1
High School Algebra 1
Scope and Sequence
Overview1
The fundamental purpose of this course is to formalize and extend the mathematics that students learned in the
middle grades. Because it is built on the middle grades standards, this is a more ambitious version of Algebra
I
than has generally been offered. The critical areas deepen and extend understanding of linear and exponential
relationships by contrasting them with each other and by applying linear models to data that exhibit a linear
trend, and students engage in methods for analyzing, solving, and using quadratic functions. The Mathematical
Practice Standards apply throughout each course and, together with the content standards, prescribe that
students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make
sense of problem situations.
Key Advances from Middle Grades2
Several standards that were previously part of the Achievement First HS Algebra 1 curriculum are now expected
to be taught in the middle grades. Therefore, at the beginning of the transition, more time may be needed to
allow for students who have not yet been through the CCSS in middle school. Listed below are examples of key
advances from middle school to High School Algebra 1.

Having already extended arithmetic from whole numbers to fractions (grades 4-6) and from fractions to
rational numbers (grade 7), students in grade 8 encountered particular irrational numbers such as
p . In Algebra I, students will begin to understand the real number system.
5 or

Students in middle grades worked with measurement units, including units obtained by multiplying and
dividing quantities. In Algebra I, students apply these skills in a more sophisticated fashion to solve
problems in which reasoning about units adds insight.

Themes beginning in middle school algebra continue and deepen during high school. As early as grades 6
and 7, students began to use the properties of operations to generate equivalent expressions (6.EE.3,
7.EE.1). By grade 7, they began to recognize that rewriting expressions in different forms could be useful
in problem solving (7.EE.2). In Algebra I, these aspects of algebra carry forward as students continue to
use properties of operations to rewrite expressions, gaining fluency and engaging in what has been called
“mindful manipulation.”

Students in grade 8 extended their prior understanding of proportional relationships to begin working
with functions, with an emphasis on linear functions. In Algebra I, students will master linear and
quadratic functions. Students encounter other kinds of functions to ensure that general principles are
perceived in generality, as well as to enrich the range of quantitative relationships considered in
problems.
1
Text of the Overview from Common Core State Standards for Mathematics Appendix A: Designing High School Mathematics Courses
Based on the Common Core State Standards
2
Text of Key Advances, Incorporation of Mathematical Practices, and Scope Clusters from PARCC Model Content Frameworks:
Mathematics, Grades 3-11, Version 2.0. Revised August 31, 2012
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High School Algebra 1
Scope and Sequence

Students in grade 8 connected their knowledge about proportional relationships, lines, and linear
equations (8.EE.5, 6). In Algebra I, students solidify their understanding of the analytic geometry of lines.
They understand that in the Cartesian coordinate plane: The graph of any linear equation in two variables
is a line. Any line is the graph of a linear equation in two variables.

As students acquire mathematical tools from their study of algebra and functions, they apply these tools
in statistical contexts (e.g., S-ID.6). In a modeling context, they might informally fit a quadratic function to
a set of data, graphing the data and the model function on the same coordinate axes. They also draw on
skills they first learned in middle school to apply basic statistics and simple probability in a modeling
context. For example, they might estimate a measure of center or variation and use it as an input for a
rough calculation.

Algebra I techniques open a huge variety of word problems that can be solved that were previously
inaccessible or very complex in grades K-8. This expands problem solving from grades K-8 dramatically.
Incorporating the Mathematical Practices
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels
should seek to develop in their students. These standards appear in every grade level and are listed below:
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning.
Two overarching practices relevant to Algebra I are:


Make sense of problems and persevere in solving them (MP.1).
Model with mathematics (MP.4). Indeed, other mathematical practices in Algebra I might be seen as contributing specific elements of these
two. The intent of the following set is not to decompose the above mathematical practices into component parts
but rather to show how the mathematical practices work together.

Reason abstractly and quantitatively (MP.2). This practice standard refers to one of the hallmarks of
3
High School Algebra 1
Scope and Sequence
algebraic reasoning, the process of decontextualization and contextualization. Much of elementary
algebra involves creating abstract algebraic models of problems (A-CED, F-BF) and then transforming the
models via algebraic calculations (A-SSE, A-APR, F-IF) to reveal properties of the problems.

Use appropriate tools strategically (MP.5). Spreadsheets, a function modeling language, graphing tools,
and many other technologies can be used strategically to gain understanding of the ideas expressed by
individual content standards and to model with mathematics.

Attend to precision (MP.6). In algebra, the habit of using precise language is not only a mechanism for
effective communication but also a tool for understanding and solving problems. Describing an idea
precisely (A-CED, A-REI) helps students understand the idea in new ways.

Look for and make use of structure (MP.7). For example, writing 49𝑥2 + 35𝑥 + 6 as (7𝑥)2 + 5(7𝑥) + 6, a
practice many teachers refer to as “chunking,’’ highlights the structural similarity between this expression
and z2 + 5z + 6, leading to a factorization of the original: ( (7x) + 3) ( (7x) + 2) (A-SSE, A-APR).

