STANDARDS FOR MATHEMATICS High School Algebra 1

STANDARDS FOR
MATHEMATICS
High School Algebra 1

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High School Overview

Conceptual Categories and Domains

Number and Quantity Statistics and Probability
• Interpreting Categorical and Quantitative Data (S-
• The Real Number System (N-RN) ID)
• Quantities (N-Q) • Making Inferences and Justifying Conclusions (S-
• The Complex Number System (N-CN) IC)
• Vector and Matrix Quantities (N-VM) • Conditional Probability and the Rules of
Probability (S-CP)
Algebra • Using Probability to Make Decisions (S-MD)

• Seeing Structure in Expressions (A-SSE) Contemporary Mathematics
• Arithmetic with Polynomials and Rational Expressions (A-APR) • Discrete Mathematics (CM-DM)
• Creating Equations (A-CED)
• Reasoning with Equations and Inequalities (A-REI)

Functions Mathematical Practices (MP)
1. Make sense of problems and persevere in solving
• Interpreting Functions (F-IF)
• Building Functions (F-BF) them.
• Linear, Quadratic, and Exponential Models (F-LE) 2. Reason abstractly and quantitatively.
• Trigonometric Functions (F-TF) 3. Construct viable arguments and critique the

Geometry reasoning of others.
4. Model with mathematics.
• Congruence (G-CO) 5. Use appropriate tools strategically.
• Similarity, Right Triangles, and Trigonometry (G-SRT) 6. Attend to precision.
• Circles (G-C) 7. Look for and make use of structure.
• Expressing Geometric Properties with Equations (G-GPE) 8. Look for and express regularity in repeated
• Geometric Measurement and Dimension (G-GMD)
• Modeling with Geometry (G-MG) reasoning.

Modeling

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Domain and Clusters

High School - Number and Quantity Overview

!

The Real Number System (N-RN) Mathematical Practices (MP)
• Extend the properties of exponents to rational exponents 1. Make sense of problems and persevere in solving them.
• Use properties of rational and irrational numbers. 2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
Quantities (N-Q)
• Reason quantitatively and use units to solve problems others.
4. Model with mathematics.
The Complex Number System (N-CN) 5. Use appropriate tools strategically.
6. Attend to precision.
• Perform arithmetic operations with complex numbers 7. Look for and make use of structure.
• Represent complex numbers and their operations on the 8. Look for and express regularity in repeated reasoning.

complex plane

• Use complex numbers in polynomial identities and equations

Vector and Matrix Quantities (N-VM)

• Represent and model with vector quantities.
• Perform operations on vectors.

• Perform operations on matrices and use matrices in
applications.

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High School - Algebra Overview

!

Seeing Structure in Expressions (A-SSE) Mathematical Practices (MP)
• Interpret the structure of expressions 1. Make sense of problems and persevere in solving them.
• Write expressions in equivalent forms to solve problems 2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
Arithmetic with Polynomials and Rational Expressions (A-APR)
• Perform arithmetic operations on polynomials others.
• Understand the relationship between zeros and factors of 4. Model with mathematics.
polynomials 5. Use appropriate tools strategically.
• Use polynomial identities to solve problems 6. Attend to precision.
• Rewrite rational expressions 7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

Creating Equations (A-CED)
• Create equations that describe numbers or relationships

Reasoning with Equations and Inequalities (A-REI)

• Understand solving equations as a process of reasoning and
explain the reasoning

• Solve equations and inequalities in one variable
• Solve systems of equations
• Represent and solve equations and inequalities graphically

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High School - Functions Overview

Interpreting Functions (F-IF) !
• Understand the concept of a function and use function
notation Mathematical Practices (MP)
• Interpret functions that arise in applications in terms of the 1. Make sense of problems and persevere in solving them.
context 2. Reason abstractly and quantitatively.
• Analyze functions using different representations 3. Construct viable arguments and critique the reasoning of

Building Functions (F-BF) others.
• Build a function that models a relationship between two 4. Model with mathematics.
quantities 5. Use appropriate tools strategically.
• Build new functions from existing functions 6. Attend to precision.
7. Look for and make use of structure.
Linear, Quadratic, and Exponential Models (F-LE) 8. Look for and express regularity in repeated reasoning.
• Construct and compare linear, quadratic, and exponential
models and solve problems
• Interpret expressions for functions in terms of the situation
they model

Trigonometric Functions (F-TF)

• Extend the domain of trigonometric functions using the unit
circle

• Model periodic phenomena with trigonometric functions

• Prove and apply trigonometric identities

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High School – Geometry Overview

!

Congruence (G-CO) Geometric Measurement and Dimension (G-GMD)
• Experiment with transformations in the plane
• Understand congruence in terms of rigid motions • Explain volume formulas and use them to solve problems
• Prove geometric theorems
• Make geometric constructions • Visualize relationships between two-dimensional and three-
dimensional objects

Similarity, Right Triangles, and Trigonometry (G-SRT) Modeling with Geometry (G-MG)
• Understand similarity in terms of similarity transformations • Apply geometric concepts in modeling situations
• Prove theorems involving similarity
• Define trigonometric ratios and solve problems involving right Mathematical Practices (MP)
triangles 1. Make sense of problems and persevere in solving them.
• Apply trigonometry to general triangles 2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
Circles (G-C)
• Understand and apply theorems about circles others.
• Find arc lengths and areas of sectors of circles 4. Model with mathematics.
5. Use appropriate tools strategically.
Expressing Geometric Properties with Equations (G-GPE) 6. Attend to precision.
• Translate between the geometric description and the 7. Look for and make use of structure.
equation for a conic section 8. Look for and express regularity in repeated reasoning.
• Use coordinates to prove simple geometric theorems
algebraically

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High School – Statistics and Probability Overview

!

Interpreting Categorical and Quantitative Data (S-ID) Mathematical Practices (MP)
• Summarize, represent, and interpret data on a single count or 1. Make sense of problems and persevere in solving them.
measurement variable 2. Reason abstractly and quantitatively.
• Summarize, represent, and interpret data on two categorical 3. Construct viable arguments and critique the reasoning of
and quantitative variables
• Interpret linear models others.
4. Model with mathematics.
Making Inferences and Justifying Conclusions (S-IC) 5. Use appropriate tools strategically.
• Understand and evaluate random processes underlying 6. Attend to precision.
statistical experiments 7. Look for and make use of structure.
• Make inferences and justify conclusions from sample 8. Look for and express regularity in repeated reasoning.
surveys, experiments and observational studies

Conditional Probability and the Rules of Probability (S-CP)

• Understand independence and conditional probability and
use them to interpret data

• Use the rules of probability to compute probabilities of
compound events in a uniform probability model

Using Probability to Make Decisions (S-MD)
• Calculate expected values and use them to solve problems
• Use probability to evaluate outcomes of decisions

High School – Contemporary Mathematics Overview

!

Discrete Mathematics (CM-DM)
• Understand and apply vertex-edge graph topics

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High School - Modeling

Modeling links classroom mathematics and statistics to everyday life, work, and decision-making. Modeling is the process of choosing and
using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. Quantities
and their relationships in physical, economic, public policy, social, and everyday situations can be modeled using mathematical and
statistical methods. When making mathematical models, technology is valuable for varying assumptions, exploring consequences, and
comparing predictions with data.

A model can be very simple, such as writing total cost as a product of unit price and number bought, or using a geometric shape to describe
a physical object like a coin. Even such simple models involve making choices. It is up to us whether to model a coin as a three-dimensional
cylinder, or whether a two-dimensional disk works well enough for our purposes. Other situations—modeling a delivery route, a production
schedule, or a comparison of loan amortizations—need more elaborate models that use other tools from the mathematical sciences. Real-
world situations are not organized and labeled for analysis; formulating tractable models, representing such models, and analyzing them is
appropriately a creative process. Like every such process, this depends on acquired expertise as well as creativity.

