Download STANDARDS FOR MATHEMATICS High School Geometry

STANDARDS FOR
MATHEMATICS
High School Geometry
1
High School Overview
Conceptual Categories and Domains
Statistics and Probability
 Interpreting Categorical and Quantitative Data (SID)
 Making Inferences and Justifying Conclusions (SIC)
 Conditional Probability and the Rules of
Probability (S-CP)
 Using Probability to Make Decisions (S-MD)
Number and Quantity
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The Real Number System (N-RN)
Quantities (N-Q)
The Complex Number System (N-CN)
Vector and Matrix Quantities (N-VM)
Algebra
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Seeing Structure in Expressions (A-SSE)
Arithmetic with Polynomials and Rational Expressions (A-APR)
Creating Equations (A-CED)
Reasoning with Equations and Inequalities (A-REI)
Contemporary Mathematics
 Discrete Mathematics (CM-DM)
Functions
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Mathematical Practices (MP)
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.
Interpreting Functions (F-IF)
Building Functions (F-BF)
Linear, Quadratic, and Exponential Models (F-LE)
Trigonometric Functions (F-TF)
Geometry
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Congruence (G-CO)
Similarity, Right Triangles, and Trigonometry (G-SRT)
Circles (G-C)
Expressing Geometric Properties with Equations (G-GPE)
Geometric Measurement and Dimension (G-GMD)
Modeling with Geometry (G-MG)
Modeling
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Domain and Clusters
High School - Number and Quantity Overview
The Real Number System (N-RN)
 Extend the properties of exponents to rational exponents
 Use properties of rational and irrational numbers.
Mathematical Practices (MP)
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.
Quantities (N-Q)
 Reason quantitatively and use units to solve problems
The Complex Number System (N-CN)
 Perform arithmetic operations with complex numbers
 Represent complex numbers and their operations on the complex
plane
 Use complex numbers in polynomial identities and equations
Vector and Matrix Quantities (N-VM)
 Represent and model with vector quantities.
 Perform operations on vectors.
 Perform operations on matrices and use matrices in applications.
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High School - Algebra Overview
Seeing Structure in Expressions (A-SSE)
 Interpret the structure of expressions
 Write expressions in equivalent forms to solve problems
Mathematical Practices (MP)
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.
Arithmetic with Polynomials and Rational Expressions (A-APR)
 Perform arithmetic operations on polynomials
 Understand the relationship between zeros and factors of
polynomials
 Use polynomial identities to solve problems
 Rewrite rational expressions
Creating Equations (A-CED)
 Create equations that describe numbers or relationships
Reasoning with Equations and Inequalities (A-REI)
 Understand solving equations as a process of reasoning and explain
the reasoning
 Solve equations and inequalities in one variable
 Solve systems of equations
 Represent and solve equations and inequalities graphically
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High School - Functions Overview
Interpreting Functions (F-IF)
 Understand the concept of a function and use function notation
 Interpret functions that arise in applications in terms of the context
 Analyze functions using different representations
Mathematical Practices (MP)
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.
Building Functions (F-BF)
 Build a function that models a relationship between two quantities
 Build new functions from existing functions
Linear, Quadratic, and Exponential Models (F-LE)
 Construct and compare linear, quadratic, and exponential models
and solve problems
 Interpret expressions for functions in terms of the situation they
model
Trigonometric Functions (F-TF)
 Extend the domain of trigonometric functions using the unit circle
 Model periodic phenomena with trigonometric functions
 Prove and apply trigonometric identities
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High School – Geometry Overview
Congruence (G-CO)
 Experiment with transformations in the plane
 Understand congruence in terms of rigid motions
 Prove geometric theorems
 Make geometric constructions
Geometric Measurement and Dimension (G-GMD)
 Explain volume formulas and use them to solve problems
 Visualize relationships between two-dimensional and threedimensional objects
Modeling with Geometry (G-MG)
 Apply geometric concepts in modeling situations
Similarity, Right Triangles, and Trigonometry (G-SRT)
 Understand similarity in terms of similarity transformations
 Prove theorems involving similarity
 Define trigonometric ratios and solve problems involving right
triangles
 Apply trigonometry to general triangles
Mathematical Practices (MP)
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.
Circles (G-C)
 Understand and apply theorems about circles
 Find arc lengths and areas of sectors of circles
Expressing Geometric Properties with Equations (G-GPE)
 Translate between the geometric description and the equation for a
conic section
 Use coordinates to prove simple geometric theorems algebraically
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High School – Statistics and Probability Overview
Interpreting Categorical and Quantitative Data (S-ID)
 Summarize, represent, and interpret data on a single count or
measurement variable
 Summarize, represent, and interpret data on two categorical and
quantitative variables
 Interpret linear models
Mathematical Practices (MP)
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.
Making Inferences and Justifying Conclusions (S-IC)
 Understand and evaluate random processes underlying statistical
experiments
 Make inferences and justify conclusions from sample surveys,
experiments and observational studies
Conditional Probability and the Rules of Probability (S-CP)
 Understand independence and conditional probability and use them
to interpret data
 Use the rules of probability to compute probabilities of compound
events in a uniform probability model
Using Probability to Make Decisions (S-MD)
 Calculate expected values and use them to solve problems
 Use probability to evaluate outcomes of decisions
High School – Contemporary Mathematics Overview
Discrete Mathematics (CM-DM)
 Understand and apply vertex-edge graph topics
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High School - Modeling
Modeling links classroom mathematics and statistics to everyday life, work, and decision-making. Modeling is the process of choosing and using
appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. Quantities and their
relationships in physical, economic, public policy, social, and everyday situations can be modeled using mathematical and statistical methods. When
making mathematical models, technology is valuable for varying assumptions, exploring consequences, and comparing predictions with data.
A model can be very simple, such as writing total cost as a product of unit price and number bought, or using a geometric shape to describe a physical
object like a coin. Even such simple models involve making choices. It is up to us whether to model a coin as a three-dimensional cylinder, or whether a
two-dimensional disk works well enough for our purposes. Other situations—modeling a delivery route, a production schedule, or a comparison of loan
amortizations—need more elaborate models that use other tools from the mathematical sciences. Real-world situations are not organized and labeled for
analysis; formulating tractable models, representing such models, and analyzing them is appropriately a creative process. Like every such process, this
depends on acquired expertise as well as creativity.
Some examples of such situations might include:
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Estimating how much water and food is needed for emergency relief in a devastated city of 3 million people, and how it might be distributed.
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Planning a table tennis tournament for 7 players at a club with 4 tables, where each player plays against each other player.
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Designing the layout of the stalls in a school fair so as to raise as much money as possible.
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Analyzing stopping distance for a car.
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Modeling savings account balance, bacterial colony growth, or investment growth.
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Engaging in critical path analysis, e.g., applied to turnaround of an aircraft at an airport.
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Analyzing risk in situations such as extreme sports, pandemics, and terrorism.
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Relating population statistics to individual predictions.
In situations like these, the models devised depend on a number of factors: How precise an answer do we want or need? What aspects of the situation do
we most need to understand, control, or optimize? What resources of time and tools do we have? The range of models that we can create and analyze is
also constrained by the limitations of our mathematical, statistical, and technical skills, and our ability to recognize significant variables and relationships
among them. Diagrams of various kinds, spreadsheets and other technology, and algebra are powerful tools for understanding and solving problems drawn
from different types of real-world situations.
One of the insights provided by mathematical modeling is that essentially the same mathematical or statistical structure can sometimes model seemingly
different situations. Models can also shed light on the mathematical structures themselves, for example, as when a model of bacterial growth makes more
vivid the explosive growth of the exponential function.
The basic modeling cycle is summarized in the diagram. It involves (1) identifying variables in the situation and selecting those that represent essential
features, (2) formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships
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between the variables, (3) analyzing and performing operations on these relationships to draw conclusions, (4) interpreting the results of the mathematics
in terms of the original situation, (5) validating the conclusions by comparing them with the situation, and then either improving the model or, if it is
acceptable, (6) reporting on the conclusions and the reasoning behind them. Choices, assumptions, and approximations are present throughout this cycle.
In descriptive modeling, a model simply describes the phenomena or summarizes them in a compact form. Graphs of observations are a familiar
descriptive model—for example, graphs of global temperature and atmospheric CO2 over time.
Analytic modeling seeks to explain data on the basis of deeper theoretical ideas, albeit with parameters that are empirically based; for example,
exponential growth of bacterial colonies (until cut-off mechanisms such as pollution or starvation intervene) follows from a constant reproduction rate.
Functions are an important tool for analyzing such problems.
