STANDARDS FOR
MATHEMATICS
High School Geometry
1
High School Overview
Conceptual Categories and Domains
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)
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)
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
2
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.
3
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
4
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. Realworld 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.
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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 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.
Explanations and Examples
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.
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.
Explanations and Examples
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.
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, twoway 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.
Explanations and Examples
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.
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.
Explanations and Examples
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.
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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Essential Questions
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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.1.
HS.G-CO.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
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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.2
HS.G-CO.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)
15
HS.G-CO.3
HS.G-CO.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.4
HS.G-CO.4
Develop definitions of
rotations, reflections, and
translations in terms of
angles, circles, perpendicular
lines, parallel lines, and line
segments.
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)
(b)
(c)
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.
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.
Connections:
ETHS-S6C1-03;
9-10.WHST.4
HS.G-CO.5
HS.G-CO.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.
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 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.
HS.MP.5. Use
appropriate tools
(Continued on next page)
16
Connections:
ETHS-S6C1-03;
9-10.WHST.3
strategically.
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
Essential Questions
 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.6
HS.G-CO.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.




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
17
appropriate tools
strategically.
Connections:
ETHS-S1C2-01;
9-10.WHST.1e
HS.MP.7. Look
for and make use
of structure.
Students may use geometric software to explore the effects of rigid motion on figures.
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.
18
HS.G-CO.7
HS.G-CO.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.
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.
Connection: 9-10.WHST.1e
HS.G-CO.8
HS.G-CO.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
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 congruent? Explain how you know.
Mathematical
Practices
Examples & Explanations
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.
Example:
 Decide whether there is enough information to prove that the two shaded triangles are
congruent.
In the figure below, ABCD is a parallelogram.
(Continued on next page)
19
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
Essential Questions
 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.9.
HS.G-CO.9
Prove theorems about lines
and angles. Theorems
include: vertical angles are
congruent; 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.
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.
 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
two-column 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)
Students may use geometric simulations (computer software or graphing calculator) to explore
theorems about lines and angles.
HS.MP.5. Use
appropriate tools
strategically.
(Continued on next page)
Connections:
ETHS-S1C2-01;
9-10.WHST.1a-1e
20
Example:

HS.G-CO.10
HS.G-CO.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.
Prove that
ÐHIB @ ÐDJG , given that
AB // DE .
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
HS.G-CO.11
21
ÐABC @ ÐACB .
HS.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.
Connection:
9-10.WHST.1a-1e
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 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)
HS.MP.5. Use
appropriate tools
strategically.
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.
2. We have Ð AMD= Ð NDM since they are opposite interior angles of the transversal
MD through parallel lines AB and CD .
(Continued on next page)
22
3. 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 .
( ) ( )
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
Essential Questions
 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.12
HS.G-CO.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.



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.
(Continued on next page)
Connection:
23

ETHS-S6C1-03
Construct the perpendicular bisector of a line segment.
Image source: http://motivate.maths.org/content/accurate-constructions
HS.G-CO.13
HS.G-CO.13
Construct an equilateral
triangle, a square, and a
regular hexagon inscribed in
a circle.
Connection: ETHS
-S6C1-03
Mathematical
Practices
Examples & Explanations
HS.MP.5. Use
appropriate tools
strategically.
Students may use geometric software to make geometric constructions.
HS.MP.6. Attend
to precision.
Example:

Construct a regular hexagon inscribed in a circle.
This construction can also be used to draw a 120° angle.
1. Keep your compasses to the same setting
throughout this construction.
2. Draw a circle.
3. Mark a point, P, on the circle.
4. Put the point of your compasses on P and draw arcs
to cut the circle at Q and U.
5. Put the point of your compasses on Q and draw an
arc to cut the circle at R.
6. Repeat with the point of the compasses at R and S to
draw arcs at S and T.
7. Join PQRSTU to form a regular hexagon.
8. 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
24
Additional Domain Information – Congruence (G-CO)
Key Vocabulary








Angle
Circle
Perpendicular Lines
Parallel Lines
Line Segment
Corresponding Angles
Midpoint
Altitude
















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
Example Resources



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.
25
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.
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.
26
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
Essential Questions
 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.1
HS.G-SRT.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
Mathematical
Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
HS.MP.5. Use
appropriate tools
strategically.