Some examples of such situations might include:
• 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.

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The basic modeling cycle is summarized in the diagram. It involves (1) identifying variables in the situation and selecting those that
represent essential features, (2) formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical
representations that describe relationships between the variables, (3) analyzing and performing operations on these relationships to draw
conclusions, (4) interpreting the results of the mathematics in terms of the original situation, (5) validating the conclusions by comparing
them with the situation, and then either improving the model or, if it is acceptable, (6) reporting on the conclusions and the reasoning behind
them. Choices, assumptions, and approximations are present throughout this cycle.

In descriptive modeling, a model simply describes the phenomena or summarizes them in a compact form. Graphs of observations are a
familiar descriptive model—for example, graphs of global temperature and atmospheric CO2 over time.
Analytic modeling seeks to explain data on the basis of deeper theoretical ideas, albeit with parameters that are empirically based; for
example, exponential growth of bacterial colonies (until cut-off mechanisms such as pollution or starvation intervene) follows from a constant
reproduction rate. Functions are an important tool for analyzing such problems.
Graphing utilities, spreadsheets, computer algebra systems, and dynamic geometry software are powerful tools that can be used to model
purely mathematical phenomena (e.g., the behavior of polynomials) as well as physical phenomena.
Modeling Standards Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making
mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards
indicated by a star symbol ( ).

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Standards for Mathematical Practice: High School

Standards for Mathematical Practice

Standards Explanations and Examples

Students are expected to: Mathematical Practices are High school students start to examine problems by explaining to themselves the meaning of a
listed throughout the grade problem and looking for entry points to its solution. They analyze givens, constraints,
HS.MP.1. Make sense of level document in the 2nd relationships, and goals. They make conjectures about the form and meaning of the solution and
problems and persevere in column to reflect the need to plan a solution pathway rather than simply jumping into a solution attempt. They consider
solving them. connect the mathematical analogous problems, and try special cases and simpler forms of the original problem in order to
practices to mathematical gain insight into its solution. They monitor and evaluate their progress and change course if
content in instruction. 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
HS.MP.2. Reason they need. By high school, students can explain correspondences between equations, verbal
abstractly and descriptions, tables, and graphs or draw diagrams of important features and relationships, graph
quantitatively. data, and search for regularity or trends. They check their answers to problems using different
methods and continually ask themselves, “Does this make sense?” They can understand the
approaches of others to solving complex problems and identify correspondences between
different approaches.
High school students seek to make sense of quantities and their relationships in problem
situations. They abstract a given situation and represent it symbolically, manipulate the
representing symbols, and pause as needed during the manipulation process in order to probe
into the referents for the symbols involved. Students use quantitative reasoning to create
coherent representations of the problem at hand; consider the units involved; attend to the
meaning of quantities, not just how to compute them; and know and flexibly use different
properties of operations and objects.

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Standards for Mathematical Practice

Standards Mathematical Practices are Explanations and Examples
Students are expected to: listed throughout the grade
level document in the 2nd High school students understand and use stated assumptions, definitions, and previously
HS.MP.3. Construct viable column to reflect the need to established results in constructing arguments. They make conjectures and build a logical
arguments and critique the connect the mathematical progression of statements to explore the truth of their conjectures. They are able to analyze
reasoning of others. practices to mathematical situations by breaking them into cases, and can recognize and use counterexamples. They
content in instruction. 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
HS.MP.4. Model with context from which the data arose. High school students are also able to compare the
mathematics. effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which
is flawed, and—if there is a flaw in an argument—explain what it is. High school students learn to
determine domains to which an argument applies, listen or read the arguments of others, decide
whether they make sense, and ask useful questions to clarify or improve the arguments.
High school students can apply the mathematics they know to solve problems arising in
everyday life, society, and the workplace. By high school, a student might use geometry to solve
a design problem or use a function to describe how one quantity of interest depends on another.
High school students making assumptions and approximations to simplify a complicated
situation, realizing that these may need revision later. They are able to identify important
quantities in a practical situation and map their relationships using such tools as diagrams, two-
way tables, graphs, flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their mathematical results in the
context of the situation and reflect on whether the results make sense, possibly improving the
model if it has not served its purpose.

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Standards for Mathematical Practice

Standards Mathematical Practices are Explanations and Examples
Students are expected to: listed throughout the grade
level document in the 2nd High school students consider the available tools when solving a mathematical problem. These
HS.MP.5. Use appropriate column to reflect the need to tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a
tools strategically. connect the mathematical spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software.
practices to mathematical High school students should be sufficiently familiar with tools appropriate for their grade or
content in instruction. 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
HS.MP.6. Attend to graphs of functions and solutions generated using a graphing calculator. They detect possible
precision. 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
HS.MP.7. Look for and varying assumptions, explore consequences, and compare predictions with data. They are able
make use of structure. to identify relevant external mathematical resources, such as digital content located on a
website, and use them to pose or solve problems. They are able to use technological tools to
explore and deepen their understanding of concepts.
High school students try to communicate precisely to others by using clear definitions in
discussion with others and in their own reasoning. They state the meaning of the symbols they
choose, specifying units of measure, and labeling axes to clarify the correspondence with
quantities in a problem. They calculate accurately and efficiently, express numerical answers
with a degree of precision appropriate for the problem context. By the time they reach high
school they have learned to examine claims and make explicit use of definitions.
By high school, students look closely to discern a pattern or structure. In the expression x2 + 9x
+ 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance
of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for
solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being
composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive
number times a square and use that to realize that its value cannot be more than 5 for any real
numbers x and y. High school students use these patterns to create equivalent expressions,
factor and solve equations, and compose functions, and transform figures.

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Standards for Mathematical Practice

Standards Mathematical Practices are Explanations and Examples
Students are expected to: listed throughout the grade
level document in the 2nd High school students notice if calculations are repeated, and look both for general methods and
HS.MP.8. Look for and column to reflect the need to for shortcuts. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x –
express regularity in connect the mathematical 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a
repeated reasoning. practices to mathematical geometric series. As they work to solve a problem, derive formulas or make generalizations, high
content in instruction. school students maintain oversight of the process, while attending to the details. They
continually evaluate the reasonableness of their intermediate results.

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High School 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. • How do you use properties of rational exponents to simplify and create
equivalent forms of numerical expressions?
• Properties of integer exponents extend to properties of rational
exponents. • Why are rational exponents and radicals related to each other?

• Properties of rational exponents are used to simplify and create • Given an expression with a rational exponent, how do you write the
equivalent forms of numerical expressions. equivalent radical expression?

• Rational exponents can be written as radicals, and radicals can be
written as rational exponents.

HS.N-RN.1

HS.N-RN.1. Mathematical Examples & Explanations

Explain how the definition of Practices In implementing the standards in curriculum, these standards should occur before discussing
the meaning of rational exponential functions with continuous domains.
exponents follows from HS.MP.2. Reason
extending the properties of abstractly and Students may explain orally or in written format.
integer exponents to those quantitatively.
values, allowing for a notation Example:
for radicals in terms of HS.MP.3. • For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to
rational exponents. For Construct viable hold, so (51/3)3 must equal 5.
example, we define 51/3 to be arguments and
the cube root of 5 because critique the
we want (51/3)3 = 5(1/3)3 to reasoning of
hold, so (51/3)3 must equal 5. others.

