Download Circles Unit Guide Geometry - circles unit guide 5 22 14_2

Circles
Unit Guide Geometry
Big Idea (Cluster):




Make geometric constructions
Understand and apply theorems about circles
Find arc lengths and areas of sectors of circles
Translate between the geometric description and the equation for a conic
section
 Use coordinates to prove simple geometric theorems algebraically
 Explain volume formulas and use them to solve problems
Edited 5/22/14
Renton School District
Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1)
Big Idea (Cluster):
Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between the
geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain volume
formulas and use them to solve problems
G.CO.13
Make geometric constructions. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
G.C.1
Understand and apply theorems about circles. Prove that all circles are similar.
G.C.2
Understand and apply theorems about circles. 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
Understand and apply theorems about circles. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
G.C.4
(+) Understand and apply theorems about circles. Construct a tangent line from a point outside a given circle to the circle.
G.C.5
Find arc lengths and areas of sectors of circles. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the
radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
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.GMD.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.
2
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Relevant Math Practices and Student Actions
MP1 Make sense of problems and persevere in solving them.
 Perseveres when solving problems
 Understands what the problem asks and the relationship among the
problem’s parts
 Looks for an entry point into solving the problem
 Makes a plan for solving a problem
 Uses multiple strategies and representations
 Explains why a solution is reasonable
MP 2 Reason abstractly and quantitatively.
 Interprets problems in context
 Uses representations to make meaning of problems
 Translates a problem from situation to equation
 Creates a situation for a given equation
 Explains connections between equation and situation
 Understands the meaning of quantities and units
 Flexibly uses properties of operations and place value
MP 3 Construct viable arguments and critique the reasoning of
others.
 Uses definitions and draws on prior mathematical knowledge when
constructing arguments
 Makes conjectures and evaluates their accuracy
 Justifies conclusions with mathematical evidence and responds to
arguments of others
 Communicates and defends mathematical reasoning using objects,
drawings, diagrams, actions, examples and counterexamples
 Asks clarifying and probing questions to improve argument
 Attends to, questions, and evaluates the reasoning and arguments of
others
MP 5 Use appropriate tools strategically.
 Selects tools strategically for visualizing, exploring, comparing,
predicting, and solving problems
 Uses technological tools and resources to pose problems, solve
problems and deepen understanding
 Justifies tool selection choices, demonstrating understanding of tools’
limitations and error ranges
MP 6 Attend to precision.
 Communicates mathematical thinking accurately both orally and in
writing
 Understands the meaning of mathematical symbols and vocabulary
and uses them appropriately
 Calculates accurately and efficiently
 Pursues a level of precision appropriate to the context of the problem
MP 7 Look for and make use of structure.
Looks for, identifies, develops and generalizes patterns and
relationships
 Makes connections to prior mathematical knowledge to solve new
problems
 Breaks down complex problems into manageable parts
 Views complicated quantities both as single objects or compilations
of several objects and uses operations to make sense of problems

MP 8 Look for and express regularity in repeated reasoning.
 Notices repeated calculations and looks for general methods and
shortcuts to solve a problem
 Identifies patterns to develop algorithm, formula, or calculation
 Evaluates reasonableness of intermediate and final results
3
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
SBAC Required Evidence (Claim 1)
All of the standards for this unit do not have an assigned SBAC assessment claim. The standards should be taught and assessed at the classroom
level.
Vocabulary
Mathematically proficient students communicate precisely by engaging in discussions about their reasoning using appropriate mathematical
language. Students should learn the following terms with increasing precision within the cluster. The bolded terms will be used on Smarter
Balanced assessment items.
Angle
Annulus
Arc
Area
Bisector
Central angle
Chord
Circumcenter
Circumference
Concentric
Congruent
Conjecture
Cyclic
Diameter
Inscribed angle
Length of an arc
Major arc
Minor arc
Perpendicular
Radius
Sector
Segment
Semicircle
Tangent
4
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1)
Big Idea (Cluster):
 Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between
the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain
volume formulas and use them to solve problems
G.CO.13
Make geometric constructions. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
See High School Flip Book page 157 for explanations and examples of this standard.
Learning Objectives

