Download High School Math 3 Unit 5: Circles

High School Math 2 Unit 7: Circles
Approximate Time Frame: 2-3 weeks
Connections to Previous Learning:
Early in Math 2, students developed a precise definition of similarity in terms of similarity transformations and used this to determine if geometric objects were
similar. In Math 1 and earlier in Math 2, students used the logical structure behind conditional statements to prove various relationships between geometric
objects. The properties of congruent and similar triangles have been used in proofs. In Grade 8, students applied the Pythagorean Theorem to find distances
between points on the coordinate plane, to find lengths, and to analyze polygons. Earlier in Math 2, students revisited the Pythagorean Theorem, using it to
solve right triangles in applied problems.
Focus of this Unit:
Students continue to expand their understanding of geometry by exploring geometric relationships pertaining to circles. As was the case in Math 1 and earlier in
Math 2, attributes of circles observed at earlier grades will now be looked at more precisely through proof. Many of the geometric relationships and properties
proven in Math 1 and Math 2, particularly those related to triangle congruence and triangle similarity, will be applied in these proofs. In this unit, students will
begin their exploration of analytic geometry, where the use of coordinates helps to connect algebra and geometry and provides students with additional analysis
and problem solving techniques. Here, students will build on their understanding of distance in coordinate systems and draw on their growing command of
algebra to discover the connection between the graph of a circle in the coordinate plane and an algebraic equation. Along with physical models, dynamic
geometry environments will provide students with tools for investigating, experimenting with, conjecturing about, and modeling geometric phenomena related
to circles.
Connections to Subsequent Learning:
The correspondence between coordinates and geometric points used here to develop the algebraic equation of a circle will be used in Math 3 as a means for
proving geometric theorems through the use algebra in the coordinate plane. The definitions and properties of geometric objects considered in this unit will
reappear in future units that deal with other geometric concepts, with analytic geometric, and with modeling. Proof will continue to be a spiraled concept
throughout subsequent units and courses as it will be used later on in both geometric and non-geometric settings. The connection between the length of the arc
of a circle, the central angle of the circle that intercepts this arc, and the radius of the circle investigated in this unit plays a critical role in the development of
trigonometric functions in Math 3.
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Adapted from UbD framework
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High School Math 2 Unit 7: Circles
Desired Outcomes
Standard(s):
Understand and apply theorems about circles.
 G.C.1 Prove that all circles are similar.
 G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed
angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
 G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
Find arc lengths and areas of sectors of circles.
 G.C.5 Derive using similarity the fact that the length of the arch intercepted by an angle is proportional to the radius, and define the radian measure of the
angle as the constant or proportionality; derive the formula for the area of a sector.
Translate between the geometric description and the equation for a conic section.
 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.
Interpret the structure of expressions.
 A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares
that can be factored as (x2 – y2)(x2 + y2).
Write expressions in equivalent forms to solve problems.
 A.SSE.3 Choose and produce equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
a) Factor a quadratic expression to reveal the zeros of the function it defines.
b) Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:
 explicit vocabulary instruction with regard to measurement units, the components, and types of circles.
 tactile and virtual tools to create and interpret relationships among angles, radii and chords.
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Adapted from UbD framework
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High School Math 2 Unit 7: Circles
Understandings: Students will understand that…
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All circles are similar.
The geometric relationships that come from proving triangles congruent or from proving triangles similar may be used to prove relationships between
geometric objects.
Different relationships among inscribed angles, radii, and chords of a circle, and between the angles of a quadrilateral inscribed in a circle are provable using
previously proven relationships between geometric objects.
A circle drawn in the coordinate plane can be represented by an algebraic equation that is dependent upon the coordinates of the center of the circle and
the radius of the circle.
The relationship between the length of the arc of a circle, the central angle of the circle that intercepts this arc, and the radius of the circle.
Essential Questions:
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What are the different relationships among inscribed angles, radii, and chords of a circle, and of the angles of a quadrilateral inscribed in a circle?
