standard_description: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
problem_activity: Activity
In the previous activity, students calculated the scaled distances they would need to create a scale drawing, but did not actually create the scale drawing. In this activity, they create two different scale drawings of the state of Utah and notice how the scale impacts the drawing. One of the reasons choice of a scale is important is that we want to see the appropriate level of detail within a fixed space.
Launch
Display the outline of Utah and ask students “What do you notice? What do you wonder?”
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Ask students to describe the shape: a rectangle with a smaller rectangle removed in the upper right corner.
Give students 5–6 minutes of quiet work time followed by whole-class discussion.
Student Facing
A rectangle around Utah is about 270 miles wide and about 350 miles tall. The upper right corner that is missing is about 110 miles wide and about 70 miles tall.
Make a scale drawing of Utah where 1 centimeter represents 50 miles.
Make a scale drawing of Utah where 1 centimeter represents 75 miles.
How do the two drawings compare? How does the choice of scale influence the drawing?
Student Response
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Anticipated Misconceptions
Some students may get a shape that is not closed or does not have right angles if they did not measure carefully enough. Prompt them to double-check their measurement for a particular side of the state if you can easily tell which side is drawn incorrectly.
Students may think that a scale of 1 centimeter to 50 miles will produce a smaller scale drawing than a scale of 1 centimeter to 75 miles (because 50 is less than 75). Ask them how many centimeters it takes to represent 75 miles if 1 centimeter represents 50 miles (1.5) and how many centimeters it takes to represent 75 miles if 1 centimeter represents 75 miles (1).
Activity Synthesis
Ask students what the two scale drawings share in common. Answers include: they both represent Utah, they both have the same shape, and they both can be used to measure distances in the actual state of Utah.
Ask students how the two scale drawings differ. The one at a scale of 1 centimeter to 50 miles is larger than the one at a scale of 1 centimeter to 75 miles.
Some students may notice that the scale drawing at a scale of 1 centimeter to 75 miles is actually a scaled copy of the other drawing, with a scale factor of 1.5. If so, ask them to share their observation linking scale drawings with scaled copies.
Representation: Internalize Comprehension.
Use color coding and annotations to highlight connections between representations in a problem. For example, annotate a display of the two scale drawings to make visible what the drawings share in common, and how they differ.
Supports accessibility for: Visual-spatial processing
Reading, Speaking: MLR3 Clarify, Critique, Correct.
Before presenting the correct scale drawings of Utah, present an incorrect drawing and written explanation. For example, present a rectangle approximately 5.4 cm wide and 7 cm tall that is missing an upper right corner which is approximately 1.4 cm wide and 2.2 cm tall, and provide the statement: “Since 1 cm represents 50 miles, I divided 110 and 70 each by 50, and got 2.2 and 1.4. The small rectangle that is missing is 2.2 cm tall and 1.4 cm wide.” Ask students to identify the error, critique the reasoning, and revise the statement so that the drawing is a scale drawing of Utah. This will remind students about the characteristics of scale drawings discussed in previous lessons and how to determine whether a drawing qualifies as a scale drawing of an actual object.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness


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