Students will be able to formally construct using a variety of tools and methods a square and a regular hexagon inscribed in a circle.
SBAC Claims 1-4 Item Specification Information
G.CO.13 does not have an assigned SBAC assessment claim. The
standard should be taught and assessed at the classroom level.
Connections to Curricular Materials
Discovering Geometry does NOT address this standard. Must
supplement.
Connections to Prior Learning
7.G.2 Draw (freehand, with ruler and protractor, and with technology)
geometric shapes with given conditions. Focus on constructing triangles
from three measures of angles or sides, noticing when the conditions
determine a unique triangle, more than one triangle, or no triangle
Additional Resources/Technology Resources
Mathematics Vision Project Secondary Mathematics I Module 6 unit
(lesson and tasks, OSPI reviewed OER)
Engage NY High School Geometry Module 1 – Congruence, Proof and
Constructions (OSPI reviewed OER)
Illustrative Mathematics Geometry tasks
Geometry Common Core teacher site (objectives, content learning, tasks)
Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives)
www.betterlesson.com
5
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1)
Big Idea (Cluster):
 Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between
the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain
volume formulas and use them to solve problems
G.C.1
Understand and apply theorems about circles. Prove that all circles are similar.
See High School Flip Book pages 171-172 for explanations and examples of this standard.
Learning Objectives



Students will be able to prove that all circles are similar by using the fact that the ratio of the diameter to circumference is the same for all circles.
Students will be able to prove that all circles are similar by showing that for a dilation centered at center of circle the pre-image and image have
equal central angle measures.
Students will be able to determine a single or sequence of similarity transformations between two circles.
SBAC Claims 1-4 Item Specification Information
G.C.1 does not have an assigned SBAC assessment claim. The
standard should be taught and assessed at the classroom level.
Connections to Prior Learning
8.G.4 Understand that a two-dimensional figure is similar to another if the
second can be obtained from the first by a sequence of rotations,
reflections, translations, and dilations; given two similar two-dimensional
figures, describe a sequence that exhibits the similarity between them.
8.G.5 Use informal arguments to establish facts about the angle sum and
exterior angle of triangles, about the angles created when parallel lines are
cut by a transversal, and the angle-angle criterion for similarity of
triangles.
Connections to Curricular Materials
Discovering Geometry does NOT address this standard. Must
supplement.
Additional Resources/Technology Resources
Mathematics Vision Project Secondary Mathematics II Module 7 unit
(lesson and tasks, OSPI reviewed OER)
Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives)
Illustrative Mathematics Geometry tasks
Geometry Common Core teacher site (objectives, content learning, tasks)
Missouri DOE Model Curriculum Unit Working with Circles
6
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1)
Big Idea (Cluster):
 Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between
the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain
volume formulas and use them to solve problems
G.C.2
Understand and apply theorems about circles. 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.
See High School Flip Book pages 173-174 for explanations and examples of this standard.
Learning Objectives



Students will be able to identify and describe relationships between central angles, inscribed angles, radii, chords, secants, tangents, and arcs.
Students will be able to find angles formed by intersections of chords, tangents and secants.
Students will be able to find segment lengths from intersections of chords, tangents and secants.
SBAC Claims 1-4 Item Specification Information
Connections to Prior Learning
G.C.2 does not have an assigned SBAC assessment claim. The
standard should be taught and assessed at the classroom level.
Connections to Curricular Materials
Discovering Geometry sections 6.1, 6.2, and 6.3 addresses this
standard.
Content missing that will need to be supplemented includes:
 Find angles formed by intersections of chords, tangents and
secants.
 Find segment lengths from intersections of chords, tangents and
secants.
Additional Resources/Technology Resources
Mathematics Vision Project Secondary Mathematics II Module 7 unit
(lesson and tasks, OSPI reviewed OER)
Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives)
Illustrative Mathematics Geometry tasks
Geometry Common Core teacher site (objectives, content learning, tasks)
www.betterlesson.com
Missouri DOE Model Curriculum Unit Working with Circles
7
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1)
Big Idea (Cluster):
 Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between
the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain
volume formulas and use them to solve problems
G.C.3
Understand and apply theorems about circles. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
See High School Flip Book pages 175-176 for explanations and examples of this standard.
Learning Objectives