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:
a. 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.
b. The dilation of a line
segment is longer or
27
shorter in the ratio given
by the scale factor.
HS.G-SRT.2
HS.G-SRT.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.3
HS.G-SRT.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
Examples & Explanations
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.
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.5. Use
appropriate tools
strategically.
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?
28
Similarity, Right Triangles, and Trigonometry (G-SRT) (Domain 2 - Cluster 2 - Standards 4, and 5)
Prove theorems involving similarity
Essential Concepts
Essential Questions
 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.
 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.4
HS.G-SRT.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.




Why does a line cutting a triangle parallel to one side of the
triangle divide the other two proportionally?
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
= ,
=
a c
b
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
29
HS.G-SRT.5
HS.G-SRT.5
Use congruence and
similarity criteria for triangles
to solve problems and to
prove relationships in
geometric figures.
Connections:
ETHS-S1C2-01;
9-10.WHST.1a-1e
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.
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.
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
Essential Questions





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.6
HS.G-SRT.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.


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)
30
sine of θ = sin θ =
opposite
hypotenuse
adjacent
cosine of θ = cos θ =
hypotenuse
opposite
tangent of θ = tan θ =
adjacent
HS.G-SRT.7
HS.G-SRT.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.8
HS.G-SRT.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.
cosecant of θ = csc θ =
hypotenuse
opposite
hypotenuse
secant of θ = sec θ =
adjacent
adjacent
cotangent of θ = cot θ =
opposite
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 problems.
(Continued on next page)
HS.MP.4. Model
with
mathematics.
31
HS.MP.5. Use
appropriate tools
strategically.
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.
Similarity, Right Triangles, and Trigonometry (G-SRT) (Domain 2 - Cluster 4 - Standards 9, 10, and 11)
Apply trigonometry to general triangles
Essential Concepts
Essential Questions




The area of any triangle can be written as a function of the lengths of
two sides and the angle between them.
The measures of the sides and angles of any triangle are related
through the Law of Sines and the Law of Cosines.
The Law of Sines and the Law of Cosines can be used to solve all
triangles in applied problems.
HS.G-SRT.9
HS.G-SRT.9
Derive the formula A = ½ ab
sin(C) for the area of a
triangle by drawing an
auxiliary line from a vertex
perpendicular to the opposite
side.
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.



How can you derive the formula A = ½ ab sin(C) for the area of
triangle?
How can you prove the Law of Sines? The Law of Cosines?
How can you use the Law of Sines or Law of Cosines to solve realworld problems?
How do you determine unknown values in any triangle from given
information?
Examples & Explanations
With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and
cosine must be extended to obtuse angles.
Connection: ETHS-S6C1-03
HS.MP.7. Look
for and make use
of structure.
Example:
Find the area of the following triangle. Generalize by finding the area in terms of the side and angle
labels instead of using the values.
32
HS.G-SRT.10
HS.G-SRT.10
Prove the Laws of Sines and
Cosines and use them to
solve problems.
Connections:
ETHS-S6C1-03;
11-12.WHST.1a-1e
Mathematical
Practices
Examples & Explanations
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.
HS.MP.6. Attend
to precision.
HS.MP.7. Look
for and make use
of structure.
HS.MP.8. Look
for and express
regularity in
repeated
reasoning.
HS.G-SRT.11
HS.G-SRT.11
Understand and apply the
Law of Sines and the Law of
Cosines to find unknown
measurements in right and
non-right triangles (e.g.,
surveying problems, resultant
forces).
Mathematical
Practices
HS.MP.1. Make
sense of
problems and
persevere in
solving them.
Law of Sines:
sin(A) sin(B ) sin(C )
.
=
=
a
b
c
Law of Cosines: c 2 = a 2 + b 2 - 2ab cos(C ) .
Examples & Explanations
Example:
 Tara wants to fix the location of a mountain by taking measurements from two positions 3
miles apart. From the first position, the angle between the mountain and the second position
is 78o. From the second position, the angle between the mountain and the first position is
53o. How can Tara determine the distance of the mountain from each position and what is
the distance from each position?
(Continued on next page)
33
Connections:
11-12.WHST.2c;
11-12.WHST.2e
HS.MP.4. Model
with
mathematics.
Additional Domain – Similarity, Right Triangles, and Trigonometry (G-SRT)
Key Vocabulary