What is the relationship between the length of the arc of a circle, the central angle of the circle that intercepts this arc, and the radius of the circle?
What is the area of a sector of a circle?
Given the coordinates of the center of the circle and the radius of that circle, what is the equation of the circle?
Given an equation for a circle drawn in the coordinate plane, what are the coordinates of the center of the circle and the radius of the circle?
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
*1. Make sense of problems and persevere in solving them. Students will recognize the hypothesis and conclusion in a proof statement and be able to
generate the requisite proof using the given information in the proof statement, along with known facts, definitions, postulates, and theorems.
*2. Reason abstractly and quantitatively. Students will be able to use figures and information pertaining to a specific geometric object as an aid in reasoning
about that geometric object in general.
*3. Construct viable arguments and critique the reasoning of others. Students will be able to create and present proofs, and be able to critiques the proofs and
deductive reasoning of others.
4. Model with mathematics.
*5. Use appropriate tools strategically. Students will be able to use physical models, drawings, and dynamic geometry environments to form conjectures about
geometric objects and to reason from information about the geometric object provided by these tools.
*6. Attend to precision. Students will recognize that incorrect initial attempts at definitions, conjectures, and theorems may be corrected through a process of
refinement.
*7. Look for and make use of structure. Students will be able to use the structure of geometric objects to gain insights into, make conjectures about, and
create proofs pertaining to these objects. Students will also use the structure of an expression to rewrite it in a form appropriate with its purpose.
8. Look for and express regularity in repeated reasoning. Students demonstrate repeated reasoning when analyzing quadratic equations and deriving the
equation of a circle given its radius and center.
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High School Math 2 Unit 7: Circles
Prerequisite Skills/Concepts:
Advanced Skills/Concepts:
Students should already be able to:
Some students may be ready to:
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Understand what constitutes a similarity transformation.
Informally understand what it means for two geometric figures to
be similar.
Perform specific reflections, rotations, translations, and dilations
to a given object.
Recall various geometric relationships proven in previous courses.
Plot points and sketch graphs in a coordinate plane that satisfy
specific conditions.
Provide statements for informal proof.
Identify and describe components of geometric objects such a
point, line, angle, triangle, parallelogram, circle, etc. and use them
to analyze geometric figures.
Recognize how counterexamples can be used to refute conjectures.
Make conjectures about and create proofs pertaining to cyclic quadrilaterals.
Construct a tangent line from a point outside a given circle to the circle.
Knowledge: Students will know…
Skills: Students will be able to …
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The definition of various geometric objects such as circle, angle,
triangle, parallel lines, perpendicular lines, parallelogram, etc.
The Pythagorean Theorem.
The formula for calculating the distance between two points in a
coordinate plane.
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Recognize if one geometric object can be transformed to another through a
sequence of rigid motions combined with a dilation.
Sketch a figure that represents specific given information.
Construct a conditional statement that represents a given conjecture.
Determine the area of a sector of a circle from the radius of the circle and the
measure of the central angle of the sector.
Use the method of completing the square to determine the coordinates of the
center of the circle and the radius of the circle, given the equation of the circle.
Use the structure of an expression to identify ways to rewrite it.
Factor a quadratic expression to reveal the zeros of the function it defines.
Complete the square in a quadratic expression to reveal the maximum or minimum
value of the function it defines.
Adapted from UbD framework
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High School Math 2 Unit 7: Circles
Academic Vocabulary:
Critical Terms:
Supplemental Terms:
Circle
Radius
Diameter
Arc
Chord
Tangent
Central angles
Inscribed angle
Circumscribed angle
Intercepted arc
Radian
Sector of a circle
Coordinate plane
Conditional statement
Hypothesis
Conclusion
Proof
Necessary Conditions
Sufficient Conditions
Postulate
Theorem
Length
Angle measure
Degree
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Adapted from UbD framework
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