STANDARDS FOR
MATHEMATICS
High School Algebra 1
1
High School Overview
Conceptual Categories and Domains
Number and Quantity




The Real Number System (N-RN)
Quantities (N-Q)
The Complex Number System (N-CN)
Vector and Matrix Quantities (N-VM)
Algebra




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




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






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
5
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
6
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
7
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:
•
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.
•
Planning a table tennis tournament for 7 players at a club with 4 tables, where each player plays against each other player.
•
Designing the layout of the stalls in a school fair so as to raise as much money as possible.
•
Analyzing stopping distance for a car.
•
Modeling savings account balance, bacterial colony growth, or investment growth.
•
Engaging in critical path analysis, e.g., applied to turnaround of an aircraft at an airport.
•
Analyzing risk in situations such as extreme sports, pandemics, and terrorism.
•
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.
8
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 (★).
9
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.
10
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.
11
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.
12
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.
13
High School Algebra 1
Conceptual Category: Number and Quantity (2 Domains, 3 Clusters)
Domain: Real Number System (2 Clusters)
The Real Number System (N-RN) (Domain 1 - Cluster 1 - Standards 1 and 2)
Extend the properties of exponents to rational exponents.
Essential Concepts
Essential Questions
 Rational exponents are exponents that are fractions.
 Properties of integer exponents extend to properties of rational
exponents.
 Properties of rational exponents are used to simplify and create
equivalent forms of numerical expressions.
 Rational exponents can be written as radicals, and radicals can be
written as rational exponents.
HS.N-RN.1
HS.N-RN.1.
Explain how the definition of
the meaning of rational
exponents follows from
extending the properties of
integer exponents to those
values, allowing for a notation
for radicals in terms of
rational exponents. For
example, we define 51/3 to be
the cube root of 5 because
we want (51/3)3 = 5(1/3)3 to
hold, so (51/3)3 must equal 5.
HS.N-RN.2
HS.N-RN.2.
Rewrite expressions involving
radicals and rational
exponents using the
properties of exponents.
Mathematical
Practices
HS.MP.2. Reason
abstractly and
quantitatively.

How do you use properties of rational exponents to simplify and create
equivalent forms of numerical expressions?
Why are rational exponents and radicals related to each other?
Given an expression with a rational exponent, how do you write the
equivalent radical expression?


Examples & Explanations
In implementing the standards in curriculum, these standards should occur before discussing
exponential functions with continuous domains.
Students may explain orally or in written format.
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.
Example:
 For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to
hold, so (51/3)3 must equal 5.
Mathematical
Practices
Examples & Explanations
HS.MP.7. Look for
and make use of
structure.
Examples:
2

3
2
5 = 53 ; 53 = 5
2
3
14
2
(Continued on next page)
4

Rewrite using fractional exponents: 5 16

Rewrite
x
x2
Solution: x
-
=
1
x

Rewrite
4
4
in at least three alternate forms.
3
2
5
= 2 = 25
2 -4
3
2
=
1
x
3
=
1
x x
Using only rational exponents.
The Real Number System (N-RN) (Domain 1 - Cluster 2 - Standard 3)
Use properties of rational and irrational numbers.
Essential Concepts
Essential Questions
 When you perform an operation with two rational numbers you will
produce a rational number.
 When you perform an operation with a nonzero rational and an irrational
number you will produce an irrational number.
HS.N-RN.3
HS.N-RN.3.
Explain why the sum or
product of two rational
numbers are rational; that the
sum of a rational number and
an irrational number is
irrational; and that the product
of a nonzero rational number
and an irrational number is
irrational.
Connection: 9-10.WHST.1e
Mathematical
Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.

Explain what type of number is produced and why when each of the
four arithmetic operations is performed on two rational numbers.
Explain what type of number is produced and why when each of the
four operations is performed on a rational number and an irrational
number.

Examples & Explanations
Since every difference is a sum and every quotient is a product, this includes differences and
quotients as well. Explaining why the four operations on rational numbers produce rational numbers
can be a review of students understanding of fractions and negative numbers. Explaining why the
sum of a rational and an irrational number is irrational, or why the product is irrational, includes
reasoning about the inverse relationship between addition and subtraction (or between multiplication
and addition).
Connect N.RN.3 to physical situations, e.g., finding the perimeter of a square of area 2.
Example:
 Explain why the number 2π must be irrational, given that π is irrational.
Answer: if 2π were rational, then half of 2π would also be rational, so π would have to be
rational as well.
Additional Domain Information – The Real Number System (N-RN)
Key Vocabulary
 Rational number


Irrational number
15
Rational exponents
Example Resources



Books
 Building Powerful Numeracy for Middle and High School Students, by Pamela Weber Harris.
 http://www.classzone.com/cz/find_book.htm?tmpState=AZ&disciplineSchool=ma_hs&state=AZ&x=24&y=24 This contains supplementary
resources for the Arizona adopted math books.
Technology
 http://nlvm.usu.edu/en/nav/topic_t_2.html - Provides teachers or students with virtual manipulatives to interact with the concepts.
 http://www.khanacademy.org/ - Khan Academy contains useful video lessons and also a galaxy of practice modules that enable students to
check answers online.
 http://www.classzone.com/books/algebra_1/page_build.cfm?content=lesson8_kh_ch11&ch=11 Help with graphing calculators and rational
functions.
 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.
 www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards.
Example Lessons
For Fractional Exponents:
 http://www.khanacademy.org/math/algebra/exponents-radicals/v/radical-equivalent-to-rational-exponents This is a short video lesson on
converting radicals to fractional exponent notation.
 http://www.purplemath.com/modules/exponent5.htm This lesson is a good basic introduction to the concept with limited examples. It
contains a useful technology extension on using calculators for building conceptual understanding of rational exponents.
 http://www.themathpage.com/alg/rational-exponents.htm#fractional This lesson includes laws of exponents but moves on to basic equations
-3
with rational exponents of the form
x5
=
1
8
For Operations Bringing Together Rational and Irrational Numbers:
 http://www.schooltube.com/video/b5ad397dc525a3795373/ This provides the steppingstones for understanding how adding or subtracting
rational and irrational numbers yields an irrational answer.
Common Student Misconceptions
1
Students may see a fractional exponent and multiply it by the base. For example, students might say
1
1
27 3 = 27· = 9 instead of 27 3 = 3 27 = 3 .
3
Students may see a negative exponent and do the same, converting the base to a negative number instead of a fraction.
16
Students may have difficulty converting between radical notation and fractional exponent notation:
b
-m
n
=
1
n
bm
. They might confuse the m
and the n.
Students tend to assume that they can combine integers and radical expressions: Example,
3+ 3 becoming 6 .
Students conversely don’t apply available laws of exponents when multiplying or dividing radicals: Example
18
3
= 6 . They don’t
understand that they can split up one radical into the product of two component radicals.
Domain: Quantities (1 Cluster)
Quantities (N-Q) (Domain 2 - Cluster 1 - Standards 1, 2 and 3)
Reason quantitatively and use units to solve problems. (Foundation for work with expressions, equations and functions.)
Essential Concepts
Essential Questions
 Units and unit relationships can be used to set up and solve multi-step
problems.
o Make sure units are compatible when creating,
simplifying/evaluating, and solving equations.
 Appropriate units or quantities need to be used when answering realworld situations.
o Use labels to put the answers into proper context.
 Working with expressions, equations, relations and functions can be
facilitated by understanding the quantities and their relationships.
 Graphs should be set up with the appropriate scales and units for the
given context.
 Level of accuracy is dependent on the limitations of measurement within
the context of the real-world problem.
HS.N-Q.1
HS.N-Q.1
Use units as a way to
understand problems and to
guide the solution of multistep problems; choose and
interpret units consistently in
formulas; choose and
interpret the scale and the
Mathematical
Practices
HS.MP.4. Model
with
mathematics.





How can you convert a given quantity in a unit rate to a different unit
rate? For example, how can you convert feet per second to miles per
hour?
Why would you want to be able to convert quantities to different units?
How can units and unit relationships be used to set up and solve
multi-step problems?
Give an example of a real-world situation and explain what unit or
quantity you expressed the answer in and why.
How can you determine which scale and unit to use when creating a
graph to represent a set of data?
Examples & Explanations
Working with quantities and the relationships between them provides grounding for work with
expressions, equations, and functions.
Include word problems where quantities are given in different units, which must be converted to
make sense of the problem.
HS.MP.5. Use
appropriate tools
(Continued on next page)
17
origin in graphs and data
displays.
strategically.
HS.MP.6. Attend
to precision.
Example:
 A problem might have one object moving 12 feet per second and another at 5 miles per
hour. To compare speeds, students convert 12 feet per second to miles per hour:
12 ft 1 mile 60sec 60min 43200 mile
·
·
·
=
» 8.18 miles per hour.
1 sec 5280 ft 1 min 1 hr
5280 hr

HS.N-Q.2
HS.N-Q.2
Define appropriate quantities
for the purpose of descriptive
modeling.
Mathematical
Practices
HS.MP.4. Model
with
mathematics.
Graphical representations and data displays include, but are not limited to: line graphs,
circle graphs, histograms, multi-line graphs, scatter plots, and multi-bar graphs.
Examples & Explanations
Examples:
 What type of measurements would you use to determine your income and expenses for one
month?
 How could you express the number of accidents in Arizona?
HS.MP.6. Attend
to precision.
HS.N-Q.3
HS.N-Q.3
Choose a level of accuracy
appropriate to limitations on
measurement when reporting
quantities.
Mathematic
al Practices
HS.MP.5. Use
appropriate
tools
strategically.
HS.MP.6.
Attend to
precision.
Examples & Explanations
The margin of error and tolerance limit varies according to the measure, tool used, and context.
Example:
 Determining price of gas by estimating to the nearest cent is appropriate because you will not
$ 3 . 479
pay in fractions of a cent, but the cost of gas is given to tenths of a cent, e.g.,
.
gallon
Additional Domain Information – Quantities (N-Q)
Key Vocabulary


Unit
Descriptive model




Unit rate
Ratio
18
Scale
Equivalent


Origin
Unit Conversion
Example Resources



Books
 Textbook
 Focus in High School Mathematics: Reasoning and Sense Making (Algebra), Chapter 3: Formal Algebra
 Uncovering Student Thinking in Mathematics Grades 6-12, How Low Can You Go pg 71
 The Xs and Whys of Algebra: Key Ideas and Common Misconceptions
Technology
 http://www.wolframalpha.com/examples/Math.html Useful for checking correct conversions.
 http://nlvm.usu.edu/en/nav/frames_asid_272_g_4_t_4.html?open=instructions&from=search.html?qt=unit+conversion Useful site with virtual
practice problems.
 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.
 www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards.
Example Lessons
 http://oakroadsystems.com/math/convert.htm#Really1 A nicely thorough introduction to the basics of unit conversion, with practice
problems at the end.
 http://www.virtualnerd.com/pre-algebra/ratios-proportions/rates-word-problem-solution.php The video lesson describes how to use unit
conversion to solve word problems, labeling each step of the process carefully.
 http://www.khanacademy.org/math/arithmetic/basic-ratios-proportions/v/unit-conversion This video lesson on unit conversion explains in
detail how metric unit breakdown is used to arrive at different units of the same quantity, and makes a great cross-curricular connection with
science.
Common Student Misconceptions
Students often have difficulty understanding how ratios expressed in different units can be equal to one. For example,
5280 ft
is simply one,
1 mile
and it is permissible to multiply by that ratio.
Students need to make sure to put the quantities in the numerator or denominator so that the terms can cancel appropriately. Example:
Convert 140 ft. to miles. In this case they often put 5280 ft in the numerator rather than in the denominator.
Students often do not understand that the scale on a graph must be marked in equal intervals. For example, if a table gives the values 1, 3, 4, 9,
then students will label constant intervals on their axis with 1, 3, 4, 9, rather than 1 through 9.
19
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.
20
High School Algebra 1
Conceptual Category: Algebra (4 Domains, 8 Clusters)
Domain: Seeing Structure in Expressions (2 Clusters)
Seeing Structure in Expressions (A-SSE) ) (Domain 1 - Cluster 1 - Standards 1 and 2)
Interpret the structure of expressions. (Linear, exponential and quadratic)
Essential Concepts
Essential Questions
 Expressions consist of terms (parts being added or subtracted).
 Terms can either be a constant, a variable with a coefficient, or a
coefficient times a variable raised to a power.
 Real-world problems with changing quantities can be represented by
expressions with variables.
 The relationship between the abstract symbolic representations of
expressions can be identified based on how they relate to the given
situation.
 Complicated expressions can be interpreted by viewing parts of the
expression as single entities.
 Structure within an expression can be identified and used to factor or
simplify the expression.
HS.A-SSE.1
HS.A-SSE.1
Interpret expressions that
represent a quantity in terms
of its context.
a.
Interpret parts of an
expression, such as terms,
factors, and coefficients.
Connection: 9-10.RST.4
b.
Interpret complicated
expressions by viewing one
or more of their parts as a
single entity. For example,
Mathematical
Practices
HS.MP.1. Make
sense of
problems and
persevere in
solving them.
HS.MP.2.
Reason
abstractly and
quantitatively.