Look for and express regularity in repeated reasoning (MP.8). Creating equations or functions to model
situations is harder for many students than working with the resulting expressions. An effective way to
help students develop the skill of describing general relationships is to work through several specific
examples and then express what they are doing with algebraic symbolism (A-CED). For example, when
comparing two different text messaging plans, many students who can compute the cost for a given
number of minutes have a hard time writing general formulas that express the cost of each plan for any
number of minutes. Constructing these formulas can be facilitated by methodically calculating the cost for
several different input values and then expressing the steps in the calculation, first in words and then in
algebraic symbols. Once such expressions are obtained, students can find the break-even point for the
two plans, graph the total cost against the number of messages sent, and make a complete analysis of the
two plans.
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High School Algebra 1
Scope and Sequence
Scope: Standards
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High School Algebra 1
Scope and Sequence
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High School Algebra 1
Scope and Sequence
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High School Algebra 1
Scope and Sequence
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High School Algebra 1
Scope and Sequence
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High School Algebra 1
Scope and Sequence
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High School Algebra 1
Scope and Sequence
Construction of the Scope
The standards included in this scope and sequence were selected by analyzing Appendix A and the PARCC
assessed curriculum document (shown in the previous table) as well a variety of scope and sequences produced
by various states and non-profit organization. Appendix A and PARCC contained the majority of standards in
common but had a few discrepancies. The table below summarizes the standards that were in only one but not
both documents.
Included in AF S&S
Standards only in PARCC Assessed Curriculum
Standards only in Appendix A
A-APR.3
F-IF.7e, F-BF.4a, A-REI.7,
N-RN.1, N-RN.2, F-BF.1b, F-IF.8b
At the time of the creation of this document, no clarification on the assessed curriculum for the Smarter Balanced
(SB) assessments had been released.
Introduction to the Sequence
The sequence organizes the standards into Interim Assessments and Units. Each unit includes an approximate
number of teaching days. The standards are written in the order in which they appear in the Common Core State
Standards for Mathematics document rather than in teaching order or order of importance.
Major, supporting, and additional designations are obtained from both the PARCC Model Content Frameworks
and the Publisher’s Criteria list of essential prerequisites for college-level work.
While the mathematical practices are not explicitly included in each unit, it is assumed that they will be woven in
throughout the course. Mathematical practices should be constantly referenced when creating unit and lesson
plans.
Teachers should note that the approximate number of teaching days covered from IA2 to IA3 is 52 days.
Therefore, they should plan to be in the middle of Unit 5 when students are taking IA2 .
This sequence appears to be similar to non-Common Core Algebra 1 courses in both unit order and topics.
However, there are some crucial changes within the units that will make this course distinct. Each unit is prefaced
with information regarding similarities and differences to previous versions of the Algebra 1 course along with
rationale.
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High School Algebra 1
Scope and Sequence
Sequence and Approximate Pacing
INTERIM ASSESSMENT 1
Unit 1: Equations and Inequalities (One-Variable)3
Because the course is designed to build off of Common Core work in the middle grades, the first unit combines
several strands that were previously taught in two or three units over a longer span of time. These standards
are also combined here to help students draw connections among the many different types of problems that
stem from solving one-variable equations.
By the end of eighth grade, students will have learned to solve linear equations in one variable and will have
applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables.
Now, students analyze and explain the process of solving an equation. Students develop fluency writing,
interpreting, and translating between various forms of linear equations and inequalities, and using them to
solve problems. In addition, students develop their understanding of quantities and use them to guide
problem solutions.
Unit 1: Equations and Inequalities (One-Variable)
21 Teaching Days
Standards
Major
Reason quantitatively and use units to solve problems.
N-Q.1 Use units as a way to understand problems and to guide the solution
of multi-step problems; choose and interpret units consistently in formulas;
choose and interpret the scale and the origin in graphs and data displays.*
N-Q.2 Define appropriate quantities for the purpose of descriptive
modeling.*
Major
N-Q.3 Choose a level of accuracy appropriate to limitations on
measurement when reporting quantities.*
Interpret the structure of expressions.
A-SSE.1 Interpret expressions that represent a quantity in terms of its
context.
A-SSE.1a Interpret parts of an expression, such as terms, factors, and
coefficients.*
A-SSE.1b Interpret complicated expressions by viewing one or more of
their parts as a single entity. For example, interpret P(1+r)^n as the
product of P and a factor not depending on P.*
Notes
Working with quantities and the relationships
between them provides grounding for work with
expressions, equations, and functions.
Standard N-Q.2 will be assessed in Algebra I by
ensuring that some modeling tasks require the
student to select a quantity of interest in the
situation being described. For example, in a
situation involving data, the student might
autonomously decide that a measure of center is
a key variable, and then choose to work with the
mean. This standard should be assessed
formatively in class as it is not suited for IAstyle assessment.