Students will be able to use inscribed angle to prove opposite angles of a cyclical quadrilateral are supplementary.
Students will be able to apply the arc addition postulate to solve for missing arcs.
SBAC Claims 1-4 Item Specification Information
Connections to Prior Learning
G.C.3 does not have an assigned SBAC assessment claim. The
standard should be taught and assessed at the classroom level.
Connections to Curricular Materials
Additional Resources/Technology Resources
Discovering Geometry sections 6.3, 6.4, and 13.6 address this standard
loosely.
Mathematics Vision Project Secondary Mathematics II Module 7 unit
(lesson and tasks, OSPI reviewed OER)
Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives)
Illustrative Mathematics Geometry tasks
Geometry Common Core teacher site (objectives, content learning, tasks)
www.betterlesson.com
Missouri DOE Model Curriculum Unit Working with Circles
Content missing that will need to be supplemented includes:
 More problems for this standard are needed.
8
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1)
Big Idea (Cluster):
 Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between
the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain
volume formulas and use them to solve problems
G.C.4
(+) Understand and apply theorems about circles. Construct a tangent line from a point outside a given circle to the circle.
Learning Objectives

Students will be able to formally construct using a variety of tools and methods a tangent line from a point outside a given circle to the circle.
SBAC Claims 1-4 Item Specification Information
Connections to Prior Learning
G.C.4 does not have an assigned SBAC assessment claim. The
standard should be taught and assessed at the classroom level.
Connections to Curricular Materials
Discovering Geometry does NOT address this standard. Must
supplement.
Additional Resources/Technology Resources
Mathematics Vision Project Secondary Mathematics II Module 7 unit
(lesson and tasks, OSPI reviewed OER)
Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives)
Illustrative Mathematics Geometry tasks
Geometry Common Core teacher site (objectives, content learning, tasks)
www.betterlesson.com
9
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1)
Big Idea (Cluster):
 Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between
the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain
volume formulas and use them to solve problems
G.C.5
Find arc lengths and areas of sectors of circles. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the
radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
See High School Flip Book pages 177-178 for explanations and examples of this standard.
Learning Objectives
 Students will be able to derive using similarity the formula for arc length using the fact that the length of the arc intercepted by an angle is
proportional to the radius.
 Students will be able to define radian measure of the angle as the constant of proportionality.
 Students will be able to explain what a radian is and convert between degrees and radius.
 Students will be able to derive using similarity the formula for the area of a sector.
SBAC Claims 1-4 Item Specification Information
Connections to Prior Learning
G.C.5 does not have an assigned SBAC assessment claim. The
standard should be taught and assessed at the classroom level.
Connections to Curricular Materials
Discovering Geometry section 6.7 and 8.6 addresses this standard
loosely.
Content missing that will need to be supplemented includes:
 Deriving the arc length formula through similarity
 Define radian measure of the angle as the constant of
proportionality
 Explain what a radian is and convert between degrees and
radius.

Deriving the area of a sector formula through similarity
Additional Resources/Technology Resources
Mathematics Vision Project Secondary Mathematics II Module 7 unit
(lesson and tasks, OSPI reviewed OER)
Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives)
Illustrative Mathematics Geometry tasks
Geometry Common Core teacher site (objectives, content learning, tasks)
www.betterlesson.com
Missouri DOE Model Curriculum Unit Working with Circles
10
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1)
Big Idea (Cluster):
 Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between
the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain
volume formulas and use them to solve problems
G.GPE.1
Translate between the geometric description and the equation for a conic section. 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.
See High School Flip Book pages 179-180 for explanations and examples of this standard.
Learning Objectives