Sine
Cosine
Tangent
Dilation







Law of Sines
Law of Cosines
Resultant Forces
Scale Factor
Pythagorean Theorem
Complementary Angles
Supplementary Angles
Example Resources



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."
34

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.
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.
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.
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?
35
HS.G-C.1
HS.G-C.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.
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.
HS.G-C.2
HS.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.
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.
HS.MP.5. Use
appropriate tools
strategically.
(Continued on next page)
36
Connections:
9-10.WHST.1c;
11-12.WHST.1c

Find the unknown length in the picture below.
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 lengths of QR and KR.
x 2 = 16 ×10
So for the example above:
x 2 = 160
x = 160 = 4 10 » 12.6

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?
37
HS.G-C.3
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.
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.
Examples & Explanations
Example:
 Given the inscribed quadrilateral below prove that
 B is supplementary to  D.
Connection:
ETHS-S6C1-03
HS.G-C.4
HS.G-C.4
Construct a tangent line from
a point outside a given circle
to the circle.
Connection: ETHS-S6C1-03
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 simulation software to make geometric constructions
Example:
 To construct a tangent line to circle C:
1. Draw a point P outside of the circle.
2. Connect center of the circle C with point P and construct the perpendicular bisector of
the segment. Label the point where the perpendicular bisector and the segment meet M.
3. With M as the center draw a half circle through P and C.
4. Construct a point at the intersection of the half
circle and circle C, label this point N.
5. Draw a line through P and N, Line PN is the
tangent to circle C.
38
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
Essential Questions
 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.5
HS.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.
Mathematical
Practices
HS.MP.2 Reason
abstractly and
quantitatively.
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.




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?
Connections:
ETHS-S1C2-01;
11-12.RST.4
39
Additional Domain Information – Circles (G-C)
Key Vocabulary





Radius
Diameter
Chord
Central Angles
Inscribed Angles








Circumscribed Angles
Tangent
Secant
Inscribed Circle
Circumscribed Circle
Arc
Radian
Sector
Example Resources

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.
40
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
Essential Questions
 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.1
HS.G-GPE.1
Derive the equation of a circle
of given center and radius
using the Pythagorean
theorem; complete the square
to find the center and radius
of a circle given by an
equation.
Connections:
ETHS-S1C2-01;
11-12.RST.4
HS.G-GPE.2
HS.G-GPE.2
Derive the equation of a
parabola given a focus and
directrix.
Connections:
ETHS-S1C2-01;
11-12.RST.4
Mathematical
Practices
HS.MP.7. Look
for and make use
of structure.
HS.MP.8. Look
for and express
regularity in
repeated
reasoning.
Mathematical
Practices
HS.MP.7. Look
for and make use
of structure.
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?
Examples & Explanations
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
The standard form of a circle is x - h + y - k
)
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.
Examples & Explanations
The directrix should be parallel to a coordinate axis.
Students may use geometric simulation software to explore parabolas.
HS.MP.8. Look
for and express
(Continued on next page)
41
regularity in
repeated
reasoning.
The conic forms of a parabola is
regular: 4p(y - k) = (x - h)2
sideways: 4p(x - h) = (y - k)2
the distance between the focus and the directrix.
, where the vertex is (h,k) and p is
Examples:
 Write and graph an equation for a parabola with focus (2, 3) and directrix y = 1.
(
) (
)
2

Given the equation 20 y - 5 = x + 3 , find the focus, vertex and directrix.

Solution: The vertex is at (-3,5) and to find the vertex we know that the constant of the
unsquared term is 20. Since 4p = 20 then p=5. The focus is 5 units above the vertex at
(-3,5+5) or (-3,10). The directrix is 5 units below the vertex so y = 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
Essential Questions
 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.