HS.N-RN.2 Mathematical Examples & Explanations (Continued on next page)

HS.N-RN.2. Practices Examples:

Rewrite expressions involving HS.MP.7. Look for 22
radicals and rational and make use of
exponents using the structure. • 3 52 = 5 3 ; 5 3 = 3 52
properties of exponents.

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4

• Rewrite using fractional exponents: 5 16 = 5 24 = 25

• Rewrite x in at least three alternate forms.
x2

Solution: −3 = 1 = 1=1
x3 x x
x2 3

x2

• Rewrite 4 2−4 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 • Explain what type of number is produced and why when each of the
produce a rational number. four arithmetic operations is performed on two rational numbers.

• When you perform an operation with a nonzero rational and an irrational • Explain what type of number is produced and why when each of the
number you will produce an irrational number. four operations is performed on a rational number and an irrational
number.
HS.N-RN.3

HS.N-RN.3. Mathematical Examples & Explanations

Explain why the sum or Practices Since every difference is a sum and every quotient is a product, this includes differences and
product of two rational quotients as well. Explaining why the four operations on rational numbers produce rational numbers
numbers are rational; that the HS.MP.2. can be a review of students understanding of fractions and negative numbers. Explaining why the
sum of a rational number and Reason sum of a rational and an irrational number is irrational, or why the product is irrational, includes
an irrational number is abstractly and reasoning about the inverse relationship between addition and subtraction (or between multiplication
irrational; and that the product quantitatively. and addition).
of a nonzero rational number
and an irrational number is HS.MP.3. Connect N.RN.3 to physical situations, e.g., finding the perimeter of a square of area 2.
irrational. Construct viable
arguments and Example:
Connection: 9-10.WHST.1e critique the • Explain why the number 2π must be irrational, given that π is irrational.
reasoning of Answer: if 2π were rational, then half of 2π would also be rational, so π would have to be
others. rational as well.

Additional Domain Information – The Real Number System (N-RN)

Key Vocabulary

• Rational number • Irrational number • Rational exponents

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Example Resources

• Books
¾ Building Powerful Numeracy for Middle and High School Students, by Pamela Weber Harris.

¾ 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
¾ http://nlvm.usu.edu/en/nav/topic_t_2.html - Provides teachers or students with virtual manipulatives to interact with the concepts.
¾ http://www.khanacademy.org/ - Khan Academy contains useful video lessons and also a galaxy of practice modules that enable students to
check answers online.
¾ http://www.classzone.com/books/algebra_1/page_build.cfm?content=lesson8_kh_ch11&ch=11 Help with graphing calculators and rational
functions.
¾ http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM
¾ 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.
¾ http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph
equations.
¾ 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:
¾ 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.
¾ 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.
¾ http://www.themathpage.com/alg/rational-exponents.htm#fractional This lesson includes laws of exponents but moves on to basic equations

with rational exponents of the form −3 1

x5 = 8

For Operations Bringing Together Rational and Irrational Numbers:

¾ 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

Students may see a fractional exponent and multiply it by the base. For example, students might say 1 = 27• 1 = 9 instead of 1 = 3 27 = 3.
3
273 273

Students may see a negative exponent and do the same, converting the base to a negative number instead of a fraction.

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−m

Students may have difficulty converting between radical notation and fractional exponent notation: b n = 1 . They might confuse the m

n bm

and the n.

3 + 3Students tend to assume that they can combine integers and radical expressions: Example, becoming 6 .

Students conversely don’t apply available laws of exponents when multiplying or dividing radicals: Example 18 = 6 . They don’t
understand that they can split up one radical into the product of two component radicals. 3

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 • How can you convert a given quantity in a unit rate to a different unit
problems. rate? For example, how can you convert feet per second to miles per
o Make sure units are compatible when creating, hour?
simplifying/evaluating, and solving equations.
• Why would you want to be able to convert quantities to different units?
• Appropriate units or quantities need to be used when answering real-
world situations. • How can units and unit relationships be used to set up and solve
o Use labels to put the answers into proper context. multi-step problems?

• Working with expressions, equations, relations and functions can be • Give an example of a real-world situation and explain what unit or
facilitated by understanding the quantities and their relationships. quantity you expressed the answer in and why.

• Graphs should be set up with the appropriate scales and units for the • How can you determine which scale and unit to use when creating a
given context. graph to represent a set of data?

• 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 Mathematical Examples & Explanations
Practices
Use units as a way to Working with quantities and the relationships between them provides grounding for work with
understand problems and to HS.MP.4. Model expressions, equations, and functions.
guide the solution of multi- with
step problems; choose and mathematics. Include word problems where quantities are given in different units, which must be converted to
interpret units consistently in make sense of the problem.
formulas; choose and HS.MP.5. Use
interpret the scale and the appropriate tools (Continued on next page)

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origin in graphs and data strategically. Example:
displays.
HS.MP.6. Attend • A problem might have one object moving 12 feet per second and another at 5 miles per
to precision. hour. To compare speeds, students convert 12 feet per second to miles per hour:

12 ft • 1 mile • 60 sec • 60min = 43200 mile ≈ 8.18 miles per hour.
1 sec 5280 ft 1 min 1 hr 5280 hr

• 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.

HS.N-Q.2 Mathematical Examples & Explanations

HS.N-Q.2 Practices Examples:

Define appropriate quantities HS.MP.4. Model • What type of measurements would you use to determine your income and expenses for one
for the purpose of descriptive with month?
modeling. mathematics.
• How could you express the number of accidents in Arizona?

HS.MP.6. Attend
to precision.

HS.N-Q.3

HS.N-Q.3 Mathematic Examples & Explanations

Choose a level of accuracy al Practices The margin of error and tolerance limit varies according to the measure, tool used, and context.
appropriate to limitations on
measurement when reporting HS.MP.5. Use Example:
quantities. appropriate • Determining price of gas by estimating to the nearest cent is appropriate because you will not
tools pay in fractions of a cent, but the cost of gas is given to tenths of a cent, e.g., $3.479 .
strategically. gallon

HS.MP.6.
Attend to
precision.

 

Additional Domain Information – Quantities (N-Q)

Key Vocabulary • Unit rate • Scale • Origin
• Ratio • Equivalent • Unit Conversion
• Unit
• Descriptive model

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Example Resources

• Books
¾ Textbook
¾ Focus in High School Mathematics: Reasoning and Sense Making (Algebra), Chapter 3: Formal Algebra
¾ Uncovering Student Thinking in Mathematics Grades 6-12, How Low Can You Go pg 71

¾ The Xs and Whys of Algebra: Key Ideas and Common Misconceptions

• Technology
¾ http://www.wolframalpha.com/examples/Math.html Useful for checking correct conversions.
¾ 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.
¾ www.classzone.com/ This is the site to access the book and extra resources online.
¾ http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM
¾ 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.
¾ http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph
equations.
¾ www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards.

• Example Lessons
¾ http://oakroadsystems.com/math/convert.htm#Really1 A nicely thorough introduction to the basics of unit conversion, with practice

problems at the end.
¾ 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.
¾ 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). • Give an example of a real-world problem and write an expression to
model the relationship, and explain how the algebraic symbols
• Terms can either be a constant, a variable with a coefficient, or a represent the words in the problem.
coefficient times a variable raised to a power.
• How are coefficients and factors related to each other?
• Real-world problems with changing quantities can be represented by
expressions with variables. • How does viewing a complicated expression by its single parts help to
interpret and solve problems?
• The relationship between the abstract symbolic representations of
expressions can be identified based on how they relate to the given • What does it mean to call something a quantity?
situation.
• How does using the structure of an expression help to simplify the
• Complicated expressions can be interpreted by viewing parts of the expression?
expression as single entities.
• Why would you want to simplify an expression?
• Structure within an expression can be identified and used to factor or
simplify the expression.

