5.6: Unit Scale
Introduction
The Bookcase
Jeremy and his Grandpa are going to build a new bookcase. Given all of the reading that Jeremy has been doing in Mrs. Henderson’s class, he is definitely in need of a new bookcase.
There is one wall in Jeremy’s room that is the perfect size for a bookcase. He and his Grandpa have decided that a bookcase that is 36” long by 12” wide by 48” high is the best size for the bookcase. Jeremy decides to draw a design of the bookcase first.
He chooses a unit scale of \begin{align*}\frac{1 \ inch}{1 \ foot}\end{align*}.
Jeremy begins to figure out how to draw the bookcase design given the unit scale. Do you know how to do it? Using this unit scale and the actual measurements of the bookcase, how big will each dimension of the drawing be?
To figure this out, you will need to understand unit scale and scale dimensions. Pay close attention during this lesson and at the end you will understand how to figure out the drawing’s dimensions.
What You Will Learn
By the end of this lesson, you will learn to demonstrate the following skills:
- Recognize unit scale as the ratio between units of measure of scale dimensions to actual dimensions.
- Use unit scale to find dimensions of scale models, drawings or maps given actual dimensions.
- Use unit scale to find actual dimensions from scale models, drawings or maps.
- Solve real-world problems involving scale models, drawings and maps using unit scale.
Teaching Time
I. Recognize Unit Scale as the Ratio between Units of Measure of Scale Dimensions to Actual Dimensions
In our last few lessons, we have been learning about ratios and proportions. Remember that a ratio compares two quantities. Ratios can be written as fractions, with a colon or with the word “to”. A proportion is created when two ratios are equivalent or equal.
There are many real-life applications that use ratios and proportions. Some examples are when we use maps, drawings, or models of real-life things.
If you think about this it makes perfect sense. We can’t really draw 500 miles on a map so we use a scale to help us to represent 500 miles in a sensible way. We can’t really draw a building that is 20 stories high, so we use a scale to help us to draw it in a practical way. We can’t really build a life-size model of a home or a car, so we use a scale to help us to build a model of that real thing.
Ratios make all of this possible. Let’s think about maps to begin.
Suppose you wanted to find your way to a place you've never been before? What would you do?
You might decide to look at a map to help you determine how to get there. A map is an example of a scale drawing.
Scale drawings and scale models allow us to represent objects accurately that are too large or too small to represent at the correct size.
Unit scale is what allows us to represent these objects. A unit scale is a ratio. It compares the dimensions of an actual object to the dimensions of a scale drawing or model. For example, 1 inch on a map of your town might actually represent a distance of 100 feet.
We could write this ratio like this \begin{align*}\frac{1 \ inch} {100 \ ft}\end{align*}.
Then if you wanted to represent 500 feet, you could use this unit scale and draw 5 inches.
If you drew a line that was 8 inches, you would know that you are representing 800 feet.
All that you have learned about ratios and proportions, applied to what you now know about scales, leads to our next section.
II. Use Unit Scale to Find Dimensions of Scale Models, Drawings or Maps Given Actual Dimensions
Once you decide on a unit scale for a scale drawing, map or model, you can use it and the actual dimensions to figure out what the measurements will be on the drawing. For example, suppose you wanted to make a scale model or a scale drawing. You would need to know the dimensions, such as the length or width, of the object you wanted to represent. Then, you would need to decide on a unit scale for your model or drawing. You would then use that unit scale to create the drawing to scale.
Example
Kim has a rectangular backyard. Its actual dimensions are 50 feet by 30 feet. Kim wants to make a scale drawing of her backyard. The scale of her drawing will be \begin{align*}\frac{1}{2} \ in. = 5 \ ft\end{align*}.
Find the dimensions of the backyard in her scale drawing.
