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

Understand congruence in terms of rigid motions.

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

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

Experiment with transformations in the plane.

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

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

Explain volume formulas and use them to solve problems.

Prove geometric theorems.

Reason with shapes and their attributes.

Apply geometric concepts in modeling situations.

Define trigonometric ratios and solve problems involving right triangles.

Understand similarity in terms of similarity transformations.

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

Make geometric constructions.

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

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

Understand and apply theorems about circles.

Use coordinates to prove simple geometric theorems algebraically.

Prove theorems involving similarity.

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

Apply trigonometry to general triangles.

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

Analyze, compare, create, and compose shapes.

Understand and apply the Pythagorean Theorem.

Find arc lengths and areas of sectors of circles.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
problem_activity: Activity
Students build their sense for the sizes of angles measured in radians by shading sectors with given central angle measures.
Launch
Arrange students in groups of 2. Distribute 1 set of pre-cut slips to each group.
Student Facing
Your teacher will give you a set of cards with angle measures on them. Place the cards upside down in a pile. Choose 1 student to go first. This student should draw a card, then on either circle shade a sector of the circle whose central angle is the measure on the card that was drawn.
Take turns repeating these steps. If you are shading in a circle that already has a shaded sector, choose a spot next to the already-shaded sectors—don’t leave any gaps. You might have to draw additional lines to break the sectors into smaller pieces.
Continue until an angle is drawn that won’t fit in any of the sectors that are still blank.
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When you’re finished, answer these questions about each circle:
What is the central angle measure for the remaining unshaded sector?
What is the central angle measure for the block of shaded sectors?
Student Response
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Anticipated Misconceptions
If students struggle to figure out the size of the angles on the cards they draw, ask them what radian measure 180 degrees is equivalent to (
\(\pi\)
radians). Then, ask them to find the measures of the central angles of each sector on the 2 circles (
\(\frac{\pi}{4}\)
and
\(\frac{\pi}{6}\)
).
Activity Synthesis
The goal is to gather strategies for understanding the sizes of angles measured in radians. Here are some questions for discussion:
“For groups that drew a card with
\(\frac{\pi}{8}\)
radians, how did you figure out what size sector to shade?” (Sample response: Each sector in the first circle had a central angle measuring
\(\frac{\pi}{4}\)
radians, so
\(\frac{\pi}{8}\)
radians is half that.)
“What is the sum of the angle measures for the shaded and unshaded sector blocks, and why?” (The angle measures add to
\(2\pi\)
, because together they make up the whole circle.)
“How could you shade a sector with central angle measure
\(\frac{\pi}{24}\)
radians in each of the circles?” (This would be one-sixth of a sector in the first circle, and one-quarter of a sector in the second circle.)


correct_option_index: 25