Curriculum  /  Math  /  10th Grade  /  Unit 6: Three-Dimensional Measurement and Application  /  Lesson 1

Three-Dimensional Measurement and Application

Lesson 1

Math

Unit 6

10th Grade

Lesson 1 of 18

Objective

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

Common Core Standards

Core Standards

  • 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.
  • 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

  • A.CED.A.4
  • 7.G.B.4

Criteria for Success

  1. 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. 
  2. 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. 
  3. Use the formulas for area and circumference generally to make comparisons based on relative size. 
  4. 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

  • 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

25-30 minutes

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, $$r$$.

  1. 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?
  2. 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

5-10 minutes

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.

Next

Calculate and justify composite area and circumference of circles.

Lesson 2
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Area and Circumference of Circles

Topic B: Three-Dimensional Concepts and General Volume

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

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

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