Limit to linear expressions and to exponential
expressions with integer exponents.
Standards A-SSE.1a and A-SSE.1b are included
again in Unit 8.
3
Introduction to each unit adapted from Five Critical Areas found in Common Core State Standards for Mathematics Appendix A:
Designing High School Mathematics Courses Based on the Common Core State Standards
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High School Algebra 1
Scope and Sequence
INTERIM ASSESSMENT 1
Major
Major
Unit 1: Equations and Inequalities (One-Variable) (continued)
21 Teaching Days
Standards
Interpret the structure of expressions.
A-SSE.1 Interpret expressions that represent a quantity in terms of its context.
A-SSE.1a Interpret parts of an expression, such as terms, factors, and
coefficients.*
A-SSE.1b Interpret complicated expressions by viewing one or more of their
parts as a single entity. For example, interpret P(1+r)^n as the product of P and a
factor not depending on P.*
Limit to linear expressions and to
exponential expressions with integer
exponents.
Create equations that describe numbers or relationship.
A-CED.1 Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and simple
rational and exponential functions.*
Limit A-CED.1 to linear
and
exponential equations, and, in the case
of exponential equations, limit to
situations requiring evaluation of
exponential functions at integer inputs.
Limit A-CED.3 to linear equations and
inequalities. Limit A-CED.4 to
formulas which are linear in the
variable of interest.
A-CED.3 Represent constraints by equations or inequalities, and by systems of
equations and/or inequalities, and interpret solutions as viable or non-viable
options in a modeling context. For example, represent inequalities describing
nutritional and cost constraints on combinations of different foods.*
A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same
reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to
highlight resistance R.*
Understand solving equations as a process of reasoning and explain the reasoning.
A-REI.1 Explain each step in solving a simple equation as following from the equality
of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a solution
method.
Major
Notes
Solve equations and inequalities in one variable.
A-REI.3 Solve linear equations and inequalities in one variable, including equations
with coefficients represented by letters.
Standards A-SSE.1a and A-SSE.1b are
included again in Unit 8.
Standard A-CED.3 is included again in
Unit 4: Linear Equations and
Inequalities (two-variables).
Students should focus on and master AREI.1 for linear equations and
be
able to extend and apply their
reasoning to other types of equations in
future courses. Students will solve
exponential equations with logarithms
in Algebra 2. This standard is also
included again in Unit 8: Expressions
and Equations.
Extend earlier work with solving linear
equations to solving linear inequalities
in one variable and to solving literal
equations that are linear in the
variable being solved for. Include
simple exponential equations that rely
only on application of the laws of
x
x
exponents, such as 5 =125 or 2 =1/16.
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High School Algebra 1
Scope and Sequence
INTERIM ASSESSMENT 1
Unit 2: Function Modeling
In earlier grades, students define, evaluate, and compare functions, and use them to model relationships
between quantities. In this unit, students will learn function notation and develop the concepts of domain and
range. They explore many examples of functions, including sequences; they interpret functions given
graphically, numerically, symbolically, and verbally, translate between representations, and understand the
limitations of various representations.
The goal of this unit is to develop students’ understanding of the broad themes of functions used to model
real-life situations: the relationship between two quantities, the domain and range over which those
quantities are defined, evaluating multiple representations of functions, and analyzing functions for given
characteristics such as increasing rates of change, maximums and minimums, and intercepts. The focus of this
unit should be on conceptual understanding of these ideas rather than on calculation.
Unit 2: Function Modeling
16 Teaching Days
Standards
Understand the concept of a function and use function notation.
F-IF.1 Understand that a function from one set (called the domain) to another set
(called the range) assigns to each element of the domain exactly one element of the
range. If f is a function and x is an element of its domain, then f(x) denotes the
output of f corresponding to the input x. The graph of f is the graph of the equation
y = f(x).
Major
F-IF.2 Use function notation, evaluate functions for inputs in their domains, and
interpret statements that use function notation in terms of a context.
F-IF.3 Recognize that sequences are functions, sometimes defined recursively,
whose domain is a subset of the integers. For example, the Fibonacci sequence is
defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1 (n is greater than
or equal to 1).
Notes
Students should experience a variety of
types of situations modeled by
functions. Detailed analysis of any
particular class of functions at this
stage is not advised. Students should
apply these concepts throughout their
future mathematics courses.
When thinking about domain and range
in context, students should also start
thinking about whether a discrete or
continuous representation makes sense,
and what that means for the D & R.
Draw examples from linear and
exponential functions. In F-IF.3, draw
connection to F-BF.2 (taught in Unit
3), which requires students to write
arithmetic and geometric sequences.
Emphasize arithmetic and geometric
sequences as examples of linear and
exponential functions. For this unit, the
emphasis should also be on explicit,
rather than recursive functions.