Graphing utilities, spreadsheets, computer algebra systems, and dynamic geometry software are powerful tools that can be used to model purely
mathematical phenomena (e.g., the behavior of polynomials) as well as physical phenomena.
Modeling Standards Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical
models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol
(★).
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Standards for Mathematical Practice: High School
Standards for Mathematical Practice
Standards
Students are expected to:
HS.MP.1. Make sense of
problems and persevere in
solving them.
HS.MP.2. Reason abstractly
and quantitatively.
Mathematical Practices are listed
throughout the grade level
document in the 2nd column to
reflect the need to connect the
mathematical practices to
mathematical content in
instruction.
Explanations and Examples
High school students start to examine problems 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. By high school, 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. They check their
answers to problems using different methods and continually ask themselves, “Does this make
sense?” They can understand the approaches of others to solving complex problems and identify
correspondences between different approaches.
High school students seek to make sense of quantities and their relationships in problem situations.
They abstract a given situation and represent it symbolically, manipulate the representing symbols,
and pause as needed during the manipulation process in order to probe into the referents for the
symbols involved. Students use quantitative reasoning to create coherent representations of the
problem at hand; consider the units involved; attend to the meaning of quantities, not just how to
compute them; and know and flexibly use different properties of operations and objects.
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Standards for Mathematical Practice
Standards
Students are expected to:
HS.MP.3. Construct viable
arguments and critique the
reasoning of others.
HS.MP.4. Model with
mathematics.
Mathematical Practices are listed
throughout the grade level
document in the 2nd column to
reflect the need to connect the
mathematical practices to
mathematical content in
instruction.
Explanations and Examples
High school 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. High
school 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. High school students learn to determine domains to which an argument applies, listen or
read the arguments of others, decide whether they make sense, and ask useful questions to clarify or
improve the arguments.
High school students can apply the mathematics they know to solve problems arising in everyday life,
society, and the workplace. 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. High school students
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.
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Standards for Mathematical Practice
Standards
Students are expected to:
HS.MP.5. Use appropriate
tools strategically.
HS.MP.6. Attend to
precision.
HS.MP.7. Look for and make
use of structure.
Mathematical Practices are listed
throughout the grade level
document in the 2nd column to
reflect the need to connect the
mathematical practices to
mathematical content in
instruction.
Explanations and Examples
High school students consider the available tools when solving a mathematical problem. These tools
might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a
computer algebra system, a statistical package, or dynamic geometry software. High school students
should be sufficiently familiar with tools appropriate for their grade or course to make sound
decisions about when each of these tools might be helpful, recognizing both the insight to be gained
and their limitations. For example, high school students analyze graphs of functions and solutions
generated using a graphing calculator. They detect possible errors by strategically using estimation
and other mathematical knowledge. When making mathematical models, they know that technology
can enable them to visualize the results of varying assumptions, explore consequences, and compare
predictions with data. They are able to identify relevant external mathematical resources, such as
digital content located on a website, and use them to pose or solve problems. They are able to use
technological tools to explore and deepen their understanding of concepts.
High school students try to communicate precisely to others by using clear definitions in discussion
with others and in their own reasoning. They state the meaning of the symbols they choose, 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. By the time they reach high school they have learned to examine claims and
make explicit use of definitions.
By high school, students look closely to discern a pattern or structure. 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. High school students use
these patterns to create equivalent expressions, factor and solve equations, and compose functions,
and transform figures.
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Standards for Mathematical Practice
Standards
Students are expected to:
HS.MP.8. Look for and
express regularity in repeated
reasoning.
Mathematical Practices are listed
throughout the grade level
document in the 2nd column to
reflect the need to connect the
mathematical practices to
mathematical content in
instruction.
Explanations and Examples
High school students notice if calculations are repeated, and look both for general methods and for
shortcuts. 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, derive formulas or make generalizations, high school
students maintain oversight of the process, while attending to the details. They continually evaluate
the reasonableness of their intermediate results.
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High School Geometry
Conceptual Category: Geometry (6 Domains, 15 Clusters)
Domain: Congruence (4 Clusters)
Congruence (G-CO) (Domain 1 - Cluster 1 - Standards 1, 2, 3, 4 and 5)
Experiment with transformations in the plane
Essential Concepts
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A point, a line, a distance along a line, and a distance around a circular arc are
undefined notions.
Definitions of angle, circle, perpendicular line, parallel line, and line segment
are based on the undefined notions of point, line, distance along a line, and
distance around a circular arc.
Transformations can be described as functions that take points in the plane as
inputs and give other points as outputs.
Some transformations preserve distance and angle, while others do not.
Some polygons can be carried onto themselves using rotations and reflections.
Rotations, reflections, and translations can be defined in terms of angles,
circles, perpendicular lines, parallel lines, and line segments.
One figure may be carried onto another through a sequence of transformations.
HS.G-CO.A.1.
HS.G-CO.A.1
Know precise definitions of
angle, circle, perpendicular line,
parallel line, and line segment,
based on the undefined notions
of point, line, distance along a
line, and distance around a
circular arc.
Connection: 9-10.RST.4
Mathematical
Practices
HS.MP.6. Attend
to precision
Essential Questions
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Why are a point, a line, a distance along a line, and a distance around a circular
arc undefined notions?
Which transformations preserve distance and angles?
Give an example of a transformation that does not preserve distance and/or angle.
Explain why this is the case.
How can you determine which rotations and reflections will carry a polygon onto
itself?
How can you determine which transformations will carry a figure onto another
figure?
Examples & Explanations
Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in
geometric concepts, e.g. translations move points a specific distance along a line parallel to a specified line;
rotations move objects along a circular arc with a specified center through a specified angle.
Angle: A figure formed by two rays or line segments that meet at a common point
Circle: The set of all points in a plane that are a fixed distance from a given point.
Perpendicular line: Two lines are perpendicular if they intersect each other at a right angle (90°).
Parallel Line: Two lines are parallel if they lie in the same plane and never intersect.
Line segment: Part of a line containing two endpoints, and all the points between those two endpoints.
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HS.G-CO.A.2
HS.G-CO.A.2
Represent transformations in the
plane using, e.g., transparencies
and geometry software; describe
transformations as functions that
take points in the plane as inputs
and give other points as outputs.
Compare transformations that
preserve distance and angle to
those that do not (e.g.,
translation versus horizontal
stretch).
Mathematical
Practices
HS.MP.5. Use
appropriate tools
strategically.
Examples & Explanations
Students may use geometry software and/or manipulatives to model and compare transformations.
In middle school students have worked with translations, reflections, and rotations and informally with dilations.
In Algebra I students work with transformations of linear, quadratic and exponential functions.
Example:
 The figure below is reflected across the y-axis and then shifted up by 4 units. Draw the transformed
figure and label the new coordinates.
What function can be used to describe these transformations in the coordinate plane?
Connection:
ETHS-S6C1-03
Solution: Function (-1x, y + 4)
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HS.G-CO.A.3
HS.G-CO.A.3
Given a rectangle,
parallelogram, trapezoid, or
regular polygon, describe the
rotations and reflections that
carry it onto itself.
Connections:
ETHS-S6C1-03;
9-10.WHST.2c
HS.G-CO.A.4
HS.G-CO.A.4
Develop definitions of rotations,
reflections, and translations in
terms of angles, circles,
perpendicular lines, parallel
lines, and line segments.
Connections:
ETHS-S6C1-03;
9-10.WHST.4
HS.G-CO.A.5
HS.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.
Connections:
ETHS-S6C1-03;
Mathematical
Practices
HS.MP.3 Construct
viable arguments
and critique the
reasoning of others.
Examples & Explanations
Students may use geometry software and/or manipulatives to model transformations.
Example:
 For each of the following shapes, describe the rotations and reflections that carry it onto itself:
(a)
HS.MP.5. Use
appropriate tools
strategically.
Mathematical
Practices
HS.MP.6. Attend
to precision.
HS.MP.7. Look for
and make use of
structure.
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of others.
HS.MP.5. Use
appropriate tools
strategically.
(b)
(c)
Examples & Explanations
Students may use geometry software and/or manipulatives to model transformations. Students may observe
patterns and develop definitions of rotations, reflections, and translations.
A rotation is a movement of all points in a figure by the same angle measure in the same direction along a circular
path around a fixed point.
Examples & Explanations
Students may use geometry software and/or manipulatives to model transformations and demonstrate a sequence
of transformations that will carry a given figure onto another.
Example:

For the diagram below, describe the sequence of transformations that was used to carry DJKL on to the
red image.
(Continued on next page)
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9-10.WHST.3
HS.MP.7. Look for
and make use of
structure.
Congruence (G-CO) (Domain 1 - Cluster 2 - Standards 6, 7, and 8)
Understand congruence in terms of rigid motions (Build on rigid motions as a familiar starting point for development of concept of geometric proof.)
Essential Concepts
 A rigid motion is a transformation of points in space consisting of a sequence
of one or more translations, reflections, and/or rotations.
 Rigid motions are assumed to preserve distances and angle measures.
 Congruent triangles have corresponding sides and corresponding angles that are
congruent.
 The criteria for triangle congruence (ASA, SAS, and SSS) follow from the
definition of congruence in terms of rigid motions.
HS.G-CO.B.6
HS.G-CO.B.6
Use geometric descriptions of
rigid motions to transform
figures and to predict the effect
of a given rigid motion on a
given figure; given two figures,
use the definition of congruence
in terms of rigid motions to
decide if they are congruent.
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of others.
Essential Questions

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

How do you determine if two figures are congruent?
What has to be true in order for two triangles to be congruent?
How do the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions?
Why does SSA not work to prove triangle congruence?
Examples & Explanations
Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties
of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and
their assumed properties can be used to establish the usual triangle congruence criteria, which can then be
used to prove other theorems.
Middle school students have worked with rigid motions and congruence of figures on a coordinate plane.
(Continued on next page)
HS.MP.5. Use
appropriate tools
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Connections:
ETHS-S1C2-01;
9-10.WHST.1e
strategically.
Students may use geometric software to explore the effects of rigid motion on figures.
HS.MP.7. Look for
and make use of
structure.
A rigid motion is a transformation of points in space consisting of a sequence of one or more translations,
reflections, and/or rotations. Rigid motions are assumed to preserve distances and angle measures.
Example:
 Determine if the figures below are congruent, if so tell what rigid motions were used.
HS.G-CO.B.7
HS.G-CO.B.7
Use the definition of congruence
in terms of rigid motions to
show that two triangles are
congruent if and only if
corresponding pairs of sides and
corresponding pairs of angles are
congruent.
Connection: 9-10.WHST.1e
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of others.
Examples & Explanations
A rigid motion is a transformation of points in space consisting of a sequence of one or more translations,
reflections, and/or rotations. Rigid motions are assumed to preserve distances and angle measures.
Congruence of triangles: Two triangles are said to be congruent if one can be exactly superimposed on the
other by a rigid motion, and the congruence theorems specify the conditions under which this can occur.
Example:
 Are the following triangles
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congruent? Explain how you know.
HS.G-CO.B.8
HS.G-CO.B.8
Explain how the criteria for
triangle congruence (ASA, SAS,
and SSS) follow from the
definition of congruence in
terms of rigid motions.
Connection: 9-10.WHST.1e
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of others.
Examples & Explanations
Example:
 Decide whether there is enough information to prove that the two shaded triangles are congruent.
In the figure below, ABCD is a parallelogram.
Solution:
The two triangles are congruent by SAS: We have AX≅CX and DX≅BX since the diagonals of a
parallelogram bisect each other, and ∠AXD≅∠BXC since they are vertical angles. Alternatively, we
could use argue via ASA: We have the opposite interior angles ∠DAX≅∠BCX and ∠ADX≅∠CBX
and AD≅BC since opposite sides of a parallelogram are congruent.
From: illustrativemathematics.org
Congruence (G-CO) (Domain 1 - Cluster 3 - Standards 9, 10, and 11)
Prove geometric theorems (Focus on validity of underlying reasoning while using a variety of ways of writing proofs.)
Essential Concepts
 A theorem is a statement that can be proven from previously known facts,
including postulates, axioms and definitions.
 A proof is a logical argument that shows that a theorem is true based on given
information.
 Properties of the sides and angles of geometric figures can be stated as
theorems and proven. Several examples are listed in the standards below.
HS.G-CO.C.9
HS.G-CO.C.9
Prove theorems about lines and
angles. Theorems include:
vertical angles are congruent;
Mathematical
Practices
HS.MP.3.
Construct viable
Essential Questions
 How is a theorem different from an axiom?
 What is the difference between an axiom and a postulate?
 How do you know when a proof is complete and valid?
Examples of essential questions for particular theorems:
 How do you prove that the interior angles of a triangle add up to 180°?
 What method would you use to prove theorems about parallelograms, and
why?
Examples & Explanations
Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in twocolumn format, and using diagrams without words. Students should be encouraged to focus on the validity of
the underlying reasoning while exploring a variety of formats for expressing that reasoning)
19
when a transversal crosses
parallel lines, alternate interior
angles are congruent and
corresponding angles are
congruent; points on a
perpendicular bisector of a line
segment are exactly those
equidistant from the segment’s
endpoints.
arguments and
critique the
reasoning of others.
Students may use geometric simulations (computer software or graphing calculator) to explore theorems about
lines and angles.
HS.MP.5. Use
appropriate tools
strategically.
Example:

Prove that
ÐHIB @ ÐDJG , given that
AB // DE .
Connections:
ETHS-S1C2-01;
9-10.WHST.1a-1e
HS.G-CO.C.10
HS.G-CO.C.10
Prove theorems about triangles.
Theorems include: measures of
interior angles of a triangle sum
to 180°; base angles of isosceles
triangles are congruent; the
segment joining midpoints of two
sides of a triangle is parallel to
the third side and half the
length; the medians of a triangle
meet at a point.
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of others.
Examples & Explanations
Students may use geometric simulations (computer software or graphing calculator) to explore theorems about
triangles.
Example:
 Given that ΔABC is isosceles, prove that
HS.MP.5. Use
appropriate tools
strategically.
Connections:
ETHS-S1C2-01;
9-10.WHST.1a-1e
20
ÐABC @ ÐACB .
HS.G-CO.C.11
HS.G-CO.C.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.
Connection:
9-10.WHST.1a-1e
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of others.
HS.MP.5. Use
appropriate tools
strategically.
Examples & Explanations
Students may use geometric simulations (computer software or graphing calculator) to explore theorems about
parallelograms.
Example:
 Suppose that ABCD is a parallelogram, and that M and N are the midpoints of AB and CD ,
respectively. Prove that MN = AD , and that the line MN is parallel to AD .
(Continued on next page)
One solution:
The diagram above consists of the given information, and one additional line segment, MD , which
we will use to demonstrate the result. We claim that triangles ΔAMD and ΔNDM are congruent by
SAS:
1. We have MD = DM by reflexivity.
Ð AMD= Ð NDM since they are opposite interior angles of the transversal
2.
We have
3.
through parallel lines AB and CD .
We have MA = ND , since M and N are midpoints of their respective sides, and opposite sides of
parallelograms are congruent: MA = 12 AB = 12 CD = ND .
( ) ( )
21
MD
Now since corresponding parts of congruent triangles are congruent, we have DA = NM as
desired. Similarly, we have congruent opposite interior angles
Ð DMN≅ Ð MDA, so
MN is
parallel to AD .
From: illustrativemathematics.org
Congruence (G-CO) (Domain 1 - Cluster 4 – Standards 12, and 13)
Make geometric constructions (Formalize and explain processes.)
Essential Concepts
 Formal geometric constructions can be made with a variety of tools and
methods.
 Equilateral triangles, squares, and regular hexagons can be inscribed in a circle
by constructions.
HS.G-CO.D.12
HS.G-CO.D.12
Make formal geometric
constructions with a variety of
tools and methods (compass and
straightedge, string, reflective
devices, paper folding, dynamic
geometric software, etc.).
Copying a segment; copying an
angle; bisecting a segment;
bisecting an angle; constructing
perpendicular lines, including
the perpendicular bisector of a
line segment; and constructing a
line parallel to a given line
through a point not on the line.
Mathematical
Practices
HS.MP.5. Use
appropriate tools
strategically.
HS.MP.6. Attend
to precision.
Essential Questions



What tools are used to create geometric construction and why?
Given a construction, how can you justify that it was done correctly?
Why do you think we use a straightedge rather than a ruler for formal
geometric constructions?
Examples & Explanations
Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain
how these constructions result in the desired objects. Some of these constructions are closely related to
previous standards and can be introduced in conjunction with them.
Students may use geometric software to make geometric constructions.
Examples:
 Construct a triangle given the lengths of two sides and the measure of the angle between the two
sides.
 Construct the circumcenter of a given triangle.