Give an example of a real-world problem and write an expression to
model the relationship, and explain how the algebraic symbols
represent the words in the problem.
How are coefficients and factors related to each other?
How does viewing a complicated expression by its single parts help to
interpret and solve problems?
What does it mean to call something a quantity?
How does using the structure of an expression help to simplify the
expression?
Why would you want to simplify an expression?
Examples & Explanations
A-SSE.1 starts by being limited to linear expressions and to exponential expressions with integer
exponents. Later in the year, focus on quadratic and exponential expressions.
A-SSE.1b starts with exponents that are integers and then extends from the integer exponents to
rational exponents, focusing on those that represent square or cube roots.
Students should understand the vocabulary for the parts that make up the whole expression and be
able to identify those parts and interpret their meanings in terms of a context.
Examples:
 Interpret P(1+r)n as the product of P and a factor not depending on P.
HS.MP.4. Model
with
(Continued on next page)
21
interpret P(1+r)n as the
product of P and a factor
not depending on P.

mathematics.
HS.MP.7. Look
for and make use
of structure.

Suppose the cost of cell phone service for a month is represented by the expression 0.40s
+ 12.95. Students can analyze how the coefficient of 0.40 represents the cost of one minute
(40¢), while the constant of 12.95 represents a fixed, monthly fee, and s stands for the
number of cell phone minutes used in the month. Similar real-world examples, such as tax
rates, can also be used to explore the meaning of expressions.
Factor 3x(x – 5) + 2(x – 5).
Solution: The “x – 5” is common to both expressions being added, so it can be factored out
by the distributive property. The factorization is (3x + 2)(x – 5).
HS.A-SSE.2
HS.A-SSE.2
Use the structure of an
expression to identify ways to
rewrite it. For example, see
x4 – y4 as (x2)2 – (y2)2, thus
recognizing it as a difference
of squares that can be
factored as
(x2 – y2)(x2 + y2).
Mathematic
al Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
HS.MP.7. Look
for and make
use of
structure.
Examples & Explanations
Students should extract the greatest common factor (whether a constant, a variable, or a combination
of each). If the remaining expression is quadratic, students should factor the expression further.
Example:
 Factor x 3 - 2x 2 - 35x .

See x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored
as (x2 – y2)(x2 + y2). Note that the first factor can be factored further.
Seeing Structure in Expressions (A-SSE) (Domain 1 - Cluster 2 - Standard 3)
Write expressions in equivalent forms to solve problems. (Quadratic and exponential)
Essential Concepts
Essential Questions
 The solutions of quadratic equations are the x-intercepts of the parabola
or zeros of quadratic functions.
 Factoring methods and the method of completing the square reveal
attributes of the graphs of quadratic functions.
 Factoring a quadratic reveals the zeros of the function.
 Completing the square in a quadratic equation reveals the maximum or
minimum value of the function.
 Properties of exponents are used to transform expressions for
exponential functions.




22
What are the solutions to a quadratic equation and how do they relate
to the graph?
What attributes of the graph will factoring and completing the square
reveal about a quadratic function?
How are properties of exponents used to transform expressions for
exponential functions?
Why would you want to transform an expression for an exponential
function?
HS.A-SSE.3
HS.A-SSE.3
Choose and produce an
equivalent form of an
expression to reveal and
explain properties of the
quantity represented by the
expression.
Connections: 9-10.WHST.1c;
11-12.WHST.1c
a. Factor a quadratic
expression to reveal the
zeros of the function it
defines.
b. Complete the square in a
quadratic expression to
reveal the maximum or
minimum value of the
function it defines.
c. Use the properties of
exponents to transform
expressions for
exponential functions. For
example the expression
1.15t can be rewritten as
(1.151/12)12t ≈ 1.01212t to
reveal the approximate
equivalent monthly
interest rate if the annual
rate is 15%.
Mathematical
Practices
HS.MP.1. Make
sense of
problems and
persevere in
solving them.
HS.MP.2.
Reason
abstractly and
quantitatively.
HS.MP.4. Model
with
mathematics.
HS.MP.7. Look
for and make use
of structure.
Examples & Explanations
It is important to balance conceptual understanding and procedural fluency in work with equivalent
expressions. For example, development of skill in factoring and completing the square goes handin-hand with understanding what different forms of a quadratic expression reveal.
Students will use the properties of operations to create equivalent expressions.
Teachers should foster the idea that changing the forms of expressions, such as factoring or
completing the square, or transforming expressions from one exponential form to another, are not
independent algorithms that are learned for the sake of symbol manipulations. They are processes
that are guided by goals (e.g., investigating properties of families of functions and solving contextual
problems).
A pair of coordinates (h, k) from the general form f(x) = a(x – h)2 + k represents the vertex of the
parabola, where h represents a horizontal shift and k represents a vertical shift of the parabola y =
x2 from its original position at the origin. A vertex (h, k) is the minimum point of the graph of the
quadratic function if a > 0 and is the maximum point of the graph of the quadratic function if a < 0.
Understanding an algorithm for completing the square provides a solid foundation for deriving a
quadratic formula.
Examples:
 Express 2(x3 – 3x2 + x – 6) – (x – 3)(x + 4) in factored form and use your answer to say for
what values of x the expression is zero.

The expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate
equivalent monthly interest rate if the annual rate is 15%.

Write the expression below as a constant multiplied by a power of x and use your answer to
decide whether the expression gets larger or smaller as x gets larger.
(2 x 3 )2 (3x 4 )
( x 2 )3
23
Additional Domain Information – Seeing Structure in Expressions (A-SSE)
Key Vocabulary




Expression
Factor
Simplify
Polynomial








Term
Exponent
Greatest Common Factor
Binomial
Coefficient
Base
Quadratic
Trinomial




Vertex
Completing the Square
Minimum
Maximum
Example Resources



Books
 Textbook
 Focus in High School Mathematics: Reasoning and Sense Making (Algebra), Chapter 3: Formal Algebra
 Uncovering Student Thinking in Mathematics Grades 6-12, How Low Can You Go pg 71
 The Xs and Whys of Algebra: Key Ideas and Common Misconceptions
Technology
 Key Curriculum Press, Exploring Algebra I with the Geometer’s Sketchpad
 www.Geogebra.org online software to create visuals
 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.
 www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards
Example Lessons
 http://illuminations.nctm.org/LessonDetail.aspx?id=L761. Predicting your financial future. Students use their knowledge of exponents to
compute an investment’s worth using a formula and a compound interest simulator. Students also use the simulator to analyze credit
card payments and debt.
 http://illuminations.nctm.org/Lessons/PowerUp/PowerUp-AS-Voltmeter.pdf. Power up. Students explore addition of signed numbers by
placing batteries end to end (in the same direction or opposite directions) and observe the sum of the batteries’ voltages.
 http://www.nctm.org/uploadedFiles/Journals_and_Books/Books/FHSM/RSM-Task/Horseshoes.pdf Students analyze the structure of
algebraic expressions and a graph to determine what information each expression readily contributes about the flight of a horseshoe.
This task is particularly relevant to students who are studying (or have studied) various quadratic expressions (or functions). The task
also illustrates a step in the mathematical modeling process that involves interpreting mathematical results in a real-world context.
 http://www.geogebra.org/cms/ Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and
using technology for more complicated cases.
 http://www.uen.org/Lessonplan/preview.cgi?LPid=26843 Students will identify linear and nonlinear relationships in a variety of contexts.
24
Common Student Misconceptions
Students will often combine terms that are not like terms. For example, 2 + 3x = 5x or 3x + 2y = 5xy.
Students sometimes forget the coefficient of 1 when adding like terms. For example, x + 2x + 3x = 5x rather than 6x.
Students will change the degree of the variable when adding/subtracting like terms. For example, 2x + 3x = 5x2 rather than 5x.
Students will forget to distribute to all terms when multiplying. For example, 6 (2x + 1) = 12x + 1 rather than 12x + 6.
Students may not follow the Order of Operations when simplifying expressions. For example, 4x2 when x = 3 may be incorrectly evaluated as 4•32
= 122 = 144, rather than 4•9 = 36. Another common mistake occurs when the distributive property should be used prior to adding/subtracting. For
example, 2 + 3( x – 1) incorrectly becomes 5(x – 1) = 5x – 5 instead of 2 + 3(x – 1) = 2 + 3x – 3 = 3x – 1.
Students fail to use the property of exponents correctly when using the distributive property. For example, 3x(2x – 1) = 6x – 3x = 3x instead of
simplifying as 3x ( 2x – 1) = 6x2 – 3x.
Students fail to understand the structure of expressions. For example, they will write 4x when x = 3 is 43 instead of 4x = 4•x so when x = 3, 4x = 4•3
= 12. In addition, students commonly misevaluate –32 = 9 rather than –32 = –9. Students routinely see –32 as the same as (–3)2 = 9. A method that may
clear up the misconception is to have students rewrite as –x2 = –1•x2 so they know to apply the exponent before the multiplication of –1.
Students frequently attempt to “solve” expressions. Many students add “= 0” to an expression they are asked to simplify. Students need to
understand the difference between an equation and an expression.
Students commonly confuse the properties of exponents, specifically the product of powers property with the power of a power property. For
example, students will often simplify (x2)3 = x5 instead of x6.
Students will incorrectly translate expressions that contain a difference of terms. For example, 8 less than 5 times a number is often incorrectly
translated as 8 – 5n rather than 5n – 8.
25
Domain: Arithmetic with Polynomials and Rational Expressions (1 Cluster)
Arithmetic with Polynomials and Rational Expressions (A-APR) ) (Domain 2 - Cluster 1 – Standard 1)
Perform arithmetic operations on polynomials. (Linear and quadratic)
Essential Concepts
Essential Questions
 Adding, subtracting and multiplying two polynomials will yield another
polynomial, thus making the system of polynomials closed.
 Addition and subtraction of polynomials is combining like terms.
 The distributive property proves why you can combine like terms.
 Multiplication of polynomials is applying the distributive property.

Why is the system of polynomials closed under addition, subtraction
and multiplication?
How is the system of polynomials similar to and different from the
system of integers?
How does the distributive property show that you can combine like
terms?
Explain how the distributive property is used to multiply any size
polynomials.



HS.A-APR.1
HS.A-APR.1
Understand that polynomials
form a system analogous to
the integers, namely, they are
closed under the operations
of addition, subtraction, and
multiplication; add, subtract,
and multiply polynomials.
Mathematical
Practices
HS.MP.8. Look for
regularity in
repeated
reasoning.
Examples & Explanations
Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive
integer power of x.
Connection: 9-10.RST.4

In arithmetic of polynomials, a central idea is the distributive property, because it is
fundamental not only in polynomial multiplication but also in polynomial addition and
subtraction. With the distributive property, there is little need to emphasize misleading
mnemonics, such as FOIL, which is relevant only when multiplying two binomials, and the
procedural reminder to “collect like terms” as a consequence of the distributive property.
For example, when adding the polynomials 3x and 2x, the result can be explained with the
distributive property as follows: 3x + 2x = (3 + 2)x = 5x.
Additional Domain Information – Arithmetic with Polynomials and Rational Expressions (A-APR)
Key Vocabulary





Expression
Simplify
Polynomial
Distributive Property
Exponential










Term
Exponent
Binomial
Trinomial
Quadratic
26
Coefficient
Base
Factor
Linear
Closure Property

Example Resources



Books
 Textbook
 Focus in High School Mathematics: Reasoning and Sense Making (Algebra), Chapter 3: Formal Algebra
 The Xs and Whys of Algebra: Key Ideas and Common Misconceptions
Technology
 Key Curriculum Press, Exploring Algebra I with the Geometer’s Sketchpad
 www.Geogebra.org online software to create visuals
 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.
 www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards
Example Lessons
 http://illuminations.nctm.org/Lessons/PowerUp/PowerUp-AS-Voltmeter.pdf Power up. Students explore addition of signed numbers by
placing batteries end to end (in the same direction or opposite directions) and observe the sum of the batteries’ voltages.
 http://wpmu.bionicteaching.com/kmspruill/2009/12/06/lesson-plan-adding-and-subtracting-polynomials/ This lesson includes a clip of how
math is used in computer graphics in the movies along with a PowerPoint presentation on adding and subtracting polynomials.
 http://www.discoveryeducation.com/teachers/free-lesson-plans/rational-number-concepts.cfm
Common Student Misconceptions
Students often forget to distribute the subtraction to terms other than the first one. For example, students will write (4x + 3) – (2x + 1) = 4x + 3 –
2x + 1 = 2x + 4 rather than 4x + 3 – 2x – 1 = 2x + 2.
Students will change the degree of the variable when adding/subtracting like terms. For example, 2x + 3x = 5x2 rather than 5x.
Students may not distribute the multiplication of polynomials correctly and only multiply like terms. For example, they will write (x + 3)(x – 2) =
x2 – 6 rather than x2 – 2x + 3x – 6.
27
Domain: Creating Equations (1 Cluster)
Creating Equations (A-CED) (Domain 3 – Cluster 1 – Standards 1, 2, 3 and 4)
Creating equations that describe numbers or relationships. [Linear, quadratic and exponential (integer inputs only); A-CED.3 linear only]
Essential Concepts
Essential Questions
 Equations and inequalities can be created to represent and solve realworld and mathematical problems.
 Relationships between two quantities can be represented through the
creation of equations in two variables and graphed on coordinate axes
with labels and scales.
 Solutions are viable or not in different situations depending upon the
constraints of the given context.
 Formulas can be rearranged and solved for a given variable using the
same reasoning as solving an equation.







