Changing Scales in Scale Drawings

5 min

Teacher Prep
Setup
Students in groups of 2. Before starting task, 1 minute to estimate students' own foot size and share with a partner.

Narrative

This Warm-up prompts students to attend to precision in measurements, which will be important in upcoming work.

Launch

Arrange students in groups of 2. Give students 1 minute of quiet think time to estimate the length of their own foot in centimeters or inches, and a moment to share their estimate with a partner. Then, ask them to complete the task. 

Student Task

  1. If a student uses a ruler like this to measure the length of their foot, which choices would be appropriate measurements? Select all that apply. Be prepared to explain your reasoning.

    The ruler measures inches and centimeters. The inches are broken into eighths and the centimeters are broken into tenths.

    1. 9149\frac14 inches

    2. 95649\frac{5}{64} inches

    3. 23.47659 centimeters

    4. 23.5 centimeters

    5. 23.48 centimeters

  2. Here is a scale drawing of an average seventh-grade student’s foot next to a scale drawing of a foot belonging to the person with the largest feet in the world. Estimate the length of the larger foot.

    <p>Scale drawing of 2 week.</p>
    Scale drawing of an average seventh-grade student's foot next to a scale drawing of a foot belonging to the person with the largest feet in the world. 

Sample Response

  1. A, D. Sample reasoning: Since the ruler is only marked in 18\frac18 inches and 110\frac{1}{10} centimeter, we could not get measurements as precise as B, C, or E.
  2. The largest foot in the world is about 1.5 times as long as the average seventh grader's foot. Sample reasoning: A seventh grader's foot is about 10 inches long, so the largest foot is about 15 inches or 1 foot and 3 inches long. 
Activity Synthesis (Teacher Notes)

Select a few students to share the measurements they think would be appropriate based on the given ruler. Consider displaying the picture of the ruler for all to see and recording students' responses on it. After each response, poll the class on whether they agree or disagree.

If students consider B, C, or E to be an appropriate measurement, ask them to share how to get such a level of precision on the ruler. Make sure students understand that reporting measurements to the nearest 164\frac{1}{64} of an inch or to the hundred-thousandths of a centimeter would not be appropriate (i.e., show that the ruler does not allow for these levels of precision).

Choice E of 23.48 cm may merit specific attention. With the ruler, it is possible to guess that the hundredths place is an 8. This may even be correct. The problem with reporting the measurement in this way is that someone who sees this might misinterpret it and imagine that an extremely accurate measuring device was used to measure the foot, rather than this ruler. The way a measurement is reported reflects how the measurement was taken.

Next, invite students to share their estimates for the length of the large foot. Since it is difficult to measure the length of these feet very precisely, these measurements should not be reported with a high level of precision; the nearest centimeter would be appropriate.

Math Community
After the Warm-up, display the revised Math Community Chart created from student responses in Exercise 3. Tell students that today they are going to monitor for two things:

  • “Doing Math” actions from the chart that they see or hear happening.
  • “Doing Math” actions that they see or hear that they think should be added to the chart.

Provide sticky notes for students to record what they see and hear during the lesson.

Anticipated Misconceptions

Some students may say the large foot is about 3123\frac12 inches or about 9 centimeters long, because they assume the ruler shown in the first question is at the same scale as the feet shown in the second question. Explain that the images are drawn at different scales.

Standards
Building On
  • 2.MD.A·Measure and estimate lengths in standard units.
  • 2.MD.A·Measure and estimate lengths in standard units.
Addressing
  • 7.G.1·Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
  • 7.G.A.1·Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
Building Toward
  • 7.RP.3·Use proportional relationships to solve multistep ratio and percent problems.
  • 7.RP.A.3·Use proportional relationships to solve multistep ratio and percent problems. <span>Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.</span>

15 min

15 min