F-IF.3 is part of the Major work in
Algebra I and will be assessed
accordingly.
14
High School Algebra 1
Scope and Sequence
INTERIM ASSESSMENT 1
Major
Major
Major
Unit 2: Function Modeling (continued)
16 Teaching Days
Standards
Interpret functions that arise in applications in terms of the context.
F-IF.4 For a function that models a relationship between two quantities,
interpret key features of graphs and tables in terms of the quantities, and sketch
graphs showing key features given a verbal description of the relationship. Key
features include: intercepts; intervals where the function is increasing,
decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.*
F-IF.5 Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives the
number of person-hours it takes to assemble n engines in a factory, then the
positive integers would be an appropriate domain for the function.*
Interpret functions that arise in applications in terms of the context.
F-IF.9 Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions). For
example, given a graph of one quadratic function and an algebraic expression for
another, say which has the larger maximum.
Build a function that models a relationship between two quantities.
F-BF.1 Write a function that describes a relationship between two quantities.*
F-BF.1b Combine standard function types using arithmetic operations.
For example, build a function that models the temperature of a cooling
body by adding a constant function to a decaying exponential, and relate
these functions to the model.
Notes
For F-IF.4 and 5, focus on linear and
exponential functions in a real-world
context.
For F-IF.9, focus on expanding the types
of functions considered to include linear
and exponential.
Standards F-IF.4, F-IF.5, and F-IF.9 are
included again in Unit 7: Introduction to
Polynomials.
Limit F-BF.1b to linear and exponential
funcitons.
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High School Algebra 1
Scope and Sequence
Supporting
Construct and compare linear, quadratic, and exponential models and solve
problems.
F-LE.1 Distinguish between situations that can be modeled with linear functions
and with exponential functions.*
F-LE.3 Observe using graphs and tables that a quantity increasing exponentially
eventually exceeds a quantity increasing linearly, quadratically, or (more
generally) as a polynomial function.*
Interpret expressions for functions in terms of the situations they model.
F-LE.5 Interpret the parameters in a linear or exponential function in terms of a
context.*
Standard F-LE.1 is included in general as
preparation for greater depth in Unit 3:
Linear and Exponential Functions and will
be assessed on IA2 rather than IA1. In this
unit, students should review the concept of
rate of change of linear functions and
should be able to compare linear to
exponential growth graphically and by
plugging in values. Students should also
learn to distinguish exponential growth
and decay.
For F-LE.3, limit to comparisons between
linear and exponential models. Standard
F-LE.3 is included again in Unit 7:
Introduction to Polynomials.
Limit exponential functions to those of the
form f(x) = bx + k.
INTERIM ASSESSMENT 2
Unit 3: Linear and Exponential Functions
In this unit, students apply their understanding from Unit 2 in a wide variety of problems involving linear and
exponential functions. Students also build on and informally extend their understanding of integer exponents
to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing
between additive and multiplicative change. They interpret arithmetic sequences as linear functions and
geometric sequences as exponential functions.
Major
Interpret functions that arise in applications in terms of the context.
F-IF.6 Calculate and interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval. Estimate the rate of change from
a graph.*
Major
Analyze functions using different representations.
F-IF.7 Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.*
F-IF.7a Graph linear and quadratic functions and show intercepts, maxima, and
minima.*
Major
Unit 3: Linear and Exponential Functions
18 Teaching Days
Standards
Analyze functions using different representations.
F-IF.7 Graph functions expressed symbolically and show key features of the graph,
by hand in simple cases and using technology for more complicated cases.*
F-IF.7e Graph exponential and logarithmic functions, showing intercepts and
end behavior, and trigonometric functions, showing period, midline, and
amplitude.*
Notes
For F-IF.6, focus on linear functions
and exponential functions whose
domain is a subset of the integers. Unit
7 in this course and the Algebra 2
course address other types of functions.
For F-IF.7a and 7e focus on linear
and exponentials functions. Include
comparisons of two functions presented
algebraically. For example, compare
the growth of two linear functions, or
two exponential functions such as y=3n
and y=1002
Standard F-IF.7a is included again in
Unit7: Quadratic Functions.
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High School Algebra 1
Supporting
Major
Scope and Sequence
Build a function that models a relationship between two quantities.
F-BF.1 Write a function that describes a relationship between two quantities.*
F-BF.1a Determine an explicit expression, a recursive process, or steps for
calculation from a context.
Limit F-BF.1a to linear
and
exponential functions.
Build a function that models a relationship between two quantities.
F-BF.2 Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.*
Limit F-BF.2 to linear and
exponential functions. In F-BF.2,
connect arithmetic sequences to linear
functions and geometric sequences to
exponential functions.
INTERIM ASSESSMENT 2
Major
Additional
Unit 3: Linear and Exponential Functions (continued)
18 Teaching Days
Standards
Build new functions from existing functions.
F-BF.4 Find inverse functions.