HS.A-SSE.1

HS.A-SSE.1 Mathematical Examples & Explanations
Practices
Interpret expressions that A-SSE.1 starts by being limited to linear expressions and to exponential expressions with integer
represent a quantity in terms HS.MP.1. Make exponents. Later in the year, focus on quadratic and exponential expressions.
of its context. sense of
problems and A-SSE.1b starts with exponents that are integers and then extends from the integer exponents to
a. Interpret parts of an persevere in rational exponents, focusing on those that represent square or cube roots.
expression, such as terms, solving them.
factors, and coefficients. Students should understand the vocabulary for the parts that make up the whole expression and be
HS.MP.2. able to identify those parts and interpret their meanings in terms of a context.
Connection: 9-10.RST.4 Reason
abstractly and Examples:
b. Interpret complicated quantitatively. • Interpret P(1+r)n as the product of P and a factor not depending on P.
expressions by viewing one
or more of their parts as a HS.MP.4. Model (Continued on next page)
single entity. For example, with

21 
 

interpret P(1+r)n as the mathematics. • Suppose the cost of cell phone service for a month is represented by the expression 0.40s
product of P and a factor + 12.95. Students can analyze how the coefficient of 0.40 represents the cost of one minute
not depending on P. HS.MP.7. Look (40¢), while the constant of 12.95 represents a fixed, monthly fee, and s stands for the
for and make use number of cell phone minutes used in the month. Similar real-world examples, such as tax
of structure. rates, can also be used to explore the meaning of expressions.

• Factor 3x(x – 5) + 2(x – 5).  

HS.A-SSE.2 Mathematic 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 al Practices
Examples & Explanations
Use the structure of an HS.MP.2.
expression to identify ways to Reason Students should extract the greatest common factor (whether a constant, a variable, or a combination
rewrite it. For example, see abstractly and of each). If the remaining expression is quadratic, students should factor the expression further.
x4 – y4 as (x2)2 – (y2)2, thus quantitatively.
recognizing it as a difference Example:
of squares that can be HS.MP.7. Look
factored as for and make • Factor x3 − 2x2 − 35x .
(x2 – y2)(x2 + y2). use of
structure. • 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 • What are the solutions to a quadratic equation and how do they relate
or zeros of quadratic functions. to the graph?

• Factoring methods and the method of completing the square reveal • What attributes of the graph will factoring and completing the square
attributes of the graphs of quadratic functions. reveal about a quadratic function?

• Factoring a quadratic reveals the zeros of the function. • How are properties of exponents used to transform expressions for
exponential functions?
• Completing the square in a quadratic equation reveals the maximum or
minimum value of the function. • Why would you want to transform an expression for an exponential
function?
• Properties of exponents are used to transform expressions for
exponential functions.

22 
 

HS.A-SSE.3 Mathematical Examples & Explanations
Practices
HS.A-SSE.3 It is important to balance conceptual understanding and procedural fluency in work with equivalent
HS.MP.1. Make expressions. For example, development of skill in factoring and completing the square goes hand-
Choose and produce an sense of in-hand with understanding what different forms of a quadratic expression reveal.
equivalent form of an problems and Students will use the properties of operations to create equivalent expressions.
expression to reveal and persevere in
explain properties of the solving them. Teachers should foster the idea that changing the forms of expressions, such as factoring or
quantity represented by the completing the square, or transforming expressions from one exponential form to another, are not
expression. HS.MP.2. independent algorithms that are learned for the sake of symbol manipulations. They are processes
Reason that are guided by goals (e.g., investigating properties of families of functions and solving contextual
Connections: 9-10.WHST.1c; abstractly and problems).
11-12.WHST.1c quantitatively.
A pair of coordinates (h, k) from the general form f(x) = a(x – h)2 + k represents the vertex of the
a. Factor a quadratic HS.MP.4. Model parabola, where h represents a horizontal shift and k represents a vertical shift of the parabola y =
expression to reveal the with x2 from its original position at the origin. A vertex (h, k) is the minimum point of the graph of the
zeros of the function it mathematics. quadratic function if a > 0 and is the maximum point of the graph of the quadratic function if a < 0.
defines. Understanding an algorithm for completing the square provides a solid foundation for deriving a
HS.MP.7. Look quadratic formula.
b. Complete the square in a for and make use
quadratic expression to of structure. Examples:
reveal the maximum or • Express 2(x3 – 3x2 + x – 6) – (x – 3)(x + 4) in factored form and use your answer to say for
minimum value of the what values of x the expression is zero.
function it defines.
• The expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate
c. Use the properties of equivalent monthly interest rate if the annual rate is 15%.
exponents to transform
expressions for • Write the expression below as a constant multiplied by a power of x and use your answer to
exponential functions. For decide whether the expression gets larger or smaller as x gets larger.
example the expression
1.15t can be rewritten as (2x3)2 (3x4 )
(1.151/12)12t ≈ 1.01212t to
reveal the approximate (x2 )3
equivalent monthly
interest rate if the annual
rate is 15%.

23 
 

Additional Domain Information – Seeing Structure in Expressions (A-SSE)

Key Vocabulary

• Expression • Term • Coefficient • Vertex
• Factor • Exponent • Base • Completing the Square
• Simplify • Greatest Common Factor • Quadratic • Minimum
• Polynomial • Binomial • Trinomial • Maximum

Example Resources

• Books
¾ Textbook
¾ Focus in High School Mathematics: Reasoning and Sense Making (Algebra), Chapter 3: Formal Algebra
¾ Uncovering Student Thinking in Mathematics Grades 6-12, How Low Can You Go pg 71
¾ The Xs and Whys of Algebra: Key Ideas and Common Misconceptions

• Technology
¾ Key Curriculum Press, Exploring Algebra I with the Geometer’s Sketchpad
¾ www.Geogebra.org online software to create visuals
¾ www.Khanacademy.org Khan Academy contains useful video lessons and also a galaxy of practice modules that enable students to
check answers online.
¾ www.classzone.com/ This is the site to access the book and extra resources online.
¾ http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM.
¾ 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.
¾ http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph
equations.
¾ www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards

• Example Lessons
¾ 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.
¾ 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.
¾ 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.
¾ 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.
¾ 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 • Why is the system of polynomials closed under addition, subtraction
polynomial, thus making the system of polynomials closed. and multiplication?

• Addition and subtraction of polynomials is combining like terms. • How is the system of polynomials similar to and different from the
• The distributive property proves why you can combine like terms. system of integers?
• Multiplication of polynomials is applying the distributive property.
• How does the distributive property show that you can combine like
HS.A-APR.1 terms?

• Explain how the distributive property is used to multiply any size
polynomials.

HS.A-APR.1 Mathematical Examples & Explanations

Understand that polynomials Practices Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive
form a system analogous to integer power of x.
the integers, namely, they are HS.MP.8. Look for
closed under the operations regularity in • In arithmetic of polynomials, a central idea is the distributive property, because it is
of addition, subtraction, and repeated fundamental not only in polynomial multiplication but also in polynomial addition and
multiplication; add, subtract, reasoning. subtraction. With the distributive property, there is little need to emphasize misleading
and multiply polynomials. 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.
Connection: 9-10.RST.4 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 • Term • Coefficient
• Simplify • Exponent • Base
• Polynomial • Binomial • Factor
• Distributive Property • Trinomial • Linear
• Exponential • Quadratic • Closure Property

26 
 

Example Resources

• Books
¾ Textbook
¾ Focus in High School Mathematics: Reasoning and Sense Making (Algebra), Chapter 3: Formal Algebra
¾ The Xs and Whys of Algebra: Key Ideas and Common Misconceptions

• Technology
¾ Key Curriculum Press, Exploring Algebra I with the Geometer’s Sketchpad
¾ www.Geogebra.org online software to create visuals
¾ www.Khanacademy.org Khan Academy contains useful video lessons and also a galaxy of practice modules that enable students to
check answers online.
¾ www.classzone.com/ This is the site to access the book and extra resources online.
¾ http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM.
¾ 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.
¾ http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph
equations.
¾ www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards

• Example Lessons
¾ 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.
¾ 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.
¾ 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 real- • How do you translate real-world situations into mathematical
world and mathematical problems. equations and inequalities?

