options: Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
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: 7