HS.A-CED.1
HS.A-CED.1
Create equations and
inequalities in one variable
and use them to solve
problems. Include equations
arising from linear, quadratic
and exponential functions.
Mathematical
Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
How do you translate real-world situations into mathematical
equations and inequalities?
How do you determine if a situation is best represented by an
equation, an inequality, a system of equations or a system of
inequalities?
Why would you want to create an equation or inequality to represent a
real-world problem?
How are graphs of equations and inequalities similar and different?
How do you determine if a given point is a viable solution to a system
of equations or inequalities, both on a graph and using the equations?
Why would you want to solve a given formula for a particular variable?
How do you solve a given formula for a particular variable?
Examples & Explanations
Limit A-CED.1 and A-CED.2 to linear and exponential equations, and, in the case of exponential
equations, limit to situations requiring evaluation of exponential functions at integer inputs.
Start with work on linear and exponential equations, then, later in the year, extend to quadratic
equations.
HS.MP.4. Model
with
mathematics.
Equations can represent real-world and mathematical problems. Include equations and inequalities
that arise when comparing the values of two different functions, such as one describing linear
growth and one describing exponential growth.
HS.MP.5. Use
appropriate tools
strategically.
Examples:
 Given that the following trapezoid has area 54 cm 2, set up an equation to find the length
of the unknown base, and solve the equation.
(Continued on next page)
28

Lava coming from the eruption of a volcano follows a parabolic path. The height h in
feet of a piece of lava t seconds after it is ejected from the volcano is given by
h(t )  16t 2  64t  936 . After how many seconds does the lava reach its maximum
height of 1000 feet?

The value of an investment over time is given by the equation A(t) = 10,000(1.03)t.
What does each part of the equation represent?
Solution: The $10,000 represents the initial value of the investment. The 1.03 means
that the investment will grow exponentially at a rate of 3% per year for t years.

You bought a car at a cost of $20,000. Each year that you own the car the value of the
car will decrease at a rate of 25%. Write an equation that can be used to find the value
of the car after t years.
Solution: C(t) = $20,000(0.75)t . The base is 1 – 0.25 = 0.75 and is between 0 and 1,
representing exponential decay. The value of $20,000 represents the initial cost of the
car.

An amount of $100 was deposited in a savings account on January 1st in each of the
years 2010, 2011, 2012, and so on to 2020, with an annual yield of 7%. What will be the
balance in the savings account at the end of the day on January 1, 2020? In your
solution, illustrate the use of a formula for a geometric series when S n represents the
value of the geometric series with the first term g, constant ratio r ≠ 1, and n + 1 terms.
Before using the formula, it might be reasonable to demonstrate the way the formula is
derived.
Solution:
Sn
Multiply by r:
rSn
Subtract:
Sn – rSn
Factor:
Sn(1 – r)
Divide by (1 – r): Sn
=
=
=
=
=
g + gr + gr2 + gr3 + … + grn
gr + gr2 + gr3 + … + grn + grn+1
g – grn+1
g(1 – rn+1)
g(1 – rn+1)/(1 – r)
The amount of the investment on January 1, 2020 can be found using: 100(1.07) 10 +
100(1.07)9 + … + 100(1.07) + 100. If the first term of this geometric series is g = 100,
the ratio is 1.07, and n = 10, the formula for the value of the geometric series gives S 10
= $1578.36 to the nearest cent.
29
HS.A-CED.2
HS.A-CED.2
Create equations in two or
more variables to represent
relationships between
quantities; graph equations
on coordinate axes with
labels and scales.
Mathematical
Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
HS.MP.4. Model
with
mathematics.
HS.MP.5. Use
appropriate tools
strategically.
HS.A-CED.3
HS.A-CED.3
Represent constraints by
equations or inequalities, and
by systems of equations
and/or inequalities, and
interpret solutions as viable or
non-viable options in a
modeling context. For
example, represent
inequalities describing
nutritional and cost
constraints on combinations
of different foods.
Mathematical
Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
HS.MP.4. Model
with
mathematics.
HS.MP.5. Use
appropriate tools
strategically.
Examples & Explanations
Limit A-CED.1 and A-CED.2 to linear and exponential equations, and, in the case of exponential
equations, limit to situations requiring evaluation of exponential functions at integer inputs.
Start with work on linear and exponential equations, then, later in the year, extend to quadratic
equations.
Example:
 The formula for the surface area of a cylinder is given by V = πr2h, where r represents the
radius of the circular cross-section of the cylinder and h represents the height. Choose a
fixed value for h and graph V vs. r. Then pick a fixed value for r and graph V vs. h.
Compare the graphs. What is the appropriate domain for r and h? Be sure to label your
graphs and use an appropriate scale.
Examples & Explanations
Limit A-CED.3 to linear equations and inequalities.
Examples:
 A club is selling hats and jackets as a fundraiser. Their budget is $1500 and they want to
order at least 250 items. They must buy at least as many hats as they buy jackets. Each hat
costs $5 and each jacket costs $8.
o Write a system of inequalities to represent the situation.
o Graph the inequalities.
o If the club buys 150 hats and 100 jackets, will the conditions be satisfied?
o What is the maximum number of jackets they can buy and still meet the conditions?

Represent inequalities describing nutritional and cost constraints on combinations of
different foods.
30
HS.A-CED.4
HS.A-CED.4
Rearrange formulas to
highlight a quantity of interest,
using the same reasoning as
in solving equations. For
example, rearrange Ohm’s
law V = IR to highlight
resistance R.
Mathematical
Practices
HS.MP.2. Reason
abstractly and
quantitatively.
HS.MP.4. Model
with mathematics.
HS.MP.5. Use
appropriate tools
strategically.
Examples & Explanations
Start by limiting A-CED.4 to formulas that are linear in the variable of interest. Later in the year,
extend to formulas involving squared variables.
Examples:
 The Pythagorean theorem expresses the relation between the legs a and b of a right
triangle and its hypotenuse c with the equation a2 + b2 = c2.
o Why might the theorem need to be solved for c?
o Solve the equation for c and write a problem situation where this form of the equation
might be useful.
4 3
p r for radius r.
3

Solve V =

Motion can be described by the formula below, where t = time elapsed, u = initial velocity,
a = acceleration, and s = distance traveled.
HS.MP.7. Look for
and make use of
structure.
s = ut+½at2
o
o

Why might the equation need to be rewritten in terms of a?
Rewrite the equation in terms of a.
Rearrange Ohm’s law V = IR to highlight resistance R.
Additional Domain Information – Creating Equations (A-CED)
Key Vocabulary





Expression
Simplify
Solution
Zeros
Graph










Equation
Linear
Exponent
X-intercept
System
Inequality
Quadratic
Y-intercept
Formula
Coordinate axes
Example Resources

Books





Textbook
Focus in High School Mathematics: Reasoning and Sense Making (Algebra), Chapter 2: Building Equations and Functions
The Xs and Whys of Algebra: Key Ideas and Common Misconceptions
Mission Mathematics II: Grades 9-12, Modeling Space-Debris Accumulation pg 28
Active Algebra: Strategies and Lessons for Successfully Teaching Linear Relationships, Section II: Linear Relations: Lessons and
Assessments
31



Technology
 Key Curriculum Press, Exploring Algebra I with the Geometer’s Sketchpad
 www.Geogebra.org online software to create visuals
 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.
 www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards
Example Lessons
 www.smartskies.nasa.gov/lineup. LineUp with Math. This simulation is an interactive online simulator featuring air traffic control
problems in a realistic route structure with 2 to 5 planes. Teacher guide, student website, and teacher website with materials included on
website.
 http://illuminations.nctm.org/LessonDetail.aspx?ID=L713. Students work collaboratively to come up with a bargaining plan to trick the raja
into feeding the village using algebra, exponential growth, and estimation.
 http://www.uen.org/Lessonplan/preview.cgi?LPid=19825 Students go car shopping online, investigate the relationship between variables
such as interest rate and monthly payment, develop two payment plans using online loan calculators, write a slope-intercept equation for
each plan, and create a graph and table for the equations using a graphing calculator.
Common Student Misconceptions
Students may interchange slope and y-intercept when creating equations. For example, a taxi cab costs $4 for a dropped flag and charges $2 per
mile. Students may fail to see that $2 is a rate of change and is slope while the $4 is the starting cost and incorrectly write the equation as y = 4x + 2
instead of y = 2x + 4.
Given a graph of a line, students use the x-intercept for b instead of the y-intercept.
Given a graph, students incorrectly compute slope as run over rise rather than rise over run. For example, they will compute slope with the
change in x over the change in y.
Students do not know when to include the “or equal to” bar when translating the graph of an inequality.
Students do not correctly identify whether a situation should be represented by a linear, quadratic, or exponential function.
Students often do not understand what the variables represent. For example, if the height h in feet of a piece of lava t seconds after it is ejected
from a volcano is given by h(t) = -16t2 + 64t + 936 and the student is asked to find the time it takes for the piece of lava to hit the ground, the student will
have difficulties understanding that h = 0 at the ground and that they need to solve for t.
32
Students have difficulties rearranging formulas to highlight a different quantity. For example, many students will not see that solving 5x = 10 by
dividing both sides by 5 is the same as solving for b in the equation ab = c by dividing both sides by a.
Domain: Reasoning with Equations and Inequalities (4 Clusters)
Reasoning with Equations and Inequalities (A-REI) (Domain 4 – Cluster 1 – Standard 1)
Understand solving equations as a process of reasoning and explain the reasoning (Master linear; learn as general principle)
Essential Concepts
Essential Questions
 Equations are solved as a process of reasoning using properties of
operations and equality, which can justify each step of the process.
 A solution to an equation can be checked, by substituting in that value
for the variable and simplifying to see if the equation holds true.




HS.A-REI.1
HS.A-REI.1
Explain each step in solving a
simple equation as following
from the equality of numbers
asserted at the previous step,
starting from the assumption
that the original equation has
a solution. Construct a viable
argument to justify a solution
method.
Mathematical
Practices
HS.MP.2. Reason
abstractly and
quantitatively.
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.
HS.MP.7. Look for
and make use of
structure.
What do you use to justify your reasoning when solving an equation?
How do you determine if an equation is solved properly?
How do you determine and justify if a solution to an equation is
correct?
Why are properties of real numbers important when solving
equations?
Examples & Explanations
Students should focus on and master A-REI.1 for linear equations and be able to extend and apply
their reasoning to other types of equations for later in the year and in future courses. Students will
solve exponential equations with logarithms in Algebra II.
Properties of operations can be used to change expressions on either side of the equation to
equivalent expressions. In addition, adding the same term to both sides of an equation or
multiplying both sides by a non-zero constant produces an equation with the same solutions. Other
operations, such as squaring both sides, may produce equations that have extraneous solutions.
Each step of solving an equation can be defended, much like providing evidence for steps of a
geometric proof.
Provide examples for how the same equation might be solved in a variety of ways as long as
equivalent quantities are added or subtracted to both sides of the equation; the order of steps taken
will not matter.
(Continued on next page)
Examples:
33

Explain why the equation x/2 + 7/3 = 5 has the same solutions as the equation 3x + 14 =
30. Does this mean that x/2 + 7/3 is equal to 3x + 14?

Show that x = 2 and x = –3 are solutions to the equation x 2  x  6 . Write the equation in
a form that shows these are the only solutions, explaining each step in your reasoning.

Transform 2x – 5 = 7 to 2x = 12 and tell what property of equality was used.
2x  5  7
Solution: 2 x  5  5  7  5 Addition property of equality .
2 x  12
Reasoning with Equations and Inequalities (A-REI) (Domain 4 – Cluster 2 – Standards 3 and 4)
Solve equations and inequalities in one variable. Linear inequalities; linear equations with letter coefficients; quadratics with real solutions
Essential Concepts
Essential Questions
 Equations and inequalities are solved using properties of operations,
equality, and inequality, which can justify each step of the process.
 A solution to an equation can be checked, by substituting in that value
for the variable and simplifying to see if the equation or inequality holds
true.
 Laws of exponents can be used to solve simple exponential equations.
 Completing the square can be used to transform a quadratic equation
into the form (x-p)2 = q.
 The quadratic formula can be derived by completing the square of ax2+
bx+c = 0.
 Quadratic equations can be solved by a variety of methods, for
example: by inspection, graphing, taking square roots, factoring,
completing the square and the quadratic formula.
 Quadratic equations can have extraneous and/or complex solutions.
HS.A-REI.3
HS.A-REI.3
Solve linear equations and
inequalities in one variable,
including equations with
coefficients represented by
letters starting from the
assumption that the original
equation has a solution.
Mathematical
Practices
HS.MP.2
Reason
abstractly and
quantitatively.






What do you use to justify your steps when solving linear and nonlinear equations and inequalities?
How do you determine and justify whether a solution to an equation or
inequality is correct?
How do operations performed on real numbers affect the relationship
between the quantities in an inequality?
Why would you want to transform a quadratic equation to the form
(x-p)2 = q?
How do you determine which method is best for solving a quadratic
equation?
Why do some quadratic equations have extraneous and/or complex
solutions?
Examples & Explanations
Extend earlier work with solving linear equations to solving linear inequalities in one variable and to
solving literal equations that are linear in the variable being solved for. Include simple exponential
equations that rely only on application of the laws of exponents, such as 5x=125 or 2x=1/16.
(Continued on next page)
HS.MP.7. Look
34
Construct a viable argument
to justify a solution method.
for and make use
of structure.
Examples:
Solve for the variable:

HS.MP.8. Look
for and express
regularity in
repeated
reasoning.