Construct the perpendicular bisector of a line segment.
Connection:
ETHS-S6C1-03
22
HS.G-CO.13
HS.G-CO.D.13
Construct an equilateral triangle,
a square, and a regular hexagon
inscribed in a circle.
Connection: ETHS
-S6C1-03
Image source: http://motivate.maths.org/content/accurate-constructions
Mathematical
Practices
HS.MP.5. Use
appropriate tools
strategically.
HS.MP.6. Attend
to precision.
Examples & Explanations
Students may use geometric software to make geometric constructions.
Example:

Construct a regular hexagon inscribed in a circle.
This construction can also be used to draw a 120° angle.
1.
2.
3.
4.
5.
6.
7.
8.
Keep your compasses to the same setting throughout this
construction.
Draw a circle.
Mark a point, P, on the circle.
Put the point of your compasses on P and draw arcs to cut
the circle at Q and U.
Put the point of your compasses on Q and draw an arc to
cut the circle at R.
Repeat with the point of the compasses at R and S to draw
arcs at S and T.
Join PQRSTU to form a regular hexagon.
Measure the lengths to check they are all equal, and the angles to check they are all 120
degrees.
Image source: http://motivate.maths.org/content/accurate-constructions
Additional Domain Information – Congruence (G-CO)
23
Key Vocabulary








Angle
Circle
Perpendicular Lines
Parallel Lines
Line Segment
Corresponding Angles
Midpoint
Altitude
Example Resources











Point
Line
Arc
Transformation
Translation
Alternate Exterior Angles
Median
Diagonals








Rotation
Reflection
Rectangle
Parallelogram
Trapezoid
Alternate Interior Angles
Isosceles Triangle
Regular Polygon







Rigid Motion
Non Rigid Motion
Congruent
Vertical Angles
Transversal
Perpendicular Bisector
Equilateral Triangle
Books
 “Patios by Madeline” p. 47-58 Springboard Geometry
 “Those Magical Midpoints” p. 102-103 Glencoe Geometry Concepts and Applications
 “Take a Shortcut” (Triangle Congruence) p. 208-209 Glencoe Geometry Concepts and Applications
 Focus in High School Mathematics: Reasoning and Sense Making, Geometry National Council of Teachers of Mathematics Publication
Technology
 http://www.mathsisfun.com/geometry/transformations.html This website gives examples of the transformations and discusses congruence.
 http://illustrativemathematics.org/standards/hs This website lists the standards and gives specific examples for the different concepts within the
standards.
 http://www.classzone.com/cz/find_book.htm?tmpState=&disciplineSchool=ma_hs&state=AZ&x=23&y=29 This website allows you to connect to the
math books used in the classroom.
 http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM
 http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be
matched to a particular textbook.
 http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph
equations.
 http://nlvm.usu.edu/en/nav/topic_t_3.html This site has a library of applets for demonstrating geometry concepts. Each applet also has
suggested on how it can be used in the classroom.
Example Lessons
 http://illuminations.nctm.org/LessonDetail.aspx?id=L727 There is a leap to be made from understanding postulates and theorems in geometry to
writing proofs using them. This lesson offers an intermediate step, in which students put together the statements and reasons to build a formal proof.
 http://illuminations.nctm.org/LessonDetail.aspx?id=L379 This lesson requires students to investigate reflections using hinged mirrors. With a
kaleidoscope, students will examine the interior angles of regular polygons.
 http://illuminations.nctm.org/LessonDetail.aspx?ID=U138 This is the first i-Math in a four-part series of i-Maths entitled Symmetries and Their
Properties. In this first i-Math you will investigate rotational symmetry. Fix a center, turn, and you have a rotation. Many objects in nature—flowers,
starfish, and crystals—and objects we use every day, such as wheels, CDs, and drinking glasses, have rotational symmetry. Here, you will learn about
the mathematical properties of rotations and have an opportunity to make your own designs. This is the link for the entire four-part unit.
 http://www.shodor.org/interactivate/lessons/TranslationsReflectionsRotations/ This lesson is designed to introduce students to translations, reflections,
and rotations. This also provides a TransmoGrapher that allows you to demonstrate translations, rotations and reflections on a coordinate grid.
24
Common Student Misconceptions
Students commonly mix up the axis that a figure is being reflected across. For example, it the figure flips vertically, they assume it was reflected across the y-axis.
Students commonly rotate figures in the wrong direction. For example, they will rotate counterclockwise as opposed to clockwise.
Students commonly mix up the order on triangle congruence. For example, students may label it as AAS instead of ASA.
Students make improper assumptions about a diagram. For example, if the sides look the same then students assume they are congruent even though no labels are
present.
Students will draw instead of construct figures. For example, students may just draw something without using proper construction methods.
Students will use properties about a specific figure and apply them to another figure. For example students may apply the square properties to a rhombus.
Students assume that axioms and postulates are the same.
25
Domain: Similarity, Right Triangles, and Trigonometry (4 Clusters)
Similarity, Right Triangles, and Trigonometry (G-SRT) (Domain 2 - Cluster 1 - Standards 1, 2, and 3)
Understand similarity in terms of similarity transformations
Essential Concepts
 When a line segment that does not pass through the center of a dilation is
dilated, a parallel line segment is formed; line segments that pass through the
center remain unchanged.
 The lengths of corresponding line segments in a figure and its dilation are
proportional in the ratio given by the scale factor.
 A dilation of a two-dimensional figure will create an image that is a similar
figure to the original multiplied by a common scale factor/ratio.
 A similarity transformation is a combination of rigid motion and dilation.
 Triangles are similar if all corresponding angles are congruent and all
corresponding sides are proportional.
 It can be shown using properties of similarity transformations that triangles are
similar if two pairs of corresponding angles are congruent.
HS.G-SRT.A.1
HS.G-SRT.A.1
Verify experimentally the
properties of dilations given by a
center and a scale factor:
Connections:
ETHS-S1C2-01;
9-10.WHST.1b;
9-10.WHST.1e
a.
b.
A dilation takes a line not
passing through the center
of the dilation to a parallel
line, and leaves a line
passing through the center
unchanged.
The dilation of a line
segment is longer or shorter
in the ratio given by the
scale factor.
Mathematical
Practices
HS.MP.2. Reason
abstractly and
quantitatively.
HS.MP.5. Use
appropriate tools
strategically.
Essential Questions






What happens to a figure once it has been dilated?
Why do dilations involve scale factors?
How can you determine if a line in a figure will be changed by a dilation?
What is the difference between similar and congruent figures?
How can you describe the relationship between sides of similar figures?
How can you show that two triangles are similar?
Examples & Explanations
Students may use geometric simulation software to model transformations. Students may observe patterns and
verify experimentally the properties of dilations.
A dilation is a transformation that moves each point along the ray through the point emanating from a fixed
center, and multiplies distances from the center by a common scale factor.
Example:
 Draw a polygon. Pick a point and construct a dilation of the polygon with that point as the center.
Identify the scale factor that you used.
One solution:
HS.G-SRT.A.2
26
HS.G-SRT.A.2
Given two figures, use the
definition of similarity in terms
of similarity transformations to
decide if they are similar;
explain using similarity
transformations the meaning of
similarity for triangles as the
equality of all corresponding
pairs of angles and the
proportionality of all
corresponding pairs of sides.
Connections:
ETHS-S1C2-01;
9-10.RST.4;
9-10.WHST.1c
HS.G-SRT.A.3
HS.G-SRT.A.3
Use the properties of similarity
transformations to establish the
AA criterion for two triangles to
be similar.
Connections:
ETHS-S1C2-01;
9-10.RST.7
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of others.
HS.MP.5. Use
appropriate tools
strategically.
Examples & Explanations
A similarity transformation is a rigid motion followed by a dilation.
Students may use geometric simulation software to model transformations and demonstrate a sequence of
transformations to show congruence or similarity of figures.
Example:

Are these two figures similar? Explain why or why not.
HS.MP.7. Look for
and make use of
structure.
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of others.
Examples & Explanations
Example:

Are all right triangles similar to one another? How do you know?
Similarity, Right Triangles, and Trigonometry (G-SRT) (Domain 2 - Cluster 2 - Standards 4, and 5)
Prove theorems involving similarity
Essential Concepts
 A theorem is a statement that can be proven from previously known facts,
including postulates, axioms and definitions.
Essential Questions

27
Why does a line cutting a triangle parallel to one side of the triangle
divide the other two proportionally?
 A proof is a logical argument that shows that a theorem is true based on given
information.
 The Pythagorean theorem can be proven using triangle similarity.
 Congruence and similarity can be used to determine missing angle measures
and side lengths in geometric figures.
HS.G-SRT.B.4
HS.G-SRT.B.4
Prove theorems about triangles.
Theorems include: a line
parallel to one side of a triangle
divides the other two
proportionally, and conversely;
the Pythagorean theorem proved
using triangle similarity.
Connections:
ETHS-S1C2-01;
9-10.WHST.1a-1e
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of others.
HS.MP.5. Use
appropriate tools
strategically.



