Lesson 12 | Circles | 10th Grade Mathematics

Objective

Describe the proportional relationship between arc length and the radius of a circle. Convert between degrees and radians to write the arc measure in radians.

Common Core Standards

Core Standards

The core standards covered in this lesson

G.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.

High School — Geometry

G.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.

Foundational Standards

The foundational standards covered in this lesson

7.RP.A.2

Ratios and Proportional Relationships

7.RP.A.2 — Recognize and represent proportional relationships between quantities.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Define radians as the number of radii needed to form the circumference. Because the measure of “radians” is not specific to the radius of the circle, the number of radians in a circle is a constant of $$2\pi$$.
Define radians as the angle of turn of the arc so that you can describe the arc measure and central angle in both radians and degrees.
Describe that the ratio of the circumference to the arc length is the same as the ratio of the central angle to the entire circle.
Describe that as the radius increases, the arc length also increases proportionally but the arc measure stays constant.
Convert between radians and degrees.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

A common misconception students may have about this objective may involve the ratio of the arc length to the entire circumference.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

How many radii do you need to go around the complete circle?

Guiding Questions

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References

GeoGebra CSS High School: Geometry (Circles) — HSG.C.B.5, Movie: Radians to Revs

CSS High School: Geometry (Circles) by Tim Brzezinski is made available by GeoGebra under the CC BY-NC-SA 3.0 license. Copyright © International GeoGebra Institute, 2013. Accessed Sept. 19, 2018, 2:30 p.m..

Problem 2

Below are three circles with center $$A$$.

The smallest circle has a radius of $$\overline{AC}$$ with a length of 1 unit.
The next circle has a radius of $$\overline{AD}$$ with a length of 3 units.
The largest circle has a radius of $$\overline{AB}$$ with a length of 5 units.

Are the radius lengths proportional to the arc lengths? Explain your reasoning.

Guiding Questions

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References

GeoGebra Geometry - 8.12 AP2

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Below is circle $$A$$:

What is the $$m\angle BAC$$ in degrees to the nearest hundredth?
What is the $$m\widehat{BC}$$ in radians to the nearest hundredth?

References

EngageNY Mathematics Geometry > Module 5 > Topic B > Lesson 10 — Exit Ticket, Question #1

Geometry > Module 5 > Topic B > Lesson 10 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Include problems where students are given the central angle and radius and asked to find the arc length.
Include problems where students are given the arc length and radius and asked to find the central angle.
Include problems such as “If the central angle of a circle with radius of $$2$$ inches has a measure of $$\frac{\pi}{6}$$ radians, what is the arc length? How many arcs of this radian measure are there in one circle? Does this change depending on the length of the radius?”
Include problems such as “Given a circle with a radius length and central angle measure of an arc, if you double the radius and halve the central angle, how will this affect the sector length?”

JMAP G.C.B.5: Arc Length 1
Mathematics Vision Project: Geometry Module 7: Circles a Geometric Perspective — Lesson 8, "Madison's Round Garden" (Only include questions about arc length)
Mathematics Vision Project: Geometry Module 7: Circles a Geometric Perspective — Lesson 9, "Rays and Radians" (especially "Set")
Kuta Software Free Algebra 2 Worksheets Arc Length and Sector Area — Questions #1-12

Lesson 11

Lesson 13

Lesson Map

Topic A: Equations of Circles

Derive the equation of a circle using the Pythagorean Theorem where the center of the circle is at the origin.

G.GPE.A.1 G.GPE.B.4

Given a circle with a center translated from the origin, write the equation of the circle and describe its features.

G.C.A.1 G.CO.A.5 G.GPE.A.1

Write an equation for a circle in standard form by completing the square. Describe the transformations of a circle.

G.CO.A.5 G.GPE.A.1

Topic B: Angle and Segment Relationships in Inscribed and Circumscribed Figures

Define a chord to derive the Chord Central Angles Conjecture and Thales’ Theorem.

G.C.A.2

Describe the relationship between inscribed and central angles in terms of their intercepted arc.

G.C.A.2

Determine the angle and length relationships between intersecting chords.

G.C.A.2

Prove properties of angles in a quadrilateral inscribed in a circle.

G.C.A.3

Define and determine properties of tangents and secants of circles to solve problems with inscribed and circumscribed triangles.

G.C.A.2 G.C.A.3

Construct tangent lines to a circle to define and describe the circumscribed angle.

G.C.A.2 G.C.A.4

Use angle and side length relationships with chords, tangents, inscribed angles, and circumscribed angles to solve problems.

G.C.A.2

Topic C: Arc Length, Radians, and Sector Area

Define, describe, and calculate arc length.

G.C.B.5

Describe the proportional relationship between arc length and the radius of a circle. Convert between degrees and radians to write the arc measure in radians.

G.C.B.5

Calculate the sector area of a circle. Identify relationships between sector area, arc angle, and radius.

G.C.B.5

Use sector area of circles to calculate the composite area of figures.

G.C.B.5 N.Q.A.2 N.Q.A.3

Circles

Objective

Common Core Standards

Core Standards

High School — Geometry

Foundational Standards

Ratios and Proportional Relationships

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Target Task

References

Additional Practice

Lesson Map

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