2.3: Scale Drawings and Maps
Introduction
Alex’s Garden Design
Now that Alex has figured out what he wants the garden to look like, he wants to make a drawing of the plot that is accurate.
What does this mean?
It means that Alex wants to use a scale to draw his design. When you use a scale, you choose a unit of measure to represent the real thing. For example, if you want to draw a picture of a ship that is 100 feet long, it doesn’t make sense to actually draw it 100 feet long. You have to choose a unit of measurement like an inch to help you.
Alex decides to use a 1” = 1 ft scale, but he is having a difficult time.
He has two pieces of paper to choose from that he wants to draw the design on. One is \begin{align*}8\frac{1}{2}” \times 11”\end{align*} and the other is \begin{align*}14\frac{1}{2}” \times 11”\end{align*}. He starts using a 1 inch scale and begins to measure the garden plot onto the \begin{align*}8\frac{1}{2}” \times 11”\end{align*} sheet of paper.
At that moment, Tania comes in from outside. She looks over Alex’s shoulder and says, “That will never fit on there. You are going to need a smaller scale or a larger sheet of paper.”
Alex is puzzled. He starts to rethink his work.
He wonders if he should use a \begin{align*}\frac{1}{2}”\end{align*} scale.
Keep in mind the measurements he figured out in the last lesson.
If he uses a 1” scale, what will the measurements be? Does he have a piece of paper that will work?
If he uses a \begin{align*}\frac{1}{2}”\end{align*} scale, what will the measurements be? Does he have a piece of paper that will work?
In this lesson you will learn all about scale and measurement, then you’ll be able to help Alex figure out his garden dilemma.
What You Will Learn
In this lesson, you will learn the following skills.
- Finding actual distances or dimensions given scale dimensions.
- Finding scale dimensions given actual dimensions.
- Solving real-world problems using scale drawings and maps.
Teaching Time
I. Finding Actual Distances or Dimensions Given Scale Dimensions
Maps represent real places. Every part of the place has been reduced to fit on a single piece of paper. A map is an accurate representation because it uses a scale. The scale is a ratio that relates the small size of a representation of a place to the real size of a place.
Maps aren’t the only places that we use a scale. Architects use a scale when designing a house. A blueprint shows a small size of what the house will look like compared to the real house. Any time a model is built, it probably uses a scale. The actual building or mountain or landmark can be built small using a scale.
We use units of measurement to create a ratio that is our scale. The ratio compares two things.
It compares the small size of the object or place to the actual size of the object or place.
A scale of 1 inch to 1 foot means that 1 inch on paper represents 1 foot in real space. If we were to write a ratio to show this we would write:
1” : 1 ft-this would be our scale.
If the distance between two points on a map is 2 inches, the scale tells us that the actual distance in real space is 2 feet.
We can make scales of any size. One inch can represent 1,000 miles if we want our map to show a very large area, such as a continent. One centimeter might represent 1 meter if the map shows a small space, such as a room.
How can we figure out actual distances or dimensions using a scale?
Let’s start by thinking about distances on a map. On a map, we have a scale that is usually found in the corner. For example, if we have a map of the state of Massachusetts, this could be a possible scale.
Here \begin{align*}\frac{3}{4}”\end{align*} is equal to 20 miles.
Example
What is the distance from Boston to Framingham?
To work on this problem, we need to use our scale to measure the distance from Boston to Framingham. We can do this by using a ruler. We know that every \begin{align*}\frac{3}{4}”\end{align*} on the ruler is equal to 20 miles.
From Boston to Framingham measures \begin{align*}\frac{3}{4}”\end{align*}, therefore the distance is 20 miles
If the scale and map were different, we could use the same calculation method. Let’s use another example that just gives us a scale.
Example
If the scale is 1”:500 miles, how far is a city that measures \begin{align*}5\frac{1}{2}”\end{align*} on a map?
We know that every inch is 500 miles. We have \begin{align*}5\frac{1}{2}”\end{align*}. Let’s start with the 5.
5 \begin{align*}\times\end{align*} 500 \begin{align*}=\end{align*} 2500 + \frac{1}{2} \times 500 \begin{align*}=\end{align*} 2750 miles
By using arithmetic, we were able to figure out the mileage.
Another way to do this is to write two ratios. We can compare the scale with the scale and the distance with the distance. Let’s look at an example that has an object in it instead of a map.
Example
If the scale is 2” : 1 ft, what is the actual measurement if a drawing shows the object as 6” long?
We can start by writing a ratio that compares the scale.
\begin{align*}\frac{1 \ ft}{2”}=\frac{x \ ft}{6”}\end{align*} Here we wrote a proportion. We don’t know how big the object really is, so we used a variable to represent the unknown quantity.
Notice that we compared the size to the scale in the first ratio and the size to the scale in the second ratio.