How many pieces of information do you need in order to solve for
the missing measures of a triangle?
How can you use congruence or similarity of pairs of figures to
determine missing information about the figures?
Why does SSA not work to prove triangle congruence?
Examples & Explanations
Students may use geometric simulation software to model transformations and demonstrate a sequence of
transformations to show congruence or similarity of figures.
Example:
 Prove that if two triangles are similar, then the ratio of corresponding altitudes is equal to the ratio of
corresponding sides.

To prove the Pythagorean theorem using triangle similarity:
We can cut a right triangle into two parts by dropping a perpendicular onto the hypotenuse. Since
these triangles and the original one have the same angles, all three are similar. Therefore
x a c-x b
= ,
=
b
c
a c
a2
b2
x = , c-x =
c
c
x+ c-x =c
(
)
a b
+
=c
c
c
a2 + b2 = c 2
2
2
From: http://www.math.ubc.ca/~cass/euclid/java/html/pythagorassimilarity.html
HS.G-SRT.B.5
HS.G-SRT.B.5
Use congruence and similarity
criteria for triangles to solve
problems and to prove
relationships in geometric
figures.
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
Examples & Explanations
Students may use geometric simulation software to model transformations and demonstrate a sequence of
transformations to show congruence or similarity of figures.
Similarity criteria include SSS, SAS, and AA.
28
Connections:
ETHS-S1C2-01;
9-10.WHST.1a-1e
reasoning of others.
Congruence criteria include SSS, SAS, ASA, AAS, and H-L.
HS.MP.5. Use
appropriate tools
strategically.
Similarity, Right Triangles, and Trigonometry (G-SRT) (Domain 2 - Cluster 3 - Standards 6, 7, and 8)
Define trigonometric ratios and solve problems involving right triangles
Essential Concepts



The ratios of the sides in any right triangle are properties of the angles in the
triangle; these are called trigonometric ratios.
The sine and cosine of complementary angles are related.
Trigonometric ratios and the Pythagorean theorem can be used to solve right
triangles in applied problems.
HS.G-SRT.C.6
HS.G-SRT.C.6
Understand that by similarity,
side ratios in right triangles are
properties of the angles in the
triangle, leading to definitions of
trigonometric ratios for acute
angles.
Connection: ETHS-S6C1-03
Mathematical
Practices
HS.MP.6. Attend
to precision.
Essential Questions




What role does similarity play in defining trigonometric ratios?
What is the relationship between the sine and cosine of complementary
angles?
How can right triangles be used to solve real-world problems?
Why do we need trigonometric ratios to solve some real-world problems
involving right triangles?
Examples & Explanations
Students may use applets to explore the range of values of the trigonometric ratios as θ ranges from 0 to 90
degrees.
hypotenuse
HS.MP.8. Look for
and express
regularity in
repeated reasoning.
opposite of θ
θ
Adjacent to θ
(Continued on next page)
sine of θ = sin θ =
opposite
hypotenuse
adjacent
cosine of θ = cos θ =
hypotenuse
opposite
tangent of θ = tan θ =
adjacent
29
cosecant of θ = csc θ =
hypotenuse
opposite
hypotenuse
secant of θ = sec θ =
adjacent
adjacent
cotangent of θ = cot θ =
opposite
HS.G-SRT.C.7
HS.G-SRT.C.7
Explain and use the relationship
between the sine and cosine of
complementary angles.
Connections:
ETHS-S1C2-01;
ETHS-S6C1-03;
9-10.WHST.1c;
9-10.WHST.1e
HS.G-SRT.C.8
HS.G-SRT.C.8
Use trigonometric ratios and the
Pythagorean theorem to solve
right triangles in applied
problems.
Connections:
ETHS-S6C2-03;
9-10.RST.7
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of others.
Mathematical
Practices
HS.MP.1. Make
sense of problems
and persevere in
solving them.
Examples & Explanations
Geometric simulation software, applets, and graphing calculators can be used to explore the relationship
between sine and cosine.
Example:
 Find the sine and cosine of angle θ in the triangle below. What do you notice?
Examples & Explanations
Students may use graphing calculators or programs, tables, spreadsheets, or computer algebra systems to solve
right triangle problem.
Example:
 Find the height of a tree to the nearest tenth if the angle of elevation of the sun is 28° and the shadow
of the tree is 50 ft.
HS.MP.4. Model
with mathematics.
HS.MP.5. Use
appropriate tools
strategically.
Additional Domain – Similarity, Right Triangles, and Trigonometry (G-SRT)
Key Vocabulary
30




Sine
Cosine
Tangent
Dilation
Example Resources




Law of Sines
Law of Cosines
Resultant Forces
Scale Factor



Pythagorean Theorem
Complementary Angles
Supplementary Angles
Books
 “Field of Vision” p. 14-22 Focus in High School Mathematics Geometry
 “Indirect Measurement” p. 632-633 Discovering Geometry
 Focus in High School Mathematics: Reasoning and Sense Making, Geometry National Council of Teachers of Mathematics Publication
 Technology
 http://illustrativemathematics.org/standards/hs This website lists the standards and gives specific examples for the different concepts within the
standards.
www.classzone.com/ This is the site to access the book and extra resources online.
 http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM.
 http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched to a
particular textbook.
 http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations.
 http://www.clarku.edu/~djoyce/trig/laws.html This site offers definitions and the Law of Sines and Law of Cosines along with little applets for
demonstration.
 Example Lessons
 http://illuminations.nctm.org/LessonDetail.aspx?id=U177 This unit includes two different lessons one on Law of Sines and one on Law of cosine
lessons. Students use right triangle trigonometry to develop the Law of Sines and right triangle trigonometry and Pythagorean theorem to develop the
law of cosines.
 http://illuminations.nctm.org/LessonDetail.aspx?id=L716 The law of cosines is an extension of the Pythagorean theorem, but seeing how –2ab cos C
fits into the picture can be difficult for students. In this lesson, students who understand the Pythagorean theorem and right triangle trigonometry will
discover the law of cosines by exploring the areas of squares on the sides of a triangle and their associated "defects."
 http://illuminations.nctm.org/LessonDetail.aspx?id=L383 This lesson offers a pair of puzzles to enforce the skills of identifying equivalent
trigonometric expressions. Additional worksheets enhance students' abilities to appreciate and use trigonometry as a tool in problem solving. This
lesson is adapted from an article by Mally Moody, which appeared in the March 1992 edition of Mathematics Teacher.
Common Student Misconceptions

