Pojoaque Valley Schools
Math CCSS Pacing Guide
Algebra 1
*Skills adapted from
Kentucky Department of Education
Math Deconstructed Standards
** Evidence of attainment/assessment,
Vocabulary, Knowledge, Skills and
Essential Elements adapted from
Wisconsin Department of Education and
Standards Insights Computer-Based Program
Version 3
2015-2016
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Pojoaque Valley Schools
ELA Common Core Pacing Guide Introduction
The Pojoaque Valley Schools pacing guide documents are intended to guide teachers’ use of Common Core State Standards (CCSS) over the course
of an instructional school year. The guides identify the focus standards by quarter. Teachers should understand that the focus standards
emphasize deep instruction for that timeframe. However, because a certain quarter does not address specific standards, it should be understood that
previously taught standards should be reinforced while working on the focus standards for any designated quarter. Some standards will recur across
all quarters due to their importance and need to be addressed on an ongoing basis.
The CCSS are not intended to be a check-list of knowledge and skills but should be used as an integrated model of literacy instruction to meet end of
year expectations.
The English Language Arts CCSS pacing guides contain the following elements:
 College and Career Readiness (CCR) Anchor Standard
 Strand: Identify the type of standard
 Cluster: Identify the sub-category of a set of standards.
 Grade Level: Identify the grade level of the intended standards
 Standard: Each grade-specific standard (as these standards are collectively referred to) corresponds to the same-numbered CCR anchor
standard. Put another way, each CCR anchor standard has an accompanying grade-specific standard translating the broader CCR statement
into grade-appropriate end-of-year expectations.
 Standards Code: Contains the strand, grade, and number (or number and letter, where applicable), so that RI.4.3, for example, stands for
Reading, Informational Text, grade 4, standard 3
 Skills and Knowledge: Identified as subsets of the standard and appear in one or more quarters. Define the skills and knowledge embedded
in the standard to meet the full intent of the standard itself.
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Version 2 of the Pojoaque Valley School District Pacing guides for Reading Language Arts and Mathematics are based on the done by staff and
teachers of the school district using the Kentucky model, and a synthesis of the excellent work done by Wisconsin Cooperative Educational Service
Agency 7 (CESA 7) School Improvement Services, Green Bay, WI. (2010), Standards Insight project.
Standards Insight was developed to give educators a tool for in depth investigation of the Common Core State Standards (CCSS). The CCSS
are “unpacked” or dissected, identifying specific knowledge, skills, vocabulary, understandings, and evidence of student attainment for each
standard. Standards Insight may be used by educators to gain a thorough grasp of the CCSS or as a powerful collaborative tool supporting
educator teams through the essential conversations necessary for developing shared responsibility for student attainment of all CCSS. . . .
serves as a high-powered vehicle to help educators examine the standards in a variety of ways.
The Version 2 Pojoaque Valley School District Pacing guides present the standard with levels of detail and then the necessary skills by quarter based
on the Kentucky model. On the second page for each standard, the synthesis of the Standards Insight project is presented in a way that further
defines and refines the standard such that teachers may use the information to refine their teaching practices.
Based on this synthesis of work and the purpose for the unpacking, the following fields were selected as most helpful to aid in understanding of the
Common Core Standards that will lead to shifts in instruction:
1. Evidence of Student Attainment: “What could students do to show attainment of the standard?”
2. Vocabulary: “What are key terms in the standard that are essential for interpretation and understanding in order for students to learn the content?”
3. Knowledge: “What does the student need to know in order to aid in attainment of this standard?”
4. Skills and Understanding: “What procedural skill(s) does the student need to demonstrate for attainment of this standard?”, and “What will
students understand to attain the standard?”
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The following fields are included in Version 2:
Evidence of Student Attainment: This field describes what the standard may look like in student work. Specific expectations are listed in
performance terms showing what students will say or do to demonstrate attainment of the standard.
Standards Vocabulary: This field lists words and phrases specific to each standard. Shared interpretation and in depth understanding of standards
vocabulary are essential for consistent instruction across and within grade levels and content areas.
Knowledge: The knowledge field lists what students will need to know in order to master each standard (facts, vocabulary, definitions).