• Relationships between two quantities can be represented through the • How do you determine if a situation is best represented by an
creation of equations in two variables and graphed on coordinate axes equation, an inequality, a system of equations or a system of
with labels and scales. inequalities?

• Solutions are viable or not in different situations depending upon the • Why would you want to create an equation or inequality to represent a
constraints of the given context. real-world problem?

• Formulas can be rearranged and solved for a given variable using the • How are graphs of equations and inequalities similar and different?
same reasoning as solving an equation.
• How do you determine if a given point is a viable solution to a system
HS.A-CED.1 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?

HS.A-CED.1 Mathematical Examples & Explanations
Practices
Create equations and Limit A-CED.1 and A-CED.2 to linear and exponential equations, and, in the case of exponential
inequalities in one variable HS.MP.2. equations, limit to situations requiring evaluation of exponential functions at integer inputs.
and use them to solve Reason
problems. Include equations abstractly and Start with work on linear and exponential equations, then, later in the year, extend to quadratic
arising from linear, quadratic quantitatively. equations.
and exponential functions.
HS.MP.4. Model Equations can represent real-world and mathematical problems. Include equations and inequalities
with that arise when comparing the values of two different functions, such as one describing linear
mathematics. growth and one describing exponential growth.

HS.MP.5. Use Examples:
appropriate tools • Given that the following trapezoid has area 54 cm2, set up an equation to find the length
strategically. 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 Sn 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: g + gr + gr2 + gr3 + … + grn
gr + gr2 + gr3 + … + grn + grn+1
Sn =
Multiply by r: rSn = g – grn+1
Subtract: Sn – rSn = g(1 – rn+1)
Factor: Sn(1 – r) = g(1 – rn+1)/(1 – r)
=
Divide by (1 – r): Sn

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 S10
= $1578.36 to the nearest cent.

29 
 

HS.A-CED.2 Mathematical Examples & Explanations
Practices
HS.A-CED.2 Limit A-CED.1 and A-CED.2 to linear and exponential equations, and, in the case of exponential
HS.MP.2. equations, limit to situations requiring evaluation of exponential functions at integer inputs.
Create equations in two or Reason
more variables to represent abstractly and Start with work on linear and exponential equations, then, later in the year, extend to quadratic
relationships between quantitatively. equations.
quantities; graph equations
on coordinate axes with HS.MP.4. Model Example:
labels and scales. with • The formula for the surface area of a cylinder is given by V = πr2h, where r represents the
mathematics. 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.
HS.A-CED.3 HS.MP.5. Use Compare the graphs. What is the appropriate domain for r and h? Be sure to label your
appropriate tools graphs and use an appropriate scale.
HS.A-CED.3 strategically.
Examples & Explanations
Represent constraints by Mathematical
equations or inequalities, and Practices Limit A-CED.3 to linear equations and inequalities.
by systems of equations
and/or inequalities, and HS.MP.2. Examples:
interpret solutions as viable or Reason • A club is selling hats and jackets as a fundraiser. Their budget is $1500 and they want to
non-viable options in a abstractly and order at least 250 items. They must buy at least as many hats as they buy jackets. Each hat
modeling context. For quantitatively. costs $5 and each jacket costs $8.
example, represent o Write a system of inequalities to represent the situation.
inequalities describing HS.MP.4. Model o Graph the inequalities.
nutritional and cost with o If the club buys 150 hats and 100 jackets, will the conditions be satisfied?
constraints on combinations mathematics. o What is the maximum number of jackets they can buy and still meet the conditions?
of different foods.
HS.MP.5. Use • Represent inequalities describing nutritional and cost constraints on combinations of
appropriate tools different foods.
strategically.

30 
 

HS.A-CED.4 Mathematical Examples & Explanations
Practices
HS.A-CED.4 Start by limiting A-CED.4 to formulas that are linear in the variable of interest. Later in the year,
HS.MP.2. Reason extend to formulas involving squared variables.
Rearrange formulas to abstractly and
highlight a quantity of interest, quantitatively. Examples:
using the same reasoning as • The Pythagorean theorem expresses the relation between the legs a and b of a right
in solving equations. For HS.MP.4. Model triangle and its hypotenuse c with the equation a2 + b2 = c2.
example, rearrange Ohm’s with mathematics. o Why might the theorem need to be solved for c?
law V = IR to highlight o Solve the equation for c and write a problem situation where this form of the equation
resistance R. HS.MP.5. Use might be useful.
appropriate tools
strategically. • Solve V = 4 π r3 for radius r.
3
HS.MP.7. Look for
and make use of • Motion can be described by the formula below, where t = time elapsed, u = initial velocity,
structure. a = acceleration, and s = distance traveled.

s = ut+½at2

o Why might the equation need to be rewritten in terms of a?
o 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 • Equation • Inequality
• Linear • Quadratic
• Expression • Exponent • Y-intercept
• Simplify • X-intercept • Formula
• Solution • System • Coordinate axes
• Zeros
• Graph

Example Resources

• Books
¾ Textbook
¾ Focus in High School Mathematics: Reasoning and Sense Making (Algebra), Chapter 2: Building Equations and Functions

¾ The Xs and Whys of Algebra: Key Ideas and Common Misconceptions
¾ Mission Mathematics II: Grades 9-12, Modeling Space-Debris Accumulation pg 28
¾ Active Algebra: Strategies and Lessons for Successfully Teaching Linear Relationships, Section II: Linear Relations: Lessons and

Assessments

31 
 

• Technology
¾ Key Curriculum Press, Exploring Algebra I with the Geometer’s Sketchpad
¾ www.Geogebra.org online software to create visuals
¾ www.Khanacademy.org Khan Academy contains useful video lessons and also a galaxy of practice modules that enable students to
check answers online.
¾ www.classzone.com/ This is the site to access the book and extra resources online.
¾ http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM.
¾ 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.
¾ http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph
equations.
¾ www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards

• Example Lessons
¾ 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.
¾ 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.
¾ 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 • What do you use to justify your reasoning when solving an equation?
operations and equality, which can justify each step of the process.
• How do you determine if an equation is solved properly?
• 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. • How do you determine and justify if a solution to an equation is
correct?

• Why are properties of real numbers important when solving
equations?

HS.A-REI.1 Mathematical Examples & Explanations
Practices
HS.A-REI.1 Students should focus on and master A-REI.1 for linear equations and be able to extend and apply
HS.MP.2. Reason their reasoning to other types of equations for later in the year and in future courses. Students will
Explain each step in solving a abstractly and solve exponential equations with logarithms in Algebra II.
simple equation as following quantitatively.
from the equality of numbers Properties of operations can be used to change expressions on either side of the equation to
asserted at the previous step, HS.MP.3. equivalent expressions. In addition, adding the same term to both sides of an equation or
starting from the assumption Construct viable multiplying both sides by a non-zero constant produces an equation with the same solutions. Other
that the original equation has arguments and operations, such as squaring both sides, may produce equations that have extraneous solutions.
a solution. Construct a viable critique the
argument to justify a solution reasoning of Each step of solving an equation can be defended, much like providing evidence for steps of a
method. others. geometric proof.