HS.A-REI.4
HS.A-REI.4
Solve quadratic equations in
one variable.
a. Use the method of
completing the square to
transform any quadratic
equation in x into an
equation of the form (x –
p)2 = q that has the same
solutions. Derive the
quadratic formula from
this form.
b.
Solve quadratic
equations by inspection
(e.g., for x2 = 49), taking
square roots, completing
the square, the quadratic
formula and factoring, as
appropriate to the initial
form of the equation.
Recognize when the
quadratic formula gives
complex solutions and
write them as a ± bi for
real numbers a and b.
Mathematical
Practices
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.
HS.MP.5. Use
appropriate tools
strategically.
HS.MP.7. Look
for and make use
of structure.
7
- y - 8 = 111
3
3x > 9
ax + 7 = 12, when a = 2
3+ x x -9
=
7
4
Solve for x: 2/3x + 9 < 18
Examples & Explanations
Students should solve by factoring, completing the square, and using the quadratic formula. The
zero product property is used to explain why the factors are set equal to zero. Students should
relate the value of the discriminant to the type of root to expect. A natural extension would be to
relate the type of solutions to ax2 + bx + c = 0 to the behavior of the graph of y = ax2 + bx + c.
Value of Discriminant
Nature of Roots
Nature of Graph
b2 – 4ac = 0
intersects x-axis once
1 real root
b2 – 4ac > 0
intersects x-axis twice
2 real roots
b2 – 4ac < 0
does not intersect x-axis
2 complex roots
Examples:
 Are the roots of 2x2 + 5 = 2x real or complex? How many roots does it have? Find all
solutions of the equation.
 What is the nature of the roots of x2 + 6x + 10 = 0? Solve the equation using the
quadratic formula and completing the square. How are the two methods related?

Projectile motion problems, in which the initial conditions establish one of the solutions as
extraneous within the context of the problem.
o An object is launched at 14.7 meters per second (m/s) from a 49-meter tall platform.
The equation for the object's height s at time t seconds after launch is s(t) = –4.9t2 +
14.7t + 49, where s is in meters. When does the object strike the ground?
0 = -4.9t 2 +14.7t + 49
Solution: 0 = t - 3t -10
2
0 = (t + 2)(t - 5)
HS.MP.8. Look
for and express
(Continued on next page
35
So the solutions for t are t = 5 or t = –2, but t = –2 does not make sense in the context of
this problem and therefore is an extraneous solution.
regularity in
repeated
reasoning.
Students should learn of the existence of the complex number system, but will not solve quadratics
with complex solutions until Algebra II.
In Algebra 1, student should be able to recognize when the solution to a quadratic equation yields a
complex solution; however writing the solution in the complex form a ± bi for real numbers a and b
will be addressed in Algebra II.
Reasoning with Equations and Inequalities (A-REI) (Domain 4 – Cluster 3 – Standards 5, 6 and 7)
Solve systems of equations (Linear-linear and linear-quadratic)
Essential Concepts
Essential Questions
 A system of linear equations can have one solution, infinitely many
solutions, or no solution.
 A system of linear equations can be solved graphically, algebraically
using elimination/linear combination, substitution, or modeling.
 Solving a system of equations algebraically yields an exact solution;
solving by graphing yields an approximate solution.
 Multiplying both sides of an equation by a non-zero constant does not
change the solution to the equation.
 Elimination/linear combination is a method of solving a system of linear
equations in which the equations are added together in order to
eliminate a variable.
 In elimination/linear combination you may need to multiply one or both
of the equations by a non-zero constant in order to be able to eliminate
one of the variables.
 Substitution is a method of solving a system of equations where one
equation is solved for a variable and then that expression is substituted
into the other equation for that variable, in order to eliminate that
variable.
 A system of a linear equation and a quadratic equation can be solved
algebraically using substitution or graphically by finding the points of
intersection.





36
How do you determine the number of solutions that a system of
equations will have?
How do you determine the best method for solving a given system of
equations?
Why would you want to multiply an equation by a constant (that is not
zero)?
Why does graphing a system of equations yield an approximate
solution as opposed to an exact solution?
How can you prove that no matter which method you choose to solve
a system of equations, you will always get the same solution?
HS.A-REI.5
HS.A-REI.5
Prove that, given a system of
two equations in two
variables, replacing one
equation by the sum of that
equation and a multiple of the
other produces a system with
the same solutions.
Mathematical
Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.
Examples & Explanations
Build on student experiences in graphing and solving systems of linear equations from middle
school to focus on justification of the methods used. Include cases where the two equations
describe the same line (yielding infinitely many solutions) and cases where two equations describe
parallel lines (yielding no solution); connect to GPE.5 when it is taught in Geometry, which requires
students to prove the slope criteria for parallel lines.
Systems of linear equations can also have one solution, infinitely many solutions or no solutions.
Students will discover these cases as they graph systems of linear equations and solve them
algebraically.
A system of linear equations whose graphs meet at one point (intersecting lines) has only one
solution, the ordered pair representing the point of intersection. A system of linear equations whose
graphs do not meet (parallel lines) has no solutions and the slopes of these lines are the same. A
system of linear equations whose graphs are coincident (the same line) has infinitely many
solutions, the set of ordered pairs representing all the points on the line.
By making connections between algebraic and graphical solutions and the context of the system of
linear equations, students are able to make sense of their solutions. Students need opportunities to
work with equations and context that include whole number and/or decimals/fractions.
Examples:
 Find x and y using elimination and then using substitution.
3x + 4y = 7
–2x + 8y = 10

HS.A-REI.6
HS.A-REI.6
Solve systems of linear
equations exactly and
approximately (e.g., with
graphs), focusing on pairs of
linear equations in two
variables.
Mathematical
Practices
HS.MP.2. Reason
abstractly and
quantitatively.
Given that the sum of two numbers is 10 and their difference is 4, what are the numbers?
Explain how your answer can be deduced from the fact that the two numbers, x and y,
satisfy the equations x + y = 10 and x – y = 4.
Examples & Explanations
The system solution methods can include but are not limited to graphical, elimination/linear
combination, substitution, and modeling. Systems can be written algebraically or can be
represented in context. Students may use graphing calculators, programs, or applets to model and
find approximate solutions for systems of equations.
(Continued on next page)
HS.MP.4. Model
37
with mathematics.
Connection: ETHS-S6C2-03
HS.MP.5. Use
appropriate tools
strategically.
Examples:
 José had 4 times as many trading cards as Phillipe. After José gave away 50 cards to his
little brother and Phillipe gave 5 cards to his friend for his birthday, they each had an equal
amount of cards. Write a system to describe the situation and solve the system.
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.A-REI.7
HS.A-REI.7
Solve a simple system
consisting of a linear equation
and a quadratic equation in
two variables algebraically
and graphically. For example,
find the points of intersection
between the line y = –3x and
the circle x2 + y2 = 3.
Mathematical
Practices
HS.MP.2
Reason abstractly
and quantitatively.
HS.MP.4. Model
with mathematics.
HS.MP.5. Use
appropriate tools
strategically.
HS.MP.6. Attend
to precision.

Solve the system of equations: x+ y = 11 and 3x – y = 5.
Use a second method to check your answer.
 Solve the system of equations:
x – 2y + 3z = 5, x + 3z = 11, 5y – 6z = 9.

The opera theater contains 1,200 seats, with three different prices. The seats cost $45 per
seat, $50 per seat, and $60 per seat. The opera needs to gross $63,750 on seat sales.
There are twice as many $60 seats as $45 seats. How many seats at each price need to
be sold?
Examples & Explanations
Include systems consisting of one linear and one quadratic equation. Include systems that lead to
work with fractions.
Examples:
 Find the points of intersection between the line y = –3x and the circle x2 + y2 = 3
algebraically.

Two friends are driving to the Grand Canyon in separate cars. Suzette has been there
before and knows the way but Andrea does not. During the trip Andrea gets ahead of
Suzette and pulls over to wait for her. Suzette is traveling at a constant rate of 65 miles per
hour. Andrea sees Suzette drive past. To catch up, Andrea accelerates at a constant rate.
The distance in miles (d) that her car travels as a function of time in hours (t) since
Suzette’s car passed is given by d = 3500t2. Write and solve a system of equations to
determine how long it takes for Andrea to catch up with Suzette.
(Continued on next page)
38
HS.MP.7. Look for
and make use of
structure.

Include systems that lead to work with fractions. For example, finding the intersections
between x2+y2=1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle,
corresponding to the Pythagorean triple 32+42=52.
HS.MP.8. Look for
and express
regularity in
repeated
reasoning.
Reasoning with Equations and Inequalities (A-REI) (Domain 4 – Cluster 4 – Standards 10, 11 and 12)
Represent and solve equations and inequalities graphically. (Linear and exponential; learn as general principle)
Essential Concepts
Essential Questions
 The graph of an equation in two variables is the set of all its solutions
plotted in the coordinate plane.
 Solving a system of equations algebraically yields an exact solution;
solving by graphing or by comparing tables of values yields an
approximate solution.
 The solutions (solution set) of a linear inequality in two variables are
represented graphically as a half-plane.
 The solution set of a system of linear inequalities in two variables is the
intersection of the corresponding half-planes.
HS.A-REI.10
HS.A-REI.10
Understand that the graph of
an equation in two variables
is the set of all its solutions
plotted in the coordinate
plane, often forming a curve
(which could be a line).
Mathematical
Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
HS.MP.4. Model
with
mathematics.




How do you determine if a given ordered pair is a solution to an
equation?
Why are the x-coordinates of the points where the graphs of the
equations y = f(x) and y = g(x) intersect equal to the solutions of the
equation f(x) = g(x)?
(Continued on next page)
When graphing a linear inequality, how do you determine which halfplane to shade in order to represent the solution set?
How do you represent the solution set of a system of linear
inequalities on a graph?
Examples & Explanations
For A-REI.10, focus on linear and exponential equations and be able to adapt and apply that
learning to other types of equations in future courses.
Examples:
 Which of the following points would be on the graph of the equation 3x+4y=24 ?
(a) (0, 6) (b) (-1, 7) (c) (4/3, 5) (d) (3, 4)

Graph the equation and determine which of the following points are on the graph of
y = 3x + 1.
(a) (2, 7) (b) (-1, 4/3) (c) (2, 10) (d) (0, 1)
39
HS.A-REI.11
HS.A-REI.11
Explain why the x-coordinates
of the points where the
graphs of the equations y =
f(x) and y = g(x) intersect are
the solutions of the equation
f(x) = g(x); find the solutions
approximately, e.g., using
technology to graph the
functions, make tables of
values, or find successive
approximations. Include
cases where f(x) and/or g(x)
are linear, polynomial,
rational, absolute value,
exponential, and logarithmic
functions.
HS.A-REI.12
HS.A-REI.12
Graph the solutions to a
linear inequality in two
variables as a half-plane
(excluding the boundary in
the case of a strict inequality),
and graph the solution set to
a system of linear inequalities
in two variables as the
intersection of the
corresponding half-planes.
Mathematical
Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
Examples & Explanations
For A-REI.11, focus on cases where f(x) and g(x) are linear, quadratic and exponential. In Algebra
II, students will extend this standard to include higher-order polynomials, rational, absolute value
and logarithmic functions.
HS.MP.4. Model
with
mathematics.
Students need to understand that numerical solution methods (data in a table used to approximate
an algebraic function) and graphical solution methods may produce approximate solutions, and
algebraic solution methods produce precise solutions that can be represented graphically or
numerically. Students may use graphing calculators or programs to generate tables of values,
graph, or solve a variety of functions.
HS.MP.5. Use
appropriate tools
strategically.
Example:
 Given the following equations, determine the x value that results in an equal output for both
functions.
f (x ) = 3x - 2
HS.MP.6. Attend
to precision.
Mathematical
Practices
HS.MP.4. Model
with
mathematics.
HS.MP.5. Use
appropriate tools
strategically.
g ( x ) = ( x + 3)2 - 1
Examples & Explanations
Students may use graphing calculators, programs, or applets to model and find solutions for
inequalities or systems of inequalities.
Examples:
 Graph the solutions: y < 2x + 3.

A publishing company publishes a total of no more than 100 magazines every year. At least
30 of these are women’s magazines, but the company always publishes at least as many
women’s magazines as men’s magazines. Find a system of inequalities that describes the
possible number of men’s and women’s magazines that the company can produce each
year consistent with these policies. Graph the solution set.
(Continued on next page)
40

Graph the system of linear inequalities below and determine if (3, 2) is a solution to the
system.
ìx - 3y > 0
ï
íx + y £ 2
ï x + 3 y > -3
î
Solution:
(3, 2) is not an element of the solution set
(graphically or by substitution).
Additional Domain Information – Reasoning with Equations and Inequalities (A-REI)
Key Vocabulary






Equation
Inequality
Quadratic formula
Extraneous solutions
Point of Intersection
Half-Plane











Solution
Laws of Exponents
Square roots
Complex solutions
Discriminant
Radical
Substitution
Completing the Square
Factoring
Elimination/Linear Combinations
System
Example Resources


Books
 Textbook
 Focus in High School Mathematics: Reasoning and Sense Making, Chapter 2: Building Equations and Functions
 The Xs and Whys of Algebra: Key Ideas and Common Misconceptions
Technology
 Key Curriculum Press, Exploring Algebra I with the Geometer’s Sketchpad
 www.Geogebra.org online software to create visuals
 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.
41
www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards
Example Lessons
 http://www.uen.org/Lessonplan/preview.cgi?LPid=19825 Students go car shopping online, investigate the relationship between variables
such as interest rate and monthly payment, develop two payment plans using online loan calculators, write a slope-intercept equation for
each plan, and create a graph and table for the equations using a graphing calculator.
 http://www.uen.org/Lessonplan/preview.cgi?LPid=20241 Students watch a short role play in which a doctor performs an operation on a
patient without first diagnosing his condition. They relate this medical malpractice to a mathematician performing an operation before
evaluating conditions or deciding what outcome he wants. Next they practice 'diagnosing' x's condition and performing the correct inverse
operation to get x by itself.
 http://www.uen.org/Lessonplan/preview.cgi?LPid=23514 Write and solve simple inequalities.


