Download Theorems, Proofs - Chariho Regional School District

Geometry, Quarter 3, Unit 3.1
Polygons: Theorems, Proofs, and Applications
On and Off the Coordinate Plane
Overview
Number of instructional days:
20
(1 day = 45 minutes)
Content to be learned
Mathematical practices to be integrated
•
Prove theorems about parallelograms.
Theorems include the following:
Make sense of problems and persevere in solving
them.
o
Opposite sides and angles are congruent.
•
o
The diagonals of a parallelogram bisect
each other.
o
Rectangles are parallelograms with
congruent diagonals.
Analyze givens, constraints, relationships, and
goals when proving theorems or solving
problems with polygons. Make conjectures
about the form and meaning of the solution and
plan a solution pathway rather than simply
jumping into a solution attempt. Monitor and
evaluate self-progress and change course if
necessary.
•
Find correspondences between equations,
verbal descriptions, and graphs or draw
diagrams of important features and
relationships and search for regularity or trends
when solving problems involving polygons.
•
Check answers to problems using a different
method, and continually ask, “Does this make
sense?”
•
Understand the approaches of others to
solving complex problems and identify
correspondences between different approaches.
•
Use coordinates to prove simple geometric
theorems algebraically. Theorems include the
following:
o
Find the coordinates of midpoint and points
along directed line segments given a ratio.
o
Use the slope criteria for parallel and
perpendicular lines to solve and prove
polygon-related problems and properties.
o
•
•
Prove or disprove that a figure defined by
four given points in the coordinate plane is
a rectangle/parallelogram/rhombus/
square/trapezoid.
Use coordinates to compute perimeters of
polygons and areas of triangles and rectangles
by employing the Pythagorean Theorem or
distance formula (from a modeling
perspective).
Reason abstractly and quantitatively.
•
Make sense of quantities and their
relationships in problem situations involving
polygons.
Describe the rotations and reflections that carry
polygons such as rectangles, parallelograms,
trapezoids, or regular polygons onto
themselves.
•
Create a coherent representation of the
problem at hand by drawing a diagram from
the verbal situation; consider the units involved
when calculating the distance; attend to the
meaning of quantities, not just how to compute
them; and know and flexibly use different
properties of operations and objects.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 21 Geometry, Quarter 3, Unit 3.1
Polygons: Theorems, Proofs, and Applications
On and Off the Coordinate Plane (20 days)
Construct viable arguments and critique the
reasoning of others.
•
Understand and use stated assumptions,
definitions, and previously established results
in constructing arguments involving problems.
•
Make conjectures and build a logical
progression of statements to explore the truth
of the conjectures when proving theorems
about polygons.
•
Analyze situations by breaking them into
cases; recognize and use counterexamples.
•
Justify conclusions, communicate them to
others, and respond to the arguments of others.
•
Compare the effectiveness of two plausible
arguments, and distinguish correct logic or
reasoning from that which is flawed. If there is
a flaw in an argument, explain what it is.
•
Listen to or read the arguments of others,
decide whether they make sense, and ask
useful questions to clarify or improve the
arguments.
Look for and make use of structure.
•
Look closely at polygons to discern a pattern
or structure.
•
Classify quadrilaterals based on their
properties and identify similarities and
differences of special quadrilaterals.
•
Recognize the significance of an existing line
in a geometric figure and use the strategy of
drawing an auxiliary line for solving problems.
Look for and express regularity in repeated
reasoning.
•
Maintain oversight of the process of proving
polygon properties algebraically, while
attending to the details. Evaluate the
reasonableness of intermediate results.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 22 Geometry, Quarter 3, Unit 3.1
Polygons: Theorems, Proofs, and Applications
On and Off the Coordinate Plane (20 days)
Essential questions
•
How do properties of polygons help establish
methods to find unknown areas?
•
How can you use the coordinate plane to prove
general relationships in polygons?
•
How can the Pythagorean Theorem help the
understanding of the distance formula?
•
How can you algebraically prove that a
polygon is a specific quadrilateral? What
information is sufficient to justify that claim?