Students commonly confuse sine, cosine and tangent. For example, students will mix up sine as adjacent over hypotenuse. Use of a mnemonic device may help
students with remembering the ratios.
Students will often use sine when asked to use trig ratios as opposed to figuring out which trig ratio is appropriate to the given information.
For example, student may be given an angle and the opposite and adjacent sides. Students will use sine instead of the tangent.
Students incorrectly apply the scale factor. For example, students will multiply instead of divide with a scale factor that reduces a figure and students will divide
instead of multiply when enlarging a figure.
31
Students will incorrectly use the Law of Cosines and Law of Sines. For example, students will apply Law of Sines with an SAS triangle when they should use Law
of Cosines.
32
Domain: Circles (G-C) (2 Clusters)
Circles (G-C) (Domain 3 - Cluster 1 - Standards 1, 2, 3, and 4)
Understand and apply theorems about circles
Essential Concepts
 It can be proven that all circles are similar.
 Central, inscribed, and circumscribed angles on a circle are related to each
other.
 The length of a chord on a circle is related to the inscribed, circumscribed, and
central angles defined by the endpoints of the chord.
 The radius of a circle is perpendicular to the tangent at the point of intersection.
 Every triangle has a unique inscribed circle and a unique circumscribed circle.
 Quadrilaterals inscribed in a circle have properties that can be stated as
theorems and proven.
 From any point outside a given circle, two tangent lines to the circle can be
constructed.
HS.G-C.A.1
HS.G-C.A.1
Prove that all circles are similar.
Connections:
ETHS-S1C2-01;
9-10.WHST.1a-1e
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of others.
Essential Questions
How can you prove that all circles are similar?
Given two points on a circle, how are the central and inscribed angles
through those points related?
 How is the length of a chord on a circle related to the central angle formed by
the endpoints?
 Why is the radius of a circle perpendicular to the tangent?
 How can you construct the inscribed and circumscribed circles of a triangle?
Example of an essential question about inscribed quadrilaterals:
 How can you prove that the sum of opposite angles in an inscribed
quadrilateral equals 180°?
 Why does a point have to be outside a circle in order to be able to construct a
tangent line from that point to the circle?


Examples & Explanations
Students may use geometric simulation software to model transformations and demonstrate a sequence of
transformations to show congruence or similarity of figures.
Example:
 Draw or find examples of several different circles. In what ways are they related? How can you describe
this relationship in terms of geometric ideas? Form a hypothesis and prove it.
HS.MP.5. Use
appropriate tools
strategically.
33
HS.G-C.A.2
HS.G-C.A.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.
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of others.
Examples & Explanations
Examples:

Given the circle below with radius of 10 and chord length of 12, find the distance from the chord to
the center of the circle.

Find the unknown length in the picture below.
HS.MP.5. Use
appropriate tools
strategically.
Connections:
9-10.WHST.1c;
11-12.WHST.1c
Solution:
The theorem for a secant segment and a tangent segment that share an endpoint not on the
circle states that for the picture below secant segment QR and tangent segment SR share an
endpoint, R, not on the circle. Then the length of SR squared is equal to the product of the
34
lengths of QR and KR.
x 2 = 16 ×10
So for the example above:
x 2 = 160
x = 160 = 4 10 » 12.6

HS.G-C.A.3
HS.G-C.A.3
Construct the inscribed and
circumscribed circles of a
triangle, and prove properties of
angles for a quadrilateral
inscribed in a circle.
Connection: ETHS-S6C1-03
Mathematical
Practices
HS.G-C.3.
Construct the
inscribed and
circumscribed
circles of a
triangle, and prove
properties of angles
for a quadrilateral
inscribed in a
circle.
How does the angle between a tangent to a circle and the line connecting the point of tangency and
the center of the circle change as you move the tangent point?
Examples & Explanations
Example:
 Given the inscribed quadrilateral below prove that
Connection:
ETHS-S6C1-03
35
 B is supplementary to  D.
Circles (G-C) (Domain 3 - Cluster 2 - Standard 5)
Find arc lengths and areas of sectors of circles (Radian introduced only as unit of measure)
Essential Concepts
 Arc lengths on a circle are proportional to the radius; this fact follows from the
similarity of circles.
 The ratio of an arc length to the radius defines a unit of measurement for the
central angle that intercepts that arc; this unit is called the radian.
 The radian measure of a angle is the constant of proportionality.
 The formula for the area of a sector can be derived by exploring the
relationship between the central angle and the area formula for a circle.
HS.G-C.B.5
HS.G-C.B.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.
Connections:
ETHS-S1C2-01;
11-12.RST.4
Mathematical
Practices
HS.MP.2 Reason
abstractly and
quantitatively.
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of others.
Essential Questions




How is an arc length on a circle related to the radius? How can you derive
this relationship?
How do you convert an angle from degree measure to radian measure? Why
does this work?
How do you find the arc length of a circle in degrees? Radians?
How can you derive the formula for the area of a sector?
Examples & Explanations
Emphasize the similarity of all circles. Note that by similarity of sectors with the same central angles, arc
lengths are proportional to the radius. Use this as a basis for introducing the radian as a unit of measure. It is
not intended that it be applied to the development of circular trigonometry in this course.
Students can use geometric simulation software to explore angle and radian measures and derive the formula
for the area of a sector.
Example:
 Find the area of the sectors below. What general formula can you develop based on this information?
36
Additional Domain Information – Circles (G-C)
Key Vocabulary





Radius
Diameter
Chord
Central Angles
Inscribed Angles
Example Resources





Circumscribed Angles
Tangent
Secant
Inscribed Circle
Circumscribed Circle



Arc
Radian
Sector
Books
 “Coming Full Circle” p. 285-286 Springboard Geometry
 Focus in High School Mathematics: Reasoning and Sense Making, Geometry National Council of Teachers of Mathematics Publication
 Technology
 http://mathforum.org/library/drmath/sets/high_circles.html This website provides banks of questions that are answered by an expert.
 http://illustrativemathematics.org/standards/hs This website lists the standards and gives specific examples for the different concepts within the
standards.
 www.classzone.com/ This is the site to access the book and extra resources online.
 http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM.
 http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched to a
particular textbook.
 http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations.
 http://www.mathopenref.com/consttangent.html Has online videos that show constructions using a compass and straight edge.
 Example Lessons
 http://illuminations.nctm.org/LessonDetail.aspx?id=L700 In many curricula, the Power of Points theorem is often taught as three separate theorems:
the Chord-Chord Power theorem, the Secant-Secant Power theorem, and the Tangent-Secant Power theorem. Using a dynamic geometry applet,
students will discover that these three theorems are related applications of the Power of Point theorem. They also use their discoveries to solve
numerical problems.
 http://mathforum.org/pow/teacher/samples/MathForumSampleGeometryPacket.pdfThis lesson focuses on Identifying relationships among inscribed
angles, radii, and chords to model a real-life situation. Students can use a variety of methods to find the size of a piece of pottery based on a sherd
collected at an archeological site.
Common Student Misconceptions

Students will commonly mix up the appropriate use of central angles and inscribed angles. For example, they will assume that the inscribed angle is equal to the
arc like a central angle.
Students will confuse the segment theorems. For example, they will use only use the two segments with the secants instead of the entire length of the secant. Students
may also use segment theorems when they should use angle theorems.
Students use the incorrect angle when finding the area of a sector. For example, the picture may show the smaller angle but the problem is asking for the area of the
larger sector.
37
Domain: Expressing Geometric Properties with Equations (2 Clusters)
Expressing Geometric Properties with Equations (G-GPE) (Domain 4 - Cluster 1 - Standards 1, and 2)
Translate between the geometric description and the equation for a conic section.
Essential Concepts
 The equation of a circle can be derived using the Pythagorean theorem.
 The center and radius of a circle can be found by completing the square.
 The equation of a parabola can be derived given its focus and directrix.
HS.G-GPE.A.1
HS.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.
Connections:
ETHS-S1C2-01;
11-12.RST.4
Essential Questions




How do you derive the equation of a circle, given a center and a radius?
How can you determine the center and radius of a circle from its equation?
How do you derive the equation of a parabola, given a focus and a directrix?
What information do you need to derive the equation of a circle or parabola,
and why?
Mathematical
Practices
Examples & Explanations
HS.MP.8. Look for
and express
regularity in
repeated reasoning.
The standard form of a circle is x - h + y - k
HS.MP.7. Look for
and make use of
structure.
G-GPE1 relates to G.GPE.4. Reasoning with triangles is limited to right triangles; e.g. derive the equation for
a line through two points using similar right triangles.
Students may use geometric simulation software to explore the connection between circles and the
Pythagorean theorem.
(
) (
2
)
2
= r 2 , where the center is (h,k) with a radius of r.
Examples:
 Write an equation for a circle with a radius of 2 units and center at (1, 3).

Write an equation for a circle given that the endpoints of the diameter are (-2, 7) and (4, -8).