HS.G-GPE.4
HS.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
Mathematical
Practices
HS.MP.3 Reason
abstractly and
quantitatively.
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?
Examples & Explanations
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.
(Continued on next page)
42
the circle centered at the
origin and containing the
point (0, 2).
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.5
HS.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).
Connection:
9-10.WHST.1a-1e
HS.G-GPE.6
HS.G-GPE.6
Find the point on a directed
line segment between two
given points that partitions the
segment in a given ratio.
Connections:
ETHS-S1C2-01;
9-10.RST.3

Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.
Prove or disprove that the point (1, √3) lies on the circle centered at the origin and
containing the point (0, 2).
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.
HS.MP.8. Look
for and express
regularity in
repeated
reasoning.
Example:
 Find the equation of a line perpendicular to 3x + 5y = 15 through the point (-3,2).
Mathematical
Practices
Examples & Explanations
HS.MP.2.
Reason
abstractly and
quantitatively.
HS.MP.8. Look
for and express
regularity in
repeated
reasoning.
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.
o
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.
43
HS.G-GPE.7
HS.G-GPE.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.
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.
HS.MP.5. Use
appropriate tools
strategically.
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)
Key Vocabulary




Circle
Parabola
Focus
Center









Completing the Square
Directrix
Vertex
Opposite
44
Directed Line Segment
Distance Formula
Parallel
Perpendicular
Reciprocal
Example Resources

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).
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)
2
2
= 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.
45
(
) (
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.
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
Essential Questions
 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.1
HS.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.
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.




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 k 2 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.
HS.MP.5. Use
appropriate tools
strategically.
(Continued on next page)
46
Examples:
 Use the diagram below to give an informal argument for the formula for finding the area of a
circle.

HS.G-GMD.3
HS.G-GMD.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)
47
Geometric Measurement and Dimension (G-GMD) (Domain 5 - Cluster 2 - Standard 4)
Visualize relationships between two-dimensional and three dimensional objects
Essential Concepts
Essential Questions
 Cross-sections of three-dimensional objects create two-dimensional
shapes.
 Three-dimensional objects can be created by rotating two-dimensional
objects.




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.
HS.G-GMD.4
HS.G-GMD.4
Identify the shapes of twodimensional cross-sections of
three-dimensional objects,
and identify threedimensional objects
Mathematical
Practices
Examples & Explanations
Students may use geometric simulation software to model figures and create cross sectional views.
HS.MP.4. Model
with
mathematics.
(Continued on next page)
48
generated by rotations of twodimensional objects.
HS.MP.5. Use
appropriate tools
strategically.
Example:


Connection: ETHS-S1C2-01
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
prism.
Additional Domain Information – Geometric Measurement and Dimension (G-GMD)
Key Vocabulary




Circumference
Area
Volume
Cavalieri’s Principle






Cylinder
Cone
Pyramid
Sphere

Prism
Two dimensional crosssection
Altitude
Example Resources



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
49
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 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.
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
Essential Questions
 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.1
HS.G-MG.1
Use geometric shapes, their
measures, and their
properties to describe objects
(e.g., modeling a tree trunk or
Mathematical
Practices
HS.MP.4. Model
with
mathematics.




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.
(Continued on next page)
50
a human torso as a cylinder).
Connections:
ETHS-S1C2-01;
9-10.WHST.2c
HS.G.MG.2
HS.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).
Connection: ETHS-S1C2-01
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.
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.
HS.G.MG.3
HS.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).
Connection: ETHS-S1C2-01
Mathematical
Practices
HS.MP.1. Make
sense of
problems and
persevere in
solving them.
Examples & Explanations
Students may use simulation software and modeling software to explore which model best
describes a set of 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.4. Model
with
mathematics.
HS.MP.5. Use
appropriate tools
(Continued on next page)
51
strategically.