We can solve this logically using mental math, or we can cross multiply to solve it.
\begin{align*}1 \times 6 &= 6\\ 2(x) &= 2x\\ 2x &= 6 \qquad \text{“What times two will give us 6?”}\\ x &= 3 \ ft\end{align*}
The object is actually 3 feet long.
This may seem more confusing, but you can use it if you need to. If it is easier to solve the problem using mental math then that is alright too.
Here are a few problems for you to try on your own.
- If the scale is 1” : 3 miles, how many miles does 5 inches represent?
- If the scale is 2” : 500 meters, how many meters does 4 inches represent?
Take a few minutes to check your work with a peer.
II. Finding Scale Dimensions Using Actual Dimensions
In the last section, we worked on figuring out actual dimensions or distances when we had been given a scale.
Now we are going to look at figuring out the scale given the actual dimensions.
To do this, we work in reverse. To make a map, for instance, we need to “shrink” actual distances down to a smaller size that we can show on a piece of paper. Again, we use the scale. Instead of solving for the actual distance, we solve for the map distance. Let’s see how this works.
Example
Suppose we are making a map of some nearby towns. We know that Trawley City and Oakton are 350 kilometers apart. We are using a scale of 1 cm : 10 km. How far apart do we draw the dots representing Trawley City and Oakton on our map?
We use the scale to write ratios that make a proportion. Then we fill in the information we know. This time we know the actual distance between the two towns, so we put that in and solve for the map distance.
\begin{align*}\frac{1 \ cm}{10 \ km}=\frac{x \ cm}{350 \ km}\end{align*} Next we cross multiply to find the number of centimeters that we would need to draw on the map.
\begin{align*}1(350) &= 10x\\ 350 &= 10x\\ 35 &= x\end{align*}
Our answer is 35 cm.
Using our scale, to draw a distance of 350 km on our map, we need to put Trawley City 35 centimeters away from Oakton.
We can figure out the scale using a model and an actual object too.
Let’s look at an example
Example
Jesse wants to build a model of a building. The building is 100 feet tall. If Jesse wants to use a scale of 1” to 25 feet, how tall will his model be?
Let’s start by looking at our scale and writing a proportion to show the measurements that we know. \begin{align*}\frac{1”}{25 \ ft}=\frac{x}{100 \ ft}\end{align*}
To solve this proportion we cross multiply.
\begin{align*}1(100) &= 25(x)\\ 100 &= 25x\\ 4 &= x\end{align*} Jesse’s model will be 4 inches tall.
Our answer is \begin{align*}4”\end{align*}.
Real Life Example Completed
Alex’s Garden Design
Now that we have learned all about scales and scale drawing, we are ready to help Alex with his garden design.
Let’s begin by looking at the problem again.
Now that Alex has figured out what he wants the garden to look like, he wants to make a drawing of the plot that is accurate.
What does this mean?
It means that Alex wants to use a scale to draw his design. When you use a scale, you choose a unit of measurement to represent the real thing. For example, if you want to draw a picture of a ship that is 100 feet long, it doesn’t make sense to actually make a drawing 100 feet long. You have to choose a unit of measurement like an inch to help you.
Alex’s decides to use a scale of 1” = 1 ft., but he is having a difficult time.
He has two pieces of paper to choose from that he wants to draw the design on. One is \begin{align*}8\frac{1}{2}” \times 11”\end{align*} and the other is \begin{align*}14 \frac{1}{2}” \times 11”\end{align*}. He starts using a 1 inch scale and begins to measure the garden plot onto the \begin{align*}8\frac{1}{2}” \times 11”\end{align*} sheet of paper.
At that moment, Tania comes in from outside. She looks over Alex’s shoulder and says, “That will never fit on there. You are going to need a smaller scale or a larger sheet of paper.”
Alex is puzzled. He starts to rethink his work.
He wonders if he should a \begin{align*}\frac{1}{2}”\end{align*} scale.
Keep in mind the measurements he figured out in the last lesson.
If he uses a 1” scale, what will the measurements be? Does he have a piece of paper that will work?
If he uses a \begin{align*}\frac{1}{2}”\end{align*} scale, what will the measurements be? Does he have a piece of paper that will work?
First, let’s begin by underlining all of the important information in the problem.
Next, let’s look at the dimensions given each scale, a 1” scale and a \begin{align*}\frac{1}{2}”\end{align*} scale.
Let’s start with the 1" scale.
First, we start by figuring out the dimensions of the square. Here is our proportion.
\begin{align*}\frac{1”}{1 \ ft} &= \frac{x \ ft}{9 \ ft}\\ 9 &= x\end{align*} To draw the square on a piece of paper using this scale, the three matching sides would each be 9 inches.
Next, we have the short side. It is one foot, so it would be 1” long on the paper.
Now we can work with the rectangle.
