Three-Dimensional Measurement and Application

Lesson 2

Math

Unit 6

10th Grade

Lesson 2 of 18

Objective


Calculate and justify composite area and circumference of circles. 

Common Core Standards


Core Standards

  • N.Q.A.1 — Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
  • N.Q.A.2 — Define appropriate quantities for the purpose of descriptive modeling.
  • N.Q.A.3 — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Foundational Standards

  • 7.G.B.4

Criteria for Success


  1. Identify the necessary formulas and measurements required to solve composite area or perimeter problems. 
  2. Write formulas modified from the base area and/or circumference formulas to model the particular situation in the problem. 
  3. Justify components of the formula and describe values of the components in the context of the problem. 
  4. Justify the level of precision used for the context of the problem and identify a reasonable range of solutions. 
  5. Use appropriate units and levels of precision of measurement in the solution of composite area and circumference problems. Describe why the answer is reasonable. 

Tips for Teachers


  • This is the first lesson out of two that involve solving application problems using the area and circumference of circles. This lesson focuses on composite area and circumference of circles, while Lesson 3 focuses on problems involving cost and other rates. 
  • 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.
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Anchor Problems


Problem 1

In rectangle $$ABCD$$, the length of $$\overline{AB}$$ is 12 inches. The two circles are congruent and touch each other in exactly one spot. Additionally, each circle touches the rectangle in exactly three points. What is the total area of the shaded regions?

Guiding Questions

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References

Cardinal Educational Consulting SAT Math: Shaded Region Problems

Problem 2

Estimate how many rotations the larger tire will make before hitting or missing the target.

The image below shows the larger tire and the target.

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

References

Divisible By 3 Rolling Tires

Rolling Tires is made available by Andrew Stadel on Divisible by 3 under the CC BY-NC-SA 3.0 license. Accessed Sept. 20, 2018, 2:01 p.m..

Target Task


The figure below is composed of eight circles, seven small circles and one large circle containing them all. Neighboring circles only share one point, and two regions between a set of smaller circles have been shaded. Each small circle has a radius of 5 centimeters.

 

  1. Calculate the area of the large circle.
  2. Calculate the area of the shaded part of the figure.
  3. Calculate the ratio of the circumference of the sum of the small circles to the circumference of the larger circle. 

References

Illustrative Mathematics Eight Circles

Eight Circles, accessed on June 2, 2017, 11:46 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

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.

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Lesson 1

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Lesson 3

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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