First, write the unit scale, \begin{align*}\frac{1}{2} \ in. = 5 \ ft\end{align*}, as a ratio. When writing a ratio, it will be easier to convert \begin{align*}\frac{1}{2}\end{align*} in. to a decimal, 0.5 in.
\begin{align*}\frac{inches}{feet} = \frac{0.5}{5}\end{align*}
The backyard has a length and a width, so the next thing is to write ratios for the length and the width. The actual length of the backyard is 50 feet. Use \begin{align*}l\end{align*} to represent the length of the backyard in the scale drawing.
\begin{align*}\frac{inches}{feet} = \frac{l}{50}\end{align*}
The actual width of the backyard is 30 feet. Use \begin{align*}w\end{align*} to represent the length of the backyard in the scale drawing.
\begin{align*}\frac{inches}{feet} = \frac{w}{30}\end{align*}
Set up and solve proportions to find the scale length, \begin{align*}l\end{align*}, and the scale width, \begin{align*}w\end{align*}. This will give us the number of inches that we will draw for the scale drawing.
\begin{align*}\frac{0.5}{5} &= \frac{l}{50} && \ \frac{0.5}{5} = \frac{w}{30}\\ 5 \cdot l &= 0.5 \cdot 50 && 5 \cdot w = 0.5 \cdot 30\\ 5l &= 25 && \ \ 5w = 15\\ \frac{5l}{5} &= \frac{25}{5} && \ \frac{5w}{5} = \frac{15}{5}\\ l &= 5 && \ \ \ w = 3\end{align*}
In the scale drawing, Kim's rectangular backyard should be 5 inches long and 3 inches wide. Her drawing might look like this example.
5L. Lesson Exercises
Use the scale \begin{align*}\frac{1}{2} \ in = 5 \ ft\end{align*} to figure out the dimensions of the scale drawing given the actual dimensions.
- If the yard is 10 feet by 20 feet.
- If the yard is 25 feet by 45 feet.
- If the yard is 60 feet by 100 feet.
Take a few minutes to check your work with a partner.
III. Use Unit Scale to Find Actual Dimensions from Scale Models, Drawings or Maps
Sometimes, you will have a scale drawing, map or model to work with first. You won’t be given the actual dimensions. Instead, you will have to use the unit scale that accompanies the scale model, drawing or map to figure out the actual dimensions.
This often happens with maps. You look at a map and try to figure out how far it is from one city to another. The scale in the bottom of the map can help you with this. If the scale says that 1” = 100 miles and the map indicates that there is 4 inches between one city and the next, then you can say that the actual distance between the two cities is 400 miles.
Example
Hector made this scale model of the Statue of Liberty. The scale height of his model, from the base to the torch, is 4.65 centimeters. Find the actual height of the Statue of Liberty if 1 cm = 10 m.
Write the unit scale as a ratio.
\begin{align*}\frac{centimeters}{meters} = \frac{1}{10}\end{align*}
The scale height of the model is 4.65 centimeters. Use \begin{align*}h\end{align*} to represent the actual height of the Statue of Liberty.
\begin{align*}\frac{centimeters}{meters} = \frac{4.65}{h}\end{align*}
Set up and solve proportions to find the actual height, \begin{align*}h\end{align*}.
\begin{align*}\frac{1}{10} &= \frac{4.65}{h}\\ 10 \cdot 4.65 &= 1 \cdot h\\ 46.5 &= h\end{align*}
The actual height of the Statue of Liberty, from the base to the torch, is 46.5 meters.
So far, we have considered scale drawings and models that are used to represent very large objects. Scale drawings or models can also help us represent very small objects.
Example
A beetle that Amelia observed was too small to draw at its actual size. So, she made this scale drawing.
In her drawing, the length of the beetle was 4.8 cm. What was the length of the actual beetle?
Write the unit scale, 0.4 cm = 1 mm, as a ratio.
\begin{align*}\frac{centimeters}{millimeters} = \frac{0.4}{1}\end{align*}
In the scale drawing, the beetle has a length of 4.8 centimeters. Use \begin{align*}m\end{align*} to represent the actual length, in millimeters, of the beetle.
\begin{align*}\frac{centimeters}{millimeters} = \frac{4.8}{m}\end{align*}
Set up and solve a proportion to find the actual length, \begin{align*}m\end{align*}.
\begin{align*}\frac{0.4}{1} &= \frac{4.8}{m}\\ 1 \cdot 4.8 &= 0.4 \cdot m\\ 4.8 &= 0.4m\\ \frac{4.8}{0.4} &= \frac{0.4m}{0.4}\\ 12 &= m\end{align*}
The actual length of the beetle Amelia observed was 12 millimeters.
IV. Solve Real-World Problems Involving Scale Models, Drawings and Maps Using Unit Scale
How does this all apply to real-world problems?
Well we can take the unit scale or the actual dimensions or the scale dimensions and figure out the missing pieces using proportions. Let’s look at how we can apply what we have learned to solve some real-world problems. Here is a complex problem for us to solve.