From: illustrativemathematics.org
Additional Domain Information – Modeling with Geometry (G-MG)
Key Vocabulary

Density


Typographic Grid System
Physical constraints
Example Resources

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.
 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
52
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.
53
High School Geometry
Conceptual Category: Statistics and Probability (2 Domains, 3 Clusters)
Domain: Conditional Probability and the Rules of Probability (2 Clusters)
Conditional Probability and the Rules of Probability (S-CP) (Domain 1 - Cluster 1 - Standards 1, 2, 3, 4 and 5)
Understand independence and conditional probability and use them to interpret data (Link to data from simulations or experiments)
Essential Concepts
Essential Questions
 A sample space is the set of all possible outcomes.
 An event is a subset of the sample space.
 An event can be the complement, union, or intersection of other events.
 Two events A and B are independent if the probability of A and B
occurring together is the product of their probabilities.
 The conditional probability of A given B is P(A ∩ B)/P(B).
 Two events A and B are independent if the conditional probability of A
given B is the same as the probability of A.
 A two-way frequency table defines a sample space, which can be used
to find conditional probabilities and determine dependence or
independence of events.
HS.S-CP.1
HS.S-CP.1
Describe events as subsets
of a sample space (the set of
outcomes) using
characteristics (or categories)
of the outcomes, or as
unions, intersections, or
complements of other events
(“or,” “and,” “not”).
Mathematical
Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
HS.MP.4. Model
with
mathematics.





Give an example of a sample space and describe several events
based on that sample space.
Explain the connection between unions, intersections, and
complements of sets and the probabilities of events.
How can you determine if two events are independent?
How can you use two-way tables to find conditional probabilities?
Explain conditional probability and independence in your own words.
Examples & Explanations
Build on work with two-way tables from S-ID.5 to develop understanding of conditional probability
and independence.
Intersection: The intersection of two sets A and B is the set of elements that are common to both
set A and set B. It is denoted by A ∩ B and is read ‘A intersect B’.


A ∩ B in the diagram below is {1, 5}
this means: BOTH/AND
Connection: 11-12.WHST.2e
HS.MP.6. Attend
to precision.
HS.MP.7. Look
(Continued on next page)
54
for and make use
of structure.
U
A
B
1
2
5
3
7
4
8
HS.S-ID.1
Union: The union of two sets A and B is the set of elements that are in A or in B or in both. It is
denoted by A U B and is read ‘A union B’.



A U B in the diagram is {1, 2, 3, 4, 5, 7}
this means: EITHER/OR/ANY
could be both
Complement: The complement of the set A U B is the set of elements that are members of the
universal set U but are not in A U B. It is denoted by (A U B )'.
(A U B )' in the diagram is {8}.
HS.S-CP.2
HS.S-CP.2
Understand that two events A
and B are independent if the
probability of A and B
occurring together is the
product of their probabilities,
and use this characterization
to determine if they are
independent.
Mathematical
Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
Examples & Explanations
Two events, A and B, are independent if the probability of them occurring together is the product of
the probabilities, P(A)•P(B). Event A doesn’t affect the probability of event B occurring.
Example:
 What is the probability of getting tails on a coin flip and then rolling a 3 on a standard 6sided die?
HS.MP.4. Model
with
(Continued on next page)
55
mathematics.
Connection: 11-12.WHST.1e
HS.MP.6. Attend
to precision.
HS.MP.7. Look
for and make use
of structure.
HS.S-CP.3
HS.S-CP.3
Understand the conditional
probability of A given B as
P(A and B)/P(B), and
interpret independence of A
and B as saying that the
conditional probability of A
given B is the same as the
probability of A, and the
conditional probability of B
given A is the same as the
probability of B.
Connections:
11-12.RST.5;
11-12.WHST.1e
HS.S-CP.4
HS.S-CP.4
Construct and interpret twoway frequency tables of data
when two categories are
associated with each object
being classified. Use the twoway table as a sample space
to decide if events are
independent and to
approximate conditional
probabilities. For example,
collect data from a random
sample of students in your
Mathematical
Practices
Examples & Explanations
HS.MP.2.
Reason
abstractly and
quantitatively.
Example:
 A drawer contains 13 blue socks and 13 red socks. If you choose two socks from the
drawer in the dark, what is the probability that the second sock will be blue, given that the
first was blue?
HS.MP.4. Model
with
mathematics.
HS.MP.6. Attend
to precision.
HS.MP.7. Look
for and make use
Mathematical
Practices
HS.MP.1. Make
sense of
problems and
persevere in
solving them.
HS.MP.2.
Reason
abstractly and
quantitatively.
Examples & Explanations
Students may use spreadsheets, graphing calculators, and simulations to create frequency tables
and conduct analyses to determine if events are independent or determine approximate conditional
probabilities.
Example:
 Collect data from a random sample of students in your school on their favorite subject
among math, science, history and English. Estimate the probability that a randomly selected
student from your school will favor science given that the student is in tenth grade. Do the
same for other subjects and compare the results.
(Continued on next page)
56
school on their favorite
subject among math, science,
and English. Estimate the
probability that a randomly
selected student from your
school will favor science
given that the student is in
tenth grade. Do the same for
other subjects and compare
the results.
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.
Connections:
ETHS-S6C2-03;
11-12.RST.4; 11-12.RST.9;
11-12.WHST.1e
HS.MP.5. Use
appropriate tools
strategically.