Students will be able to use the Pythagorean Theorem, the coordinates of a circle center, and the circle’s radius to derive the equation of a circle.
Students will be able to find the center and radius of a circle given an equation by completing the square.
SBAC Claims 1-4 Item Specification Information
Connections to Prior Learning
G.GPE.1 does not have an assigned SBAC assessment claim. The
standard should be taught and assessed at the classroom level.
Connections to Curricular Materials
Discovering Geometry section 9.5 addresses this standard.
Content missing that will need to be supplemented includes:
 Find the center and radius of a circle given an equation by
completing the square
Additional Resources/Technology Resources
Mathematics Vision Project Secondary Mathematics II Module 8 unit
(lesson and tasks, OSPI reviewed OER)
Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives)
Illustrative Mathematics Geometry tasks
Geometry Common Core teacher site (objectives, content learning, tasks)
www.betterlesson.com
Missouri DOE Model Curriculum Unit Working with Circles
11
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1)
Big Idea (Cluster):
 Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between
the geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain
volume formulas and use them to solve problems
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).
See High School Flip Book pages 182-183 for explanations and examples of this standard.
Learning Objectives

Students will be able to use distance formula or Pythagorean Theorem to decide if a point is inside the circle, outside the circle or on the circle.
SBAC Claims 1-4 Item Specification Information
Connections to Prior Learning
G.GPE.4 does not have an assigned SBAC assessment claim. The
standard should be taught and assessed at the classroom level.
Connections to Curricular Materials
Discovering Geometry does NOT address this standard. Must
supplement.
Additional Resources/Technology Resources
Mathematics Vision Project Secondary Mathematics II Module 7 unit
(lesson and tasks, OSPI reviewed OER)
Mathematics Vision Project Secondary Mathematics II Module 8 unit
(lesson and tasks, OSPI reviewed OER)
Geometry Common Core teacher site (objectives, content learning, tasks)
Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives)
Missouri DOE Model Curriculum Unit Working with Circles
Missouri DOE Model Curriculum Unit Coordinate Geometry
Geometry Common Core district site HCPSS Unit 3 (tasks, videos, objectives)
Illustrative Mathematics Geometry tasks
www.betterlesson.com
12
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
Circles Unit Guide (G.CO.13, G.C.1, G.C.2, G.C.3, G.C.4, G.C.5, G.GPE.1, G.GPE.4, G.GMD.1)
Big Idea (Cluster):
Make geometric constructions, understand and apply theorems about circles, find arc lengths and areas of sectors of circles, translate between the
geometric description and the equation for a conic section, use coordinates to prove simple geometric theorems algebraically, and explain volume
formulas and use them to solve problems
Standard G.GMD.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.
See High School Flip Book pages 188-190 for explanations and examples of this standard.
Learning Objectives

Students will be able to explain the formula for circumference and area of a circle by determining the meaning of each term or factor of the
formulas.
SBAC Claims 1-4 Item Specification Information
G.GMD.1 does not have an assigned SBAC assessment claim. The
standard should be taught and assessed at the classroom level.
Connections to Curricular Materials
Discovering Geometry section 8.5, 6.5, and 6.6 addresses this standard
loosely.
Content missing that will need to be supplemented includes:
 The formal arguments for the formulas for the circumference of
a circle and area of a circle.
Connections to Prior Learning
7.G.4 Know the formulas for the area and circumference of a circle and use
them to solve problems; give an informal derivation of the relationship
between the circumference and area of a circle.
Additional Resources/Technology Resources
Computing Pi applet
Graphing circumference and area of circle ratios (NCTM)
Estimating Circumference of a Circle
Area of circle informal argument-Rabbi Abraham
Area of circle informal argument – Leonardo Da Vinci
Hard copies of these lessons can be found at http://staff.rentonschools.us/renton/secondary-math. The following resources was used to create this
curriculum guide: High School Common Core State Standards Flip Book compiled by Trish Hart. A further unpacking of the Geometry Common
Core State Standards by Hung-Hsi Wu (2013) titled “Teaching Geometry in Grade 8 and High School According to the Common Core Standards”
can be found at http://math.berkeley.edu/~wu/CCSS-Geometry.pdf.
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.
13
14
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment.