YOUNGSTOWN CITY SCHOOLS
MATH: GEOMETRY
UNIT 5B: CIRCLES WITH AND WITHOUT COORDINATES – PART 2 (2 WEEKS) 2013-2014
Synopsis: Students will derive the equation of a circle from the definition and the Pythagorean theorem. They will
then complete the square to find the radius and center and perform proves involving coordinate geometry and circles.
STANDARDS
G.GPE.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.
G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined
by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at
the origin and containing the point (0, 2).
G.GPE. 4
Include simple proofs involving circles
MATH PRACTICES
1. Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning
LITERACY STANDARDS:
L.1
L.2
L.4
L.5
L.7
Learn to read mathematical text (including textbooks, articles, problems, problem explanations)
Communicate using correct mathematical terminology
Listen to and critique peer explanations of reasoning
Justify orally and in writing mathematical reasoning
Read appropriate text, providing explanation for mathematical concepts, reason or procedures
MOTIVATION
TEACHER NOTES
1. Preview expectations for end of Unit
2. Have students set both personal and academic goals for this Unit.
3. Discuss the location of cell phone towers. Cell phone companies locate towers so they service
several communities. Suppose there are three communities and they are modeled on a coordinate
system with City A at (1, 10), city B at (5, 6) and C at (-3, 2) with each unit representing 100 miles.
Determine the location of the tower equidistant from all three cities. Ask students where that location
would be and how they might be able to find it. Engage students by having them place the three
points on graph paper and trying to estimate that point. Explain to them by the time they have
completed this unit, they will be able to calculate that point.
TEACHING-LEARNING
6/30/2013
TEACHER NOTES
YCS Geometry Unit 5B Circles With and Without Coordinates Part II 2013-2014 1
TEACHING-LEARNING
Vocabulary:
Locus of points
Equation of a circle
Tangent
Perpendicular bisectors
Conic section
TEACHER NOTES
Circumscribed
Equidistant
1. Discuss the definition of a circle: locus of points equidistant from a given point (center). Use
this definition to write the equation of a circle. To write the equation of a conic section,
choose an arbitrary point on the curve (x, y) then use the definition to write the equation. A
circle is a cross-section of a cone sliced parallel to the base, hence a conic section. If a circle
has center (3, 4) and radius 5, have students draw a picture depicting this situation. They
should be plotting the point (3, 4), go out 5 units, draw the radius, label the point (x,y) at the
endpoint of the radius and draw the circle. Ask the students about the definition of a circle
and how it can be used with this drawing. They should respond with the distance from the
center (3,4) to (x, y) is 5 and the distance formula can be used to represent this.
or squaring both sides gives
,
which is the equation of the circle. Have students generalize for obtain the formula for a
circle: (x-a)2 + (y-b)2 = r2. Have students write the equation of a circle whose center is (-4,4)
and each axis is tangent to the circle. Reinforce by writing equations of various circles given
the radius and center. (G.GPE.1, MP.1, MP.2, MP.4, MP.5, MP.8, L.2)
2. Students should be able to find the radius and center of a circle given the equation in various
forms such as (x-3)2 + (y+7)2 = 10 and x2 – 2x + y2 + 6y = 15. With the latter students will
have to complete the square twice:
x2 – 2x +____ + y2 + 6y+ ____ = 15 + ____ + ____
x2 – 2x +__1__ + y2 + 6y+ _9___ = 15 + _1___ + __9__
x2 – 2x + 1 + y2 + 6y + 9 = 25, factor and simplify to get
(x-1)2 + (y + 3)2 = 25 so the center is (1, -3) and the radius is 5.
From this the student can then graph the circle. Reinforce with more problems using
completing he square. (G.GPE.1, MP.2, MP.4, MP.5, MP.6, MP.8, L.2)
3. Review circle circumscribed about triangles and how to construct the circle about the triangle:
have students research this and write down the steps for constructing a circle about the
triangle. Once they have the steps listed, discuss how to write the equation of this circle.
Going back to the motivation, find the perpendicular bisectors of two sides of the triangle.
Depending on the class, this can be done on graph paper with a compass, on geometer
sketchpad or geogebra or the student can write the equations of the two perpendicular
bisectors and find their intersection algebraically. This is the center of the circle. Now find
the distance from the center to a vertex of the triangle using the distance formula. This is the
radius. You can now write its equation. (G.GPE.1, MP.2, MP.4, MP.5, MP.6, MP.8, L.1, L.2,
L.7)
4. Engage students in coordinate geometry proofs involving circles. Have them explain their
reasoning to the class and have the class critique the proofs. (G.GPE.4, MP.1, MP.2, MP.3,
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YCS Geometry Unit 5B Circles With and Without Coordinates Part II 2013-2014 2
TEACHING-LEARNING
MP.4, MP.5, MP.8, L.1, L.2, L.4, L.5)
TEACHER NOTES
 Example 1: Prove or disprove the point
lies on the circle centered at the origin
and containing the point (0, 2). Have students draw a picture on graph paper showing
the circle and the point. Discuss as a class the steps needed for this proof. Then have
the students write up the proofs perhaps working in groups at first. Remind students
that if a point satisfies an equation of a certain figure then the point is on the figure.
 Example 2: Given a circle with equation (x+2)2 + (y-3)2 = 16. a. Prove that (-2, 1) is
inside the circle. b. Prove (1, -2) is outside the circle.
 Example 3: Write the equation of a line tangent to the circle whose equation is x 2+y2 =
100 at the point of tangency (6, 8). Explain why this is the tangent.
TRADITIONAL ASSESSMENT
TEACHER NOTES
1. Paper-pencil test with M-C questions
TEACHER CLASSROOM ASSESSMENT
TEACHER NOTES
1. Paper-pencil test with M-C questions
AUTHENTIC ASSESSMENT
TEACHER NOTES
1. Students evaluate goals they set at beginning of unit or on a weekly basis.
2. A garden is to be in the form of two intersecting circles and the walkway is to be the
common chord between the circles. You are working for the landscape company and your
boss has asked you to use graph paper and draw two intersecting circles and the common
chord which is not to be parallel to either axis. Write the equations of the circles and find the
points of intersection algebraically. Check your answer by examining your graph. Lastly, find
the equation of the common chord. This can now be entered into a drawing program for the
company to complete the landscaping work. (G.GPE.1)
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YCS Geometry Unit 5B Circles With and Without Coordinates Part II 2013-2014 3
RUBRIC
ELEMENTS OF THE
0
PROJECT
Draw two
Did not attempt
intersecting circles
and the common
chord on graph
paper
Write the equations
Did not attempt
of both circles
Find points of
intersection
algebraically
Find equation of the
common chord
6/30/2013
Did not attempt
Did not attempt
1
2
3
Did not use graph
paper
Drew the circles but
not the chord on
graph paper
Drew the circles and
chord not parallel to
either axis on graph
paper
Wrote equation of one
circle
Wrote equations of
both circles with
errors
Found points of
intersection with
errors
NA
Wrote equations of
both circles correctly
Found points of
intersection
graphically
Found equation of the
chord with errors
Found points of
intersection
algebraically correct
Found equation of
chord correctly
YCS Geometry Unit 5B Circles With and Without Coordinates Part II 2013-2014 4