In previous lessons, students learned to express scales with or without units that can be the same or different. In this lesson, they 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.
This lesson is also the culmination of students' work on scaling and area. Students have seen many examples of the relationship between scaled area and actual area, and now they must use this realization to find the area of an irregularly-shaped pool (MP7, MP8).
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.
- Generate a scale without units that is equivalent to a given scale with units, or vice versa.
- Justify (orally and in writing) that scales are equivalent, including scales with and without units.
Let's use different scales to describe the same drawing.
Note: This lesson contains optional activities. Decide which activities you will do before preparing the materials!
- For the Card Sort: Scales activity, print and cut the slips from the blackline master, so that each group of 3–4 students gets one complete set. If possible, copy each complete set on a different color of paper, so that a stray slip can quickly be put back.
- For the Pondering Pools activity, prepare one copy of the blackline master for every two students.
Ensure students have access to geometry toolkits. It is also recommended that a conversion chart for metric and customary units of length be provided while students are working on the activities in this lesson.
- I can tell whether two scales are equivalent.
- I can write scales with units as scales without units.
A scale tells how the measurements in a scale drawing represent the actual measurements of the object.
For example, the scale on this floor plan tells us that 1 inch on the drawing represents 8 feet in the actual room. This means that 2 inches would represent 16 feet, and \(\frac12\) inch would represent 4 feet.
A scale drawing represents an actual place or object. All the measurements in the drawing correspond to the measurements of the actual object by the same scale.