•
How can you classify quadrilaterals? What
properties do they share, and how may they
differ? (not on the coordinate plane)
Written Curriculum
Common Core State Standards for Mathematical Content
Congruence
G-CO
Prove geometric theorems [Focus on validity of underlying reasoning while using variety of ways of
writing proofs]
G-CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent,
opposite angles are congruent, the diagonals of a parallelogram bisect each other, and
conversely, rectangles are parallelograms with congruent diagonals.
Expressing Geometric Properties with Equations
G-GPE
Use coordinates to prove simple geometric theorems algebraically [Include distance formula; relate to
Pythagorean theorem]
G-GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric
problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes
through a given point).
G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment
in a given ratio.
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.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g.,
★
using the distance formula.
Congruence
G-CO
Experiment with transformations in the plane
G-CO.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and
reflections that carry it onto itself.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 23 Geometry, Quarter 3, Unit 3.1
Polygons: Theorems, Proofs, and Applications
On and Off the Coordinate Plane (20 days)
Common Core Standards for Mathematical Practice
1
Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and
looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They
make conjectures about the form and meaning of the solution and plan a solution pathway rather than
simply jumping into a solution attempt. They consider analogous problems, and try special cases and
simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate
their progress and change course if necessary. Older students might, depending on the context of the
problem, transform algebraic expressions or change the viewing window on their graphing calculator to
get the information they need. Mathematically proficient students can explain correspondences between
equations, verbal descriptions, tables, and graphs or draw diagrams of important features and
relationships, graph data, and search for regularity or trends. Younger students might rely on using
concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students
check their answers to problems using a different method, and they continually ask themselves, “Does
this make sense?” They can understand the approaches of others to solving complex problems and
identify correspondences between different approaches.
2
Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations.
They bring two complementary abilities to bear on problems involving quantitative relationships: the
ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own, without necessarily attending to their referents—
and the ability to contextualize, to pause as needed during the manipulation process in order to probe into
the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent
representation of the problem at hand; considering the units involved; attending to the meaning of
quantities, not just how to compute them; and knowing and flexibly using different properties of
operations and objects.
3
Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously
established results in constructing arguments. They make conjectures and build a logical progression of
statements to explore the truth of their conjectures. They are able to analyze situations by breaking them
into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them
to others, and respond to the arguments of others. They reason inductively about data, making plausible
arguments that take into account the context from which the data arose. Mathematically proficient
students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or
reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.
Elementary students can construct arguments using concrete referents such as objects, drawings,
diagrams, and actions. Such arguments can make sense and be correct, even though they are not
generalized or made formal until later grades. Later, students learn to determine domains to which an
argument applies. Students at all grades can listen or read the arguments of others, decide whether they
make sense, and ask useful questions to clarify or improve the arguments.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 24 Geometry, Quarter 3, Unit 3.1
7
Polygons: Theorems, Proofs, and Applications
On and Off the Coordinate Plane (20 days)
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they may
sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8
equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In
the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the
significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line
for solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed of several
objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that
to realize that its value cannot be more than 5 for any real numbers x and y.
8
Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods
and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating
the same calculations over and over again, and conclude they have a repeating decimal. By paying attention
to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope
3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way
terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them
to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically
proficient students maintain oversight of the process, while attending to the details. They continually
evaluate the reasonableness of their intermediate results.
Clarifying the Standards
Prior Learning
In grade 2, students recognized and identified triangles, quadrilaterals, pentagons, hexagons, and circles.