F-BF.4a Solve an equation of the form f(x) = c for a simple function f that has an
inverse and write an expression for the inverse. For example, f(x) =2(x^3) or f(x)
= (x+1)/(x-1) for x ≠ 1 (x not equal to 1).
Notes
For F.BF.4a, focus on linear functions
but consider simple situations where
the domain of the function must be
restricted in order for the inverse to
exist, such as f(x) = x2, x>0.
Construct and compare linear, quadratic, and exponential models and solve
problems.
F-LE.1 Distinguish between situations that can be modeled with linear functions and
with exponential functions.*
F-LE.1a Prove that linear functions grow by equal differences over equal
intervals and that exponential functions grow by equal factors over equal
intervals.*
F-LE.1b Recognize situations in which one quantity changes at a constant rate
per unit interval relative to another.*
F-LE.1c Recognize situations in which a quantity grows or decays by a constant
percent rate per unit interval relative to another.*
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High School Algebra 1
Supporting
Scope and Sequence
F-LE.2 Construct linear and exponential functions, including arithmetic and
geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table).*
In constructing linear functions in FLE.2, draw on and consolidate
previous work in Grade 8 on finding
equations for lines and linear functions
(8.EE.6, 8.F.4)
INTERIM ASSESSMENT 2
Unit 4: Equations and Inequalities (Two-Variables)
In Unit 4, students explore systems of equations and inequalities, and they find and interpret their solutions.
This unit is similar to a non-Common Core systems unit with the addition of systems of both linear and
exponential equations as well as increased focus on the reasoning behind elimination.
Unit 4: Equations and Inequalities (two-variables)
18 Teaching Days
Standards
Major
Create equations that describe numbers or relationship.
A-CED.2 Create equations in two or more variables to represent relationships
between quantities; graph equations on coordinate axes with labels and scales.*
A-CED.3 Represent constraints by equations or inequalities, and by systems of
equations and/or inequalities, and interpret solutions as viable or non-viable
options in a modeling context. For example, represent inequalities describing
nutritional and cost constraints on combinations of different foods.*
Notes
Limit A-CED.2 to linear
and exponential
equations, and, in the case of exponential
equations, limit
to situations requiring
evaluation of exponential functions at
integer inputs.
Limit A-CED.3 to linear equations and
inequalities and for this unit, include twovariable equations.
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High School Algebra 1
Scope and Sequence
Additional
Solve systems of equations.
A-REI.5 Prove that, given a system of two equations in two variables, replacing
one equation by the sum of that equation and a multiple of the other produces a
system with the same solutions.
A-REI.6 Solve systems of linear equations exactly and approximately (e.g., with
graphs), focusing on pairs of linear equations in two variables.
Major
Represent and solve equations and inequalities graphically.
A-REI.10 Understand that the graph of an equation in two variables is the set of
all its solutions plotted in the coordinate plane, often forming a curve (which
could be a line).
A-REI.11 Explain why the x-coordinates of the points where the graphs of the
equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) =
g(x); find the solutions approximately, e.g., using technology to graph the
functions, make tables of values, or find successive approximations. Include
cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value,
exponential, and logarithmic functions.*
Build on student experiences
graphing
and solving systems of linear equations
from middle school to focus on
justification of the methods used. Include
cases where the two equations describe the
same line (yielding infinitely many
solutions) and cases where two equations
describe parallel lines (yielding no
solution).
For A-REI.6, tasks have a real-world
context. Tasks have hallmarks of modeling
as a mathematical practice (less defined
tasks, more of the modeling cycle, etc.).
For A-REI.10, focus on linear and
exponential equations and be able to adapt
and apply that learning to other types of
equations in future courses. For A-REI.11,
focus on cases where f(x) and g(x) are
linear or exponential and finding the
solutions approximately is limited to cases
where f(x) and g(x) are polynomial
functions.
A-REI.12 Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and graph the
solution set to a system of linear inequalities in two variables as the intersection
of the corresponding half-planes.
INTERIM ASSESSMENT 2
M ASSESSM
Unit 5: Two-variable Statistics
Units 5 and 6 build upon students’ prior experiences with data, providing students with more formal means of
assessing how a model fits data. Students use regression techniques to describe approximately linear
relationships between quantities. They use graphical representations and knowledge of the context to make
judgments about the appropriateness of linear models. With linear models, they look at residuals to analyze
the goodness of fit. The two statistics units start with two-variable data in order to reinforce connections with
work that students have done on linear functions in Units 2 and 3.
Unit 5A: Two-variable Statistics
8 Teaching Days
Standards
Notes
19
High School Algebra 1
Supporting
Scope and Sequence
Summarize, represent, and interpret data on two categorical and quantitative
variables.
S-ID.5 Summarize categorical data for two categories in two-way frequency tables.
