𝑥, 𝑥 = !

!, or even 𝑘 = !

!. Encourage students to try any of these approaches in the

next exercise.

Exercise (10 minutes)

Hold a brief discussion of the problem as a class, and identify how to find the answer. Guide students to identify the following big ideas to address as they solve the problem:

§ We need to find the relationship between the lengths in the scale drawing and the corresponding actual lengths.

§ Use this relationship to calculate the width of the actual mall entrance.

§ Compare this with the width of the panels.

Scale Length Width

Picture, 𝒙 𝟎.𝟐𝟓 in. 𝟏 𝟏𝟒 in. 𝟏 𝟏𝟖 in.

Actual Advertisement, 𝒚 𝟏 in. 𝟓 in. 𝟒 𝟏𝟐 in.

Scaffolding: § The map distance of the

mall entrance could be noted so that students would not need to measure.

§ When determining what unit to use when measuring, look at the given scale.

𝟏𝟏𝟖 in.

𝟏𝟏𝟒 in

.



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Allow time for students to measure and complete the problem (see the measurement on the diagram below). Encourage students to check with a partner to ensure that their measurements match each other’s.

Sample responses shown below include work for two different approaches. Students do not need to apply both and will receive credit for using either method.

Exercise

1. Students from the high school are going to perform one of the acts from their upcoming musical at the atrium in the mall. The students want to bring some of the set with them so that the audience can get a better feel for the whole production. The backdrop that they want to bring has panels that measure 𝟏𝟎 feet by 𝟏𝟎 feet. The students are not sure if they will be able to fit these panels through the entrance of the mall since the panels need to be transported flat (horizontal). They obtain a copy of the mall floor plan, shown below, from the city planning office. Use this diagram to decide if the panels will fit through the entrance. Use a ruler to measure.

Answer the following questions.

a. Find the actual width of the mall entrance, and determine whether the set panels will fit.

Step 1: Relationship between lengths in drawing and lengths in actual

Scale: 𝟒𝟏𝟐 ft.𝟏𝟖 in.

= 𝟗𝟐𝟏𝟖=

𝟗𝟐 ∙𝟖𝟏𝟖 ∙𝟖

= 𝟑𝟔𝟏 feet to inches

Scale factor: First convert 4 𝟏𝟐 feet to 𝟓𝟒 inches since the units of measurement must be the same.

k = 𝟓𝟒𝟏𝟖= 𝟓𝟒 ∙𝟖

𝟏𝟖 ∙𝟖

= 𝟒𝟑𝟐, an enlargement

Step 2: Find the actual distance of entrance

Use the given scale: 𝟑𝟖∙ 𝟑𝟔𝟏

= 𝟏𝟑 𝟏𝟐 feet wide

OR

Using scale factor: 𝟑𝟖∙ 𝟒𝟑𝟐𝟏

= 𝟏𝟔𝟐 inches, or 𝟏𝟑 𝟏𝟐 feet wide

Yes, the set panels which are 𝟏𝟎 ft. ×𝟏𝟎 ft. will fit (lying flat) through the mall entrance.

To Atrium

and

Additional

Stores

Store 1 Store 2

Mall Entrance

𝟑𝟖 in.

Scale:

𝟏𝟖 inch on the

drawing represents

𝟒𝟏𝟐 feet of actual length



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b. What is the scale factor? What does it tell us?

The scale factor is 𝟒𝟑𝟐. Each length on the scale drawing is 𝟏𝟒𝟑𝟐

of the actual length. The actual lengths are

𝟒𝟑𝟐 times larger than the lengths in the scale drawing.

Closing (1 minute)

§ What does the scale factor tell us about the relationship between the actual picture and the scale drawing?

ú It gives us an understanding of how much larger or smaller the scale drawing is compared to the actual picture.

§ How does a scale drawing differ from other drawings?

ú In a scale drawing, there exists a constant ratio of scale drawing length to actual length, whereas other drawings may not have a constant scale ratio between all corresponding lengths of the drawing and the actual picture or object.

Exit Ticket (5 minutes)

Lesson Summary

A scale factor is the ratio between two images when the units of measure are the same. The scale factor is the value of the ratio of the second image to the first image.

A scale is a ratio that relates corresponding measures between two images.

An actual length can be found by multiplying a scale drawing length by the scale factor or by multiplying a scale drawing length by a value calculated from the scale.

Scaling by factors greater than 𝟏 enlarge the segment, and scaling by factors less than 𝟏 reduce the segment.



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Name ___________________________________________________ Date____________________

Lesson 18: Computing Actual Lengths from a Scale Drawing

Exit Ticket

A drawing of a surfboard in a catalog shows its length as 8 !! inches. Find the actual length of the surfboard if

!! inch

length on the drawing corresponds to !! foot of actual length.

Circle one:

I’m on my way. I’ve got it. I can teach it!



