HHH
CCSS
Grade Algebra 1
Units of Study
2013-2014
8th
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Resources Used:
Engage NY - A Story of Functions - Algebra 1 Curriculum Overview
PARCC Model Content Frameworks – Mathematics, Algebra I
Engage NY – Algebra 1 Materials (Exam Format, Reference Sheet, Sample
Test Items)
Explorations in Core Math – Algebra 1, Holt McDougal Mathematics
Chicago Public Schools Content Framework – Algebra 1 Planning Guide
The Common Core Toolbox – Charles A. Dana Center @ the University of
Texas – Algebra 1 Sequenced Units
Big Ideas MATH - Algebra 1: A Common Core Curriculum, by Ron Larson
and Laurie Boswell
Algebra 1: Common Core Edition – Glencoe
Please Note:
Throughout this document,
text that is italicized is in the Common Core Math 8 curriculum,
but not in the Common Core Algebra 1 curriculum.
Common Core Mathematical Practices to be Addressed
Throughout the Course
Overarching Algebra 1 Standards to be Addressed
Throughout the Course
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 multistep 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.
N.Q .3: Choose a level of accuracy appropriate to limitations on measurement when
reporting quantities.
Unit 1: The Real Number System & Solving Equations
Standards: N-Q.1, N-RN.3, A-APR.1, A-SSE.1a, A-REI.1, A-REI.3, A-CED.4,
A-CED.1, A-CED.3, 8.EE.7a, 8.EE.7b
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Rational vs. Irrational Numbers
o Explain why the sum or product of two 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.
Simplifying radicals with integer radicands
Properties of Real Numbers
o Commutative
o Associative
o Identity
o Inverse
o Distributive
o Closure
 Solving Linear Equations in One Variable
o Interpreting parts of equations: terms, factors, coefficients
o Solving equations involving the distributive property, combining like
terms, and variables on both sides; justify/explain solution method
o Recognizing the difference between equations with one solution
(x=a), infinitely many solutions (a=a), or no solutions (a=b).
 Geometry Applications:
o Angles Formed by Parallel Lines Cut by a Transversal
o Triangles
 Triangle Sum Theorem
 Exterior Angle Theorem
 Angle-Angle Criterion for Triangle Similarity
 Unit Analysis to Guide the Solution of Multi-Step Problems
 Translating Verbal and Algebraic Equations Representing Real -World
Contexts
 Solving Literal Equations (Coefficients Represented by Letters) and Rearranging Formulas to Highlight a Particular Quantity
o Example: Re-arrange Ohm’s Law, V = IR, to highlight resistance, R
Unit 2: Solving/Graphing Linear Inequalities in One Variable
Standards: A-REI.3, A-CED.1, A-CED.3,
 Graphing Inequalities on a Number Line
 Modeling, Solving, and Graphing Inequalities in One Variable
o Multiplying and dividing by a negative
o Solving multi-step inequalities with variables on both sides
 Compound Inequalities
o Graphing on a number line
o Solving compound inequalities
Units 3 & 4:
Understanding/Identifying Functions and Graphing Linear Equations
Unit 3: Functions and Graphing Linear Equations
Part 1: Understanding and Identifying Functions
Standards: F-IF.1, F-IF.2, 8.F.1
 Domain/Range & Input/Output
 Determining if a Relation is a Function
o Understand that a function is a rule that assigns to each input exactly one
output.
o The graph of a function is the set of ordered pairs consisting of an input
and the corresponding output.
 Function Notation & Evaluating Functions
 Recognize that Sequences are Functions
o Example: Fibonacci Sequence
Part 2: Graphing Linear Equations
Standards: F-LE.1a, F-LE.1b, N-Q.1, A-REI.10, F-IF.6, F-IF.5, F-BF.3, F-LE.5,
A-CED.3, 8.F.3, 8.F.4
 Graphing Linear Equations
o Understanding that linear functions grow by equal differences over equal
intervals (connect to real-world situations)
o Choose and interpret an appropriate scale for the function
o Slope/rate of change (and interpret)
 Use similar triangles to explain why the slope is the same between
any two distinct points on a non-vertical line.