Interpret relative frequencies in the context of the data (including joint, marginal,
and conditional relative frequencies). Recognize possible associations and trends in
the data.*
S-ID.6 Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.*
S-ID.6a Fit a function to the data; use functions fitted to data to solve problems
in the context of the data. Use given functions or choose a function suggested by
the context. Emphasize linear, quadratic, and exponential models.*
Major
S-ID.6c Fit a linear function for a scatter plot that suggests a linear association.*
Interpret linear models.
S-ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a
linear model in the context of the data.*
S-ID.9 Distinguish between correlation and causation.*
Students use frequency tables and
relative frequency tables to summarize
and interpret data.
S-ID.6a should be focused on linear
models, but may be used to preview
quadratic functions in Unit 7 of this
course.
S-ID.6 overlaps with F-LE from Units
2 and 3. Here students can review rates
of change but can also look more
informally for the shape of the data.
Both S-ID.6 and F-LE should be
spiraled back in during Unit 7 as
students learn quadratic functions.
Build on students’ work with linear
relationships in eighth grade.The focus
for the first half of Unit 5 is
determining the line of best fit and the
meaning of the parameters for that
line.
ENT 2
INTERIM ASSESSMENT 3
Unit 5B: Two-variable Statistics
10 Teaching Days
Standards
Notes
20
High School Algebra 1
Scope and Sequence
Supporting
Summarize, represent, and interpret data on two categorical and quantitative
variables.
S-ID.6 Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.*
S-ID.6a Fit a function to the data; use functions fitted to data to solve problems
in the context of the data. Use given functions or choose a function suggested by
the context. Emphasize linear, quadratic, and exponential models.*
S-ID.6b Informally assess the fit of a function by plotting and analyzing
residuals.*
S-ID.6c Fit a linear function for a scatter plot that suggests a linear association.*
Major
Interpret linear models.
S-ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a
linear model in the context of the data.*
S-ID.8 Compute (using technology) and interpret the correlation coefficient of a
linear fit.*
S-ID.9 Distinguish between correlation and causation.*
Students take a more sophisticated look
at using a linear function to model the
relationship between two numerical
variables. In addition to fitting a line to
data, students assess how well the
model fits by analyzing residuals.
S-ID.6a should be focused on linear
models, but may be used to preview
quadratic functions in Unit 7 of this
course.
S-ID.6 overlaps with F-LE from Units
2 and 3. Here students can review rates
of change but can also look more
informally for the shape of the data.
Both S-ID.6 and F-LE should be
spiraled back in during Unit 7 as
students learn quadratic functions.
Build on students’ work with linear
relationships in eighth grade and
introduce the correlation coefficient.
The focus here is on the computation
and interpretation of the correlation
coefficient as a measure of how well
the data fit the relationship. The
important distinction between a
statistical relationship and a causeand-effect relationship arises in S-ID.9.
INTERIM ASSESSMENT 3
Unit 6: Single-Variable Statistics
In this unit, students build on the understanding of key ideas for describing distributions—shape, center, and
spread—that they learned in the middle grades. This enhanced understanding allows them to give more
precise answers to deeper questions, often involving comparisons of data sets. Students use shape and the
21
High School Algebra 1
Scope and Sequence
question(s) to be answered to decide on the median or mean as the more appropriate measure of center and
to justify their choice through statistical reasoning. They also add a key measure of variation to their toolkits.
Unit 6: Single-variable Statistics
10 Teaching Days
Standards
Major
Additional
Summarize, represent, and interpret data on a single count or measurement
variable.
S-ID.1 Represent data with plots on the real number line (dot plots, histograms, and
box plots).*
S-ID.3 Interpret differences in shape, center, and spread in the context of the data
sets, accounting for possible effects of extreme data points (outliers).*
Notes
In grades 6 – 8, students describe
center and spread in a data
distribution. Here they choose a
summary statistic appropriate to the
characteristics of the data distribution,
such as the shape of the distribution or
the existence of extreme data points.
S-ID.2 Use statistics appropriate to the shape of the data distribution to compare
center (median, mean) and spread (interquartile range, standard deviation) of two
or more different data sets.*
INTERIM ASSESSMENT 3
Unit 7: Introduction to Polynomials
In this unit, students are introduced to polynomial functions. They compare the key characteristics of
polynomial functions, with a heavy focus on quadratics, to those of linear and exponential functions. They
select from among these functions to model phenomena. Students should apply their learning from Unit 2 to
22
High School Algebra 1
Scope and Sequence
this new class of functions. They also expand their experience with functions to include more specialized
functions—absolute value, step, and those that are piecewise-defined.
For this unit, students should be able to examine and identify intercepts and maxima or minima graphically as
well as in context. They should also be able to relate the factored form of a quadratic function to its zeros on
the graph as well as note the effect of the leading coefficient in both standard and factored forms. However,
they do not need to be able to solve algebraically for these characteristics until Unit 8.
Supporting
Unit 7: Introduction to Polynomials
20 Teaching Days
Standards
Understand the relationship between zeros and factors of polynomials.
