options: Understand and apply theorems about circles.

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

Understand congruence in terms of rigid motions.

Find arc lengths and areas of sectors of circles.

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

Understand similarity in terms of similarity transformations.

Make geometric constructions.

Understand and apply the Pythagorean Theorem.

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

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

Explain volume formulas and use them to solve problems.

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

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

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

Analyze, compare, create, and compose shapes.

Apply geometric concepts in modeling situations.

Define trigonometric ratios and solve problems involving right triangles.

Prove theorems involving similarity.

Experiment with transformations in the plane.

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

Apply trigonometry to general triangles.

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

Use coordinates to prove simple geometric theorems algebraically.

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

Prove geometric theorems.

Reason with shapes and their attributes.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
problem_activity: Warm-up
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 size 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 Facing
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.
[IMG]
\(9\frac14\)
inches
\(9\frac{5}{64}\)
inches
23.47659 centimeters
23.5 centimeters
23.48 centimeters
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.
[IMG]
Student Response
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Anticipated Misconceptions
Some students may say the large foot is about
\(3\frac12\)
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.
Activity Synthesis
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
\(\frac{1}{64}\)
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.


correct_option_index: 8