Skills and Understanding: The skills field identifies the procedural knowledge students apply in order to master each standard (actions,
applications, strategies), as well as the overarching understanding that connects the standard, knowledge, and skills. Understandings included in
Standards Insight synthesize ideas and have lasting value.
Instructional Achievement Level Descriptors: This field lists, by level what a teacher can expect to see in a student who achieves at a particular
level. Additionally teachers can use this filed to differentiate instruction to provide further growth for student’s in moving from one level to another.
This field can be used to provide specific teaching approaches to the standard in question.
A Note About High School Standards: The high school standards are listed in conceptual categories. Conceptual categories portray a coherent
view of high school instruction that crosses traditional course boundaries. We have done everything possible, with teacher input, to link individual
standards to the appropriate pacing guides,
References to Tables: References to tables within the standards in the Standards Insight tool refer to Tables 1-5 found in the glossary of the
Mathematics Common Core State Standards document found at www.corestandards.o
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Quarterly View of Standards
Algebra I Pacing Guide
Quarter
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.
N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
A.SSE.1a Interpret expressions that represent a quantity in terms of its context.*(*Modeling standard) a. Interpret parts of an expression, such as terms,
factors, and coefficients.
A.SSE.1b Interpret expressions that represent a quantity in terms of its context.* (Modeling standard) b. Interpret complicated expressions by viewing
one or more of their parts as a single entity. For example, interpret
as the product of P and a factor not depending on P.
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.
A.CED.2 Create equations in two or more variables to represent relationships between quantities, graph equations on a 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 nonviable
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.
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.
A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
N.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values,
1/3
1/3 3
(1/3)3
allowing for a notation for radicals in terms of rational exponents. For example, we define 5
to be the cube root of 5 because we want (5 ) = 5
to
1/3 3
hold, so (5 ) must equal 5.
N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
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.
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).
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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.* (Modeling standard)
A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph
the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
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).
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.
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.*(*Modeling standard)
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.*(*Modeling standard)
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.*(Modeling standard)
F.IF.7e Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated
cases.*(Modeling standard)
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing
period, midline, and amplitude.
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.
F.BF.1a Write a function that describes a relationship between two quantities.*(Modeling standard) a. Determine an explicit expression, a recursive
process, or steps for calculation from a context.
F.BF.1b Write a function that describes a relationship between two quantities.*(Modeling standard)
b. Combine standard function types using arithmetic
operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate
these functions to the model.
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.*(*Modeling standard)
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.
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F.LE.1a Distinguish between situations that can be modeled with linear functions and with exponential functions. a. 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 Distinguish between situations that can be modeled with linear functions and with exponential functions. b. Recognize situations in which one
quantity changes at a constant rate per unit interval relative to another.
F.LE.1c Distinguish between situations that can be modeled with linear functions and with exponential functions. c. Recognize situations in which a
quantity grows or decays by a constant percent rate per unit interval relative to another.
F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two
input-output pairs (include reading these from a table).
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.
F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots). (Statistics and Probability is a Modeling Conceptual
Category.)
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.
(Statistics and Probability is a Modeling Conceptual Category.)
S.ID.3 Interpret differences in shape, center and spread in the context of data sets, accounting for possible effects of extreme data points (outliers).
(Statistics and Probability is a Modeling Conceptual Category.)
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.(Statistics and Probability is a Modeling
Conceptual Category.)
S.ID.6a Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use
functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear
and exponential models.
(Statistics and Probability is a Modeling Conceptual Category.)
S.ID.6b Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. b. Informally assess the fit of a function
by plotting and analyzing residuals.
(Statistics and Probability is a Modeling Conceptual Category.)
S.ID.6c Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. c. Fit a linear function for a scatter plot
that suggests a linear association.
(Statistics and Probability is a Modeling Conceptual Category.)
S.ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit. (Statistics and Probability is a Modeling Conceptual Category.)
S.ID.9 Distinguish between correlation and causation. (Statistics and Probability is a Modeling Conceptual Category.)
A.SSE.1a Interpret expressions that represent a quantity in terms of its context.* (*Modeling standard) a. Interpret parts of an expression, such as terms,
factors, and coefficients.
