Unit 7, Lesson 13
Part to Whole
Integrated Math 2
13.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
HSG-C.B.5
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Instructional Routines
NoneMaterials
NoneActivity Narrative
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.
Activity
NoneA circle has radius 10 centimeters. Suppose an arc on the circle has length centimeters. What is the measure of the central angle whose radii define the arc?
Student Response
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Building on Student Thinking
If students are stuck, suggest they find the circumference of the circle.
Activity Synthesis
The goal is to discuss students’ strategies for solving. Ask students how they began the problem. (Students may start 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 select another student who used proportions or equations.
13.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
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.
Engagement: Develop Effort and Persistence. Differentiate the degree of difficulty or complexity. Begin with an example with more accessible values. For example, show a square with half or a quarter of the area shaded, and give the area of the shaded region. Highlight connections between representations by showing that, in squares or circles, we can find the whole area and other measurements if we know what fraction the part is of the whole.
Supports accessibility for: Conceptual Processing, Memory
Supports accessibility for: Conceptual Processing, Memory
Activity
None13.3
Activity
Instructional Routines
MLR4: Information Gap CardsActivity Narrative
This 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 Information 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).
This activity uses the Information Gap math language routine, which facilitates meaningful interactions by positioning some students as holders of information that is needed by other students, creating a need to communicate.
Lesson Synthesis
The goal is to discuss relationships between arc length and the central angle of a circle. This discussion 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 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 and to label the radius as 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 is approximately 1.57 units. An arc with length 1 unit is shorter than an arc with length 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.)
Student Lesson Summary
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 units that is defined by a 90-degree central angle. This arc makes up the entire circumference of the circle because . So we can multiply the arc’s length by 4 to find that the entire circumference measures units. The circumference of a circle can be calculated using the expression , so we know that . The value of must be 14 units.
For more difficult problems, we can write equations. Suppose a circle has a sector with area square units and a central angle of 216 degrees. Let 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 and set it equal to the area of the sector.
Solve this equation to find that the circle’s area is square units. This tells us that . The value of must be the positive number that squares to make 225, or 15 units.