o Y-intercept and x-intercept (and interpret)
o Graphing lines using slope-intercept form; understanding the effect of m
and b in y=mx+b
 Contrast to functions that are not linear, such as A = s2
o Graphing when equation is in standard form – isolate “y”
o Transforming Linear Functions
o Identify the effect on the graph of replacing f(x) with…
 f(x) + k
 k  f(x)
 f(kx)
 f(x + k)
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Real-World Considerations…
o Interpreting the parameters in a linear function
o Identifying an appropriate domain
o Interpret solutions as viable or non-viable options
Unit 4: Writing the Equation of a Line & Comparing Linear Functions
Standards: F-LE.2, A-CED.1, A-CED.2, F-IF.4, F-IF.9, F-BF.1a, 8.F.2,
8.F.5, 8.EE.5, 8.EE.6
 Direct Variation Relationships/Proportional Relationships
o Interpret unit rate as the slope of the graph of a proportional
relationship.
o Compare 2 different proportional relationships represented in
different ways.
 Writing Linear Equations that Represent a Relationship Between 2
Quantities
o Construct the function from a verbal description, two ordered pairs,
a table of values, and a graph.
o Write in slope-intercept form and point-slope form.
o Write arithmetic sequences.
 Analyzing/Comparing Linear Functions (Rate of Change and Initial Value)
Represented in Different Ways
o Algebraically
o Graphically
o Numerically in tables
o Verbal Descriptions
 Real-World “Motion” Graphs
o Describe the relationship between 2 quantities by analyzing a graph.
o Sketch a graph that exhibits the features of a function that has been
described verbally.
o Qualitative features include where the function is increasing or
decreasing, linear or nonlinear
Units 5 & 6:
Solving Systems of Linear Equations
Unit 5: Solving Linear Systems of Equations Graphically
Standards: A-REI.6, A-REI.11, 8.EE.8a, 8.EE.8c
 Solving Systems Graphically and Meaning-Making
o Understand that solutions to a system of two linear equations in two
variables correspond to points of intersection of their graphs,
because points of intersection satisfy both equations simultaneously.
o Estimate the solution to a system of two linear equations
graphically.
o Find the solution to a system of two linear equations graphically.
o Special systems – zero solutions (parallel lines) and infinitely many
solutions (same line)
 Solving Simple Cases by Inspection
o Example: 3x + 2y = 5 and 3x + 2y = 6 have no solution because
the expression 3x + 2y cannot simultaneously have the values of 5
and 6.
 Real-World and Mathematical Problems
o In real-world problems - interpret solutions as viable or non-viable
options in a modeling context
o Example of a mathematical problem – Given coordinates for two
pairs of points, determine whether the line through the first pair of
points intersects the line through the second pair of points.
Unit 6: Solving Linear Systems of Equations Algebraically
Standards: A-REI.5, A-REI.6, A-CED.3, 8.EE.8b, 8.EE.8c
 Solving Algebraically
o Substitution
o Elimination
o Special systems – zero solutions & infinitely many solutions
 Real-World and Mathematical Problems
o In real-world problems - interpret solutions as viable or non-viable
options in a modeling context
o Example of a mathematical problem – Given coordinates for two
pairs of points, determine whether the line through the first pair of
points intersects the line through the second pair of points.
Unit 7: Graphing Linear Inequalities & Systems of Linear Inequalities
on the Coordinate Plane
Standard: A-REI.12
 Graphing Linear Inequalities on the Coordinate Plane
o Vocabulary: half-plane, boundary
o Real-world applications
 Graphing Systems of Linear Inequalities on the Coordinate Plane
o Real-world applications
 Interpret Solutions as Viable or Non-Viable Options in a Modeling Context
o Example: Represent inequalities describing nutritional and cost
constraints on combinations of different foods
Unit 8: Data Analysis
Standards: S-ID.1, S-ID.2, S-ID.3, S-ID.5, S-ID.6, S-ID.7, S-ID.8, SID.9, 8.SP.1, 8.SP.2, 8.SP.3, 8.SP.4
 Summarize, Represent and Interpret Single Variable Data on a Number Line
o Dot plots
o Histograms
o Box plots
 Compare 2 or More Data Sets Based On:
o Measures of Central Tendency (mean, median and mode)
o Spread (range, interquartile range, standard deviation)
o Effects of extreme data points (outliers)
 Summarize Bivariate Categorical Data Using a Two-Way Frequency Table
o Calculate relative frequencies:
 Joint relative frequency
 Marginal relative frequency
 Conditional relative frequency
o Recognize possible associations/trends in the data
 Summarize Bivariate Measurement Data Using Scatter Plots
o Identify scatter plots as representing linear or nonlinear associations
(quadratic, exponential)
o Line of best fit
 Assess the fit of the linear model by judging the closeness of the
data points and by analyzing residuals (observed value –
predicted value)
 Linear regression equation
 Interpret the slope and y-intercept in terms of the real-world
context
 Compute (using the graphing calculator) and interpret the
correlation coefficient
 Correlation/association versus causation
Unit 9: Powers and Polynomials
Standards: 8.EE.1, 8.EE.2, 8.EE.3, 8.EE.4, A-SSE.1, A-ARR.1, A-SSE.2, AAPR.1
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Powers:
o Zero and negative exponents
o Laws of exponents
o Scientific notation
 Use numbers written in scientific notation to estimate very large or
very small quantities.