A.SSE.1b Interpret expressions that represent a quantity in terms of its context.* (Modeling standard)
b. Interpret complicated expressions by viewing
one or more of their parts as a single entity. For example, interpret
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as the product of P and a factor not depending on P.
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F A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x – y as (x ) – (y ) , thus recognizing it as a difference of
2 2 2
2
squares that can be factored as (x – y )(x + y ).
A.SSE.3a Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the
expression.*(Modeling standard)
a. Factor a quadratic expression to reveal the zeros of the function it defines.
A.SSE.3b Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.*
(Modeling standard)
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
A.SSE.3c Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.*
t
(*Modeling standard)
c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15 can be
1/12 12t
t
rewritten as (1.15
)
≈ 1.01212 to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
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.
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.
A.CED.2 Create equations in two or more variables to represent relationships between quantities, graph equations on a coordinate axes with labels and
scales.
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.
A.REI.4a Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an
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equation of the form (x-p) =q that has the same solutions. Derive the quadratic formula from this form.
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A.REI.4b Solve quadratic equations in one variable.
b. Solve quadratic equations by inspection (e.g., for x = 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.
N.RN.3 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.
A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find
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the points of intersection between the line y = –3x and the circle x + y = 3.
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.*(Modeling standard)
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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.*(Modeling standard)
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.*(Modeling standard)
F.IF.7a Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated
cases.*(Modeling standard)
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
F.IF.7b Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated
cases.*(Modeling standard)
b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
F.IF.8a Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
a. Use the
process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in
terms of a context.
F.IF.8b Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function:
b. Use the
t
properties of exponents to interpret expressions for exponential functions. For example: identify percent rate of change in functions such as y= (1.02) ,
t
12t
t/10
y=(.97) , y=(1.01) , y=(1.2)
, and classify them as representing exponential growth or decay.
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.
F.BF.1a Write a function that describes a relationship between two quantities.*(Modeling standard) a. Determine an explicit expression, a recursive
process, or steps for calculation from a context.
F.BF.1b Write a function that describes a relationship between two quantities.*(Modeling standard)
b. Combine standard function types using
arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential,
and relate these functions to the model.
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.
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CCSS Math Algebra 1 Pacing Guide
Grade Level/ Course: Algebra 1 Unit 1
Standard: 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.
Domain: Quantities* (*Modeling
Domain)
Cluster: Reason quantitatively and use units to solve problems.
Quarter 1:
Calculate unit conversions.
Quarter 2:
Quarter 3:
Quarter 4:
Recognize units given or needed to solve
problem.
Use given units and the context of a
problem as a way to determine if the
solution to a multi-step problem is
reasonable (e.g. length problems dictate
different units than problems dealing with
a measure such as slope)
Choose appropriate units to represent a
problem when using formulas or graphing.
Interpret units or scales used in formulas
or represented in graphs.
Use units as a way to understand problems
and to guide the solution of multi-step
problems.
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Make sense of
Reason abstractly
problems and
and
persevere in solving quantitatively.
them.
Evidence of Student
Attainment/Assessment
Students:
Interpret and make sense of
problems through analyzing
units for agreement using
dimensional analysis (e.g.,
knowing that there are 5,280 feet
in a mile we might change 20
miles/hour to inches per second
by 20 miles/hour x 5280
feet/mile x 12 inches/foot x 1
hour/60 minutes x 1 minute/60
seconds to yield just over 352
inches per second),
Model contextual problem
situations with appropriately
chosen units or derived units,
analyze the data using those
units, and interpret the solution
(e.g., problems involving per
capita income, person hours,
heat degree days, or currency
conversions),
Interpret and evaluate, with and
without appropriate technology,
graphical and tabular data
displays for consistency with the
data and precisely determine and
interpret a scale and origin that
Vocabulary
Construct viable
arguments and
critique the reasoning
of others.
Model with
mathematics.
Use appropriate
tools strategically.
Attend to precision.
Look for and
make use of
structure.
Look for and
express regularity
in repeated
reasoning.
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
Techniques for dimensional
analysis,
Choose the appropriate known
conversions to perform dimensional
analysis to convert units,
Level IV Students will:
EEN-Q.1-3. Express solutions to problems using the appropriate precision of
measurements.
Ex. Determine elapsed time (watch a TV show that starts at 8 p.m. and ends at 8:30
p.m.).