(2.G.1) In grade 3, they recognized rhombi, rectangles, and squares as examples of quadrilaterals. (3.G.1)
Students also solved real-world problems involving perimeters of polygons. (3.MD.8)
In grade 4, students drew and identified points, lines, line segments, rays, angles (right, acute, obtuse),
and perpendicular and parallel lines in two-dimensional figures. They classified two-dimensional figures
based on the presence or absence of parallel and perpendicular lines. (4.G.1 and 2) In grade 5, students
classified two-dimensional figures into categories based on their properties. (5.G.3 and 4) In addition,
they graphed and identified points on the coordinate plane to solve real-world and mathematical
problems. (5.G.1 and 2)
In grade 6, students found the area of triangles, quadrilaterals, and polygons. (6.G.1) They drew polygons
on the coordinate plane, given the vertices, and used the coordinates to find the length of a side. (6.G.3)
In grade 7, students used the area of two-dimensional figures composed of triangles, quadrilaterals, and
polygons to solve real-world problems. (7.G.6) They also identified the unit rate from graphs, tables,
equations, diagrams, and verbal descriptions. (7.RP.2)
In grade 8, students interpreted the unit rate as the slope of a graph. (8.EE. 5) In Unit 1.3 of Algebra 1,
students used the slope formula to determine whether two lines are parallel or perpendicular. In Unit 1.2
of this course, students experimented with transformations in the plane. (G.CO.2-5)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 25 Geometry, Quarter 3, Unit 3.1
Polygons: Theorems, Proofs, and Applications
On and Off the Coordinate Plane (20 days)
Current Learning
Students prove theorems about parallelograms. They find the coordinates of midpoint and points along
directed line segments given a ratio. (See Writing Team Notes for clarification.) Students utilize the slope
criteria for parallel and perpendicular lines to solve and prove geometric problems. They use coordinates
to prove simple geometric theorems algebraically. Students use the coordinates to compute perimeters of
polygons and areas of triangles and rectangles (Pythagorean Theorem or the distance formula). They
describe the rotations and reflections that carry a polygon onto itself. Students apply geometric methods
to solve design problems.
Future Learning
Students will build on their understanding of distance in coordinate systems and draw on their command
of algebra to connect equations and graphs of conic sections. They will apply properties of quadrilaterals
when inscribed in a circle. Students will also construct equilateral triangles, squares, and regular hexagons
inscribed in a circle. (Unit 4.1) They will build upon and utilize the properties of special quadrilaterals in
professions such as construction, carpentry, civil/structural engineering, and architecture.
Additional Findings
According to Principles and Standards for School Mathematics, “In grades 9–12 all students should
analyze properties and determine attributes of two-dimensional objects. They should explore relationships
among classes of two-dimensional geometric objects, make and test conjectures about them, and solve
problems involving them. They should investigate conjectures and solve problems involving twodimensional objects represented with Cartesian coordinates.” (p. 308)
According to A Research Companion to Principles and Standards for School Mathematics, although
students in grade 8 “believe that parallel lines should not intersect and should be equidistant, they also
believe that parallel segments must be aligned and that curves might be parallel.” (p. 164) Research also
states, “As students develop notions of two-dimensional space, they must learn to construct, select, and
use increasingly coordinated reference systems as frameworks for spatial organization, the foundation of
spatial-geometric thinking.” (p. 166)
Writing Team Notes
To see one interpretation of partitioning a segment, refer to page 223 in On Core Mathematics Geometry.
Example: Find the coordinates of the Point P that lies along the directed line segment from A(3, 4) to
B(6, 10) and partitions the segment in ratio 3:2.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 26 Geometry, Quarter 3, Unit 3.2
2-D and 3-D Measurements and Modeling
Overview
Number of instructional days:
10
(1 day = 45 minutes)
Content to be learned
Mathematical practices to be integrated
•
Model with mathematics.
•
•
Describe objects using geometric shapes, their
measures, and their properties (e.g., tree trunk
as a cylinder).
Make informal arguments about formulas for
(1) circumference and area of a circle and
(2) volume of cylinder, pyramid, and cone.
(Use dissection arguments, Cavalieri’s
Principle, and informal limit arguments.)
Use volume formulas for cylinders, pyramids,
cones, and spheres to solve problems.
•
Apply concepts of density, based on area and
volume in modeling situations.
•
Apply geometric methods to solve design
problems such as (1) designing an object or
structure to satisfy physical constraints or
minimize cost or (2) working with typographic
grid systems based on ratios.
•
•
Apply the area and volume formulas to
everyday situations.
•
Model situations to apply concepts of
population density based on area (persons per
square mile).
•
Model situations to apply concepts of density
based on volume (BTUs per cubic foot).
•
Understand that when developing
mathematical models, there is often a trade-off
between a model that is more precise and one
that is easier to work with (e.g., choosing a
trapezoid versus a more complex polygon
when calculating area).