Common Student Misconceptions
Students often forget to flip the direction of the inequality sign when multiplying or dividing by a negative number.
Students commonly believe that there is only one correct method to solve an equation or inequality.
Students often perform inverse operations on the same side of the equation, including combining non-like terms. For example,
3x + 2 = 8
instead of 3x + 2 = 8
-2 -2
-2 -2
x=8
3x = 6
Students will often drop the negative sign at the end or just stop solving for x ,if the coefficient of x is –1. For example,
3x + 2 – 4x = 5
–x + 2 = 5
–x = 3, so a student will say that the solution is 3 rather than –3.
Students will be confused as to when there is no real number solution or when the solution set is all real numbers for an equation or system of
equations. For example, students will be confused when they try to solve an equation and end up with –2 = 9, which means there are no solutions, or
when the solution to a system of equations gives x = x, which means the solution set is all real numbers.
Students fail to see that there may still be real number solutions even though a quadratic doesn’t factor. For example, x2 + x – 4 = 0 does not
factor, but students should use the quadratic formula to find the exact solutions.
Students will substitute incorrectly in a system. There is a variety of ways that students may incorrectly solve for a variable to be substituted in to the
other equation in a system. In addition, students may erroneously substitute back into the same equation and believe the solution set is infinite.
Students often fail to give both solutions when solving x2 = b type equations. For example, given x2 = 49, students often only state x = 7 rather
than x = ± 7.
Students will believe that the variable can only be on the left side of the equation.
42
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.
43
High School Algebra 1
Conceptual Category: Functions (3 Domains, 7 Clusters)
Domain: Interpreting Functions (3 Clusters)
Interpreting Functions (F-IF) (Domain 1 – Cluster 1 – Standards 1, 2 and 3)
Understand the concept of a function and use function notation. (Learn as a general principle; focus on linear and exponential and on arithmetic and
geometric sequences.)
Essential Concepts
Essential Questions
 A function is a rule that assigns each input exactly one output.
 In function notation, f(x) denotes that f is a function of x.
 The set of all inputs (x) for a function is called the domain; the set of all
outputs (f(x)) for a function is called the range.
 The domain and range of a function can be expressed as a set of
numbers using set notation, an inequality, or as a graphed solution.
 The graph of a function f is the graph of the equation y = f(x).
 Algebraic equations, written in function notation, can be used to
evaluate functions for a given input.
 For a function f(x), f(a) represents the value of the function when x = a.
 Sequences are functions that have a discrete domain, which is a subset
of the integers.
 A recursive sequence is a sequence in which each term is built upon the
previous term.
HS.F-IF.1
HS.F-IF.1
Understand that a function
from one set (called the
domain) to another set (called
the range) assigns to each
element of the domain exactly
one element of the range. If f
is a function and x is an
element of its domain, then
f(x) denotes the output of f
corresponding to the input x.
The graph of f is the graph of
the equation y = f(x).
Mathematical
Practices
HS.MP.2
Reason
abstractly and
quantitatively.





Given a table or graph, how do you determine if it represents a
function?
How is a graph related to its algebraic function?
How could you use function notation to represent a specific output
of a function?
How can the Fibonacci sequence be used to explain a recursive
pattern?
How can you describe a sequence as a function?
Examples & Explanations
Students should experience a variety of types of situations modeled by functions. Detailed analysis
of any particular class of functions at this stage is not advised. Students should apply these
concepts throughout their future mathematics courses. Draw examples from linear, quadratic, and
exponential functions.
The domain is the set of all inputs (x values); the range is the set of all outputs (y = f(x)).
The domain of a function may be given by an algebraic expression. Unless otherwise specified, it is
the largest possible domain.
(Continued on next page)
44
For example, the rule that takes x as input and gives x2+5x+4 as output is a function. Using y to
stand for the output we can represent this function with the equation y = x2+5x+4, and the graph of
the equation is the graph of the function. Students are expected to use function notation such as
f(x) = x2+5x+4.
Example:
 Determine which of the following tables represent a function and explain why.
A
B
x
f(x)
x
f(x)
0
1
0
0
1
2
1
2
2
2
1
3
3
4
4
5
Solution: A represents a function because for each element in the domain there is exactly
one element in the range.
B does NOT represent a function because when x = 1, there are two values for f(x): 2 and 3.
HS.F-IF.2
HS.F-IF.2
Use function notation,
evaluate functions for inputs
in their domains, and interpret
statements that use function
notation in terms of a context.
Mathematical
Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
Connection: 9-10.RST.4
HS.F-IF.3
HS.F-IF.3
Recognize that sequences
are functions, sometimes
defined recursively, whose
domain is a subset of the
integers. For example, the
Fibonacci sequence is
defined recursively by f(0) =
f(1) = 1, f(n+1) = f(n) + f(n-1)
for n ≥ 1.
Mathematical
Practices
HS.MP.8. Look
for and express
regularity in
repeated
reasoning.
Examples & Explanations
Examples:
 If f ( x ) = x 2 + 4x - 12 , find f (2).


1
) , f (a ) , and f (a - h).
2
If P(t) is the population of Tucson t years after 2000, interpret the statements P(0) = 487,000
and P(10)-P(9) = 5,900.
Let f ( x ) = 2( x + 3)2 . Find f (3) , f (-
Examples & Explanations
In F-IF.3, draw a connection to F.BF.2, which requires students to write arithmetic and geometric
sequences. Emphasize arithmetic and geometric sequences as examples of linear and exponential
functions.
Example:
 The Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
45
Interpreting Functions (F-IF) (Domain 1 – Cluster 2 – Standards 4, 5 and 6)
Interpret functions that arise in applications in terms of a context. (Linear, exponential, and quadratic)
Essential Concepts
Essential Questions
 Key features of a graph or table may include intercepts, intervals in
which the function is increasing, decreasing or constant, intervals in
which the function is positive, negative or zero, symmetry, maxima,
minima, and end behavior.
 Given a verbal description of a relationship that can be modeled by a
function, a table or graph can be constructed and used to interpret key
features of that function.
 The meaning of the key features of a graph or table, such as domain,
range, rate of change and intercepts, can be interpreted in the context
of a problem.
 The intervals over which a function is increasing, decreasing or
constant, positive, negative or zero are subsets of the function’s
domain.
 The appropriate domain for a function describing a real-life situation
may be smaller than the largest possible domain.
 The average rate of change of a function y = f(x) over an interval [a,b] is
Dy f (b) - f (a)
.
=
Dx
b-a
HS.F-IF.4
HS.F-IF.4
For a function that models a
relationship between two
quantities, interpret key
features of graphs and tables
in terms of the quantities, and
sketch graphs showing key
features given a verbal
description of the relationship.
Key features include:
intercepts; intervals where the
function is increasing,
decreasing, positive, or
negative; relative maxima and
minima; symmetries; end
behavior; and periodicity.
Mathematical
Practices





How can you describe the shape of a graph?
How can you relate the shape of a graph to the meaning of the
relationship it represents?
How would you determine the appropriate domain for a function
describing a real-life situation?
Given a function that describes a real-life situation, what can the
average rate of change of the function tell you?
How do the parts of a graph of a function relate to its real world
context?
Examples & Explanations
HS.MP.2. Reason
abstractly and
quantitatively.
Start F-IF.4 and 5 by focusing on linear and exponential functions. Later in the year, focus on
quadratic functions and compare them with linear and exponential functions. In Algebra II, students
will extend this standard to include higher order polynomials, rational, absolute value, and
trigonometric functions.
HS.MP.4. Model
with mathematics.
Students may be given graphs to interpret or produce graphs given an expression or table for the
function, by hand or using technology.
HS.MP.5. Use
appropriate tools
strategically.
Examples:
 A rocket is launched from 180 feet above the ground at time t = 0. The function that models
this situation is given by h = – 16t2 + 96t + 180, where t is measured in seconds and h is
height above the ground measured in feet.
o What is a reasonable domain restriction for t in this context?
o Determine the height of the rocket two seconds after it was launched.
(Continued on next page)
HS.MP.6. Attend
to precision.
46
o
o
o
o
Connections:
ETHS-S6C2.03;
9-10.RST.7; 11-12.RST.7
HS.F-IF.5
HS.F-IF.5
Relate the domain of a
function to its graph and,
where applicable, to the
quantitative relationship it
describes. For example, if the
function h(n) gives the
number of person-hours it
takes to assemble n engines
in a factory, then the positive
integers would be an
appropriate domain for the
function.
HS.F-IF.6
HS.F-IF.6
Calculate and interpret the
average rate of change of a
function (presented
symbolically or as a table)
over a specified interval.
Estimate the rate of change
from a graph.
Mathematical
Practices
HS.MP.2. Reason
abstractly and
quantitatively.
HS.MP.4. Model
with mathematics.
Determine the maximum height obtained by the rocket.
Determine the time when the rocket is 100 feet above the ground.
Determine the time at which the rocket hits the ground.
How would you refine your answer to the first question based on your response to
the second and fifth questions?

Compare the graphs of y = 3x2 and y = 3x.

Let f (x) = -x 2 - 5x +1. Graph the function and identify end behavior and any intervals of
constancy, increase, and decrease.

It started raining lightly at 5am, then the rainfall became heavier at 7am. By 10am the
storm was over, with a total rainfall of 3 inches. It didn’t rain for the rest of the day. Sketch
a possible graph for the number of inches of rain as a function of time, from midnight to
midday.
Examples & Explanations
Start F-IF.4 and 5 by focusing on linear and exponential functions. Later in the year, focus on
quadratic functions and compare them with linear and exponential functions.
Students may explain the existing relationships orally or in written format.
Example:
 If the function h(n) gives the number of person-hours it takes to assemble n engines in a
factory, then the positive integers would be an appropriate domain for the function.
HS.MP.6. Attend
to precision.
Mathematic
al Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
Examples & Explanations
Start F-IF.6 by focusing on linear functions and exponential functions whose domain is a subset of the
integers. Later in the year, focus on quadratic functions and compare them with linear and exponential
functions. The Algebra II course will address other types of functions.
(Continued on next page)
47
Connections:
ETHS-S1C2-01;
9-10.RST.3
HS.MP.4.
Model with
mathematics.
HS.MP.5. Use
appropriate
tools
strategically.
The average rate of change of a function y = f(x) over an interval [a,b] is
Dy f (b) - f (a)
. In
=
Dx
b-a
addition to finding average rates of change from functions given symbolically, graphically, or in a table,
students may collect data from experiments or simulations (ex. falling ball, velocity of a car, etc.) and
find average rates of change for the function modeling the situation.
Examples:
 Use the following table to find the average rate of change of g over the intervals [-2, -1] and
[0,2]:
x
-2
-1
0
2

g(x)
2
-1
-4
-10
The table below shows the elapsed time when two different cars pass a 10, 20, 30, 40 and 50
meter mark on a test track.
o
For car 1, what is the average velocity (change in distance divided by change in time)
between the 0 and 10 meter mark? Between the 0 and 50 meter mark? Between the
20 and 30 meter mark? Analyze the data to describe the motion of car 1.
o
How does the velocity of car 1 compare to that of car 2?
d
10
20
30
40
50
48
Car 1
T
4.472
6.325
7.746
8.944
10
Car 2
t
1.742
2.899
3.831
4.633
5.348
Interpreting Functions (F-IF) (Domain 1 – Cluster 3 – Standards 7, 8 and 9)
Analyze functions using different representations. (Linear, exponential, quadratic, absolute value, step and piecewise-defined)
Essential Concepts
Essential Questions
 To graph a function you can create a table of values, analyze the
equation, or use a graphing calculator.
 Key features of a graph or table may include intercepts, intervals in
which the function is increasing, decreasing or constant, intervals in
which the function is positive, negative or zero, symmetry, maxima,
minima, and end behavior.
 A linear function can be written in point-slope, slope-intercept or
standard form.
 A quadratic function can be written in vertex or standard form.
 Factoring a quadratic function will help to determine the zeros.
 Completing the square will help determine the vertex of the graph.