Ex. Using a measuring tape, determine if a large item purchased in a store will fit in
the car to take it home.
Ex. If it takes 30 minutes to get home, will I be home by 6:00 p.m. if I leave at 5:45
p.m.?
Uses of technology in
producing graphs of data,
Correctly use graphing window and
other technology features to precisely
Criteria for selecting different determine features of interest in a graph.
displays for data (e.g.,
knowing how to select the
The relationships of units to each other
window on a graphing
and how using a chain of conversions
calculator to be able to see the allows one to reach a desired unit or
important parts of the graph). rate.
The relationships of units to each other
and how using a chain of conversions
allows one to reach a desired unit or
rate.
Level III Students will:
EEN-Q.1-3. Express quantities to the appropriate precision of measurement.
Ex. Measure the length of an object to the nearest half and quarter of an inch.
Ex. Measure time in hours (e.g., determine elapsed time when watching a TV show
that starts at 8:00 p.m. and ends at 9:00 p.m.).
Ex. Measure ingredients for a recipe accurately.
Level II Students will:
EEN-Q.1-3. Select the appropriate type of unit as a measurement tool.
Ex. What label would you use to describe the length of a football field (inches, yards,
or miles)?
Ex. When you want to know how much ground meat you have, what kind of
measuring do you need to do? (Weight, length, and temperature).
Ex. What unit of measure would you use to measure the length of the room? (Length,
weight, volume).
Ex. What unit of measurement would you use to measure produce at the grocery
store? (Weight, volume, length).
Ex. Which is best to describe your weight – pounds or inches?
Ex. Record the daily temperature for a week using degrees.
Ex. Match a thermometer to two non-standard units of measurement.
Level I Students will:
EEN-Q.1-3. Identify measurement tools. Identify the attribute to be measured
(weight, length, and temperature).
Ex. Of these items, which is a measurement tool? (pencil, ruler, can)
Ex. If I wanted to measure the desk, would I use a ruler or a pen?
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Ex. Match units of measurement to measurement tools (days and hours measure time,
inches and feet measure length).
is useful in examining the
problem of interest.
Grade Level/ Course: Algebra 1 Unit 1
Standard with code: N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.
Domain: Quantities* (*Modeling
Domain)
Quarter 1:
Define descriptive modeling.
Determine appropriate quantities for
the purpose of descriptive modeling.
Make sense Reason abstractly and
of problems quantitatively.
and
persevere
in solving
them.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
Use appropriate tools Attend to precision.
strategically.
Look for and make
use of structure.
Look for and express
regularity in
repeated reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Quantities
Students know:
Students understand/are able to:
Choose appropriate quantities
for descriptively modeling
important features of the
phenomenon being investigated
(e.g., find a good measure of
overall highway safety: propose
and debate such measures as
fatalities per year, fatalities per
driver per year, or fatalities per
vehicle mile driven).
Descriptive modeling
descriptive models
Determine when a descriptive model
accurately portrays the phenomenon it
was chosen to model,
Level IV Students will:
EEN-Q.1-3. Express solutions to problems using the appropriate precision of
measurements.
Ex. Determine elapsed time (watch a TV show that starts at 8 p.m. and ends at
8:30 p.m.).
Ex. Using a measuring tape, determine if a large item purchased in a store
will fit in the car to take it home.
Ex. If it takes 30 minutes to get home, will I be home by 6:00 p.m. if I leave
at 5:45 p.m.?
Justify their selection of model and
choice of quantities in the context of
the situation modeled and critique the
arguments of others concerning the
same situation.
Different models reveal different
features of the phenomenon that is
being modeled.
Different models reveal different
features of the phenomenon that is
being modeled.
Level III Students will:
EEN-Q.1-3. Express quantities to the appropriate precision of measurement.
Ex. Measure the length of an object to the nearest half and quarter of an inch.
Ex. Measure time in hours (e.g., determine elapsed time when watching a TV
show that starts at 8:00 p.m. and ends at 9:00 p.m.).
Ex. Measure ingredients for a recipe accurately.
Level II Students will:
EEN-Q.1-3. Select the appropriate type of unit as a measurement tool.
Ex. What label would you use to describe the length of a football field
(inches, yards, or miles)?