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Exit Ticket Sample Solutions

A drawing of a surfboard in a catalog shows its length as 𝟖 𝟒𝟗 inches. Find the actual length of the surfboard if

𝟏𝟐 inch

length on the drawing corresponds to 𝟑𝟖 foot of actual length.

Using the scale: 𝟑𝟖𝒇𝒕.𝟏𝟐𝒊𝒏.

= 𝟔𝟖𝟏= 𝟑

𝟒 feet to inches

Scale Equivalent Scale

Ratio Surfboard

Scale Drawing Length, 𝒙 𝟏𝟐 inch

𝟏 inch 𝟖𝟒𝟗 inches

Actual Length, 𝒚 𝟑𝟖 foot

𝟔𝟖 ft.

or 𝟑𝟒 ft.

?

𝒚 = 𝒌𝒙

=𝟑𝟒 ∙ 𝟖

𝟒𝟗

=𝟏𝟗𝟑 𝑶𝑹 𝟔

𝟏𝟑 𝒇𝒕.

Using the scale factor: First convert 𝟑𝟖 feet to 𝟒 𝟏

𝟐 inches since the units of

measurement must be the same.

k = 𝟒𝟏𝟐𝟏𝟐= 𝟗

𝟏= 𝟗, an enlargement

𝒚 = 𝟗𝒙

= 𝟗 ∙ 𝟖𝟒𝟗

= 𝟗 ∙𝟕𝟔𝟗

= 𝟕𝟔 𝒊𝒏.𝑶𝑹 𝟔𝟏𝟑 𝒇𝒕.

The actual surfboard measures 𝟔 𝟏

𝟑 feet long.

Note: Students could also use an equation where 𝑦 represents the scale drawing and 𝑥 represents the actual measurement, in which case, 𝑘 would equal !

! if representing the scale or k would be equal to !

! if representing the scale

factor.

Model Problem An electrical contractor was hired to rewire the living room of an apartment. Electrical boxes are to be mounted in the

walls according to a blueprint provided by the architect. The scale on the blueprint indicates that !! inch represents 1

foot of actual distance. If an electrical box is located 1 !! inches from the corner of the room on the blueprint, use the

scale factor to determine the actual distance from the corner that the electrical box should be installed. Give the result in feet. Then, use the scale to confirm this same distance. Which approach is easier? Explain your response.

∙𝟑𝟒

Scaffolding:

Students do not need to solve the problems using both a scale and scale factor. Either is acceptable.



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Solution: Note: The problem can be solved using either a scale or scale factor. Both are not necessary. The scale factor will represent an enlargement from the blueprint length to the actual length. It will be a value greater

than 1. When using a scale factor, the units of measurement must be the same. Thus, !!inch on the blueprint

corresponds to 12 inches in the actual room. This give a scale factor of 48.

𝑘 =1214=

12 ∙ 414 ∙ 4

= 48

Using the scale factor, the equation for this relationship is y = 48x where x is the distance on the blueprint, 1 !!

inches, and y is the actual distance in inches. 𝑦 = 48 ∙ 1 !!

𝑦 = 48 ∙32

𝑦 = 72 𝑖𝑛𝑐ℎ𝑒𝑠 OR 6 feet When the scale is used, the units of measure can be maintained.

1𝑓𝑜𝑜𝑡14 𝑖𝑛𝑐ℎ

=1 ∙ 414 ∙ 4

=41= 4 𝑓𝑒𝑒𝑡 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 1 𝑖𝑛𝑐ℎ

The equation for this relationship is y = 4x where x is the distance on the blueprint, 1 !! inches and y is the actual

distance in feet. 𝑦 = 4 ∙ 1 !!

𝑦 = 4 ∙32

𝑦 = 6 𝑓𝑒𝑒𝑡 This confirms that the distance from the corner to the electrical box is 6 feet.

Problem Set Sample Solutions

1. A toy company is redesigning their packaging for model cars. The graphic design team needs to

take the old image shown below and resize it so that 𝟏𝟐 inch on the old packaging represents

𝟏𝟑 inch

on the new package. Find the length of the image on the new package. Car image length on old packaging measures 𝟐 inches

The length of the image on the new package will be 𝟒𝟑 inches. The scale is

𝟏𝟐 to

𝟏𝟑 and the length of

the original figure is 𝟐, which is 𝟒 halves, so in the scale drawing the length will be 𝟒 thirds.

Scaffolding:

Help students to understand the reasoning demonstrated in the suggested solution to problem 1. This supports MP.1, making sense of problems, rather than a reliance on procedural steps to calculate a new package length.



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2. The city of St. Louis is creating a welcome sign on a billboard for visitors to see as they enter the city. The following

picture needs to be enlarged so that 𝟏𝟐 inch represents 𝟕 feet on the actual billboard. Will it fit on a billboard that

measures 𝟏𝟒 feet in height?

Yes, if 𝟏𝟐 inch represents 𝟕 feet on the actual billboard, then 𝟏 inch in height will correspond to 𝟏𝟒 feet on the actual

billboard.