A two-way frequency table is shown below displaying the relationship between age and
baldness. We took a sample of 100 male subjects and determined who is or is not bald. We
also recorded the age of the male subjects by categories.
Bald
HS.MP.4. Model
with
mathematics.
No
Yes
Total
Two-way Frequency Table
Age
Younger than 45
45 or older
35
11
24
30
59
41
Total
46
54
100
What is the probability that a man from the sample is bald, given that he is under 45?
HS.MP.6. Attend
to precision.
HS.MP.7. Look
for and make use
of structure.
HS.MP.8. Look
for and express
regularity in
repeated
reasoning.
HS.S-CP.5
HS.S-CP.5
Recognize and explain the
concepts of conditional
probability and independence
in everyday language and
everyday situations. For
example, compare the
chance of having lung cancer
if you are a smoker with the
chance of being a smoker if
you have lung cancer.
Mathematical
Practices
HS.MP.1. Make
sense of
problems and
persevere in
solving them.
HS.MP.4. Model
with
mathematics.
Examples & Explanations
Examples:
 What is the probability of drawing a heart from a standard deck of cards on a second draw,
given that a heart was drawn on the first draw and not replaced? Are these events
independent or dependent?

At Johnson School, the probability that a student takes computer science and French is
0.062. The probability that a student takes computer science is 0.43. What is the probability
that a student takes French given that the student is taking computer science?
(Continued on next page)
57
Connections: 11-12.RST.4;
11-12.RST.5;
11-12.WHST.1e
HS.MP.6. Attend
to precision.
HS.MP.8. Look
for and express
regularity in
repeated
reasoning.
Conditional Probability and the Rules of Probability (S-CP) (Domain 1 - Cluster 2 - Standards 6, 7, 8 and 9)
Use the rules of probability to compute probabilities of compound events in a uniform probability model.
Essential Concepts
Essential Questions
 The conditional probability of A given B is the fraction of B’s outcomes
that also belong to A.
 The Addition Rule states that P(A U B) = P(A) + P(B) – P(A ∩ B).
 The Multiplication Rule states that P(A U B) = P(A)P(B|A) = P(B)P(A|B).
 Permutations and combinations can be used to compute probabilities
and solve problems.
HS.S-CP.6
HS.S-CP.6
Find the conditional
probability of A given B as the
fraction of B’s outcomes that
also belong to A, and
interpret the answer in terms
of the model.
Connections:
ETHS-S1C2-01;
ETHS-S6C2-03;
11-12.RST.9;
11-12.WHST.1b;
11-12.WHST.1e;
Mathematical
Practices
HS.MP.1. Make
sense of
problems and
persevere in
solving them.