Attend to precision.
Explain volume formulas and give an informal
argument using Cavalieri’s Principle for
formulas for volumes of spheres and other solid
figures.
•
Communicate convincing arguments and
accurate responses in multiple formats,
including technological, written, and oral
forms.
•
Express numerical answers with a specific
degree of precision appropriate for the problem
context.
•
Specify units of measure when calculating
circumference, area, and volume.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 27 Geometry, Quarter 3, Unit 3.2
2-D and 3-D Measurements and Modeling (10 days)
Essential questions
•
How can you use geometric shapes, their
measures, and their properties to describe and
model objects in the world around you?
•
How can you use Cavalieri’s Principle to give
an informal argument for the volume of
different solids?
•
How can you use an informal limit argument to
determine the circumference of a circle?
•
•
How can you use an informal dissection
argument to determine the area of a circle?
What are some examples of how you can use
geometry to design a structure, given specific
constraints?
•
•
How have you utilized the geometry concepts
learned in this unit for solving and modeling
real-world problems?
Given a map and the population of Rhode
Island, how you would estimate the number of
persons per square mile?
Written Curriculum
Common Core State Standards for Mathematical Content
Modeling with Geometry
G-MG
Apply geometric concepts in modeling situations
G-MG.1
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a
★
tree trunk or a human torso as a cylinder).
Geometric Measurement and Dimension
G-GMD
Explain volume formulas and use them to solve problems
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. Use dissection arguments, Cavalieri’s principle, and
informal limit arguments.
★
G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Modeling with Geometry
G-MG
Apply geometric concepts in modeling situations
G-MG.2
Apply concepts of density based on area and volume in modeling situations (e.g., persons per
★
square mile, BTUs per cubic foot).
G-MG.3
Apply geometric methods to solve design problems (e.g., designing an object or structure to
satisfy physical constraints or minimize cost; working with typographic grid systems based on
★
ratios).
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 28 Geometry, Quarter 3, Unit 3.2
2-D and 3-D Measurements and Modeling (10 days)
Geometric Measurement and Dimension
G-GMD
Explain volume formulas and use them to solve problems
G-GMD.2 (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a
sphere and other solid figures.
Common Core Standards for Mathematical Practice
4
Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in
everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition
equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a
school event or analyze a problem in the community. By high school, a student might use geometry to
solve a design problem or use a function to describe how one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are comfortable making assumptions
and approximations to simplify a complicated situation, realizing that these may need revision later. They
are able to identify important quantities in a practical situation and map their relationships using such
tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their mathematical results in the context of
the situation and reflect on whether the results make sense, possibly improving the model if it has not
served its purpose.
6
Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the symbols
they choose, including using the equal sign consistently and appropriately. They are careful about
specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.
They calculate accurately and efficiently, express numerical answers with a degree of precision
appropriate for the problem context. In the elementary grades, students give carefully formulated
explanations to each other. By the time they reach high school they have learned to examine claims and
make explicit use of definitions.
Clarifying the Standards
Prior Learning
In kindergarten, students not only identified and described 2-D and 3-D shapes, they also analyzed,
compared, created, and composed them. (K.G.3–6) In grades 1–3, students reasoned with shapes and their
attributes and partitioned shapes into equal parts. (1.G.1–3, 2.G.1–3, 3.G.1–2) In grade 3, students solved
problems involving measurement and estimation of liquid volumes and masses of objects. They also
distinguished between linear and area measures and the concepts of area in relation to multiplication and
addition. (3.MD.2, 5–8)
In grade 4, students applied area formulas for rectangles in real-world and mathematical problems.