If the rectangle is 12 ft \begin{align*}\times\end{align*} 8 ft and every foot is measured with 1”, then the dimensions of the rectangle are 12” \begin{align*}\times\end{align*} 8”.
You would think that this would fit on either piece of paper, but it won’t because remember that Alex decided to put the two garden plots next to each other.
If one side of the square is 9” and the length of the rectangle is 12” that equals 21”. 21 inches will not fit on a piece of \begin{align*}8\frac{1}{2}” \times 11”\end{align*} paper or \begin{align*}14\frac{1}{2}” \times 11”\end{align*} paper.
Let’s see what happens if we use a \begin{align*}\frac{1}{2}” = 1\end{align*} foot scale.
We already figured out a lot of the dimensions here.
We can use common sense and divide the measurements from the first example in half since \begin{align*}\frac{1}{2}”\end{align*} is half of 1”.
The square would be 4.5” on each of the three matching sides.
The short side of the square would be \begin{align*}\frac{1}{2}”\end{align*}.
The length of the rectangle would be 6”. The width of the rectangle would be 4”.
With the square and the rectangle side-by-side, the length of Alex's drawing would be 10.5". This will fit on either piece of paper.
Use your notebook to draw Alex’s garden design.
Use a ruler and draw it to scale.
The scale is \begin{align*}\frac{1}{2}” = 1\end{align*} foot.
When you have finished, check your work with a peer.
Vocabulary
Here are the vocabulary words from this lesson.
- Scale
- a ratio that compares a small size to a larger actual size. One measurement represents another measurement in a scale.
- Ratio
- the comparison of two things
- Proportion
- a pair of equal ratios, we cross multiply to solve a proportion
Technology Integration
Khan Academy Scale and Indirect Measurement
Other Videos
http://www.teachertube.com/viewVideo.php?video_id=79418&title=PSSA_Grade_7_Math_19_Map_Scale – You will need to register with this website. This is a video about solving a ratio and proportion problem.
Time to Practice
Directions: Use the given scale to determine the actual distance.
Given: Scale 1” = 100 miles
1. How many miles is 2” on the map?
2. How many miles is \begin{align*}2\frac{1}{2}”\end{align*} on the map?
3. How many miles is \begin{align*}\frac{1}{4}”\end{align*} on the map?
4. How many miles is \begin{align*}\frac{1}{2}”\end{align*} on the map?
5. How many miles is \begin{align*}5 \frac{1}{4}”\end{align*} on the map?
Given: 1 cm = 20 mi
6. How many miles is 2 cm on the map?
7. How many miles is 4 cm on the map?
8. How many miles is \begin{align*}\frac{1}{2}”\end{align*} cm on the map?
9. How many miles is \begin{align*}1 \frac{1}{2}\end{align*} cm on the map?
10. How many miles is \begin{align*}4 \frac{1}{4}\end{align*} cm on the map?
Directions: Use the given scale to determine the scale measurement given the actual distance.
Given: Scale 2” = 150 miles
11. How many scale inches would 300 miles be?
12. How many scale inches would 450 miles be?
13. How many scale inches would 75 miles be?
14. How many scale inches would 600 miles be?
15. How many scale inches would 900 miles be?
Directions: Use the given scale to determine the scale measurement for the following dimensions.
Given: Scale 1” = 1 foot
16. What is the scale measurement for a room that is 8’ \begin{align*}\times\end{align*} 12’?
17. What is the scale measurement for a tree that is 1 yard high?
18. What is the scale measurement for a tower that is 360 feet high?
19. How many feet is that?
20. What is the scale measurement for a room that is \begin{align*}12’ \times 16 \frac{1}{2}’\end{align*}?
Directions: Use what you have learned about scale and measurement to answer each of the following questions.
21. Joaquin is building the model of a tower. He is going to use a scale of 1” = 1 foot. How big will his tower be in inchesif the actual tower if 480 feet tall?
22. How many feet high will the model be?
23. Is this a realistic scale for this model? Why or why not?
24. If Joaquin decided to use a scale of \begin{align*}\frac{1}{2}” = 1\end{align*} foot, what would the new height of the model be in inches?
25. How many feet tall will the model be?
26. If Joaquin decided to use a scale that was \begin{align*}\frac{1}{4}”\end{align*} for every 1 foot, how many feet high would his model be?
27. What scale would Joaquin need to use if he wanted his model to be 5 feet tall?
28. How tall would the model be if Joaquin decided to use \begin{align*}\frac{1}{16}” = 1\end{align*} foot?
29. If Joaquin’s model ends up being shorter than \begin{align*}2 \frac{1}{2}\end{align*} feet tall, did he use a scale that is smaller or larger than \begin{align*}\frac{1}{8}” = 1\end{align*} foot?
30. If Joaquin wants his model to be half the size of the real model, will it fit in his classroom or will he need to build it outside?