Ex. When you want to know how much ground meat you have, what kind of
measuring do you need to do? (Weight, length, and temperature).
Ex. What unit of measure would you use to measure the length of the room?
(Length, weight, volume).
Ex. What unit of measurement would you use to measure produce at the
grocery store? (Weight, volume, length).
Ex. Which is best to describe your weight – pounds or inches?
Ex. Record the daily temperature for a week using degrees.
Ex. Match a thermometer to two non-standard units of measurement.
Level I Students will:
EEN-Q.1-3. Identify measurement tools. Identify the attribute to be measured
(weight, length, and temperature).
Ex. Of these items, which is a measurement tool? (pencil, ruler, can)
Ex. If I wanted to measure the desk, would I use a ruler or a pen?
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Ex. Match units of measurement to measurement tools (days and hours
measure time, inches and feet measure length).
Grade Level/ Course: Algebra 1 Unit 1
Standard with code: N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Domain: Quantities* (*Modeling
Domain)
Cluster: Reason quantitatively and use units to solve problems.
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Identify appropriate units of measurement
to report quantities.
Determine the limitations of different
measurement tools.
Choose and justify a level of accuracy
and/or precision appropriate to limitations
on measurement when reporting quantities.
Identify important quantities in a problem or
real-world context.
Make sense of
Reason abstractly
problems and
and quantitatively.
persevere in
solving them.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
Use appropriate
tools strategically.
Attend to
precision.
Look for and make use
of structure.
Look for and
express regularity
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given contextual situations
involving measurements,
Report direct measurements and
measurements gained by
combining direct measurements
to levels of accuracy allowed by
the units on the quantities and
will not report combined or
converted results with accuracy
beyond that of the original
measurements (e.g., if one side of
a rectangle is measured to the
nearest meter, and the other side
to the nearest centimeter, the
perimeter can only be accurate to
the nearest meter).
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Accuracy
Students know:
Students understand/are able to:
Attributes of measurements
including precision and
accuracy and techniques for
determining each.
Determine and distinguish the
accuracy and precision of
measurements.
Level IV Students will:
EEN-Q.1-3. Express solutions to problems using the appropriate precision of
measurements.
Ex. Determine elapsed time (watch a TV show that starts at 8 p.m. and ends at
8:30 p.m.).
Ex. Using a measuring tape, determine if a large item purchased in a store will
fit in the car to take it home.
Ex. If it takes 30 minutes to get home, will I be home by 6:00 p.m. if I leave at
5:45 p.m.?
Calculations involving measurements
can't produce results more accurate
than the least accuracy in the original
measurements,
Level III Students will:
The margin of error in a measurement, EEN-Q.1-3. Express quantities to the appropriate precision of measurement.
(often expressed as a tolerance limit), Ex. Measure the length of an object to the nearest half and quarter of an inch.
varies according to the measurement, Ex. Measure time in hours (e.g., determine elapsed time when watching a TV
show that starts at 8:00 p.m. and ends at 9:00 p.m.).
tool used, and problem context.
Ex. Measure ingredients for a recipe accurately.
Level II Students will:
EEN-Q.1-3. Select the appropriate type of unit as a measurement tool.
Ex. What label would you use to describe the length of a football field (inches,
yards, or miles)?
Ex. When you want to know how much ground meat you have, what kind of
measuring do you need to do? (Weight, length, and temperature).
Ex. What unit of measure would you use to measure the length of the room?
(Length, weight, volume).
Ex. What unit of measurement would you use to measure produce at the
grocery store? (Weight, volume, length).
Ex. Which is best to describe your weight – pounds or inches?
Ex. Record the daily temperature for a week using degrees.
Ex. Match a thermometer to two non-standard units of measurement.
Level I Students will:
EEN-Q.1-3. Identify measurement tools. Identify the attribute to be measured
(weight, length, and temperature).
Ex. Of these items, which is a measurement tool? (pencil, ruler, can)
Ex. If I wanted to measure the desk, would I use a ruler or a pen?
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Ex. Match units of measurement to measurement tools (days and hours
measure time, inches and feet measure length).