Example
The map below shows the distances between three towns.
a. On the map, the distance between Smithville and Frankton is \begin{align*}2 \frac{1}{4}\end{align*} inches. Find the actual straight-line distance between Smithville and Frankton.
b. On the map, the distance between Frankton and Blair is \begin{align*}1 \frac{1}{2}\end{align*} inches. Find the actual straight-line distance between Frankton and Blair.
c. How many miles closer is Frankton to Blair than to Smithville?
First, let’s look at the information that we have been given. Then we can use this information to solve each part of the problem. Notice that there are three parts, \begin{align*}a, b\end{align*} and \begin{align*}c\end{align*}.
The unit scale is \begin{align*}\frac{1}{4} \ in. = 2 \ mi\end{align*}. Express that ratio as a fraction. Since \begin{align*}\frac{1}{4} = 0.25\end{align*}, use 0.25 mile as the first term of the ratio.
\begin{align*}\frac{inches}{miles} = \frac{0.25}{2}\end{align*}
Now, consider part \begin{align*}a\end{align*}.
You know that the scale distance between Smithville and Frankton is \begin{align*}2 \frac{1}{4}\end{align*} inches. The actual distance between those two towns is unknown, so use \begin{align*}x\end{align*} to represent that distance. Write a ratio to represent this. Since \begin{align*}2 \frac{1}{4} = 2.25\end{align*}, use 2.25 as the first term.
\begin{align*}\frac{inches}{miles} = \frac{2.25}{x}\end{align*}
Set up a proportion using the unit scale and the ratio above and solve for \begin{align*}x\end{align*}.
\begin{align*}\frac{0.25}{2} &= \frac{2.25}{x}\\ 2 \cdot 2.25 &= 0.25 \cdot x\\ 4.5 &= 0.25x\\ \frac{4.5}{0.25} &= \frac{0.25x}{0.25}\\ 18 &= x\end{align*} The actual distance between Smithville and Frankton is 18 miles.
Next, consider part \begin{align*}b\end{align*}.
You know that the scale distance between Frankton and Blair is \begin{align*}1 \frac{1}{2}\end{align*} inches. The actual distance between those two towns is unknown, so use \begin{align*}x\end{align*} to represent that distance. Write a ratio to represent this. Since \begin{align*}1 \frac{1}{2} = 1.5\end{align*}, use 1.5 as the first term.
\begin{align*}\frac{inches}{miles} = \frac{1.5}{x}\end{align*}
Set up a proportion using the unit scale and the ratio above and solve for \begin{align*}x\end{align*}.
\begin{align*}\frac{0.25}{2} &= \frac{1.5}{x}\\ 2 \cdot 1.5 &= 0.25 \cdot x\\ 3 &= 0.25x\\ \frac{3}{0.25} &= \frac{0.25x}{0.25}\\ 12 &= x\end{align*} The actual distance between Frankton and Blair is 12 miles.
Finally, consider part \begin{align*}c\end{align*}.
We know that Smithville is 18 miles from Frankton, and that Blair is 12 miles from Frankton. So, we can subtract to find out how much closer Frankton is to Blair than to Smithville.
\begin{align*}18 - 12 = 6\end{align*}
Frankton is 6 miles closer to Blair than to Smithville.
Real Life Example Completed
The Bookcase
Here is the original problem once again. Reread it and then underline any important information.
Jeremy and his Grandpa are going to build him a new bookcase. Given all of the reading that Jeremy has been doing in Mrs. Henderson’s class, he is definitely in need of a new bookcase.
There is one wall in Jeremy’s room that is the perfect size for bookcase. He and his Grandpa have decided that a bookcase that is 36” long by 36” wide by 48” high is the best size for the bookcase. Jeremy decides to draw a design of the bookcase first.
He chooses a unit scale of \begin{align*}\frac{1 \ inch}{1 \ foot}\end{align*}.
Jeremy begins to figure out how to draw the bookcase design given the unit scale. Do you know how to do it? Using this unit scale and the actual measurements of the bookcase, how big will each dimension of the drawing be?
With a unit scale of 1” : 1 foot, the first thing to notice is that the actual measurements of the bookcase are in inches. To figure out how many inches we need to draw, we first need to convert the number of inches of the bookcase’s actual dimensions into feet.
We can do this by dividing by 12.