How can you use the rules of probability to solve real-world problems?
How can you use permutations and combinations to compute
probabilities?
Examples & Explanations
Students could use graphing calculators, simulations, or applets to model probability experiments
and interpret the outcomes.
Example:
 A teacher gave her class two quizzes. 30% of the class passed both quizzes and 60% of
the class passed the first quiz. What percent of those who passed the first quiz also passed
the second quiz?
HS.MP.4. Model
with
mathematics.
HS.MP.5. Use
appropriate tools
strategically.
HS.MP.7. Look
for and make use
of structure.
58
HS.S-CP.7
HS.S-CP.7
Apply the Addition Rule, P(A
or B) = P(A) + P(B) – P(A and
B), and interpret the answer
in terms of the model.
Connections:
ETHS-S1C2-01;
ETHS-S6C2-03; 11-12.RST.9
Mathematical
Practices
HS.MP.4. Model
with
mathematics.
HS.MP.5. Use
appropriate tools
strategically.
Examples & Explanations
Students could use graphing calculators, simulations, or applets to model probability experiments
and interpret the outcomes.
Example:
 In a math class of 32 students, 18 are boys and 14 are girls. On a unit test, 5 boys and 7
girls made an A grade. If a student is chosen at random from the class, what is the
probability of choosing a girl or an A student?
HS.MP.6. Attend
to precision.
HS.MP.7. Look
for and make use
of structure.
HS.S-CP.8
HS.S-CP.8
Apply the general
Multiplication Rule in a
uniform probability model,
P(A and B) = P(A)P(B|A) =
P(B)P(A|B), and interpret the
answer in terms of the model.
Connections:
ETHS-S1C2-01;
ETHS-S6C2-03;
11-12.RST.9
Mathematical
Practices
HS.MP.4. Model
with
mathematics.
HS.MP.5. Use
appropriate tools
strategically.
Examples & Explanations
Students could use graphing calculators, simulations, or applets to model probability experiments
and interpret the outcomes.
Example:
 In a certain town, all of the children eat either peanut butter sandwiches or cheese
sandwiches for lunch. 4/9 of the boys prefer cheese and 1/4 of the girls prefer peanut
butter. 3/5 of the children are boys. What is the probability that a child chosen at random
will be a girl who prefers peanut butter?
HS.MP.6. Attend
to precision.
HS.MP.7. Look
for and make use
of structure.
HS.S-CP.9
HS.S-CP.9
Use permutations and
combinations to compute
probabilities of compound
Mathematical
Practices
Examples & Explanations
Students may use calculators or computers to determine sample spaces and probabilities.
HS.MP.1. Make
sense of
(Continued on next page)
59
events and solve problems.
Connections:
ETHS-S1C2-01;
ETHS-S6C2-03; 11-12.RST.
problems and
persevere in
solving them.
Example:
 You and two friends go to the grocery store and you each buy a soda. If there are five
different kinds of soda, and each of you is equally likely to buy each variety, what is the
probability that no one buys the same kind?
HS.MP.2.
Reason
abstractly and
quantitatively.
HS.MP.4. Model
with
mathematics.
HS.MP.5. Use
appropriate tools
strategically.
HS.MP.7. Look
for and make use
of structure.
Domain: Using Probability to Make Decisions (1 Cluster)
Using Probability to Make Decisions (S-MD) (Domain 2 - Cluster 1 - Standards 6 and 7)
Use probability to evaluate outcomes of decisions (Introductory; apply counting rules.)
Essential Concepts
Essential Questions
 Probabilities can be used to make fair decisions.
 Probabilities can be used to analyze and evaluate decisions and
strategies.
HS.S-MD.6
HS.S-MD.6
Use probabilities to make fair
decisions (e.g., drawing by
lots, using a random number
generator).
Mathematical
Practices
HS.MP.1. Make
sense of problems
and persevere in
solving them.




