Lesson 1 | Three-Dimensional Measurement and Application | 10th Grade Mathematics

Objective

Describe and use the formulas for area and circumference of circles to solve problems.

Common Core Standards

Core Standards

The core standards covered in this lesson

G.GMD.A.1 — Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

Geometric Measurement and Dimension

G.GMD.A.1 — Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

N.Q.A.3 — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

High School — Number and Quantity

N.Q.A.3 — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

A.SSE.A.1 — Interpret expressions that represent a quantity in terms of its context Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Seeing Structure in Expressions

A.SSE.A.1 — Interpret expressions that represent a quantity in terms of its context Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

The foundational standards covered in this lesson

A.CED.A.4

Creating Equations

A.CED.A.4 — Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.
7.G.B.4

Geometry

7.G.B.4 — Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Criteria for Success

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

Dissect an approximation of a circle and reorganize the pieces into a parallelogram to describe the derivation of the area and circumference formulas of a circle.
Use the knowledge that area is a square dimension and circumference is a linear dimension to describe how area will increase at a faster rate than circumference as the radius increases.
Use the formulas for area and circumference generally to make comparisons based on relative size.
Justify the use of 3.14 as an approximation for $$\pi$$ and note that the area or circumference when $$\pi$$ is calculated out is an approximation.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Students may need to review the formulas for area and circumference of circles before they can fully access this lesson. It is recommended to spend time outside of class building this specific skill.
Some of the Problem Set Guidance is from Grade 7 (7.G.4). However, these questions will help the students build procedural fluency with area and circumference of circles through their conceptual understanding.
EXTENSION: Try this activity, "Lawnmower Math" by Desmos, involving estimation with trial and error, leading students to determine the important variables to find the best solution.

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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

Below are four stages of an informal derivation of the area and circumference formulas of a circle:

Stage 1:

Stage 2:

Stage 3:

Stage 4:

Explain what these stages are deriving and what each stage explains.

Guiding Questions

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References

GeoGebra Area of a Circle Animation (Parallelogram)

Area of a Circle Animation (Parallelogram) by Mr Hardin is made available by GeoGebra under the CC BY-NC-SA 3.0 license. Copyright © International GeoGebra Institute, 2013. Accessed June 2, 2017, 11:37 a.m..

Modified by Fishtank Learning, Inc.

Problem 2

The area of a circle is approximately 56.55 square inches. What is the circumference?

Guiding Questions

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Problem 3

Below is a large circle with radius, $$r$$, and two smaller circles that each have a diameter the length of the radius of the larger circle, $$r$$.

Write a formula to find the combined area of the two smaller circles. Using this formula, what is the relationship between the combined area of the two smaller circles and the area of the larger circle?
Write a formula to find the combined circumference of the two smaller circles. Using this formula, what is the relationship between the combined circumference of the two smaller circles and the circumference of the larger circle?

Guiding Questions

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Target Task

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

a. Write a formula for finding the circumference of a circle in terms of the area.

b. Using your formula from part (a), find the circumference when the area of a circle is $$10\pi$$ square units.

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 can become procedurally fluent through calculation.
Include problems where students must determine which circle is bigger given different dimensions.

MARS Summative Assessment Tasks for High School Circle Pattern
Open Middle Area/Circumference of Circles
Illustrative Mathematics Area of a Circle
Open Middle What Measurements Make a Circle?
Open Middle Which Circle is Bigger? (Middle School)
Open Middle Circle Radius
Illustrative Mathematics Wedges of a Circle — (Although this is listed under 7.G.4, it also addresses G-GMD.1 with an informal proof for the area of a circle)
Smarter Balanced Assessment Consortium: Sample Items Item MAT.07.ER.2.0000G.F.501

Lesson 2

Lesson Map

Topic A: Area and Circumference of Circles

Describe and use the formulas for area and circumference of circles to solve problems.

A.SSE.A.1 G.GMD.A.1 N.Q.A.3

Calculate and justify composite area and circumference of circles.

N.Q.A.1 N.Q.A.2 N.Q.A.3

Solve multistep area and circumference of circles problems involving cost and other rates.

N.Q.A.1 N.Q.A.2 N.Q.A.3

Topic B: Three-Dimensional Concepts and General Volume

Describe the terms point, line, and plane. Define and classify polyhedrons, specifically prisms and pyramids.

G.CO.A.1

Define a general cylinder and general cone. Identify two-dimensional shapes that when revolved will form a cylinder.

G.CO.A.1 G.GMD.B.4

Use volume concepts and formulas to analyze and solve multistep problems with cylinders and prisms.

G.GMD.A.1 G.GMD.A.3

Define and calculate the volume of pyramids and cones. Describe the relationship between general cylinders and general cones with the same base area.

G.GMD.A.1 G.GMD.A.3

Use the Pythagorean Theorem to find missing measurements and calculate volume of pyramids, prisms, and compound shapes comprised of pyramids and prisms.

G.GMD.A.3 G.GMD.B.4 G.SRT.C.8

Topic C: Cavalieri's Principle, Spheres, and Composite Volume

Describe the cross-sections of prisms and cylinders and make conjectures about volume from the cross-sections.

G.GMD.A.3 G.GMD.B.4

Describe Cavalieri’s principle relating equal area cross-sections and volume, and how this relates to the formulas for volume. Derive the volume of a sphere using Cavalieri’s principle.

G.GMD.A.1 G.GMD.A.2

Identify cross-sections of pyramids and use the relationships between the cross-sections to determine the volume of truncated cones and pyramids.

G.GMD.A.2 G.GMD.A.3

Calculate the volume of a sphere and use this in the solution of problems.

G.GMD.A.1 G.GMD.A.2 G.GMD.A.3 N.Q.A.3

Calculate the volume of compound objects and those with subtracted solids. Determine how the volume will be affected by scaling one or more dimensions.

G.GMD.A.3 G.GMD.B.4 N.Q.A.3

Topic D: Surface Area, Scaling, and Modeling with Geometry

Use lateral surface area formulas to solve problems.

N.Q.A.2

Use the surface area and volume to solve application problems.

G.GMD.A.3 G.GMD.B.4 G.MG.A.1 G.MG.A.3

Solve multistep volume and surface area problems with rates and unit conversions.

G.GMD.A.3 N.Q.A.2 N.Q.A.3

Apply density concepts to surface area and volume problems.

G.GMD.A.3 G.MG.A.2 N.Q.A.2 N.Q.A.3

Apply constraints on volume, surface area, or cost to solve design problems with three-dimensional figures.

G.GMD.A.3 G.MG.A.2 G.MG.A.3 N.Q.A.2 N.Q.A.3

Three-Dimensional Measurement and Application

Objective

Common Core Standards

Core Standards

Geometric Measurement and Dimension

High School — Number and Quantity

Seeing Structure in Expressions

Foundational Standards

Creating Equations

Geometry

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Target Task

Additional Practice

Lesson Map

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