Lesson 9
Part to Whole
9.1: What’s Your Angle? (5 minutes)
Warm-up
Students begin thinking about the relationships between radius, central angle, and arc length. This leads to a deeper analysis of these relationships in upcoming activities. Monitor for a variety of ways students solve this problem. Students may use mental math, they may set up proportions or equations, or they may walk through a logical series of reasoning steps.
Launch
Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.
Student Facing
A circle has radius 10 centimeters. Suppose an arc on the circle has length \(\pi\) centimeters. What is the measure of the central angle whose radii define the arc?
Student Response
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Anticipated Misconceptions
If students are stuck, suggest they find the circumference of the circle.
Activity Synthesis
The goal is to discuss students’ strategies for solving this problem. Ask students how they began the problem. (Sample response: I started by finding the circumference of the circle.)
Select previously identified students to share their strategies. If possible, select a student who used mental math, and one who used a strategy such as proportions or equations.
9.2: Enough Information? (10 minutes)
Activity
In this activity, students work backward from the area and central angle of a sector to find the area, radius, and circumference of the circle as well as the arc length of the initial sector.
Launch
Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.
Supports accessibility for: Conceptual processing
Student Facing
The central angle of this shaded sector measures 45 degrees, and the sector’s area is \(32\pi\) square inches.
Kiran says, “We can find the area of the whole circle, the arc length of the sector, and the circumference of the circle with this information.”
Priya says, “But how? We don’t know the circle’s radius!”
Do you agree with either of them? Explain or show your reasoning. Calculate as many of the values Kiran mentioned as possible.
Student Response
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Student Facing
Are you ready for more?
A Greek mathematician named Eratosthenes calculated the circumference of Earth more than 2,000 years ago. He used information similar to what is given here:
Seattle and San Francisco are on the same longitude line and 680 miles apart. At noon on a summer day, an upright meter stick in Seattle casts a shadow with an angle of 2 degrees. On the same day at the same time, an upright meter stick in San Francisco casts a shadow with an angle of 7.83 degrees.
image not to scale
Use this information to calculate the circumference of Earth. Explain or show your reasoning.
Student Response
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Anticipated Misconceptions
If students are stuck, ask them what fraction the sector represents of the entire circle.
Activity Synthesis
The purpose is to analyze the relationships between a sector and the full circle. Here are some questions for discussion:
- “What fraction of the circle is the sector?” (It is \(\frac18\) of the circle.)
- “What if the radius of the circle were increased from 16 inches to 25 inches, but the central angle of the sector stayed the same. How would this fraction change?” (It would not change. The sector would still be \(\frac18\) of the circle.)
- “What if the central angle of the sector were increased to 135 degrees, but the radius stayed the same. How would the fraction change?” (The fraction would increase to \(\frac38\).)
9.3: Info Gap: From Sector to Circle (20 minutes)
Activity
This info gap activity gives students an opportunity to determine and request the information needed to work backward from sector area or arc length to find information about a circle.
The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).
Here is the text of the cards for reference and planning:
Launch
Tell students they will continue to work with sector areas and arc lengths. Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it.
Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give them the cards for a second problem and instruct them to switch roles.
Design Principle(s): Cultivate Conversation
Supports accessibility for: Memory; Organization
Student Facing
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the data card:
- Silently read the information on your card.
- Ask your partner, “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
- Before telling your partner the information, ask, “Why do you need to know (that piece of information)?”
- Read the problem card, and solve the problem independently.
- Share the data card, and discuss your reasoning.
If your teacher gives you the problem card:
- Silently read your card and think about what information you need to answer the question.
- Ask your partner for the specific information that you need.
- Explain to your partner how you are using the information to solve the problem.
- When you have enough information, share the problem card with your partner, and solve the problem independently.
- Read the data card, and discuss your reasoning.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
Student Response
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Activity Synthesis
After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:
- “Did you ask for any information your partner didn’t have? If so, what did you do next?” (Sample response: I asked for the radius of the circle, but my partner didn’t have that information. After that, I asked for information about the sector.)
- “What information did you use to find the answer for problem card 1?” (Sample response: I used the sector area and the central angle measurement.)
- “What information did you use to find the answer for problem card 2?” (Sample response: I used the arc length and the central angle measurement.)
Lesson Synthesis
Lesson Synthesis
The goal is to discuss relationships between arc length and the central angle of a circle. This will start to connect the work students did in this lesson with the idea of radians that will be developed in upcoming lessons. Ask students:
- “Suppose an arc in a circle has arc length \(\frac{\pi}{2}\) units and central angle 90 degrees. What is the radius of the circle?” (The radius measures 1 unit.)
Instruct students to sketch the circle and the central angle, marking the radius 1 unit. Then ask:
- “Suppose the same circle has an arc with length 1 unit. Is this central angle larger or smaller than that of the previous arc?” (The number \(\frac{\pi}{2}\) is approximately 1.57 units. An arc with length 1 unit is shorter than an arc with length \(\frac{\pi}{2}\) units and therefore must have a smaller central angle.)
- “Sketch what this arc and central angle would look like. About how many degrees does this angle measure? How do you know?” (Students should sketch a central angle of approximately 60 degrees. The correct angle measure is approximately 57.3 degrees, but an exact angle measurement is not expected here. Students may use proportional reasoning to find the measure of the angle. It is also acceptable to simply estimate the value.)
9.4: Cool-down - Find the Radius (5 minutes)
Cool-Down
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Student Lesson Summary
Student Facing
We can work backward from information about a sector or arc to get information about the whole circle.
Suppose a circle has an arc with length \(7\pi\) units that is defined by a 90 degree central angle. This arc makes up \(\frac14\) of the entire circumference of the circle because \(360\div 90=4\). So, we can multiply the arc’s length by 4 to find that the entire circumference measures \(28\pi\) units. The circumference of a circle can be calculated using the expression \(2\pi r\), so we know that \(2\pi r=28\pi\). The value of \(r\) must be 14 units.
For more difficult problems, we can write equations. Suppose a circle has a sector with area \(135\pi\) square units and a central angle of 216 degrees. Let \(x\) stand for the area of the whole circle. Dividing 216 by 360 gives the fraction of the circle represented by the sector. Multiply that fraction by \(x\) and set it equal to the area of the sector.
\(\frac{216}{360}x = 135\pi\)
Solve this equation to find that the circle’s area is \(225\pi\) square units. This tells us that \(\pi r^2 = 225\pi\). The value of \(r\) must be the positive number that squares to make 225, or 15 units.