In previous lessons, students worked with scales that associated two distinct measurements—one for the distance on a drawing and one for actual distance. The units used in the two measurements are often different (centimeter and meter, inch and foot, etc.). In this lesson, students see that a scale can be expressed without units. For example, consider the scale 1 to 60. This means that every unit of length on the scale drawing represents an actual length that is 60 times its size, whatever the unit may be (inches, centimeters, etc.).
Expressing the scale as 1 to 60 highlights the scale factor relating the scale drawing to the actual object. Each measurement on the scale drawing is multiplied by 60 to find the corresponding measurement on the actual object. This relates closely to the scaled copies that were examined earlier in the unit in which each copy was related to the original by a scale factor. Students gain a better understanding of both scaled copies and scale drawings as they examine the common underlying structure (MP7).
Students then analyze various scales and find that sometimes it is helpful to rewrite scales with units as scales without units in order to compare them. They see that equivalent scales relate scaled and actual measurements by the same scale factor, even though the scales may be expressed differently. For example, the scale 1 inch to 2.5 feet is equivalent to the scale 5 m to 150 m, because they are both at a scale of 1 to 30.
Here is some information about equal lengths that students may want to refer to during these activities.
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
- Comprehend that the phrase “equivalent scales” refers to different scales that relate scaled and actual measurements by the same scale factor.
- Explain (orally and in writing) how to use scales without units to determine scaled or actual distances.
- Interpret scales expressed without units, e.g., “1 to 50,” (in spoken and written language).
- Justify (orally and in writing) that scales are equivalent, including scales with and without units.
You will need the Apollo Lunar Module blackline master for this lesson. Prepare one copy per student.
Ensure students have access to geometry toolkits, especially rulers and graph paper.
- I can use scales without units to find scaled distances or actual distances.
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