 Use numbers written in scientific notation to express how many times
larger one quantity is than another.
 Perform operations with numbers expressed in scientific notation.
 Solve problems where both decimal and scientific notation are used.
 Use scientific notation and choose units of appropriate size for
measurements of very large or very small quantities.
 Interpret scientific notation that has been generated by technology
(scientific calculator).
 Polynomials:
o Standard Form, Terms, Degrees, Coefficients, Leading Coefficients
o Closure:
 Understand that the system of polynomials is closed under
addition, subtraction, and multiplication.
o Adding and Subtracting Polynomials
o Multiplying Polynomials (Including Special Products)
Unit 10: Powers and Polynomials
Standards: A-SSE.2
o Factoring: Use the structure of a polynomial to identify how to re-write it in
an equivalent form by factoring.
 Example: x4 – y4 = (x2)2 – (y2)2 = (x2 + y2)(x2 – y2)
o Factoring by GCF
o Factoring x2 + bx + c
o Factoring ax2 + bx + c, where a ≠ 1
o Factoring Special Products
o Factoring Completely
Unit 11 : Quadratic Functions
Standards: F-IF.7, A-ARR-3, F-IF.4, F-IF.5, A-REI.4, F-IF.6, F-BF.3, FIF.8a, F-IF.9, F-BF.1, A-APR.3
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Graphing Quadratic Functions: y = ax2 + bx + c
Identifying How Changes in the Equation Affect the Graph:
o Identifying how changes in “a” and “c” affect the parabola.
o Identifying the effect on the graph of replacing f(x) with…
 f(x) + k
 k  f(x)
 f(kx)
 f(x + k)
Identify Quadratic Relationships from a Table
Characteristics of a Quadratic Function (Graph/Table):
o Intercepts
o Maximum and minimum
o Intervals where the function is increasing/decreasing
o Symmetry
Calculate/Interpret the Average Rate of Change Over a Specified Interval
o Symbolically
o Table
o Graph (estimate)
Real-World Considerations…
o Identifying an appropriate domain
o Interpreting solutions as viable or non-viable options
Unit 12 : Solving Quadratic Equations
Standards: A-REI.4, F-IF.8a, F-IF.9, F-BF.1, A-APR.3
 Solving quadratic equations by:
o Graphing
o Inspection
o Taking square roots
o Factoring
o Completing the square
o Quadratic formula
o Using the process of factoring and completing the square to show
zeros, extreme values, and symmetry; interpreting these in terms of a
context.