Grade Level/ Course (HS): Algebra 1 Unit 1
Standard with code: A.SSE.1a Interpret expressions that represent a quantity in terms of its context.*(*Modeling standard) a. Interpret parts of an expression, such as terms, factors,
and coefficients.
Domain: Seeing Structure in
Expressions
Cluster: Interpret the structure of expressions
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
For expressions that represent a contextual
quantity, define and recognize parts of an
expression, such as terms, factors, and
coefficients.
For expressions that represent a contextual
quantity, interpret parts of an expression,
such as terms, factors, and coefficients in
terms of the context.
Make sense of
Reason abstractly
problems and
and quantitatively.
persevere in solving
them.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
Use appropriate
tools strategically.
Attend to
precision.
Look for and make
use of structure.
Look for and express
regularity in
repeated reasoning.
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Evidence of Student
Vocabulary
Attainment/Assessment
Students:
Terms
Given a contextual situation and
an expression that does model it, Factors
Connect each part of the
expression to the corresponding
piece of the situation,
Interpret parts of the expression
such as terms, factors, and
coefficients.
Coefficients
Knowledge
Skills
Students know:
Students understand/are able to:
Instructional Achievement Level Descriptors
Level IV Students will:
EEA-SSE.1. Write or match an algebraic expression for a given word
Interpretations of parts of
Produce mathematical expressions that expression involving more than one operation.
Ex. Write an expression to represent the problem, six weeks minus two weeks
algebraic expressions such as
model given contexts,
plus four weeks, to find the total number of weeks you are working?
terms, factors, and coefficients.
Ex. How would you represent five dogs plus two cats plus one mouse to find
Provide a context that a given
mathematical expression accurately fits, the total number of animals in a pet store?
Ex. Shown pictures representing two expressions, select the one for two
drinks, plus three slices of pizza, plus two salads if d represents drinks, s
Explain the reasoning for selecting a
represents salad, and p represents pizza?
particular algebraic expression by
Ex. Match two dimes, three nickels, and four pennies to an expression when d
connecting the quantities in the
expression to the physical situation that represents dimes, n represents nickels, and p represents pennies.
Ex. Match 2r + 3b + 4y with two red disks, three blue disks, and four yellow
produced them, (e.g., the formula for
the area of a trapezoid can be explained when given colored disks.
as the average of the two bases
Level III Students will:
multiplied by height).
EEA-SSE.1. Match an algebraic expression involving one operation to
represent a given word expression with an illustration.
Physical situations can be represented
Ex. Match the correct algebraic expression to a picture of three boys and two
by algebraic expressions which
girls if b represents boys and g represents girls (3b + 4g) when asked, “Which
combine numbers from the context,
is the correct way to express three boys and two girls if b represents the
variables representing unknown
quantities, and operations indicated by number of boys and g represents the number of girls in the classroom?”
Ex. Shown a picture of three hamburgers at $4 each, match an expression to
the context,
the picture given two expressions when asked, “Which is the correct way to
express the cost of three hamburgers if each hamburger is $4.00? (three
Different but equivalent algebraic
hamburgers x $4).
expressions can be formed by
Ex. Shown two drinks plus three slices of pizza, match an expression to the
approaching the context from a different
picture given two expressions when asked, “Which one shows two drinks plus
perspective.
three slices of pizza if d represents drinks and p represents pizza?”
Ex. Match two dimes and three nickels to an expression where d represents
dimes and n represents nickels.
Ex. Match the expression of 2r + 3b with two red disks and three blue disks
when given an assortment of colored disks.
Level II Students will:
EEA-SSE.1. Identify the operation used for word expressions as indicated by
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an illustration.
Ex. Nancy has 10 balloons. She gives three away to her friend. What operation
(addition or subtraction) do you use to find how many are left as indicated by
an illustration or manipulatives.
Ex. Dave has 10 cookies. His friend gives him two more cookies. What
operation (addition or subtraction) should Dave use to determine how many
cookies he has in all as indicated by an illustration or manipulatives?
Ex. Jose has three times as many baseball cards as his brother. What operation
(addition or multiplication) do you use to find how many baseball cards Jose
has as indicated by an illustration?
Ex. One box has six books in it and another box only has two. How many
books are there together?
Ex. Match words (and, more, take away, times) to (addition, subtraction,
multiplication).