HS.MP.7. Look for Provide examples for how the same equation might be solved in a variety of ways as long as
and make use of equivalent quantities are added or subtracted to both sides of the equation; the order of steps taken
structure. 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: 2x − 5 + 5 = 7 + 5 Addition property of equality .
2x = 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, • What do you use to justify your steps when solving linear and non-
equality, and inequality, which can justify each step of the process. linear equations and inequalities?

• A solution to an equation can be checked, by substituting in that value • How do you determine and justify whether a solution to an equation or
for the variable and simplifying to see if the equation or inequality holds inequality is correct?
true.
• How do operations performed on real numbers affect the relationship
• Laws of exponents can be used to solve simple exponential equations. between the quantities in an inequality?

• Completing the square can be used to transform a quadratic equation • Why would you want to transform a quadratic equation to the form
into the form (x-p)2 = q. (x-p)2 = q?

• The quadratic formula can be derived by completing the square of ax2+ • How do you determine which method is best for solving a quadratic
bx+c = 0. equation?

• Quadratic equations can be solved by a variety of methods, for • Why do some quadratic equations have extraneous and/or complex
example: by inspection, graphing, taking square roots, factoring, solutions?
completing the square and the quadratic formula.

• Quadratic equations can have extraneous and/or complex solutions.

HS.A-REI.3

HS.A-REI.3 Mathematical Examples & Explanations
Practices
Solve linear equations and Extend earlier work with solving linear equations to solving linear inequalities in one variable and to
inequalities in one variable, HS.MP.2 solving literal equations that are linear in the variable being solved for. Include simple exponential
including equations with Reason equations that rely only on application of the laws of exponents, such as 5x=125 or 2x=1/16.
coefficients represented by abstractly and
letters starting from the quantitatively. (Continued on next page)
assumption that the original
equation has a solution. HS.MP.7. Look

34 
 

Construct a viable argument for and make use Examples:
to justify a solution method. of structure.
Solve for the variable:
HS.A-REI.4 HS.MP.8. Look
HS.A-REI.4 for and express • − 7 y − 8 = 111
regularity in 3
Solve quadratic equations in repeated
one variable. reasoning. • 3x > 9

a. Use the method of Mathematical • ax + 7 = 12, when a = 2
completing the square to Practices
transform any quadratic • 3+x = x−9
equation in x into an HS.MP.2. 7 4
equation of the form (x – Reason
p)2 = q that has the same abstractly and • Solve for x: 2/3x + 9 < 18
solutions. Derive the quantitatively.
quadratic formula from Examples & Explanations
this form. HS.MP.3.
Construct viable Students should solve by factoring, completing the square, and using the quadratic formula. The
b. Solve quadratic arguments and zero product property is used to explain why the factors are set equal to zero. Students should
equations by inspection critique the relate the value of the discriminant to the type of root to expect. A natural extension would be to
(e.g., for x2 = 49), taking reasoning of relate the type of solutions to ax2 + bx + c = 0 to the behavior of the graph of y = ax2 + bx + c.
square roots, completing others.
the square, the quadratic Value of Discriminant Nature of Roots Nature of Graph
formula and factoring, as HS.MP.4. Model b2 – 4ac = 0 1 real root intersects x-axis once
appropriate to the initial with b2 – 4ac > 0
form of the equation. mathematics. b2 – 4ac < 0 2 real roots intersects x-axis twice
Recognize when the
quadratic formula gives HS.MP.5. Use 2 complex roots does not intersect x-axis
complex solutions and appropriate tools
write them as a ± bi for strategically. Examples:
real numbers a and b. • Are the roots of 2x2 + 5 = 2x real or complex? How many roots does it have? Find all
HS.MP.7. Look
for and make use solutions of the equation.
of structure. • What is the nature of the roots of x2 + 6x + 10 = 0? Solve the equation using the

HS.MP.8. Look quadratic formula and completing the square. How are the two methods related?
for and express
• 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 2 − 3t −10

0 = (t + 2)(t − 5)

(Continued on next page

35 
 

regularity in So the solutions for t are t = 5 or t = –2, but t = –2 does not make sense in the context of
repeated this problem and therefore is an extraneous solution.
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 • How do you determine the number of solutions that a system of
solutions, or no solution. equations will have?

• A system of linear equations can be solved graphically, algebraically • How do you determine the best method for solving a given system of
using elimination/linear combination, substitution, or modeling. equations?

• Solving a system of equations algebraically yields an exact solution; • Why would you want to multiply an equation by a constant (that is not
solving by graphing yields an approximate solution. zero)?

• Multiplying both sides of an equation by a non-zero constant does not • Why does graphing a system of equations yield an approximate
change the solution to the equation. solution as opposed to an exact solution?

• Elimination/linear combination is a method of solving a system of linear • How can you prove that no matter which method you choose to solve
equations in which the equations are added together in order to a system of equations, you will always get the same solution?
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 
 

HS.A-REI.5 Mathematical Examples & Explanations

HS.A-REI.5 Practices 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
Prove that, given a system of HS.MP.2. describe the same line (yielding infinitely many solutions) and cases where two equations describe
two equations in two Reason parallel lines (yielding no solution); connect to GPE.5 when it is taught in Geometry, which requires
variables, replacing one abstractly and students to prove the slope criteria for parallel lines.
equation by the sum of that quantitatively.
equation and a multiple of the Systems of linear equations can also have one solution, infinitely many solutions or no solutions.
other produces a system with HS.MP.3. Students will discover these cases as they graph systems of linear equations and solve them
the same solutions. Construct viable algebraically.
arguments and
critique the A system of linear equations whose graphs meet at one point (intersecting lines) has only one
reasoning of solution, the ordered pair representing the point of intersection. A system of linear equations whose
others. 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

• 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.

HS.A-REI.6 Mathematical Examples & Explanations
Practices
HS.A-REI.6 The system solution methods can include but are not limited to graphical, elimination/linear
HS.MP.2. Reason combination, substitution, and modeling. Systems can be written algebraically or can be
Solve systems of linear abstractly and represented in context. Students may use graphing calculators, programs, or applets to model and
equations exactly and quantitatively. find approximate solutions for systems of equations.
approximately (e.g., with
graphs), focusing on pairs of HS.MP.4. Model (Continued on next page)
linear equations in two
variables.

37 
 

Connection: ETHS-S6C2-03 with mathematics. Examples:

HS.MP.5. Use • José had 4 times as many trading cards as Phillipe. After José gave away 50 cards to his
appropriate tools little brother and Phillipe gave 5 cards to his friend for his birthday, they each had an equal
strategically. 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 • Solve the system of equations: x+ y = 11 and 3x – y = 5.
and make use of Use a second method to check your answer.
structure.
• Solve the system of equations:
HS.MP.8. Look for x – 2y + 3z = 5, x + 3z = 11, 5y – 6z = 9.
and express
regularity in • The opera theater contains 1,200 seats, with three different prices. The seats cost $45 per
repeated seat, $50 per seat, and $60 per seat. The opera needs to gross $63,750 on seat sales.
reasoning. There are twice as many $60 seats as $45 seats. How many seats at each price need to
be sold?