 Using the Discriminant to Identify the Number of Solutions
 Compare Properties of 2 Functions Represented in Different Ways
 Sketching Quadratics:
o According to a verbal description of key features
o Using the zeros
 Writing a Quadratic Function that Describes a Relationship Between 2
Quantities
o Use an explicit expression, a recursive process, or steps for calculation
 Solving Linear-Quadratic Systems Graphically
o Include real-world applications
Unit 13: Exponential Functions
Standards: F-LE.1b, F-LE.1c, F-LE.2, F-LE.3, F-LE.5, A-CED.2, A-CED.3,
A-REI.11, F-IF.3, F-IF.4, F-IF.6, F-IF.9, F-BF.1, F-BF.3, A-SSE.1, ASSE.3c, F-IF.5
 Understanding Exponential Functions:
o Understand that exponential functions grow by equal factors over
equal intervals
o Understand that a quantity increasing exponentially eventually
exceeds a quantity increasing linearly or quadratically (from
graphs/tables)
 Graphing Exponential Functions Expressed Symbolically and Given a Verbal
Description
 Interpreting Key Features of Graphs/Tables
 Transforming Exponential Functions Using Technology
o Experiment with identifying the effect on the graph of replacing f(x)
with…
 f(x) + k
 k  f(x)
 f(kx)
 f(x + k)
 Real-World Considerations…
o Interpreting the parameters of an exponential function
o Identifying an appropriate domain
o Interpreting solutions as viable or non-viable options
 Calculating/Interpreting the Average Rate of Change Over a Specified
Interval
o Symbolically
o Table
o Graph (Estimate)
 Approximate the Solution to a System of Equations Involving an
Exponential Function
 Comparing Properties of Two Functions Represented in Different Ways
 Writing an Exponential Function that Describes a Relationship Between 2
Quantities
o Use an explicit expression, a recursive process, or steps for
calculation
o Write a function rule for geometric sequences
 Exponential Growth and Decay
o Recognize situations in which a quantity grows or decays by a
constant percent rate per unit interval.
o Include compound interest that is compounded monthly
o Interpret complicated expressions by viewing one or more of their
parts as a single entity
 Example: Interpret P(1+r)n as the product of P and a factor
not depending on P.
Unit 14: Piecewise Functions (Absolute Value and Step Functions)
Standards: F-IF.4, F-IF.7b
 Absolute Value Functions
o Graphing
o Transforming Absolute Value Functions Using Technology
 Experiment with identifying the effect on the graph of
replacing f(x) with…
 f(x) + k
 k  f(x)
 f(kx)
 f(x + k)
 Step Functions: f(x) = [[x]], where f(x) is the greatest integer not greater
than x.
o Example: [[6.8]] = 6 because 6 is the greatest integer that is not
greater than 6.8
o Graph by creating a table; identify the domain and range
 Piecewise-Defined Functions
o Example: f(x) = -2x, if x > 1
x + 3, if x ≤ 1
o Graph; identify the domain and range
Unit 15: Square Root and Cube Root Functions
Standards: F-IF.4, F-IF.7b
 Square and Cube Roots
 Square Root Functions
o Graphing Square Root Functions
o State the Domain and Range
o Transforming Square Root Functions Using Technology
 Experiment with identifying the effect on the graph of
replacing f(x) with…
 f(x) + k
 k  f(x)
 f(kx)
 f(x + k)
 Cube Root Functions
o Graphing Cube Root Functions
o State the Domain and Range
o Transforming Cube Root Functions Using Technology
 Experiment with identifying the effect on the graph of
replacing f(x) with…
 f(x) + k
 k  f(x)
 f(kx)
 f(x + k)
Math 8 Units NOT Explicitly Taught in the 2013-2014 School Year
Transformations Unit
Standards: 8.G.1, 8.G.2, 8.G.3, 8.G.4
 Graphing Transformations on the Coordinate Plane
o Translations, reflections, rotations, dilations
o Describe the effect of translations, reflections, rotations, and dilations
on two-dimensional figures using coordinates.
 Identifying the Properties Preserved Under a Rotation, Reflection, and
Translation:
o Lines are taken to lines.
o Line segments are taken to line segments of the same length.
o Angles are taken to angles of the same measure.
o Parallel lines are taken to parallel lines.
 Understanding Congruence in a Sequence of Transformations
o Understand that a two-dimensional figure is congruent to another if the
second can be obtained from the first by a sequence of rotations,
reflections, and translations.
o Describe a sequence of transformations that exhibits the congruence
between two figures.
 Understanding Similarity in a Sequence of Transformations
o Understand that a two-dimensional figure is similar to another if the
second can be obtained from the first by a sequence of rotations,
reflections, translations, and dilations.
o Describe a sequence of transformations that exhibit the similarity
between two figures.
Three-Dimensional Geometry Unit
Standard: 8.G.9
 Volume of Three-Dimensional Figures
o Cones
o Cylinders
o Spheres