Ex. Given a word problem (June has four marbles and Cho has two marbles.
How many marbles do they have all together?) Student will identify if they
should add or subtract to find the answer as indicated by an illustration.
Ex. When given a pictorial number sentence, complete an algebraic
representation of the pictures by placing/drawing in the correct sign for the
operation.
Level I Students will:
EEA-SSE.1. Recognize the symbol for an operation.
Ex. What does this mean? + means add.
Ex. What does this mean? – means subtract or take away.
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Grade Level/ Course (HS): Algebra 1 Unit 1
Standard with code: A.SSE.1b Interpret expressions that represent a quantity in terms of its context.* (Modeling standard) b. Interpret complicated expressions by viewing one or
more of their parts as a single entity. For example, interpret
as the product of P and a factor not depending on P.
Domain: Seeing Structure in Expressions
Cluster: Interpret the structure of expressions
Quarter 1:
Quarter 2:
Quarter 3:
Construct viable
Model with
arguments and
mathematics.
critique the
reasoning of others.
Use appropriate
tools strategically.
Quarter 4:
For expressions that represent a contextual
quantity, interpret complicated expressions,
in terms of the context, by viewing one or
more of their parts as a single entity.
Make sense of
Reason abstractly
problems and
and quantitatively.
persevere in solving
them.
Attend to
precision.
Look for and make use
of structure.
Look for and
express regularity in
repeated reasoning.
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Evidence of Student
Vocabulary
Attainment/Assessment
Students:
Terms
Given a contextual situation and
an expression that does model it, Factors
Connect each part of the
expression to the corresponding
piece of the situation,
Interpret parts of the expression
such as terms, factors, and
coefficients.
Coefficients
Knowledge
Skills
Students know:
Students understand/are able to:
Interpretations of parts of
algebraic expressions such as
terms, factors, and
coefficients.
Instructional Achievement Level Descriptors
Level IV Students will:
EEA-SSE.1. Write or match an algebraic expression for a given word
Produce mathematical expressions that expression involving more than one operation.
Ex. Write an expression to represent the problem, six weeks minus two weeks
model given contexts,
plus four weeks, to find the total number of weeks you are working?
Ex. How would you represent five dogs plus two cats plus one mouse to find
Provide a context that a given
the total number of animals in a pet store?
mathematical expression accurately
Ex. Shown pictures representing two expressions, select the one for two drinks,
fits,
plus three slices of pizza, plus two salads if d represents drinks, s represents
salad, and p represents pizza?
Explain the reasoning for selecting a
Ex. Match two dimes, three nickels, and four pennies to an expression when d
particular algebraic expression by
represents dimes, n represents nickels, and p represents pennies.
connecting the quantities in the
Ex. Match 2r + 3b + 4y with two red disks, three blue disks, and four yellow
expression to the physical situation
that produced them, (e.g., the formula when given colored disks.
for the area of a trapezoid can be
Level III Students will:
explained as the average of the two
EEA-SSE.1. Match an algebraic expression involving one operation to
bases multiplied by height).
represent a given word expression with an illustration.
Physical situations can be represented Ex. Match the correct algebraic expression to a picture of three boys and two
girls if b represents boys and g represents girls (3b + 4g) when asked, “Which is
by algebraic expressions which
the correct way to express three boys and two girls if b represents the number of
combine numbers from the context,
boys and g represents the number of girls in the classroom?”
variables representing unknown
quantities, and operations indicated by Ex. Shown a picture of three hamburgers at $4 each, match an expression to the
picture given two expressions when asked, “Which is the correct way to express
the context,
the cost of three hamburgers if each hamburger is $4.00? (three hamburgers x
$4).
Different but equivalent algebraic
Ex. Shown two drinks plus three slices of pizza, match an expression to the
expressions can be formed by
picture given two expressions when asked, “Which one shows two drinks plus
approaching the context from a
three slices of pizza if d represents drinks and p represents pizza?”
different perspective.
Ex. Match two dimes and three nickels to an expression where d represents
dimes and n represents nickels.
Ex. Match the expression of 2r + 3b with two red disks and three blue disks
when given an assortment of colored disks.
Level II Students will:
EEA-SSE.1. Identify the operation used for word expressions as indicated by
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an illustration.