Find the center and radius of the circle 4x2 + 4y2 - 4x + 2y – 1 = 0.
Expressing Geometric Properties with Equations (G-GPE) (Domain 4 - Cluster 2 - Standards 4, 5, 6 and 7)
Use coordinates to prove simple geometric theorems algebraically (Include distance formula; relate to Pythagorean theorem)
Essential Concepts
 Coordinate geometry can be used to solve simple geometric theorems
algebraically.
 Coordinate geometry can be used to prove that parallel lines have the same
slope and perpendicular lines have opposite reciprocal slopes.
 The formula for finding the distance between two points on a coordinate plane
can be derived using the Pythagorean theorem.
 The area and perimeter of a polygon can be found by placing the polygon on a
coordinate plane.
Essential Questions





38
How do you find the point on a directed line segment between two given
points that partitions the segment in a given ratio?
How do you determine if two lines are parallel, perpendicular or neither?
Given an equation of a line and a point not on the line, how do you write the
equation of a line that is parallel to the given line and through the given point?
Perpendicular?
How are geometry and algebra related to each other?
How is the distance formula related to the Pythagorean theorem?
HS.G-GPE.B.4
HS.G-GPE.B.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).
Mathematical
Practices
HS.MP.3 Reason
abstractly and
quantitatively.
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).
Connection:
9-10.WHST.1a-1e
HS.G-GPE.B.6
HS.G-GPE.B.6
Find the point on a directed line
segment between two given
points that partitions the segment
in a given ratio.
Connections:
Relate work on parallel lines in G-GPE.7 to work on A-REI.5 in High School Algebra 1 involving systems of
equations having no solutions or infinitely many solutions.
Students may use geometric simulation software to model figures and prove simple geometric theorems.
Examples:
 Use slope and distance formula to verify the polygon formed by connecting the points (-3, -2), (5, 3),
(9, 9), (1, 4) is a parallelogram.
 Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle.

Connections:
ETHS-S1C2-01;
9-10.WHST.1a-1e;
11-12.WHST.1a-1e
HS.G-GPE.B.5
HS.G-GPE.B.5
Examples & Explanations
Prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the
point (0, 2).

Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of others.
HS.MP.8. Look for
and express
regularity in
repeated reasoning.
Mathematical
Practices
HS.MP.2. Reason
abstractly and
quantitatively.
HS.MP.8. Look for
Examples & Explanations
Relate work on parallel lines in G-GPE.5 to work on A-REI.5 involving systems of equations having no
solution or infinitely many solutions.
Lines can be horizontal, vertical, or neither.
Students may use a variety of different methods to construct a parallel or perpendicular line to a given line and
calculate the slopes to compare the relationships.
Example:
 Find the equation of a line perpendicular to 3x + 5y = 15 through the point (-3,2).
Examples & Explanations
Students may use geometric simulation software to model figures or line segments.
Examples:

Given A(3, 2) and B(6, 11),
o Find the point that divides the line segment AB two-thirds of the way from A to B.
39
ETHS-S1C2-01;
9-10.RST.3
and express
regularity in
repeated reasoning.
o
HS.G-GPE.B.7
HS.G-GPE.B.7
Use coordinates to compute
perimeters of polygons and areas
of triangles and rectangles, e.g.,
using the distance formula.
Connections:
ETHS-S1C2-01;
9-10.RST.3;
11-12.RST.3
Mathematical
Practices
HS.MP.2. Reason
abstractly and
quantitatively.
HS.MP.5. Use
appropriate tools
strategically.
The point two-thirds of the way from A to B has x-coordinate two-thirds of the way from 3 to 6
and y coordinate two-thirds of the way from 2 to 11.
So, (5, 8) is the point that is two-thirds from point A to point B.
Find the midpoint of line segment AB.
Examples & Explanations
This standard provides practice with the distance formula and its connection with the Pythagorean theorem.
Students may use geometric simulation software to model figures.
Example:
 Find the area and perimeter for the figure below.
HS.MP.6. Attend
to precision.
Additional Domain Information – Expressing Geometric Properties with Equations (G-GPE)
40
Key Vocabulary




Circle
Parabola
Focus
Center




Completing the Square
Directrix
Vertex
Opposite





Example Resources


Directed Line Segment
Distance Formula
Parallel
Perpendicular
Reciprocal
Books
 “Coordinate Proof” p. 712-717 Discovering Geometry
 Connected Mathematics Project 2 “Looking for Pythagoras”
 Focus in High School Mathematics: Reasoning and Sense Making, Geometry National Council of Teachers of Mathematics Publication
Technology
 http://www.learnzillion.com/lessons/280-derive-the-equation-of-a-circle-using-the-pythagorean-theorem A short video demonstrating how to derive
the equation of a circle using the Pythagorean theorem.
 http://www.themathpage.com/alg/pythagorean-distance.htm#origin This provides a resource for connecting the Pythagorean theorem and distance
formula.
 http://illustrativemathematics.org/standards/hs This website lists the standards and gives specific examples for the different concepts within the
standards.
 www.classzone.com/ This is the site to access the book and extra resources online.
 http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM.
 http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched to a
particular textbook.
 http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations.
 http://www.mathwarehouse.com/geometry/circle/equation-of-a-circle.php This site provides explanations and examples for working with circles.
 http://www.purplemath.com/modules/sqrcircle.htm Using completing the square with circles.
 http://www.purplemath.com/modules/parabola.htm This site provides information on parabolas.
Example Lessons
 http://www.nctm.org/uploadedFiles/Journals_and_Books/Books/FHSM/RSM-Task/Crow.pdf This activity shows how the distance formula and the
Pythagorean theorem are related in a real-life scenario. Students will find the distance between two cities by using coordinate points.
 http://www.winpossible.com/lessons/Geometry_Equation_of_a_circle_in_standard_form.html
This lesson includes a video and examples of graphing and finding equations of a circle.
Common Student Misconceptions

(
) (
2
)
2
Students commonly swap h and k when working with the equations for the circle. For example, if the are given the equation x - 3 + y - 4 = 25 , student will
say that the center is at (4,3).
41
(
) (
2
)
2
Students commonly forget to change the sign of h and k when finding the center of a circle. For example, if the are given the equation x - 3 + y - 4 = 25 ,
student will say that the center is at (-3,-4).
Students will make similar mistakes with h and k, when finding the vertex of a parabola.
(
) (
2
)
2
Students commonly forget to the square root of the constant to find the radius. For example, if the are given the equation x - 3 + y - 4 = 25 , student will say
that the radius is 25 instead of 5.
Students will often misuse the negative reciprocal slope with perpendicular lines. For example, they will take the reciprocal and forget the negative or change the
sign and forget the reciprocal.
42
Domain: Geometric Measurement and Dimension (2 Clusters)
Geometric Measurement and Dimension (G-GMD) (Domain 5 - Cluster 1 - Standards 1 and 3)
Explain volume formulas and use them to solve problems
Essential Concepts
 Informal arguments can be used to explain the formulas for the circumference
and area of a circle, volume of a cylinder, pyramid, and cone.
 Cavalieri’s principle can be used to explain the formulas for the volume of a
cylinder and other solid figures.
 Volume formulas for cylinders, pyramids, cones and spheres can be used to
solve real-world problems.
HS.G-GMD.A.1
HS.G-GMD.A.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.
Connections: 9-10.RST.4;
9-10.WHST.1c;
9-10.WHST.1e;
11-12.RST.4;
11-12.WHST.1c;
11-12.WHST.1e
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of others.
HS.MP.4. Model
with mathematics.
HS.MP.5. Use
appropriate tools
strategically.
Essential Questions




What constitutes an informal argument?
How is the formula for the circumference of a circle related to the formula for
the area of a circle?
What is Cavalieri’s principle and how is it used in informal arguments?
What is a real-world situation in which you would need to calculate the
volume of a cylinder? Pyramid? Cone? Sphere?
Examples & Explanations
Informal arguments for area and volume formulas can make use of the way in which area and volume scale
under similarity transformations: when one figure in the plane results from another by applying a similarity
transformation with scale factor k, its area is k2 times the area of the first. Similarly, volumes of solid figures
scale by k3 under a similarity transformation with scale factor k.
In middle school students work with informally deriving the formula for the area of a circle from the
circumference. Students also do work with finding the volume and surface area of prisms and cylinders. In 8th
grade students are finding volumes of cylinders, cones and spheres.
Cavalieri’s principle is if two solids have the same height and the same cross-sectional area at every level,
then they have the same volume.
Examples:
 Use the diagram below to give an informal argument for the formula for finding the area of a circle.
43