(4.MD.3). In grade 5, students studied basic concepts of volume measurement. They measured volume
using unit cubes and applied volume formulas for rectangular prisms. (5.MD.3–5) In grade 6, students
solved real-world and mathematical problems involving area, surface area, and volume for right
rectangular prisms. (6.G.1–2)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 29 Geometry, Quarter 3, Unit 3.2
2-D and 3-D Measurements and Modeling (10 days)
In grade 7, students solved problems involving circles using area and circumference formulas. Problem
solving was expanded to include area, volume, and surface area of 2-D and 3-D objects composed of
triangles, quadrilaterals, polygons, cubes, and right prisms. (7.G.4, 6) In grade 8, students solved realworld and mathematical problems involving volume of cylinders, cones, and spheres. (8.G.9) In Unit 3.1
of this course, students worked with various polygons and used coordinates to compute perimeters and
areas of triangles and rectangles using the distance formula. (G.GPE.7)
Current Learning
Students use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a
tree trunk or a human torso as a cylinder). Students explain and use volume formulas to solve problems.
They give an informal argument for the formulas for the circumference of a circle, area of a circle, and
volume of a cylinder, pyramid, and cone. Students use dissection arguments, Cavalieri’s Principle, and
informal limit arguments. They use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems. Students apply concepts of density based on area and volume in modeling situations (e.g.,
persons per square mile, BTUs per cubic foot). They apply geometric methods to solve design problems.
Students give an informal argument using Cavalieri’s Principle for the formulas for the volume of a
sphere and other solid figures.
Future Learning
Students will use the concepts of circumference and area of circle to find arc length and area of a sector.
(Unit 3.3) They will identify the shapes of 2-D objects, cross sections of 3-D objects, and 3-D objects
generated by rotations of 2-D objects. (Unit 4.1) Students will use their understanding of 2-D and 3-D
models in calculus to solve implicit differentiation and related rate problems. They will use their
understanding of 2-D and 3-D models in future college courses related to applied calculus, civil
engineering, graphic design, and architecture.
Additional Findings
Benchmarks for Science Literacy (American Association for the Advancement of Science) states, “In the
upper grades considerable emphasis should be placed on the mathematical modeling, because it optimizes
the nature and power of models and provides a context for integrating knowledge from many different
domains.” (p. 270)
Writing Team Notes
Students should be able to answer questions like the following to have mastered G.MG.2:
A spherical gas tank has diameter 1.5 ft. When filled with propane, it provides 4506.2 BTU. How
many BTUs does one cubic foot of propane yield?
Students should be able to answer questions like the following to have mastered G.MG.3:
You want to build a box from a piece of wood that is 6 feet by 6 feet. You must use 6 pieces (for the
top, bottom, and sides of the box) and you must make cuts that are parallel and perpendicular to the
edges of the wood. How can you describe three possible designs and what do you think is the
maximum possible volume for the box?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 30 Geometry, Quarter 3, Unit 3.3
Relationships in Circles: Angles, Arcs,
Segments, and Similarity
Overview
Number of instructional days:
10
(1 day = 45 minutes)
Content to be learned
Mathematical practices to be integrated
•
Construct viable arguments.
Identify and describe relationships among
inscribed angles, radii, and chords. This
includes the following:
o
Relationships between central, inscribed,
and circumscribed angles.
o
Inscribed angles on a diameter are right
angles.
o
The radius of a circle is perpendicular to
the tangent where the radius intersects the
circle.
•
Prove that all circles are similar.
•
Find arc lengths and areas of sectors of circles.
•
Introduce radians as a unit of measure and
define it as the constant of proportionality.
•
Derive, using similarity, the fact that the
length of the arc intercepted by an angle is
proportional to the radius.
•
Derive the formula for the area of a sector.
•
Create logical arguments regarding the
similarity of all circles and communicate them
to others.
•
Make and justify conclusions about the
relationships among inscribed angles, radii,
and chords.
•
Recognize flaws in arguments.
Model with mathematics.
•
Solve real-world problems using arc lengths
and areas of sectors.
Attend to precision.
•
Solve problems involving circles with
inscribed angles, radii, and chords accurately
and efficiently.
•
Recognize when it is appropriate to
approximate answers or leave in terms of pi.
Essential questions
•
Why are all circles similar?
•
•
What kinds of angles are formed within
circles? How do they relate to the measure of
their intercepted arcs?
Under what conditions is a line tangent to a
circle? How can this be proven?
•
How can calculating the area of a sector be
useful when estimating area in a real situation?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 31 Geometry, Quarter 3, Unit 3.3
Relationships in Circles: Angles, Arcs, Segments, and Similarity (10 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Circles
G-C
Understand and apply theorems about circles
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.1
Prove that all circles are similar.