HS.A-REI.7 Mathematical Examples & Explanations
Practices
HS.A-REI.7 Include systems consisting of one linear and one quadratic equation. Include systems that lead to
HS.MP.2 work with fractions.
Solve a simple system Reason abstractly
consisting of a linear equation and quantitatively. Examples:
and a quadratic equation in • Find the points of intersection between the line y = –3x and the circle x2 + y2 = 3
two variables algebraically HS.MP.4. Model algebraically.
and graphically. For example, with mathematics.
find the points of intersection • Two friends are driving to the Grand Canyon in separate cars. Suzette has been there
between the line y = –3x and HS.MP.5. Use before and knows the way but Andrea does not. During the trip Andrea gets ahead of
the circle x2 + y2 = 3. appropriate tools Suzette and pulls over to wait for her. Suzette is traveling at a constant rate of 65 miles per
strategically. 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
HS.MP.6. Attend Suzette’s car passed is given by d = 3500t2. Write and solve a system of equations to
to precision. determine how long it takes for Andrea to catch up with Suzette.
(Continued on next page)

38 
 

HS.MP.7. Look for • Include systems that lead to work with fractions. For example, finding the intersections
and make use of between x2+y2=1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle,
structure. 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 • How do you determine if a given ordered pair is a solution to an
plotted in the coordinate plane.
equation?
• Solving a system of equations algebraically yields an exact solution;
solving by graphing or by comparing tables of values yields an • Why are the x-coordinates of the points where the graphs of the
approximate solution.
equations y = f(x) and y = g(x) intersect equal to the solutions of the
• The solutions (solution set) of a linear inequality in two variables are
represented graphically as a half-plane. equation f(x) = g(x)? (Continued on next page)

• The solution set of a system of linear inequalities in two variables is the • When graphing a linear inequality, how do you determine which half-
intersection of the corresponding half-planes.
plane to shade in order to represent the solution set?
HS.A-REI.10
• How do you represent the solution set of a system of linear

inequalities on a graph?

HS.A-REI.10 Mathematical Examples & Explanations

Understand that the graph of Practices For A-REI.10, focus on linear and exponential equations and be able to adapt and apply that
an equation in two variables learning to other types of equations in future courses.
is the set of all its solutions HS.MP.2.
plotted in the coordinate Reason Examples:
plane, often forming a curve abstractly and • Which of the following points would be on the graph of the equation 3x+4y=24 ?
(which could be a line). quantitatively. (a) (0, 6) (b) (-1, 7) (c) (4/3, 5) (d) (3, 4)

HS.MP.4. Model • Graph the equation and determine which of the following points are on the graph of
with y = 3x + 1.
mathematics. (a) (2, 7) (b) (-1, 4/3) (c) (2, 10) (d) (0, 1)

39 
 

HS.A-REI.11 Mathematical Examples & Explanations
Practices
HS.A-REI.11 For A-REI.11, focus on cases where f(x) and g(x) are linear, quadratic and exponential. In Algebra
HS.MP.2. II, students will extend this standard to include higher-order polynomials, rational, absolute value
Explain why the x-coordinates Reason and logarithmic functions.
of the points where the abstractly and
graphs of the equations y = quantitatively. Students need to understand that numerical solution methods (data in a table used to approximate
f(x) and y = g(x) intersect are an algebraic function) and graphical solution methods may produce approximate solutions, and
the solutions of the equation HS.MP.4. Model algebraic solution methods produce precise solutions that can be represented graphically or
f(x) = g(x); find the solutions with numerically. Students may use graphing calculators or programs to generate tables of values,
approximately, e.g., using mathematics. graph, or solve a variety of functions.
technology to graph the
functions, make tables of HS.MP.5. Use Example:
values, or find successive appropriate tools • Given the following equations, determine the x value that results in an equal output for both
approximations. Include strategically. functions.
cases where f(x) and/or g(x)
are linear, polynomial, HS.MP.6. Attend f(x) = 3x − 2
rational, absolute value, to precision.
exponential, and logarithmic g(x) = (x + 3)2 − 1
functions.

HS.A-REI.12 Mathematical Examples & Explanations
Practices
HS.A-REI.12 Students may use graphing calculators, programs, or applets to model and find solutions for
HS.MP.4. Model inequalities or systems of inequalities.
Graph the solutions to a with
linear inequality in two mathematics. Examples:
variables as a half-plane • Graph the solutions: y < 2x + 3.
(excluding the boundary in HS.MP.5. Use
the case of a strict inequality), appropriate tools • A publishing company publishes a total of no more than 100 magazines every year. At least
and graph the solution set to strategically. 30 of these are women’s magazines, but the company always publishes at least as many
a system of linear inequalities women’s magazines as men’s magazines. Find a system of inequalities that describes the
in two variables as the possible number of men’s and women’s magazines that the company can produce each
intersection of the year consistent with these policies. Graph the solution set.
corresponding half-planes.

(Continued on next page)

40 
 

• Graph the system of linear inequalities below and determine if (3, 2) is a solution to the

system.

⎧x − 3y > 0
⎪
⎨ x + y ≤ 2

⎩⎪x + 3y > −3

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 • Solution • Substitution
• Inequality • Laws of Exponents • Completing the Square
• Quadratic formula • Square roots • Factoring
• Extraneous solutions • Complex solutions • Elimination/Linear Combinations
• Point of Intersection • Discriminant • System
• Half-Plane • Radical

Example Resources

• Books
¾ Textbook
¾ Focus in High School Mathematics: Reasoning and Sense Making, Chapter 2: Building Equations and Functions

¾ The Xs and Whys of Algebra: Key Ideas and Common Misconceptions

• Technology
¾ Key Curriculum Press, Exploring Algebra I with the Geometer’s Sketchpad
¾ www.Geogebra.org online software to create visuals
¾ www.Khanacademy.org Khan Academy contains useful video lessons and also a galaxy of practice modules that enable students to
check answers online.
¾ www.classzone.com/ This is the site to access the book and extra resources online.
¾ http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM.
¾ 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.
¾ 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 
 

¾ www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the
standards

• Example Lessons
¾ 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.
¾ 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.
¾ 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. • Given a table or graph, how do you determine if it represents a
function?
• In function notation, f(x) denotes that f is a function of x.
• How is a graph related to its algebraic function?
• 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. • How could you use function notation to represent a specific output
of a function?
• 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. • How can the Fibonacci sequence be used to explain a recursive
pattern?
• The graph of a function f is the graph of the equation y = f(x).
• How can you describe a sequence as a function?
• 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 Mathematical Examples & Explanations

Understand that a function Practices Students should experience a variety of types of situations modeled by functions. Detailed analysis
from one set (called the of any particular class of functions at this stage is not advised. Students should apply these
domain) to another set (called HS.MP.2 concepts throughout their future mathematics courses. Draw examples from linear, quadratic, and
the range) assigns to each Reason exponential functions.
element of the domain exactly abstractly and
one element of the range. If f quantitatively. The domain is the set of all inputs (x values); the range is the set of all outputs (y = f(x)).
is a function and x is an The domain of a function may be given by an algebraic expression. Unless otherwise specified, it is
element of its domain, then the largest possible domain.
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).

(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.