How can you compare properties of two functions if they are
represented in different ways?
How can you determine which form of a function is best for a given
situation?
Why might you need to complete the square?
How could you determine if a function represents exponential growth
or exponential decay?
 For a function of the form f (t) = a × b , if b>1 the function represents
exponential growth; if b<1 the function represents exponential decay.
t
HS.F-IF.7
HS.F-IF.7
Graph functions expressed
symbolically and show key
features of the graph, by
hand in simple cases and
using technology for more
complicated cases.
a.
Graph linear and
quadratic functions and
show intercepts, maxima,
and minima.
Connections:
ETHS-S6C1-03;
ETHS-S6C2-03
b.
Graph square root,
cube root, and piecewisedefined functions, including
step functions and absolute
value functions.
Mathematical
Practices
HS.MP.5. Use
appropriate tools
strategically.
HS.MP.6. Attend
to precision.
Examples & Explanations
For F-IF.7a, 7e, focus only on linear and exponential functions. Include comparisons of two
functions presented algebraically. For example, compare the growth of two linear functions, or two
exponential functions such as y=3n and y=100*2n.
For F-IF.7b, compare and contrast absolute value, step and piecewise-defined functions with
linear, quadratic, and exponential functions. Highlight issues of domain, range, and usefulness
when examining piecewise-defined functions.
For F-IF.7e focus only on exponential functions.
Key characteristics include but are not limited to maxima, minima, intercepts, symmetry, end
behavior, and asymptotes. Students may use graphing calculators or programs, spreadsheets, or
computer algebra systems to graph functions.
Examples:
 Graph the function f(x) = │x – 3│ + 5 and describe key characteristics of the graph

Graph the function f(x) = 2x by creating a table of values. Identify the key characteristics of
the graph.
(Continued on next page)
49
Connections:
ETHS-S6C1-03;
ETHS-S6C2-03

Sketch the graph and identify the key characteristics of the function described below.
 x  2for x  0
F ( x)   2
 x for x  1
e.
Graph exponential
and logarithmic functions,
showing intercepts and end
behavior, and trigonometric
functions, showing period,
midline, and amplitude.
Solution:
Connections:
ETHS-S6C1-03;
ETHS-S6C2-03
HS.F-IF.8
HS.F-IF.8
Write a function defined by an
expression in different but
equivalent forms to reveal
and explain different
properties of the function.
Connection: 11-12.RST.7
a. Use the process of
factoring and completing
the square in a quadratic
function to show zeros,
extreme values, and
symmetry of the graph,
and interpret these in
terms of a context.
Mathematical
Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
HS.MP.7. Look
for and make use
of structure.
Examples & Explanations
F-IF.8a will be taught later in the year, because it focuses on quadratic factoring and completing the
square. In Algebra II students will extend their work on F-IF.8 to focus on applications and how key
features relate to characteristics of a situation, making selection of a particular type of function
model appropriate.
F-IF.8b is extending work done earlier in the year on exponential functions with integer exponents.
Therefore this will need to be taught later in the year.
Example:
 Factor the following quadratic to identify its zeros: x2 + 2x - 8 = 0

Complete the square for the quadratic and identify its vertex: x2 + 6x +19 = 0

Identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t,
y = (1.2)t/10, and classify them as representing exponential growth or decay.
Connection: 11-12.RST.7
b. Use the properties of
exponents to interpret
expressions for
(Continued on next page)
50
exponential functions. For
example, identify percent
rate of change in
functions such as y =
(1.02)t, y = (0.97)t, y =
(1.01)12t, y = (1.2)t/10, and
classify them as
representing exponential
growth or decay.
Connection: 11-12.RST.7
HS.F-IF.9
HS.F-IF.9. Compare
properties of two functions
each represented in a
different way (algebraically,
graphically, numerically in
tables, or by verbal
descriptions). For example,
given a graph of one
quadratic function and an
algebraic expression for
another, say which has the
larger maximum.
Connections:
ETHS-S6C1-03;
ETHS-S6C2-03;
9-10.RST.7
Mathematical
Practices
HS.MP.6. Attend
to precision.
HS.MP.7. Look
for and make use
of structure.
Examples & Explanations
Start F-IF.9 by focusing on linear and exponential functions. Include comparisons of two functions
presented algebraically. Later in the year focus on expanding the types of functions to include linear,
exponential, and quadratic. Extend work with quadratics to include the relationship between
coefficients and roots, and once roots are known, a quadratic equation can be factored.
Example:
 Given a graph of one quadratic function and an algebraic expression for another, say which
has the larger maximum.

Examine the functions below. Which function has the larger maximum? How do you know?
f (x) = -2x 2 -8x + 20
51
Additional Domain Information – Interpreting Functions (F-IF)
Key Vocabulary








Average Rate of Change
Function
Input/Domain
Output/Range
Linear Function
Piecewise Function
Quadratic Function
Exponential Function


















Trigonometric Function
Period
Midline
Amplitude
Sequence
Fibonacci Sequence
Recursive
Arithmetic Sequence
Geometric Sequence
Maxima
Minima
End Behavior
Point-Slope Form
Slope-Intercept Form
Standard Form
Vertex Form
x-intercept
y-intercept
Example Resources
 Books
 Developing Essential Understanding of Functions: Grades 9-12 by NCTM
 The X’s and Why’s of Algebra: Key Ideas and Common Misconceptions by Anne Collins & Linda Dacey
 Technology
Interpreting functions:
 http://www.brightstorm.com/math/algebra/graphs-and-functions/interpreting-graphs-problem-1/
Video on interpreting graphs for students in Algebra. Shows students how to look at what graphs mean and how to interpret slopes using
real world situations.
 http://www.khanacademy.org/
Conceptual videos: Linear, Quadratic, Exponential Functions Basic Linear Function, Recognizing Linear Functions, Linear Function
Graphs, Graphing a Quadratic Function, Applying Quadratic Functions 2, Quadratic Functions 1, Applying Quadratic Functions 3,
Applying Quadratic Functions 1, Quadratic Functions 2, Quadratic Functions 3, Exponential Growth Functions, Exponential Decay
Functions, See all videos from topic ck12.org Algebra 1 Examples, Graphing Exponential Functions
 http://www.schooltube.com/video/f44d3551932486f18d81/
Provides examples on how to build & analyze functions.
 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.illustrativemathematics.org/ This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards.
52
 Example Lessons
 http://www.illustrativemathematics.org/standards/hs
Understand the concept of a function and use function notation.
Interpret functions that arise in applications in terms of the context. F-IF: Interpret functions that arise in applications in terms of the context.
Common Student Misconceptions
Students commonly have difficulty understanding the function notation Y = f(x). For example, f(2) = 5 is the function notation for the point (2, 5).
Students can have difficulty identifying the x and y values for a given point from the function notation.
Students often confuse the domain with the range.
Students commonly have difficulty identifying the zeros of a function. Students think that the zero of the function is when x = 0, as opposed to when
y = 0.
Students misunderstand how to write the interval where a function is increasing or decreasing. Often students write the answer in terms of yinterval rather than in terms of the x-interval.
Students misunderstand how to identify the parameters (function transformation) to build new functions from existing functions. For example,
to shift the graph of y = x2 left by 6 units, they may write y = (x - 6)2 rather than y = (x + 6)2.
Students believe that the domain of a function is the same, regardless of the contextual problem that it is modeling. For example, the domain of
a quadratic includes negative values, but for a quadratic modeling the height of a falling object as a function of time t, the domain should be t ≥ 0.
Students often confuse the independent variable with the dependent variable.
Students often have difficulty identifying the quantities to be represented by variables, in modeling problems.
53
Domain: Building Functions (2 Clusters)
Building Functions (F-BF) (Domain 2 – Cluster 1 – Standards 1 and 2)
Build a function that models a relationship between two quantities. (For F-BF.1 and 2, linear, exponential and quadratic)
Essential Concepts
Essential Questions
 A function is a relationship between two quantities.
 The function representing a given situation may be a combination of
more than one standard function.
 Standard functions may be combined through arithmetic operations.
 Arithmetic and geometric sequences can be written both recursively and
with an explicit formula.
 A recursive formula for a sequence describes how to determine the next
term from the previous term(s).
 An explicit formula for a sequence describes how to determine any term
in the sequence.
 Arithmetic sequences can be described by linear functions.
 Geometric sequences can be described by exponential functions.
 Sequences model situations in which the domain is a set of integers.
HS.F-BF.1
HS.F-BF.1
Write a function that
describes a relationship
between two quantities.
Connections:
ETHS-S6C1-03;
ETHS-S6C2-03
a. Determine an explicit
expression, a recursive
process, or steps for
calculation from a
context.
Connections:
ETHS-S6C1-03;
ETHS-S6C2-03;
9-10.RST.7; 11-12.RST.7
Mathematical
Practices
HS.MP.1. Make
sense of problems
and persevere in
solving them.
HS.MP.2. Reason
abstractly and
quantitatively.
HS.MP.4. Model
with mathematics.
HS.MP.5. Use
appropriate tools
strategically.





What data would you need to write a linear, basic quadratic, or basic
exponential function?
How do you translate a description of the relationship between two
quantities into an algebraic equation or inequality?
Why are arithmetic sequences described by linear functions?
Why are geometric sequences described by exponential functions?
How could you translate a recursive formula for a sequence into an
explicit formula? Vice versa?
Examples & Explanations
Start by limiting F-BF.1a, 1b to linear and exponential functions. Later in the year extend this work
to focus on situations that exhibit a quadratic relationship.
Students will analyze a given problem to determine the function expressed by identifying patterns
in the function’s rate of change. They will specify intervals of increase, decrease, constancy, and, if
possible, relate them to the function’s description in words or graphically. Students may use
graphing calculators or programs, spreadsheets, or computer algebra systems to model functions.
Examples:
 You buy a $10,000 car with an annual interest rate of 6 percent compounded annually and
make monthly payments of $250. Express the amount remaining to be paid off as a
function of the number of months, using a recursion equation.

A cup of coffee is initially at a temperature of 93º F. The difference between its temperature
and the room temperature of 68º F decreases by 9% each minute. Write a function
describing the temperature of the coffee as a function of time.
(Continued on next page)
HS.MP.6. Attend
54
b. Combine standard
function types using
arithmetic operations. For
example, build a function
that models the
temperature of a cooling
body by adding a
constant function to a
decaying exponential,
and relate these functions
to the model.
to precision.
HS.MP.7. Look for
and make use of
structure.
HS.MP.8. Look for
and express
regularity in
repeated
reasoning.
Connections:
ETHS-S6C1-03;
ETHS-S6C2-03
HS.F-BF.2
HS.F-BF.2
Write arithmetic and
geometric sequences both
recursively and with an
explicit formula, use them to
model situations, and
translate between the two
forms.
Mathematical
Practices
HS.MP.4. Model
with mathematics.
HS.MP.5. Use
appropriate tools
strategically.
HS.MP.8. Look for
and express
regularity in
repeated
reasoning.
Examples & Explanations
For F-BF.2 limit to linear and exponential functions. Connect arithmetic sequences to linear
functions and geometric sequences to exponential functions.
An explicit rule for the nth term of a sequence gives an as an expression in the term’s position n; a
recursive rule gives the first term of a sequence, and a recursive equation relates an to the
preceding term(s). Both methods of presenting a sequence describe an as a function of n.
Examples:
 Generate the 5th-11th terms of a sequence if A1= 2 and A( n 1)  ( An )2  1

Use the formula: An = A1 + d(n - 1) where d is the common difference to generate a
sequence whose first three terms are: -7, -4, and -1.

There are 2,500 fish in a pond. Each year the population decreases by 25 percent, but
1,000 fish are added to the pond at the end of the year. Find the population in five
years. Also, find the long-term population.
Given the formula An= 2n - 1, find the 17th term of the sequence. What is the 9th term in
the sequence 3, 5, 7, 9, ?


Given a1 = 4 and an = an-1 + 3, write the explicit formula.
55
Building Functions (F-FB) (Domain 2 – Cluster 2 – Standards 3 and 4)
Build new functions from existing functions. (Linear, exponential, quadratic and absolute value; for F-BF.4a, linear only)
Essential Concepts
Essential Questions
 f(x) + k will translate the graph of the function f(x) up or down by k units.
 k f(x) will expand or contract the graph of the function f(x) vertically by a
factor of k. If k<0 the graph will reflect across the x-axis.
 f(kx) will expand or contract the graph of the function f(x) horizontally by
a factor of k. If k<0 the graph will reflect across the y-axis.
 f(x + k) will translate the graph of the function f(x) left or right by k units.
 If f(-x) = f(x) then the function is even, therefore its graph is symmetrical
across the y-axis.
 If f(-x) = - f(x) then the function is odd, therefore its graph is symmetrical
across the origin.
 Two functions f and g are inverses of one another if for all values of x in
the domain of f, f(x)=y and g(y)=x.
 Not all functions have an inverse.
HS.F-BF.3
HS.F-BF.3
Identify the effect on the
graph of replacing f(x) by f(x)
+ k, k f(x), f(kx), and f(x + k)
for specific values of k (both
positive and negative); find
the value of k given the
graphs. Experiment with
cases and illustrate an
explanation of the effects on
the graph using technology.
Include recognizing even and
odd functions from their
graphs and algebraic
expressions for them.
Connections:
ETHS-S6C2-03;
11-12.WHST.2e
Mathematical
Practices
HS.MP.4. Model
with
mathematics.
HS.MP.5. Use
appropriate tools
strategically.
HS.MP.7. Look
for and make use
of structure.







Create a graph and explain what transformation(s) were done on the
parent function to create that graph.
What are the transformations that can be done to a graph and how
can they be represented algebraically?
How do you determine if a graph is odd, even, or neither?
Why are the two descriptions of an even function equivalent?
Why are the two descriptions of an odd function equivalent?
How do you determine if two functions are inverses of one another?
Given a function, how do you find its inverse?
Examples & Explanations
Start by focusing on vertical translations of graphs of linear and exponential functions. Relate the
vertical translation of a linear function to its y-intercept. While applying other transformations to a
linear graph is appropriate at this level, it may be difficult for students to identify or distinguish
between the effects of the other transformations included in this standard. Later in the year, extend
this work to quadratic functions, and consider including absolute value functions.
Students will apply transformations to functions and recognize functions as even and odd. Students
may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph
functions.
Examples:
 Compare the graphs of f(x)=3x with those of g(x)=3x+2 and h(x)=3x -1 to see that parallel
lines have the same slope AND to explore the effect of the transformation of the function,
f(x)=3x such that g(x)=f(x)+2 and h(x)=f(x) – 1.

Explore the relationship between f(x)=3x, g(x)= 5x, and h( x ) 
1
x with a calculator to
2
develop a relationship between the coefficient on x and the slope.
(Continued on next page)
56
HS.F-BF.4
HS.F-BF.4
Find inverse functions.
Connection:
ETHS-S6C2-03
a. Solve an equation of the
form f(x) = c for a simple
function f that has an
inverse and write an
expression for the
inverse.
Mathematical
Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
HS.MP.4. Model
with
mathematics.
HS.MP.5. Use

Describe the effect of varying the parameters a, h, and k on the shape and position of the
graph f(x) = ab(x + h) + k, orally or in written format. What effect do values between 0 and 1
have? What effect do negative values have?