Length \begin{align*}= 36” \div 12 = 3 \ feet\end{align*}
Width \begin{align*}= 12” \div 12 = 1 \ foot\end{align*}
Height \begin{align*}48” \div 12 = 4 \ feet\end{align*}
Now we know that Jeremy is going to draw 1” for every foot. That means that the length of the bookcase in the drawing will be 1 inch, the width of the bookcase will be 3 inches and the height of the bookcase will be 4 inches.
Jeremy completes his drawing and decides that it is too small. He decides to change his unit scale. Here is the new unit scale.
\begin{align*}\frac{2 \ inches}{1 \ foot}\end{align*}
This means that the new dimensions will be double the old ones. The drawing will have the following dimensions now.
6” length \begin{align*}\times\end{align*} 2” width \begin{align*}\times\end{align*} 8 inches height
Jeremy is pleased with the adjustment and heads off to show his Grandpa his design for the bookcase.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Ratio
- a comparison between two quantities. Ratios can be written in fraction form, with a colon or by using the word “to”.
- Proportion
- when two ratios are equal or equivalent, they form a proportion.
- Scale Drawing
- a smaller drawing that is used to represent a larger, life-size building or model.
- Unit Scale
- a measurement meant to represent the actual measurements of a larger, life-size building, map or other item. For example 1” = 2 feet would be a unit scale.
Technology Integration
Time to Practice
Directions: Solve each problem. Pay attention because some problems have more than one part.
1. Haley made a scale model of her new school building. The scale she used for her model was 1 inch = 6 feet. The actual height of her school building is 30 feet. What was the height of the school building in her scale model?
2. If the width of Haley’s school is 120 feet, what would be the width in the scale model?
3. If the length of Haley’s school is 180 feet, what would be the length in the scale model?
4. Eddie drew a map of Main Street in his hometown. The scale he used for his map was 1 centimeter = 8 meters. The actual distance between the post office and City Hall, both of which are on Main Street, is 56 meters. What is the distance between those two places on Eddie's map?
5. If the distance from the post office to the library is twice the distance from the post office to the City Hall, what is the distance on Eddie’s map?
6. If the distance from the library to the school is three times the distance as from the post office to the City Hall, what is the distance on Eddie’s map?
7. Asharah built a model of a car. The actual length of the car is 12 feet. The scale of her model is \begin{align*}\frac{1}{4} \ inch = 1 \ foot\end{align*}. What is the length of her model car?
8. Kenya built a model of the same car. His scale of the model is \begin{align*}\frac{1}{2}” = 1 \ foot\end{align*}. What is the length of his model?
9. Jonah observed a spider that was too small to draw at its actual size. So he made a scale drawing, using the scale 0.5 centimeter = 4 millimeters. The actual length of the spider's body, not including its legs, was 16 millimeters. What is the length of the spider's body, not including its legs, in Jonah's drawing?
10. If Jonah made a drawing that is half the size of this one, what would length of the spider’s body be in the new drawing?
11. Alyssa made a scale drawing of her rectangular classroom. She used the scale \begin{align*}\frac{1}{2} \ inch = 4 \ feet\end{align*}. Her actual classroom has dimensions of 32 feet by 28 feet. What are the dimensions of her classroom in the scale drawing?
12. Below is a scale drawing of a circular fountain. In the scale drawing, the diameter of the fountain measures 3 centimeters. What is the actual diameter of the fountain?
13. On a map, Brandon measured the straight-line distance between Los Angeles, California and San Francisco, California to be 2 inches. The scale on the map shows that \begin{align*}\frac{1}{4} \ inch = 43 \ miles\end{align*}. What is the actual straight-line distance between Los Angeles and San Francisco?
14. A butterfly that Adriana observed was too small to draw at its actual size. So, she made this scale drawing.
In the drawing, the wingspan of the butterfly measures 4.5 centimeters. What was the actual wingspan of the butterfly Adriana observed?
15. Jeremy made this scale model of Taipei 101, one of the tallest buildings in the world. The scale height of his model is \begin{align*}2\frac{1}{2}\end{align*} inches. Find the actual height of Taipei 101.
16. Below is a map of Camp Cardinal, where Mia is a counselor.
a. This morning, Mia walked directly from her bunk to the dining hall for breakfast. How many meters did she walk from the bunk to the dining hall?
b. After breakfast, Mia walked directly from the dining hall to the arts and crafts center. How many meters did she walk from the dining hall to the arts and crafts center?
c. What is the total distance that Mia walked from her bunk to the dining hall and then to the arts and crafts center this morning?
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