Why is drawing by lot or using a random number generator a fair way
to make decisions?
How can you determine if a game is fair?
How can you evaluate the results of a diagnostic test?
Give an example of an event in which probabilities are used to make a
decision, and explain how the probability is used.
Examples & Explanations
This work sets the stage for work in Algebra 2, where the ideas of statistical inference are
introduced. Evaluating the risks associated with conclusions drawn from sample data (i.e.
incomplete information) requires an understanding of probability concepts.
(Continued on next page)
60
Connections:
ETHS-S1C2-01;
ETHS-S6C2-03;
11-12.RST.3; 11-12.RST.9;
11-12.WHST.1b;
11-12.WHST.1e
HS.MP.2. Reason
abstractly and
quantitatively.
HS.MP.3. Construct
viable arguments
and critique the
reasoning of others.
HS.MP.4. Model
with mathematics.
Students may use graphing calculators or programs, spreadsheets, or computer algebra systems
to model and interpret parameters in linear, quadratic or exponential functions.
Example:
 There are 80 freshmen at a high school that want to participate in a free throw contest
during halftime at the next basketball game. There will only be time for five students to
participate. Besides numbering students and pulling a number from a hat, in what other
way(s) can the five students be chosen fairly?
HS.MP.5. Use
appropriate tools
strategically.
HS.MP.7. Look for
and make use of
structure.
HS.S-MD.7
HS.S-MD.7
Analyze decisions and
strategies using probability
concepts (e.g., product
testing, medical testing,
pulling a hockey goalie at the
end of a game).
Connections:
ETHS-S1C2-01;
ETHS-S6C2-03
Mathematical
Practices
HS.MP.1. Make
sense of problems
and persevere in
solving them.
HS.MP.2. Reason
abstractly and
quantitatively.
Examples & Explanations
Students may use graphing calculators or programs, spreadsheets, or computer algebra systems
to model and interpret parameters in linear, quadratic or exponential functions.
Example:
 Suppose that a blood test indicates the presence of a particular disease 97% of the time
when the disease is actually present. The same test gives false positive results 0.25% of
the time. Suppose that one percent of the population actually has the disease. If our
blood test is positive, how likely is it that we actually have the disease?
HS.MP.3. Construct
viable arguments
and critique the
reasoning of others.
HS.MP.4. Model
with mathematics.
(Continued on next page)
61
HS.MP.5. Use
appropriate tools
strategically.
HS.MP.7. Look for
and make use of
structure.
Additional Information – Probability
Key Vocabulary





Sample space
Complement
Dependent event
Outcomes
Combinations








Event
Union
Independent event
Conditional probability
Subset
Intersection
Venn diagram
Permutations
Example Resources



Books
 Textbook
 Focus in High School Mathematics: Reasoning and Sense Making in Statistics and Probability, 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.
 http://mathforum.org/probstat/probstat.lessons.html This website has a bank of questions that are answered.
 http://www.socr.ucla.edu/
 www.Khanacademy.org Khan Academy contains useful video lessons and also a galaxy of practice modules that enable students to
check answers online.
 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.
Example Lessons
 http://www.shodor.org/interactivate/lessons/ConditionalProb/ This lesson is based on several interesting problems. Each problem has a
somewhat unexpected answer; in fact, many people have a hard time accepting experimental results for these problems, as the results
may seem counterintuitive. This very difference in expectations and actual results leads to a deeper consideration of the related
mathematics and to acquiring new tools for solving problems, namely the ideas and formulas connected with conditional probability and
probability of simultaneous events.
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http://www.shodor.org/interactivate/lessons/SetsTheVennDiagram/ This lesson is designed to introduce students to the idea of a set and
what it means to be contained in a set. Students will experiment with sets in conjunction with the Venn Diagram.
 http://www.shodor.org/interactivate/lessons/ProbabilityGeometry/ The activity and two discussions of this lesson connect probability and
geometry. The Polyhedra discussion leads to platonic solids, and the Probability and Geometry discussion leads to connections between
angles, areas and probability. The subtle difference between defining probability by counting outcomes and defining probability by
measuring proportions of geometrical characteristics is brought to light.
Common Student Misconceptions
Students commonly do not use the addition rule properly. In the formula P(A U B) = P(A) + P(B) – P(A ∩ B), students tend to leave off the last term.
Students commonly do not understand the difference between, and when to use, permutations or combinations. Students commonly calculate
the number of permutations instead of the combinations and vice versa.
Students commonly do not understand how sample sets can be used to answer probability questions.
Students commonly do not understand the difference between dependent and independent events. For example, when you have to spin a
spinner to get blue and get a 5 on a dice, they do not see these as dependent events because of the word and.
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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