AB

x f(x) x f(x)

01 00

12 12

22 13

34 45

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 Mathematical Examples & Explanations
Practices
HS.F-IF.2 Examples:
HS.MP.2.
Use function notation, Reason • If f (x) = x2 + 4x − 12 , find f (2).
evaluate functions for inputs abstractly and
in their domains, and interpret quantitatively. • Let f (x) = 2(x + 3)2 . Find f (3) , f (− 1) , f (a), and f (a − h).
statements that use function 2
notation in terms of a context. Mathematical
Practices • If P(t) is the population of Tucson t years after 2000, interpret the statements P(0) = 487,000
Connection: 9-10.RST.4
HS.MP.8. Look and P(10)-P(9) = 5,900.
HS.F-IF.3 for and express
regularity in Examples & Explanations
HS.F-IF.3 repeated
reasoning. In F-IF.3, draw a connection to F.BF.2, which requires students to write arithmetic and geometric
Recognize that sequences sequences. Emphasize arithmetic and geometric sequences as examples of linear and exponential
are functions, sometimes functions.
defined recursively, whose
domain is a subset of the Example:
integers. For example, the • The Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
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 • How can you describe the shape of a graph?
which the function is increasing, decreasing or constant, intervals in • How can you relate the shape of a graph to the meaning of the
which the function is positive, negative or zero, symmetry, maxima,
minima, and end behavior. relationship it represents?
• How would you determine the appropriate domain for a function
• 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 describing a real-life situation?
features of that function.
• Given a function that describes a real-life situation, what can the
• 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 average rate of change of the function tell you?
of a problem. • How do the parts of a graph of a function relate to its real world

• The intervals over which a function is increasing, decreasing or context?
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

Δy = f (b) − f (a) .
Δx b − a
HS.F-IF.4

HS.F-IF.4 Mathematical Examples & Explanations
Practices
For a function that models a Start F-IF.4 and 5 by focusing on linear and exponential functions. Later in the year, focus on
relationship between two HS.MP.2. Reason quadratic functions and compare them with linear and exponential functions. In Algebra II, students
quantities, interpret key abstractly and will extend this standard to include higher order polynomials, rational, absolute value, and
features of graphs and tables quantitatively. trigonometric functions.
in terms of the quantities, and
sketch graphs showing key HS.MP.4. Model Students may be given graphs to interpret or produce graphs given an expression or table for the
features given a verbal with mathematics. function, by hand or using technology.
description of the relationship.
HS.MP.5. Use Examples:
Key features include: appropriate tools • A rocket is launched from 180 feet above the ground at time t = 0. The function that models
intercepts; intervals where the strategically. this situation is given by h = – 16t2 + 96t + 180, where t is measured in seconds and h is
function is increasing, height above the ground measured in feet.
decreasing, positive, or HS.MP.6. Attend o What is a reasonable domain restriction for t in this context?
negative; relative maxima and to precision. o Determine the height of the rocket two seconds after it was launched.
minima; symmetries; end (Continued on next page)
behavior; and periodicity.

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Connections: o Determine the maximum height obtained by the rocket.
ETHS-S6C2.03; o Determine the time when the rocket is 100 feet above the ground.
9-10.RST.7; 11-12.RST.7 o Determine the time at which the rocket hits the ground.
o 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) = −x2 − 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.

HS.F-IF.5 Mathematical Examples & Explanations
Practices
HS.F-IF.5 Start F-IF.4 and 5 by focusing on linear and exponential functions. Later in the year, focus on
HS.MP.2. Reason quadratic functions and compare them with linear and exponential functions.
Relate the domain of a abstractly and
function to its graph and, quantitatively. Students may explain the existing relationships orally or in written format.
where applicable, to the
quantitative relationship it HS.MP.4. Model Example:
describes. For example, if the with mathematics. • If the function h(n) gives the number of person-hours it takes to assemble n engines in a
function h(n) gives the factory, then the positive integers would be an appropriate domain for the function.
number of person-hours it HS.MP.6. Attend
takes to assemble n engines to precision.
in a factory, then the positive
integers would be an Mathematic Examples & Explanations
appropriate domain for the
function. al Practices 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
HS.F-IF.6 HS.MP.2. functions. The Algebra II course will address other types of functions.
Reason
HS.F-IF.6 abstractly and (Continued on next page)
quantitatively.
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.

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Connections: HS.MP.4. The average rate of change of a function y = f(x) over an interval [a,b] is Δy = f (b) − f (a) . In
ETHS-S1C2-01; Model with Δx b − a
9-10.RST.3 mathematics.
addition to finding average rates of change from functions given symbolically, graphically, or in a table,
HS.MP.5. Use students may collect data from experiments or simulations (ex. falling ball, velocity of a car, etc.) and
appropriate find average rates of change for the function modeling the situation.
tools
strategically. Examples:
• Use the following table to find the average rate of change of g over the intervals [-2, -1] and
[0,2]:

x g(x)
-2 2
-1 -1
0 -4
2 -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?

Car 1 Car 2

dT t

10 4.472 1.742

20 6.325 2.899

30 7.746 3.831

40 8.944 4.633

50 10 5.348

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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 • How can you compare properties of two functions if they are
equation, or use a graphing calculator. represented in different ways?

• Key features of a graph or table may include intercepts, intervals in • How can you determine which form of a function is best for a given
which the function is increasing, decreasing or constant, intervals in situation?
which the function is positive, negative or zero, symmetry, maxima,
minima, and end behavior. • Why might you need to complete the square?

• A linear function can be written in point-slope, slope-intercept or • How could you determine if a function represents exponential growth
standard form. or exponential decay?

• 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.

• For a function of the form f (t) = a ⋅ bt , if b>1 the function represents

exponential growth; if b<1 the function represents exponential decay.

HS.F-IF.7

HS.F-IF.7 Mathematical Examples & Explanations
Practices
Graph functions expressed For F-IF.7a, 7e, focus only on linear and exponential functions. Include comparisons of two
symbolically and show key HS.MP.5. Use functions presented algebraically. For example, compare the growth of two linear functions, or two
features of the graph, by appropriate tools exponential functions such as y=3n and y=100*2n.
hand in simple cases and strategically.
using technology for more For F-IF.7b, compare and contrast absolute value, step and piecewise-defined functions with
complicated cases. HS.MP.6. Attend linear, quadratic, and exponential functions. Highlight issues of domain, range, and usefulness
to precision. when examining piecewise-defined functions.
a. Graph linear and
quadratic functions and For F-IF.7e focus only on exponential functions.
show intercepts, maxima,
and minima. 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
Connections: computer algebra systems to graph functions.
ETHS-S6C1-03;
ETHS-S6C2-03 Examples:
• Graph the function f(x) = │x – 3│ + 5 and describe key characteristics of the graph
b. Graph square root,
cube root, and piecewise- • Graph the function f(x) = 2x by creating a table of values. Identify the key characteristics of
defined functions, including the graph.
step functions and absolute (Continued on next page)
value functions.

49 
 

Connections: • Sketch the graph and identify the key characteristics of the function described below.
ETHS-S6C1-03;
ETHS-S6C2-03 ⎧x + 2 for x ≥ 0
F (x) = ⎩⎨−x2for x < −1  
e. Graph exponential
and logarithmic functions, Solution:
showing intercepts and end
behavior, and trigonometric
functions, showing period,
midline, and amplitude.

Connections:
ETHS-S6C1-03;
ETHS-S6C2-03

HS.F-IF.8 Mathematical Examples & Explanations
Practices
HS.F-IF.8 F-IF.8a will be taught later in the year, because it focuses on quadratic factoring and completing the
HS.MP.2. square. In Algebra II students will extend their work on F-IF.8 to focus on applications and how key
Write a function defined by an Reason features relate to characteristics of a situation, making selection of a particular type of function
expression in different but abstractly and model appropriate.
equivalent forms to reveal quantitatively.
and explain different F-IF.8b is extending work done earlier in the year on exponential functions with integer exponents.
properties of the function. HS.MP.7. Look Therefore this will need to be taught later in the year.
for and make use
Connection: 11-12.RST.7 of structure. Example:
• Factor the following quadratic to identify its zeros: x2 + 2x - 8 = 0
a. Use the process of
factoring and completing • Complete the square for the quadratic and identify its vertex: x2 + 6x +19 = 0
the square in a quadratic
function to show zeros, • Identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t,
extreme values, and y = (1.2)t/10, and classify them as representing exponential growth or decay.
symmetry of the graph,
and interpret these in
terms of a context.

Connection: 11-12.RST.7

b. Use the properties of (Continued on next page)
exponents to interpret
expressions for

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