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Units in Scale Drawings

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Unit 1

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Learning Targets

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

Notes

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. The same scale without units is 1:1,000,000, 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 see that these two scales are equivalent, notice that there are 1,000 millimeters in 1 meter and 1,000 meters in 1 kilometer. This means there are 1,000 ⋅ 1,000 or 1,000,000 millimeters in 1 kilometer. So the actual distances in Nebraska are 1,000,000 times as far as the distances on the map.

A scale tells us how a length on a drawing corresponds to an actual length, and it also tells us how an area on a drawing corresponds to an actual area.

For example, if 1 centimeter on a scale drawing represents 2 meters in actual distance, what does 1 square centimeter on the drawing represent in actual area? The square on the left shows a square with side lengths 1 cm, so its area is 1 square cm.

The square on the right shows the actual dimensions represented by the square on the left. Because each side length in the actual square is 2 m, the actual square has an area of 2² or 4 square meters.

We can use this relationship to find the actual area of any region represented on this drawing. If a room has an area of 18 cm2on the drawing, we know that it has an actual area of 18 ⋅ 4 = 72 or 72 m².

In general, if 1 unit on the drawing represents n actual units, then one square unit on the drawing represents n² actual square units.

Activities

12.1 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.

12.3 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.

Complete the table.
Complete each scale with the value that makes it equivalent to the scale of 1 to 2,000. Explain or show your reasoning. *Note* All these scales are equivalent because in each scale, a factor of 2,000 relates scaled distances to actual distances!
1. 1. - 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
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?

Add To Your Notes

A scale does not have to be expressed in terms of 1 scaled unit...but 1 is often chosen because it makes the scale factor easier to see and our calculations are more efficient! Scales can be expressed in many different ways, including using different units or not using any units.

Example:

How can we express the scale 1 inch to 5 miles without units? There are 12 inches in a foot and 5,280 feet in a mile… This is the same as 1 inch to 63,360 inches.

A scale tells us how a distance on a scale drawing corresponds to an actual distance...and it can also tell us how an area on a drawing corresponds to an actual area.

Example:

If a map uses the scale 1 inch to 5 miles: How can we find the actual area of a region represented on the map? Find the area on the map in square inches and multiply by 25, because 1 square inch represents 25 square miles.
If a map uses the scale 1 inch to 5 miles: How can we find a region’s scaled area if we know its actual area? Multiply the area of the actual region by 1/25.

Summary

Assignment

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