Is f(x) = x3 - 3x2 + 2x + 1 even, odd, or neither? Explain your answer orally or in written
format.

Compare the shape and position of the graphs of f (x) = x 2 and g(x) = 2x 2 , and explain
the differences in terms of the algebraic expressions for the functions.

Describe the effect of varying the parameters a, h, and k have on the shape and position of
the graph of f(x) = a(x-h)2 + k.
Examples & Explanations
For F-BF.4a, focus on linear functions but consider simple situations where the domain of the
function must be restricted in order for the inverse to exist, such as f(x) = x 2, x>0. This work will be
extended in Algebra II to include simple rational, simple radical and simple exponential functions.
Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to
model functions.
Examples:
 For the function h(x) = 3(x – 2), defined on the domain of all real numbers, find the inverse
function if it exists or explain why it doesn’t exist.
(Continued on next page)
57
For example, f(x) =2x3 or
f(x) = (x+1)/(x-1) for x ≠ 1.
appropriate tools
strategically.

Graph h(x) and h-1(x) and explain how they relate to each other graphically for a linear
function.
HS.MP.7. Look
for and make use
of structure.

Consider simple situations where the domain of the function must be restricted in order for
the inverse to exist, such as f(x) = x2, x>0.
Additional Domain Information – Building Functions (F-BF)
Key Vocabulary




Domain
Range
Rate of change
Transformation









Expand/contract
Linear function
Exponential function
Quadratic function
Translate
Recursive formula
Explicit formula
Even/odd function
Inverse
Example Resources

Books
 SpringBoard by College Board [activity 2.2, 4.2]
 Algebra I by McDougal and Littell [8.2, 8.5, 8.6]
 Developing Essential Understanding of Functions: Grades 9-12 by NCTM

Technology
 http://www.khanacademy.org/#algebra-functions provides an extensive list of function video lectures
 http://www.purplemath.com/modules/index.htm provides resources about teaching functions
 http://a4a.learnport.org/page/comparing-functions provides multiple interactive applet investigations: introduction to recursive notation, etc.
 http://www.padowan.dk/graph/ A free downloadable graph generator
 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.illustrativemathematics.org/ This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards.
Example Lessons
 http://www.regentsprep.org/regents/math/algebra/AE7/PennyLabSheet.pdf Exponential growth and decay: Penny Activity
 http://illuminations.nctm.org/LessonDetail.aspx?ID=U142 Population of Trout Pond 4 lessons: recursion, numerical analysis, graphical
analysis, and symbolic analysis.

58
Common Student Misconceptions
Students have difficulty identifying that a variable squared with a coefficient does not mean to also square the coefficient. For example, 3x2 is
misrepresented as 9x2.
Students have difficulty applying rules of exponents when the exponent is rational. For example, adding/subtracting rational exponents that do not
have like denominators.
Students confuse power functions and exponential functions. For example, students think that n2 is an exponential function because it contains an
exponent.
Domain: Linear, Quadratic and Exponential Models (2 Clusters)
Linear, Quadratic and Exponential Models (F-LE) (Domain 3 – Cluster 1 – Standards 1, 2 and 3)
Construct and compare linear, quadratic, and exponential models and solve problems.
Essential Concepts
Essential Questions
 Linear functions grow by equal differences over equal intervals.
 Exponential functions grow by equal factors over equal intervals.
 Linear functions have an additive recursive pattern; exponential
functions have a multiplicative recursive pattern.
 Linear and exponential functions can be constructed given a graph, a
description of a relationship, or a set of input-output pairs (which may be
given in a table).
 An exponential growth model will eventually exceed in quantity any
linear or quadratic growth model.
HS.F-LE.1
HS.F-LE.1
Distinguish between
situations that can be
modeled with linear functions
and with exponential
functions.
Connections:
ETHS-S6C2-03;
SSHS-S5C5-03
a. Prove that linear
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.
HS.MP.4. Model
with mathematics.



How do you determine if a given situation is modeled by a linear or
exponential function?
How do you construct an exponential function given a graph? Table?
Description of a relationship?
How do you know an exponential growth model will eventually exceed
in quantity any linear or quadratic growth model?
Examples & Explanations
Students may use graphing calculators or programs, spreadsheets, or computer algebra systems
to model and compare linear and exponential functions.
Examples:
 A cell phone company has three plans. Graph the equation for each plan, and analyze the
change as the number of minutes used increases. When is it beneficial to enroll in Plan 1?
Plan 2? Plan 3?
1. $59.95/month for 700 minutes and $0.25 for each additional minute,
2. $39.95/month for 400 minutes and $0.15 for each additional minute, and
3. $89.95/month for 1,400 minutes and $0.05 for each additional minute.
(Continued on next page)
59
functions grow by equal
differences over equal
intervals, and that
exponential functions
grow by equal factors
over equal intervals.
Connection:
11-12.WHST.1a-1e
b. Recognize situations in
which one quantity
changes at a constant
rate per unit interval
relative to another.
HS.MP.5. Use
appropriate tools
strategically.
HS.MP.7. Look for
and make use of
structure.
HS.MP.8. Look for
and express
regularity in
repeated
reasoning.
Connection: 11-12.RST.4
c.
Recognize situations in
which a quantity grows or
decays by a constant
percent rate per unit
interval relative to
another.

A computer store sells about 200 computers at the price of $1,000 per computer. For each
$50 increase in price, about ten fewer computers are sold. How much should the computer
store charge per computer in order to maximize their profit?
Students can investigate functions and graphs modeling different situations involving simple and
compound interest. Students can compare interest rates with different periods of compounding
(monthly, daily) and compare them with the corresponding annual percentage rate. Spreadsheets
and applets can be used to explore and model different interest rates and loan terms.

A couple wants to buy a house in five years. They need to save a down payment of $8,000.
They deposit $1,000 in a bank account earning 3.25% interest, compounded quarterly.
How much will they need to save each month in order to meet their goal?

Sketch and analyze the graphs of the following two situations. What information can you
conclude about the types of growth each type of interest has?
o Lee borrows $9,000 from his mother to buy a car. His mom charges him 5% interest a
year, but she does not compound the interest.
o Lee borrows $9,000 from a bank to buy a car. The bank charges 5% interest
compounded annually.
Connections:
ETHS-S6C1-03;
ETHS-S6C2-03; 11-12.RST.4
HS.F-LE.2
HS.F-LE.2
Construct linear and
exponential functions,
including arithmetic and
geometric sequences, given a
graph, a description of a
relationship, or two inputoutput pairs (include reading
these from a table).
Mathematical
Practices
HS.MP.4. Model
with
mathematics.
HS.MP.8. Look
for and express
regularity in
Examples & Explanations
In constructing linear functions in F-LE.2, draw on and consolidate previous work in Grade 8 on
finding equations for lines and linear functions (8.EE.6, 8.F.4). In 8th grade students should do work
with identifying slope and unit rates for linear functions given two points, a table or a graph.
Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to
construct linear and exponential functions.
(Continued on next page)
60
Connections:
ETHS-S6C1-03;
ETHS-S6C2-03;
11-12.RST.4; SSHS-S5C5-03
repeated
reasoning.
Examples:
 Determine an exponential function of the form f(x) = abx using data points from the table.
Graph the function and identify the key characteristics of the graph.
x
0
1
3


HS.F-LE.3
HS.F-LE.3
Observe using graphs and
tables that a quantity
increasing exponentially
eventually exceeds a quantity
increasing linearly,
quadratically, or (more
generally) as a polynomial
function.
Mathematic
al Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
f(x)
1
3
27
Sara’s starting salary is $32,500. Each year she receives a $700 raise. Write a sequence in
explicit form to describe the situation.
Solve the equation 2x = 300.
Possible solution using a graphing calculator: enter y = 2x and y = 300 into a graphing
calculator and find where the graphs intersect, by viewing the table to see where the
function values are about the same.
Examples & Explanations
Start F-LE.3 by limiting to comparisons between linear and exponential models. Later in the year
compare linear and exponential growth to quadratic growth.
Example:
 Contrast the growth of the functions f(x)=3x, f(x)=3x and f(x) = x2 + 3.
61
Linear, Quadratic and Exponential Models (F-LE) (Domain 3 – Cluster 2 – Standard 5)
Interpret expressions for functions in terms of the situation they model. (Linear and exponential of the form f(x)=bx+k)
Essential Concepts
 A given situation will set parameters for any linear or exponential
function that models the situation.
HS.F-LE.5
HS.F-LE.5
Interpret the parameters in a
linear or exponential function
in terms of a context.
Mathematical
Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
Connections:
ETHS-S6C1-03;
ETHS-S6C2-03;
SSHS-S5C5-03;
11-12.WHST.2e
HS.MP.4.
Model with
mathematics.
Essential Questions
 Create an example of a linear situation and give the function that can
be used to model the situation. What does each part of the function
represent in the context of the problem? What are the parameters for
this function?
 Create an example of an exponential situation and give the function
that can be used to model the situation. What does each part of the
function represent in the context of the problem? What are the
parameters for this function?
Examples & Explanations
Limit exponential functions to those of the form f(x) = b x + k.
Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to
model and interpret parameters in linear, quadratic or exponential functions.
Examples:
 The total cost for a plumber who charges $50 for a house call and $85 per hour would be
expressed as the function y = 85x + 50. If the rate were raised to $90 per hour, how would
the function change?

The equation y = 8,000(1.04)x models the rising population of a city with 8,000 residents when
the annual growth rate is 4%.
o What would be the effect on the equation if the city’s population were 12,000 instead
of 8,000?
o What would happen to the population over 25 years if the growth rate were 6%
instead of 4%?

A function of the form f(n) = P(1 + r)n is used to model the amount of money in a savings
account that earns 5% interest, compounded annually, where n is the number of years since
the initial deposit. What is the value of r? What is the meaning of the constant P in terms of
the savings account? Explain either orally or in written format.
62
Additional Domain Information – Linear, Quadratic and Exponential Models (F-LE)
Key Vocabulary



Linear function
Exponential function
Quadratic function





Explicit formula
Even/odd function
Principal
Interest rate
Growth/decay rate
Example Resources

Books





Algebra I by McDougal and Littell [9.8, 11.3]
SpringBoard Algebra I by CollegeBoard [2.2]
Developing Essential Understanding of Functions: Grades 9-12 by NCTM
Technology
 http://www.ixl.com/math/algebra-1/write-linear-quadratic-and-exponential-functions Activities in creating functions based on data tables.
 http://a4a.learnport.org/page/comparing-functions Apply a linear and exponential model to a small data set in order to explore the
differences in growth pattern, depending on which model is selected.
 http://www.padowan.dk/graph/ Free downloadable graph generator.
 eltrevoogmath.weebly.com/uploads/6/6/8/2/6682813/9-8.pdf This PDF is a note-taking guide for 9.8 McDougal Littell.
 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.illustrativemathematics.org/ This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards.
Example Lessons
 http://www.nsa.gov/academia/_files/collected_learning/high_school/algebra/matchstick_math.pdf Use matchsticks and straws to create
models of the three types of functions and analyze them.
 http://www.yummymath.com/2011/harry-potter/ Using the income and expense data from the Harry Potter films, create a function and
predict the income of a hypothetical ninth movie.
 Technology Time SpringBoard [2.2 pg 80] Given 6 different types of functions, determine the domain and range of the functions.
 Investigation 3 Thinking with Mathematical Models, Connected Math 2 [pg 47] Inverse Variation
63
Common Student Misconceptions
Students tend to draw all graphs of functions as linear. For example, they may graph x2 as 2x.
Students misunderstand how to identify the parameters (function transformation) to build new functions from existing functions. For example,
to shift the graph of y = x2 left by 6 units, they may write y = (x - 6)2 rather than y = (x + 6)2.
Students often confuse the independent variable with the dependent variable.
Students often have difficulty identifying the quantities to be represented by variables, in modeling problems,.
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.
64
High School Algebra 1
Conceptual Category: Statistics and Probability (1 Domain, 3 Clusters)
Domain: Interpreting Categorical and Quantitative Data (3 Clusters)
Interpreting Categorical and Quantitative Data (S-ID) (Domain 1 – Cluster 1 – Standards 1, 2 and 3)
Summarize, represent, and interpret data on a single count or measurement variable.
Essential Concepts
Essential Questions
 Sets of data can be represented on number lines via dot plots,
histograms, and box plots, in order to look at and compare the overall
shape of the data, measures of center and spread.
 Extreme data points (outliers) can affect the shape, measures of center,
and spread of a given data set.
 The measure of center or variability that best interprets a data set will
depend upon the shape of the data distribution and context of data
collection.
HS.S-ID.1
HS.S-ID.1
Represent data with plots on
the real number line (dot
plots, histograms, and box
plots).
Connections:
SCHS-S1C1-04;
SCHS-S1C2-03;
SCHS-S1C2-05;
SCHS-S1C4-02;
SCHS-S2C1-04;
ETHS-S6C2-03;
SSHS-S1C1-04;
9-10.RST.7
Mathematical
Practices
HS.MP.4. Model
with
mathematics.
HS.MP.5. Use
appropriate tools
strategically.