A-APR.3 Identify zeros of polynomials when suitable factorizations are available, and
use the zeros to construct a rough graph of the function defined by the polynomial.
Major
Major
Interpret functions that arise in applications in terms of the context.
F-IF.4 For a function that models a relationship between two quantities, interpret key
features of graphs and tables in terms of the quantities, and sketch graphs showing
key features given a verbal description of the relationship. Key features include:
intercepts; intervals where the function is increasing, decreasing, positive, or negative;
relative maximums and minimums; symmetries; end behavior; and periodicity.*
Notes
For standard A-APR.3 tasks are
limited to quadratic and cubic
polynomials in which linear and
quadratic factors are available. For
example, find the zeros of (x - 2)(x2 9).
Focus on quadratic functions; compare
with linear and exponential functions
studied in Units 2 and 3.
F-IF.5 Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives the
number of person-hours it takes to assemble n engines in a factory, then the positive
integers would be an appropriate domain for the function.*
F-IF.6 Calculate and interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval. Estimate the rate of change from a
graph.*
Analyze functions using different representations.
F-IF.7 Graph functions expressed symbolically and show key features of the graph, by
hand in simple cases and using technology for more complicated cases.*
F-IF.7a Graph linear and quadratic functions and show intercepts, maxima, and
minima.*
F-IF.7b Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.*
For F-IF.7b, focus on absolute value,
step and piecewise- defined functions
as compared and contrasted to linear,
quadratic, and exponential functions.
Highlight issues of domain, range, and
usefulness when examining piecewisedefined functions.
INTERIM ASSESSMENT 3
Unit 7: Introduction to Polynomials (continued)
20 Teaching Days
Standards
Notes
23
High School Algebra 1
Supporting
Additional
Major
Scope and Sequence
Analyze functions using different representations.
F-IF.9 Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions). For
example, given a graph of one quadratic function and an algebraic expression for
another, say which has the larger maximum.
For F-IF.9, focus on expanding the
types of functions considered to
include, linear, exponential, and
quadratic.
Build new functions from existing functions.
F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x
+ k) for specific values of k (both positive and negative); find the value of k given the
graphs. Experiment with cases and illustrate an explanation of the effects on the
graph using technology. Include recognizing even and odd functions from their
graphs and algebraic expressions for them.
For F-BF.3, focus on quadratic
functions, and consider including
absolute value functions. In Algebra 1,
tasks do not involve recognizing even
and odd functions.
Construct and compare linear, quadratic, and exponential models and solve
problems.
F-LE.3 Observe using graphs and tables that a quantity increasing exponentially
eventually exceeds a quantity increasing linearly, quadratically, or (more generally)
as a polynomial function.*
Compare linear and exponential
growth to quadratic growth.
24
High School Algebra 1
Scope and Sequence
INTERIM ASSESSMENT 4
Unit 8: Quadratic Functions and Equations
The focus of this unit is on transforming quadratic expressions and equations into different forms. Students
create and solve equations, inequalities, and systems of equations involving quadratic expressions.
Throughout this unit, draw connections to quadratic functions and graphs.
Major
Unit 8: Quadratic Functions and Equations
24 Teaching Days
Standards
Interpret the structure of expressions.
A-SSE.1 Interpret expressions that represent a quantity in terms of its context.
A-SSE.1a Interpret parts of an expression, such as terms, factors, and
coefficients.*
A-SSE.1b Interpret complicated expressions by viewing one or more of their
parts as a single entity. For example, interpret P(1+r)^n as the product of P and
a factor not depending on P.*
A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For
example, see x^4 – y^4 as (x^2)^2 – (y^2)^2, thus recognizing it as a difference of
squares that can be factored as (x^2 – y^2)(x^2 + y^2).
Notes
Focus on quadratic and exponential
expressions.
For standard A-SSE.2, tasks are limited
to numerical expressions and polynomial
expressions in one variable. Examples:
2
Recognize 532
as a difference of
squares and see an opportunity to
rewrite it in the easier-to-evaluate form
It is important to balance conceptual
understanding and procedural fluency in
work with equivalent expressions. For
example, development of skill in
factoring and completing the square goes
hand-in-hand with understanding what
different forms of a quadratic expression
reveal.
Major
Perform arithmetic operations on polynomials.
A-APR.1 Understand that polynomials form a system analogous to the integers,
namely, they are closed under the operations of addition, subtraction, and
multiplication; add, subtract, and multiply polynomials.
Focus on polynomial expressions that
simplify to forms that are linear or
quadratic in a positive integer power of
x.
Understand solving equations as a process of reasoning and explain the
reasoning.
A-REI.1 Explain each step in solving a simple equation as following from the
equality of numbers asserted at the previous step, starting from the assumption
that the original equation has a solution. Construct a viable argument to justify a
solution method.
Major
Write expressions in equivalent forms to solve problems.
