options: Solve real-world and mathematical problems involving area, surface area, and volume.

Apply geometric concepts in modeling situations.

Visualize relationships between two-dimensional and three-dimensional objects.

Apply trigonometry to general triangles.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Understand similarity in terms of similarity transformations.

Analyze, compare, create, and compose shapes.

Translate between the geometric description and the equation for a conic section.

Understand and apply theorems about circles.

Explain volume formulas and use them to solve problems.

Prove theorems involving similarity.

Find arc lengths and areas of sectors of circles.

Reason with shapes and their attributes.

Understand congruence in terms of rigid motions.

Understand and apply the Pythagorean Theorem.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Use coordinates to prove simple geometric theorems algebraically.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Define trigonometric ratios and solve problems involving right triangles.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Make geometric constructions.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Classify two-dimensional figures into categories based on their properties.

Experiment with transformations in the plane.

Prove geometric theorems.
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?”
[IMG]
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


correct_option_index: 15