Curriculum  /  Math  /  10th Grade  /  Unit 7: Circles  /  Lesson 3

Circles

Lesson 3

Math

Unit 7

10th Grade

Lesson 3 of 14

Objective

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

Common Core Standards

Core Standards

  • G.CO.A.5 — Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
  • G.GPE.A.1 — Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Foundational Standards

  • A.REI.B.4.A
  • A.SSE.B.3.B

Criteria for Success

  1. Write an equation for a circle in standard from, $$(x-h)^2+(y-k)^2=r^2$$, from the general form, $$x^2+y^2+Dx+Ey+F=0$$, by completing the square.
  2. Given the equation of a circle in the general form, identify the center and radius.
  3. Given the equations of two circles, describe the translation that maps the centers onto each other. 
  4. Given the equations of two circles, describe the dilation that maps the radii onto each other.
  5. Compare the features of two circles. 

Tips for Teachers

  • Students will need to review completing the square with quadratic equations before they can fully access this lesson. It is recommended to spend time outside of class building this specific skill.
  • This lesson will focus on circle equations with even values for the coefficients $$D$$ and $$E$$ to ensure that students understand the concept. 
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Anchor Problems

25-30 minutes

Problem 1

Both equations below model the same circle:

Equation A: $$x^2+y^2-2x+6y=15$$
Equation B: $$(x-1)^2+(y+3)^2=25$$

Equation A is the general form of the circle while Equation B is the standard form of the circle. Describe how you could transform Equation A to get Equation B.

Guiding Questions

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Problem 2

Write the following equation in standard form to identify the center and the radius of the circle. 

$$x^2+y^2+10x-8y-6=2$$

Guiding Questions

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References

EngageNY Mathematics Geometry > Module 5 > Topic D > Lesson 18 — Exit Ticket, Question #1

Geometry > Module 5 > Topic D > Lesson 18 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.

Target Task

5-10 minutes

Describe the translations and dilations you could use to map circle $$A$$ to circle $$B$$.

Circle $$A$$: $$(x+4)^2+(y+3)^2=16$$
Circle $$B$$: $$ x^2+y^2-4y-2x+2=1$$

 

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 one or two points on a coordinate plane and asked to write the equation of a circle where these points are insides the circle but not on the circle. 
  • Include problems where students are asked if a circle passes through particular points where they must explain their reasoning algebraically.
  • Include problems where students convert an equation from standard form to general form and vice versa. 
  • Include problems where students describe why a circle is not a function. 
  • Include problems where students must identify equations that will not form a circle. 
  • EXTENSION: Include problems that have a system of a linear function and a circle. 

Next

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

Lesson 4
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Lesson Map

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

Topic A: Equations of Circles

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

Topic C: Arc Length, Radians, and Sector Area

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