For a given data set, which measure of center or variability best
describes the data and why?
How can extreme data points affect the shape, measures of center
and spread of a data set?
What types of data would you want to display on a number line and
why?
Examples & Explanations
In grades 6 – 8, students describe center and spread in a data distribution. Here they choose a
summary statistic appropriate to the characteristics of the data distribution, such as the shape of the
distribution or the existence of extreme data points.
A statistical process is a problem-solving process consisting of four steps:
1. formulating a question that can be answered by data;
2. designing and implementing a plan that collects appropriate data;
3. analyzing the data by graphical and/or numerical methods;
4. and interpreting the analysis in the context of the original question.
Example:
 What measure of center or variability would best represent the data distribution for the
height of basketball players on this team? Why?
(Continued on next page)
65
Are there any extreme data points that may skew the data?
Basketball Team – Height of Players in inches for 2010-2011 Season
75, 73, 76, 78, 79, 78, 79, 81, 80, 82, 81, 84, 82, 84, 80, 84
HS.S-ID.2
HS.S-ID.2
Use statistics appropriate to
the shape of the data
distribution to compare center
(median, mean) and spread
(interquartile range, standard
deviation) of two or more
different data sets.
Connections:
SCHS-S1C3-06;
ETHS-S6C2-03;
SSHS-S1C1-01
Mathematical
Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.
Examples & Explanations
Students may use spreadsheets, graphing calculators and statistical software for calculations,
summaries, and comparisons of data sets.
Examples:
 The two data sets below depict the housing prices sold in the King River area and Toby
Ranch areas of Pinal County, Arizona. Based on the prices below, which price range can be
expected for a home purchased in Toby Ranch? In the King River area? In Pinal County?
o King River area {1.2 million, 242000, 265500, 140000, 281000, 265000, 211000}
o Toby Ranch homes {5million, 154000, 250000, 250000, 200000, 160000, 190000}

Given a set of test scores: 99, 96, 94, 93, 90, 88, 86, 77, 70, 68, find the mean, median and
standard deviation. Explain how the values vary about the mean and median. What
information does this give the teacher?
HS.MP.4. Model
with
mathematics.HS.
MP.5. Use
appropriate tools
strategically.
HS.MP.7. Look
for and make use
of structure.
66
HS.S-ID.3
HS.S-ID.3
Interpret differences in shape,
center, and spread in the
context of the data sets,
accounting for possible
effects of extreme data points
(outliers).
Mathematical
Practices
HS.MP.2.
Reason
abstractly and
quantitatively.
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.
Examples & Explanations
Students may use spreadsheets, graphing calculators and statistical software to statistically identify
outliers and analyze data sets with and without outliers as appropriate.
Comparing two data sets using a histogram. Not only can the shape of the distribution be observed,
but so can the distribution's location and spread. Figure 16 shows how a mean has increased -- a
transition from the distribution shown at the left (blue) to the one shown on the right (green). Figure
17 shows a different method of comparing distributions. The original data set (shown in green) has
greater variability than the later data set (the blue histogram superimposed over the original data
set).
HS.MP.4. Model
with
mathematics.
HS.MP.5. Use
appropriate tools
strategically.
HS.MP.7. Look
for and make use
of structure.
Figure 16: Histogram of two data sets, one with increased mean
(Continued on next page)
67
Figure 17: Histogram of two data sets, one with increased variability
From: http://illuminae.info/matec/index.php?option=com_content&view=article&id=12:quality-toolsand-spc-charts&catid=9
Interpreting Categorical and Quantitative Data (S-ID) (Domain 1 – Cluster 2 – Standards 5 and 6)
Summarize, represent, and interpret data on two categorical and quantitative variables. (Linear focus, discuss general principles)
Essential Concepts
Essential Questions
 Two-way frequency tables can be used to interpret joint, marginal and
conditional relative frequencies of categorical data.
 Two-way frequency tables and scatter plots of categorical data can be
used to identify possible associations and trends in the data.
 Scatter plots of data sets can be used to identify the type of function that
best represents the shape of the data (linear, quadratic or exponential).
 Residuals (lines of regressions) are drawn on scatter plots in order to
informally assess the fit of a function to a data set.
 If a scatter plot has a linear association, then a line of best fit can be
drawn to interpret the data set.
 How are two-way frequency tables used to interpret joint, marginal and
conditional relative frequencies of categorical data?
 How do you use a two-way frequency table or scatter plot to identify
associations or trends in a data set?
 Why would you want to identify trends or associations in a data set?
 Why would you want to informally assess and identify a type of function
to fit a data set?
68
HS.S-ID.5
HS.S-ID.5
Summarize categorical data
for two categories in two-way
frequency tables. Interpret
relative frequencies in the
context of the data (including
joint, marginal, and
conditional relative
frequencies). Recognize
possible associations and
trends in the data.
Connections:
ETHS-S1C2-01;
ETHS-S6C2-03;
11-12.RST.9;
11-12.WHST.1a-1b;
11-12.WHST.1e
Mathematical
Practices
HS.MP.1. Make
sense of
problems and
persevere in
solving them.
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.
HS.MP.5. Use
appropriate tools
strategically.
HS.MP.8. Look
for and express
regularity in
repeated
reasoning.
HS.S-ID.6
HS.S-ID.6
Represent data on two
quantitative variables on a
scatter plot, and describe how
Mathematic
al Practices
HS.MP.2.
Reason
Examples & Explanations
Students may use spreadsheets, graphing calculators, and statistical software to create frequency
tables and determine associations or trends in the data.
Examples:
 Two-way Frequency Table
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
No
Yes
Total
Two-way Frequency Table
Age
Younger than 45
45 or older
35
11
24
30
59
41
Total
46
54
100
The total row and total column entries in the table above report the marginal frequencies, while
entries in the body of the table are the joint frequencies.

Two-way Relative Frequency Table
The relative frequencies in the body of the table are called conditional relative frequencies.
Bald
No
Yes
Total
Two-way Relative Frequency Table
Age
Total
Younger than 45
45 or older
0.35
0.11
0.46
0.24
0.30
0.54
0.59
0.41
1.00
Examples & Explanations
Students take a more sophisticated look at using a linear function to model the relationship between
two numerical variables. In addition to fitting a line to data, students assess how well the model fits by
analyzing residuals.
(Continued on next page)
69
the variables are related.
Connections:
SCHS-S1C2-05;
SCHS-S1C3-01;
ETHS-S1C2-01;
ETHS-S1C3-01;
ETHS-S6C2-03
a. Fit a function to the data;
use functions fitted to
data to solve problems in
the context of the data.
Use given functions or
choose a function
suggested by the context.
Emphasize linear,
quadratic, and
exponential models.
Connection: 11-12.RST.7
b. Informally assess the fit
of a function by plotting
and analyzing residuals.
Connections: 11-12.RST.7;
11-12.WHST.1b-1c
c.
Fit a linear function for a
scatter plot that suggests
a linear association.
abstractly and
quantitatively.
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.7. Look
for and make
use of
structure.
S.ID.6b should be focused on linear models, but may be used to preview quadratic functions for later
in the year.
This concept can be explained through the use of technology to model the idea and allow students to
explore this standard.
The residual in a regression model is the difference between the observed and the predicted y for
some x (where y is the dependent variable and x is the independent variable). So if we have a model
y = ax + b and a data point (x i, y i ) , the residual for this point is ri = y i - (ax i + b) .
Students may use spreadsheets, graphing calculators, and statistical software to represent data,
describe how the variables are related, fit functions to data, perform regressions, and calculate
residuals.
Example:

Measure the wrist and neck size of each person in your class and make a scatter plot. Find
the least squares regression line. Calculate and interpret the correlation coefficient for this
linear regression model. Graph the residuals and evaluate the fit of the linear equation. Use
the line of best fit to predict the wrist size for a person not in your class.
HS.MP.8. Look
for and express
regularity in
repeated
reasoning.
Connection: 11-12.RST.7
70
Interpreting Categorical and Quantitative Data (S-ID) (Domain 1 – Cluster 3 – Standards 7, 8 and 9)
Interpret linear models
Essential Concepts
Essential Questions
 If a scatter plot has a linear association, then a linear model can be
drawn and used to identify and interpret the meaning of the slope
(constant rate of change) and the intercept (constant term) between the
data sets.
 Technology is used to compute and interpret the correlation coefficient
(the slope) of a linear model.
 A correlation does not necessarily mean there is causation.
HS.S-ID.7
HS.S-ID.7
Interpret the slope (rate of
change) and the intercept
(constant term) of a linear
model in the context of the
data.
Connections:
SCHS-S5C2-01;
ETHS-S1C2-01;
ETHS-S6C2-03;
9-10.RST.4; 9-10.RST.7;
9-10.WHST.2f
Mathematical
Practices
HS.MP.1. Make
sense of
problems and
persevere in
solving them.






How do you interpret the meaning of a slope in a linear model in
context?
What is the meaning of an intercept in terms of a linear model for a
given data set?
How are the slope and correlation coefficient related?
How do you use technology to compute the correlation coefficient?
What is the difference between a correlation and causation?
Give an example of a relationship that has a correlation but is not
causation and explain why.
Examples & Explanations
Build on students’ work with linear relationships in eighth grade and introduce the correlation
coefficient.
Students may use spreadsheets or graphing calculators to create representations of data sets and
create linear models.
Example:
HS.MP.2.
Reason
abstractly and
quantitatively.
HS.MP.4. Model
with
mathematics.
HS.MP.5. Use
appropriate tools
strategically.

Lisa lights a candle and records its height in inches every hour. The results recorded as
(time, height) are (0, 20), (1, 18.3), (2, 16.6), (3, 14.9), (4, 13.2), (5, 11.5), (7, 8.1), (9, 4.7),
and (10, 3). Express the candle’s height (h) as a function of time (t) and state the meaning
of the slope and the intercept in terms of the burning candle.
Solution:
h = -1.7t + 20
Slope: The candle’s height decreases by 1.7 inches for each hour it is burning.
Intercept: Before the candle begins to burn, its height is 20 inches.
HS.MP.6. Attend
to precision.
71
HS.S-ID.8
HS.S-ID.8
Compute (using technology)
and interpret the correlation
coefficient of a linear fit.
Connections:
ETHS-S1C2-01;
ETHS-S6C2-03;
11-12.RST.5;
11-12.WHST.2e
Mathematical
Practices
HS.MP.4. Model
with
mathematics.
HS.MP.5. Use
appropriate tools
strategically.
HS.MP.8. Look
for and express
regularity in
repeated
reasoning.
HS.S-ID.9
HS.S-ID.9
Distinguish between
correlation and causation.
Connection: 9-10.RST.9
Mathematical
Practices
HS.MP.3.
Construct viable
arguments and
critique the
reasoning of
others.
HS.MP.4. Model
with
mathematics.
HS.MP.6. Attend
to precision.
Examples & Explanations
The focus here is on the computation and interpretation of the correlation coefficient as a measure
of how well the data fit the relationship.
Students may use spreadsheets, graphing calculators, and statistical software to represent data,
describe how the variables are related, fit functions to data, perform regressions, and calculate
residuals and correlation coefficients.
Example:
 Collect height, shoe-size, and wrist circumference data for each student. Determine the
best way to display the data. Answer the following questions: Is there a correlation between
any two of the three indicators? Is there a correlation between all three indicators? What
patterns and trends are apparent in the data? What inferences can be made from the data?
Examples & Explanations
The important distinction between a statistical relationship and a cause-and-effect relationship
arises in S-ID.9.
Some data leads observers to believe that there is a cause and effect relationship when a strong
relationship is observed. Students should be careful not to assume that correlation implies
causation. The determination that one thing causes another requires a controlled randomized
experiment.
Example:

Diane did a study for a health class about the effects of a student’s end-of-year math test
scores on height. Based on a graph of her data, she found that there was a direct
relationship between students’ math scores and height. She concluded “doing well on your
end-of-year math tests makes you tall.” Is this conclusion justified? Explain any flaws in
Diane’s reasoning.
72
Additional Domain Information – Interpreting Categorical and Quantitative Data (S-ID)
Key Vocabulary





Outliers
Quartile intervals
Frequency
Two-way frequency tables
Correlation coefficient




Standard deviation
Interval
Scatter plot
Causation


Line of regression
(residual)
Data distribution



Line of best fit
Distribution
Correlation
Example Resources



Books
 Textbooks (Pending)
 Focus in High School Mathematics: Reasoning and Sense Making in Statistics and Probability, National Council of Teachers of Mathematics
Publication
Technology
 http://www.shodor.org/interactivate/lessons/LinearRegressionCorrelation/ Within a complete lesson with categorical and quantitative data
concepts, with vocabulary, there is a useful tool for scatter plots where the student is able to plot the data, find the best line of fit, and analyze
residuals.
 http://nlvm.usu.edu/en/nav/topic_t_5.html This page contains a variety of data analysis formats to present, analyze and predict data.
 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
 www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards.
Example Lessons
 www.malamaaina.org/files/mathematics/lesson7.pdf This lesson is a series of activities designed to use previous data gathered at a
higher level with each succeeding activity.
 http://www.indiana.edu/~iucme/mathmodeling/lessons.htm - A series of 40 problem-based activities to develop data-gathering techniques
from 7th – 12th grade.
Common Student Misconceptions
Students confuse the measures of center and how an outlier can have different impacts on the mean and median. For example, students think
that including an outlier changes the median, when in fact it changes the mean.
Students tend to have difficulty with the distinction between experimental and predicted values.
Students sometimes have difficulty connecting lines to functions. For example, they may have difficulty in predicting that a steeper line will change
more quickly than a gradual one.
73
Students have more difficulty with negative correlations than with positive correlations.
Students may have difficulty identifying when a group of data represent a linear or nonlinear function.
Students have difficulty placing lines of best fit. For example, they may try to connect all of the points on a scatter plot, or may not draw the line
through the middle of the data.
Students may believe that a correlation implies causation.
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.
74
75
76
77
78
79
80
81
82
83
84
85