Have you ever built a bookcase?

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?

**This Concept is all about unit scale. Pay close attention and you will know how to help Jeremy at the end of the Concept.**

### Guidance

Previously we worked on ** ratios** and

**. Remember that a**

*proportions***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 of ratios and proportions. One such application is when we use maps or drawings or models of real-life things.**

If you think about this it makes perfect sense. We can’t really draw 500 miles 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 sensible 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 with.

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 draw 500 feet, you could use this unit scale to draw 5 inches.

If you drew a line that was 8 inches, you would know that you are representing 800 feet.

**All of this is thanks to ratios.**

Once you decide on a unit scale for a scale drawing, map or model, you can use it and the actual dimension 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.

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.

Now it's time for you to practice. 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.

#### Example A

If the yard is 10 feet by 20 feet.

**Solution: 1 inch x 2 inches**

#### Example B

If the yard is 25 feet by 45 feet.

**Solution: 1.5" x 4.5"**

#### Example C

If the yard is 60 feet by 100 feet.

**Solution: 6" x 10"**

Here is the original problem once again.

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*}= 36'' \div 12 = 3 \ feet\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 3 inches, 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*} 6” 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

- 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.

### Guided Practice

Here is one for you to try on your own.

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?

**Answer**

First, let's use the scale to figure out these answers.

You can see that .5 cm = 14 m. This means that 1 cm = 28 meters.

Mia walked 4 cm on the map.

\begin{align*}28 \times 4 = 112\end{align*}

**She walked 112 meters to the dining hall.**

When she walked to the arts and crafts center, she walked 2.25 cm.

\begin{align*}28 \times 2.25 = 63\end{align*}

**Mia walked 63 meters to the arts and crafts center.**

Now we add them together for our last answer.

\begin{align*}112 + 63 = 175\end{align*}

**Mia walked 175 meters in all.**

### Video Review

- This is a James Sousa video on unit scale.

### Practice

Directions: Solve each problem using scale and measurement.

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.