A-SSE.3 Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
A-SSE.3a Factor a quadratic expression to reveal the zeros of the function it
defines.*
A-SSE.3b Complete the square in a quadratic expression to reveal the
maximum or minimum value of the function it defines.*
Major
rewrite a2
25
High School Algebra 1
Scope and Sequence
INTERIM ASSESSMENT 4
Major
Additional
Major
Unit 8: Expressions and Equations (continued)
24 Teaching Days
Standards
Notes
Solve equations and inequalities in one variable.
A-REI.4 Solve quadratic equations in one variable.
A-REI.4a Use the method of completing the square to transform any quadratic
equation in x into an equation of the form (x – p)^2 = q that has the same solutions.
Derive the quadratic formula from this form.
A-REI.4b Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square
roots, completing the square, the quadratic formula and factoring, as appropriate
to the initial form of the equation. Recognize when the quadratic formula gives
complex solutions and write them as a ± bi for real numbers a and b.
Students should learn of the
existence of the complex number
system and be able to recognize
cases in which a quadratic equation
has no real solutions, but will not
solve quadratics with complex
solutions until Algebra 2.
Solve systems of equations.
A-REI.7 Solve a system consisting of a linear equation and a quadratic equation in two
variables algebraically and graphically. For example, find the points of intersection
between the line y=-3x and the circle x^2 + y^2 = 3.
Include systems consisting of one
linear and one quadratic equation.
Include systems that lead to work
with fractions. Students won’t work
with circles until Geometry so focus
here on quadratic equations of the
type that students have been
learning.
Students should connect their
learning from Unit 7 and Unit 8 to
form a coherent picture of
quadratic functions and equations.
Analyze functions using different representations.
F-IF.8 Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
F-IF.8a Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and interpret
these in terms of a context.
Students should also solve
quadratic equations in context to
consider when the situation
determines that only one solution
(positive or negative) is necessary.
26
High School Algebra 1
Scope and Sequence
INTERIM ASSESSMENT 4
Unit 9: Extended Exponential Functions and Equations
In this unit, students will learn rational exponents. They will extend what they learned about exponential
functions from the first three units using rational exponents. Students will also transform more complicated
exponential expressions into different forms in order to solve problems.
Major
Major
Major
Major
Unit 9: Extended Exponential Functions and Expressions
10 Teaching Days
Standards
Extend the properties of exponents to rational exponents.
N-RN.1 Explain how the definition of the meaning of rational exponents follows
from extending the properties of integer exponents to those values, allowing for a
notation for radicals in terms of rational exponents. For example, we define
5^(1/3) to be the cube root of 5 because we want [5^(1/3)]^3 = 5^[(1/3) x 3] to
hold, so [5^(1/3)]^3 must equal 5.
N-RN.2 Rewrite expressions involving radicals and rational exponents using the
properties of exponents.
N-RN.3 Explain why the sum or product of rational numbers is rational; that the
sum of a rational number and an irrational number is irrational; and that the
product of a nonzero rational number and an irrational number is irrational.
Interpret the structure of expressions.
A-SSE.1 Interpret expressions that represent a quantity in terms of its context.
A-SSE.1a Interpret parts of an expression, such as terms, factors, and
coefficients.*
A-SSE.1b Interpret complicated expressions by viewing one or more of their
parts as a single entity. For example, interpret P(1+r)^n as the product of P and
a factor not depending on P.*
Write expressions in equivalent forms to solve problems.
A-SSE.3 Choose and produce an equivalent form of an expression to reveal and
explain properties of the quantity represented by the expression.
A-SSE.3c Use the properties of exponents to transform expressions for
exponential functions. For example the expression 1.15^t can be rewritten as
[1.15^(1/12)]^(12t) ≈ 1.012^(12t) to reveal the approximate equivalent
monthly interest rate if the annual rate is 15%.*
Notes
Students begin working with rational
exponents to reinforce and extend their
understanding of exponents. Students
should revisit exponent rules that they
learned in 8th grade now adding rational
exponents.
Connect N-RN.3 to physical situations,
e.g., finding the perimeter of a square of
area 2.
Focus on quadratic and exponential
expressions. For A.SSE.1b, exponents
are extended from the integer exponents
found in Unit 1 to rational exponents
focusing on those that represent square
or cube roots.
For standard A-SSE.3c tasks have a realworld context. As described in the
standard, there is an interplay between
the mathematical structure of the
expression and the structure of the
situation such that choosing and
producing an equivalent form of the
expression reveals something about the
situation. Tasks are limited to
exponential expressions with integer
exponents.
Analyze functions using different representations.
F-IF.8 Write a function defined by an expression in different but equivalent forms
to reveal and explain different properties of the function.
F-IF.8b Use the properties of exponents to interpret expressions for
exponential functions. For example, identify percent rate of change in
functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^(12t), y = (1.2)^(t/10),
and classify them as representing exponential growth and decay.
27