Lesson 12Units in Scale Drawings

Learning Goal

Let’s use different scales to describe the same drawing.

Learning Targets

I can tell whether two scales are equivalent.
I can write scales with units as scales without units.

Lesson Terms

scale
scale drawing

Warm Up: Centimeters in a Mile

There are 2.54 cm in an inch, 12 inches in a foot, and 5,280 feet in a mile. Which expression gives the number of centimeters in a mile? Explain your reasoning.

$\frac{2.54}{12 \cdot 5, 280}$
$5, 280 \cdot 12 \cdot (2.54)$
$\frac{1}{5, 280 \cdot 12 \cdot (2.54)}$
$5, 280 + 12 + 2.54$
$\frac{5, 280 \cdot 12}{2.54}$

Activity 1: Scales Card Sort

Your teacher will give you some cards with a scale on each card.

Sort the cards into sets of equivalent scales. Be prepared to explain how you know that the scales in each set are equivalent. Each set should have at least two cards.
Trade places with another group and check each other’s work. If you disagree about how the scales should be sorted, work to reach an agreement.
Pause here so your teacher can review your work.
Next, record one of the sets with three equivalent scales and explain why they are equivalent.

Activity 2: The World’s Largest Flag

As of 2016, Tunisia holds the world record for the largest version of a national flag. It was almost as long as four soccer fields. The flag has a circle in the center, a crescent moon inside the circle, and a star inside the crescent moon.

Problem 1

Complete the table. Explain or show your reasoning.

	flag length	flag height	height of crescent moon
actual	$396 m$		$99 m$
at 1 to 2,000 scale		$13.2 cm$

Problem 2

Complete each scale with the value that makes it equivalent to the scale of 1 to 2,000. Explain or show your reasoning.

1 cm to cm
1 cm to m
1 cm to km
2 m to m
5 cm to m
cm to 1,000 m
mm to 20 m

Problem 3

What is the area of the large flag?
What is the area of the smaller flag?
The area of the large flag is how many times the area of the smaller flag?

Activity 3: Pondering Pools

Your teacher will give you a floor plan of a recreation center.

What is the scale of the floor plan if the actual side length of the square pool is 15 m? Express your answer both as a scale with units and without units.
Find the actual area of the large rectangular pool. Show your reasoning.
The kidney-shaped pool has an area of 3.2 cm squared on the drawing. What is its actual area? Explain or show your reasoning.

Are you ready for more?

Problem 1

Square $A$ is a scaled copy of Square $B$ with scale factor 2. If the area of Square $A$ is 10 units $^{2}$ , what is the area of Square $B$ ?

Problem 2

Cube $A$ is a scaled copy of Cube $B$ with scale factor 2. If the volume of Cube $A$ is 10 units $^{3}$ , what is the volume of Cube $B$ ?

Problem 3

The four-dimensional Hypercube $A$ is a scaled copy of Hypercube $B$ with scale factor 2. If the “volume” of Hypercube $A$ is 10 units $^{4}$ , what do you think the “volume” of Hypercube $B$ is?

Lesson Summary

Sometimes scales come with units, and sometimes they don’t. For example, a map of Nebraska may have a scale of 1 mm to 1 km. This means that each millimeter of distance on the map represents 1 kilometer of distance in Nebraska. Notice that there are 1,000 millimeters in 1 meter and 1,000 meters in 1 kilometer. This means there are $1, 000 \cdot 1, 000$ or 1,000,000 millimeters in 1 kilometer. So, the same scale without units is 1 to 1,000,000, which means that each unit of distance on the map represents 1,000,000 units of distance in Nebraska. This is true for any choice of unit to express the scale of this map.

Sometimes when a scale comes with units, it is useful to rewrite it without units. For example, let’s say we have a different map of Rhode Island, and we want to use the two maps to compare the size of Nebraska and Rhode Island. It is important to know if the maps are at the same scale. The scale of the map of Rhode Island is 1 inch to 10 miles. There are 5,280 feet in 1 mile, and 12 inches in 1 foot, so there are 63,360 inches in 1 mile (because $5, 280 \cdot 12 = 63, 360$ ). Therefore, there are 633,600 inches in 10 miles. The scale of the map of Rhode Island without units is 1 to 633,600. The two maps are not at the same scale, so we should not use these maps to compare the size of Nebraska to the size of Rhode Island.

Here is some information about equal lengths that you may find useful.

Customary Units	Metric Units
1 foot (ft) = 12 inches (in) 1 yard (yd) = 36 inches 1 yard = 3 feet 1 mile = 5,280 feet	1 meter (m) = 1,000 millimeters (mm) 1 meter = 100 centimeters 1 kilometer (km) = 1,000 meters

Equal Lengths in Different Systems
1 inch = 2.54 centimeters 1 foot $\approx$ 0.30 meter 1 mile $\approx$ 1.61 kilometers	1 centimeter $\approx$ 0.39 inch 1 meter $\approx$ 39.37 inches 1 kilometer $\approx$ 0.62 mile