HS.G-GMD.A.3
HS.G-GMD.A.3
Use volume formulas for
cylinders, pyramids, cones, and
spheres to solve problems.
Connection: 9-10.RST.4
Mathematical
Practices
HS.MP.1. Make
sense of problems
and persevere in
solving them.
Prove that the right cylinder and the oblique cylinder have the same volume.
Examples & Explanations
Missing measures can include but are not limited to slant height, altitude, height, diagonal of a prism, edge
length, and radius.
Example:
 Determine the volume of the figure below.
HS.MP.2. Reason
abstractly and
quantitatively.
(Continued on next page)
44
Geometric Measurement and Dimension (G-GMD) (Domain 5 - Cluster 2 - Standard 4)
Visualize relationships between two-dimensional and three dimensional objects
Essential Concepts
 Cross-sections of three-dimensional objects create two-dimensional shapes.
 Three-dimensional objects can be created by rotating two-dimensional objects.
HS.G-GMD.B.4
HS.G-GMD.B.4
Identify the shapes of twodimensional cross-sections of
three-dimensional objects, and
identify three-dimensional
objects generated by rotations of
two-dimensional objects.
Mathematical
Practices
HS.MP.4. Model
with mathematics.
HS.MP.5. Use
appropriate tools
strategically.
Essential Questions




What two-dimensional shapes are created by taking cross-sections of a given
three-dimensional object?
What three-dimensional objects can be sliced to create a trapezoid, and how?
How do you create a circle or parabola by slicing a three-dimensional figure?
Give an example and explain how a three-dimensional object is created from
a two-dimensional object.
Examples & Explanations
Students may use geometric simulation software to model figures and create cross sectional views.
Example:


Identify the shape of the vertical, horizontal, and other cross sections of a cylinder.
Identify the shape of the vertical, horizontal, and other cross sections of a rectangular
45
Connection: ETHS-S1C2-01
prism.
Additional Domain Information – Geometric Measurement and Dimension (G-GMD)
Key Vocabulary




Circumference
Area
Volume
Cavalieri’s Principle
Example Resources







Cylinder
Cone
Pyramid
Sphere



Prism
Two dimensional crosssection
Altitude
Books
 “What’s the Capacity?” p. 122-125 Uncovering Student Thinking in Mathematics Grades 6-12
 “Developing a Formula for Volume” p. 63-76 Focus in High School Mathematics Geometry
 Focus in High School Mathematics: Reasoning and Sense Making, Geometry National Council of Teachers of Mathematics Publication
Technology
 http://illustrativemathematics.org/standards/hs This website lists the standards and gives specific examples for the different concepts within the
standards.
 www.classzone.com/ This is the site to access the book and extra resources online.
 http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM.
 http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched to a
particular textbook.
 http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations.
 http://math2.org/math/algebra/conics.htm This site offers a visual of how to create the conic sections by slicing a cone.
Example Lessons
 http://illuminations.nctm.org/LessonDetail.aspx?id=L797 This lesson can be used for students to discover the relationship between dimension and
volume. Students create two rectangular prisms and two cylinders to determine which holds more popcorn. Students then justify their observation by
analyzing the formulas and identifying the dimension(s) with the largest impact on the volume.
 http://www.shodor.org/interactivate/lessons/VolumeRectangular/ This lesson is designed to introduce students to the concept of volume and how to
find the volume of rectangular prisms.9999
 http://www.shodor.org/interactivate/lessons/VolumePrisms/ This lesson is designed to introduce students to the concept of volume and how to find the
46
volume of triangular and other shape prisms. This lesson is designed to follow the Volume of a Rectangular Prism lesson.
http://www.shodor.org/interactivate/lessons/ConicFlyer/ This lesson utilizes the geometric interpretations of the various conic sections to explain their
equations.
 http://www.shodor.org/interactivate/lessons/CrossSections/ This lesson utilizes the concepts of cross-sections of three-dimensional models to
demonstrate the derivation of two-dimensional shapes.
Common Student Misconceptions

Students will use the incorrect formula for a figure. For example, when finding the volume of a cone, students will drop the 1/3 and will end up finding the volume
of a cylinder.
Students confuse the notation used in the formulas. For example, students will confuse B (area of the base) for b (base dimension) in the formula for finding the
volume of a prism or pyramid.
Students will sometimes incorrectly use
p.
For example, students will eliminate the
p when the answer should be 5p .
Students will have difficulty identifying the base of a figure. For example, in a triangular prism they will use the rectangular side instead of the triangular base.
47
Domain: Modeling with Geometry (1 Cluster)
Modeling with Geometry (G-MG) (Domain 6 - Cluster 1 - Standards 1, 2 and 3)
Apply geometric concepts in modeling situations
Essential Concepts
 Geometric shapes and their properties can be used to model real-world objects.
 Density is the measure of a quantity per area or volume.
 Design and structure problems can be solved with geometry.
HS.G.MG.A.1
HS.G-MG.A.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).
Connections:
ETHS-S1C2-01;
9-10.WHST.2c
HS.G.MG.A.2
HS.G-MG.A.2
Apply concepts of density based
on area and volume in modeling
situations (e.g., persons per
square mile, BTUs per cubic
foot).
Connection: ETHS-S1C2-01
HS.G.MG.A.3
HS.G-MG.A.3
Apply geometric methods to
Mathematical
Practices
HS.MP.4. Model
with mathematics.
HS.MP.5. Use
appropriate tools
strategically.
HS.MP.7. Look for
and make use of
structure.
Mathematical
Practices
HS.MP.4. Model
with mathematics.
HS.MP.5. Use
appropriate tools
strategically.
Essential Questions




How can you model objects in your classroom as geometric shapes?
How is density related to area and volume?
How is a typographic grid used in design?
Give an example of a real-world problem that can be solved using geometry.
Examples & Explanations
Focus on situations that require relating two- and three- dimensional objects, determining and using volume,
and the trigonometry of general triangles.
Focus on situations in which the analysis of circles is required.
Students may use simulation software and modeling software to explore which model best describes a set of
data or situation.
Example:
 A cylinder can model a tree trunk or a human torso.

How can you model objects in your classroom as geometric shapes?
Examples & Explanations
Students may use simulation software and modeling software to explore which model best describes a set of
data or situation.
Example:
 Tucson has about one million people within approximately 195 square miles. What is Tucson’s
population density?
HS.MP.7. Look for
and make use of
structure.
Mathematical
Practices
Examples & Explanations
Students may use simulation software and modeling software to explore which model best describes a set of
48
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).
Connection: ETHS-S1C2-01
HS.MP.1. Make
sense of problems
and persevere in
solving them.
HS.MP.4. Model
with mathematics.
data or situation.
Examples:
 Design an object or structure to satisfy physical constraints or minimize cost.
Work with typographic grid systems based on ratios.

HS.MP.5. Use
appropriate tools
strategically.

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From: illustrativemathematics.org
Additional Domain Information – Modeling with Geometry (G-MG)
Key Vocabulary

Density

Typographic Grid System

Example Resources


Physical constraints
Books
 Focus in High School Mathematics: Reasoning and Sense Making, Geometry National Council of Teachers of Mathematics Publication
Technology
 http://illustrativemathematics.org/standards/hs This website lists the standards and gives specific examples for the different concepts within the standards.
 www.classzone.com/ This is the site to access the book and extra resources online.
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http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM
http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched to a particular
textbook.
 http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations.
 Example Lessons
 http://illuminations.nctm.org/LessonDetail.aspx?id=U175 This lesson takes a student’s prior knowledge on circumference and area to solve a real-life
situation.
Common Student Misconceptions


Students will use incorrect units of measurement. For example, they will use linear miles instead of square miles.
Students will combine different units of measure. For example, they may add something that is measured in square miles with something in square feet without
acknowledging the difference in units.
Students will refer to the wrong variable in a modeling problem. For example, students will maximize the area instead of the cost.
Assessment
Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments, daily checks for
understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation; summative assessments (e.g., unit
assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All district-adopted resources contain multiple assessment
tools and include online resources that can be used for the purposes delineated above.
PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills, concepts, and
understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The performance measure will be
administered as close to the end of the school year as possible. The second, an end of the year machine-scorable summative assessment, will be administered after
approximately 90% of the school year. These assessments are to begin during the SY 2014-2015.
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