3. Your mom is repainting your younger brother’s room. She is going to project the image shown below onto his wall so that she can paint an enlarged version as a mural. Use a ruler to determine the length of the image of the train.

Then determine how long the mural will be if the projector uses a scale where 𝟏 inch of the image represents 𝟐 𝟏𝟐 feet on the wall.

Using the scale of 𝟐 𝟏

𝟐 feet to inches:

𝒚 = 𝟐𝟏𝟐×𝟐

𝒚 =𝟓𝟐×𝟐

𝒚 = 5 feet

Using the scale factor:

𝒌 = 𝟑𝟎

𝟏 (inches to inches)

𝒚 = 𝟑𝟎×𝟐

𝒚 = 𝟔𝟎 𝒊𝒏𝒄𝒉𝒆𝒔 or 5 feet

Scaffolding:

Students do not need to demonstrate use of a scale and scale factor to find a length. Either approach is acceptable. Using a scale might be easier for struggling students since converting units is not necessary.



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4. A model of a skyscraper is made so that 𝟏 inch represents 𝟕𝟓 feet. What is the height of the actual building if the

height of the model is 𝟏𝟖 𝟑𝟓 inches?

Using the scale, 𝟏𝟖 𝟑

𝟓×𝟕𝟓 = 𝟏𝟑𝟗𝟓 feet.

Using the scale factor, 𝟏𝟖 𝟑

𝟓×𝟗𝟎𝟎 = 𝟏𝟔𝟕𝟒𝟎 𝒊𝒏𝒄𝒉𝒆𝒔 𝑶𝑹 𝟏𝟑𝟗𝟓 feet. (Note: 𝟕𝟓 feet = 𝟗𝟎𝟎 inches)

5. The portrait company that takes little league baseball team photos is offering an option where a portrait of your

baseball pose can be enlarged to be used as a wall decal (sticker). Your height in the portrait measures 𝟑 𝟏𝟐 inches. If the company uses a scale where 𝟏 inch on the portrait represents 𝟐𝟎 inches on the wall decal, find the height on the wall decal. Your actual height is 𝟓𝟓 inches. If you stand next to the wall decal, will it be larger or smaller than you?

Using the scale, 𝟑 𝟏

𝟐×𝟐𝟎 = 𝟕𝟎. Your height on the wall decal is 𝟕𝟎 inches. The wall decal will be larger than your

actual height (when you stand next to it).

6. The sponsor of a 𝟓K run/walk for charity wishes to create a stamp of its billboard to commemorate the event. If the sponsor uses a scale where 𝟏 inch represents 𝟒 feet, and the billboard is a rectangle with a width of 𝟏𝟒 feet and a length of 𝟒𝟖 feet, what will be the shape and size of the stamp?

Using the scale, 𝟏𝟒×𝟏𝟒 = 𝟑 𝟏

𝟐 and 𝟏

𝟒×𝟒𝟖 = 𝟏𝟐. The stamp will be a rectangle measuring

𝟑 𝟏𝟐 inches by 𝟏𝟐 inches.

7. Danielle is creating a scale drawing of her room. The rectangular room measures 𝟐𝟎 𝟏𝟐 ft. by 𝟐𝟓 ft. If her drawing uses the scale where 𝟏 inch represents 𝟐 feet of the actual room,

will her drawing fit on an 𝟖 𝟏𝟐 in. by 𝟏𝟏 in. piece of paper?

No, the drawing will not fit. Using the scale, 𝟏𝟐×𝟐𝟎 𝟏

𝟐= 𝟏𝟎 𝟏

𝟒 and 𝟏

𝟐×𝟐𝟓 = 𝟏𝟐 𝟏

𝟐. The

drawing would be 𝟏𝟎 𝟏𝟒 inches by 𝟏𝟐𝟏𝟐 inches, which is larger than the piece of paper.

8. A model of an apartment is shown below where 𝟏𝟒 inch represents 𝟒 feet in the actual apartment. Use a ruler to

measure the drawing and find the actual length and width of the bedroom.

Ruler measurements: 𝟏 𝟏𝟖 inches by 𝟗𝟏𝟔

inches.

The scale is 𝟒𝟏𝟒

= 𝟒∙𝟒𝟏𝟒 ∙𝟒

= 𝟏𝟔 feet for each inch on the drawing.

Using the scale, 𝟏𝟔×𝟏 𝟏

𝟖= 𝟏𝟖 and 𝟏𝟔× 𝟗

𝟏𝟔= 𝟗. The actual length would be 𝟏𝟖 feet and the actual width

would be 𝟗 feet.

Bedroom

Scaffolding:

In # 6, check that students understand that the scale will be a value less than 1 because the stamp size is a reduction from the billboard size. This is also true in #7 where the scale drawing of the bedroom is smaller than the actual room.

Lesson 18: Computing Actual Lengths from a Scale Drawing

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