Find arc lengths and areas of sectors of circles [Radian introduced only as unit of measure]
G-C.5
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.
Common Core Standards for Mathematical Practice
3
Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously
established results in constructing arguments. They make conjectures and build a logical progression of
statements to explore the truth of their conjectures. They are able to analyze situations by breaking them
into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them
to others, and respond to the arguments of others. They reason inductively about data, making plausible
arguments that take into account the context from which the data arose. Mathematically proficient
students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or
reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.
Elementary students can construct arguments using concrete referents such as objects, drawings,
diagrams, and actions. Such arguments can make sense and be correct, even though they are not
generalized or made formal until later grades. Later, students learn to determine domains to which an
argument applies. Students at all grades can listen or read the arguments of others, decide whether they
make sense, and ask useful questions to clarify or improve the arguments.
4
Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in
everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition
equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a
school event or analyze a problem in the community. By high school, a student might use geometry to
solve a design problem or use a function to describe how one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are comfortable making assumptions
and approximations to simplify a complicated situation, realizing that these may need revision later. They
are able to identify important quantities in a practical situation and map their relationships using such
tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their mathematical results in the context of
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 32 Geometry, Quarter 3, Unit 3.3
Relationships in Circles: Angles, Arcs, Segments, and Similarity (10 days)
the situation and reflect on whether the results make sense, possibly improving the model if it has not
served its purpose.
6
Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the symbols
they choose, including using the equal sign consistently and appropriately. They are careful about
specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.
They calculate accurately and efficiently, express numerical answers with a degree of precision
appropriate for the problem context. In the elementary grades, students give carefully formulated
explanations to each other. By the time they reach high school they have learned to examine claims and
make explicit use of definitions.
Clarifying the Standards
Prior Learning
In kindergarten, students identified and described circles regardless of their orientation and overall size.
(K.G.1–2) In grades 1 and 2, students partitioned circles into equal shares such as halves, thirds, and
fourths. (1.G.3, 2.G.3) In grade 3, students recognized area as additive and found the areas of
quadrilaterals. (3.MD.7)
In grade 4, students recognized, measured, and drew points, lines, line segments, rays, and angles.
(4.MD.5) In grade 7, students used the formulas for the area and circumference of a circle to solve
problems. They used facts about supplementary, complementary, vertical, and adjacent angles to solve
multistep problems. (7.G.4–5)
In Unit 2.2 of this course, students used the definition of similarity transformations to decide if two
figures are similar. (G.SRT.2) In Unit 2.3, they used trigonometric ratios and the Pythagorean Theorem to
solve right triangles in applied problems. (G.SRT.8)
Current Learning
Students understand and apply theorems about circles. They identify and describe relationships among
inscribed angles, radii, and chords. Students know the relationship between central, inscribed, and
circumscribed angles. They recognize that inscribed angles on a diameter are right angles. Students learn
that the radius of a circle is perpendicular to the tangent where the radius intersects the circle. They prove
that all circles are similar. Students find arc lengths and areas of sectors of circles. They derive, using
similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius. Students
define radians as a unit of measure and as the constant of proportionality. Finally, they derive the formula
for the area of a sector.
Future Learning
Students will continue to learn about circles in Unit 4.1. They will use their understanding of arc length in
both Algebra II and Precalculus when studying radian measure and the unit circle.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 33 Geometry, Quarter 3, Unit 3.3
Relationships in Circles: Angles, Arcs, Segments, and Similarity (10 days)
Additional Findings
According to Principles and Standards for School Mathematics, “Working to understand orientation and
drawings in a 3-D rectangular coordinate system will afford opportunities for student to think and reason
spatially.” In addition, “Schooling should provide rich mathematical settings in which they [students] can
hone their visualization skills. Visualizing a building represented in architectural plans, the shape of the
cross section formed when a plane slices through a cone (conic section) or another solid object, or the
shape of the solid swept out when a plane figure is rotated about an axis become easier when students
work with physical models, drawings, and software capable of manipulating 3-D representations.”
(pp. 315 and 316)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 34