Ex. Nancy has 10 balloons. She gives three away to her friend. What operation
(addition or subtraction) do you use to find how many are left as indicated by an
illustration or manipulatives.
Ex. Dave has 10 cookies. His friend gives him two more cookies. What
operation (addition or subtraction) should Dave use to determine how many
cookies he has in all as indicated by an illustration or manipulatives?
Ex. Jose has three times as many baseball cards as his brother. What operation
(addition or multiplication) do you use to find how many baseball cards Jose
has as indicated by an illustration?
Ex. One box has six books in it and another box only has two. How many books
are there together?
Ex. Match words (and, more, take away, times) to (addition, subtraction,
multiplication).
Ex. Given a word problem (June has four marbles and Cho has two marbles.
How many marbles do they have all together?) Student will identify if they
should add or subtract to find the answer as indicated by an illustration.
Ex. When given a pictorial number sentence, complete an algebraic
representation of the pictures by placing/drawing in the correct sign for the
operation.
Level I Students will:
EEA-SSE.1. Recognize the symbol for an operation.
Ex. What does this mean? + means add.
Ex. What does this mean? – means subtract or take away.
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Grade Level/ Course: Algebra 1 Unit 1
Standard with code 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.
Domain: Creating Equations*
(*Modeling Domain)
Cluster: Create equations that describe numbers or relationships
Quarter 1:
Quarter 2:
Quarter 3:
Solve linear and exponential equations in one
variable.
Solve inequalities in one variable.
Describe the relationships between the
quantities in the problem (for example, how the
quantities are changing or growing with respect
to each other); express these relationships using
mathematical operations to create an appropriate
equation to solve.
Create equations (linear) in one variable and
use them to solve problems.
Create equations in one variable to model realworld situations.
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and quantitatively.
Compare and contrast problems that can
be solved by different types of equations
(linear & exponential).
Construct viable
Model with
arguments and
mathematics.
critique the
reasoning of others.
Quarter 4:
Describe the relationships between the
quantities in the problem (for example, how
the quantities are changing or growing with
respect to each other); express these
relationships using mathematical operations
to create an appropriate equation or
inequality to solve.
Create equations (linear and exponential)
and inequalities in one variable and use
them to solve problems.
Create equations and inequalities in one
variable to model real-world situations.
Use appropriate
tools strategically.
Attend to precision. Look for and make
use of structure.
Look for and express
regularity in repeated
reasoning.
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Evidence of Student
Vocabulary
Attainment/Assessment
Students:
Given a contextual situation that
may include linear, quadratic,
exponential, or rational functional
relationships in one variable,
Model the relationship with
equations or inequalities and
solve the problem presented in
the contextual situation for the
given variable.
(Please Note: This standard must
be taught in conjunction with the
standard that follows).
Knowledge
Skills
Student know:
Students understand/are able to:
When the situation presented
in a contextual problem is
most accurately modeled by a
linear, quadratic, exponential,
or rational functional
relationship.
Instructional Achievement Level Descriptors
Level IV Students will:
EEA-CED.1. Solve an algebraic expression with more than one variable.
Ex. If I have two bills, one of them is a $5 and one of them is unknown.
Write equations or inequalities in one
variable that accurately model contextual What is the value of the unknown bill if I have $10 total?
Ex. If I have some money in my pocket and some money in the other pocket
situations.
and I still need $3 more to buy the bird that cost $10, how much money is in
Features of a contextual problem can be my pockets?
used to create a mathematical model for
Level III Students will:
that problem.
EEA-SSE.3. Solve an algebraic expression using subtraction.
Ex. If I need $10 and I have $5, how much more money do I need?
Ex. If I have two bills, one of them is a $5 and one of them is a $1, how
much money do I need to have $10?
Level II Students will:
EEA-SSE.3. Solve simple equations with unknown/missing values (without
variables).
Ex. If I have three dogs and one runs away, how many dogs are left?
Ex. I walked to the store to buy a book. I gave the cashier $10 and she gives
me back $7. How much was the book?
Ex. If I have two pens in my backpack when I get to school and I left home
with five pens, how many pens were given away on the trip from home to
school?
Ex. 5 – [__] = 2.
Ex. [__] x 2 = 8.
Level I Students will:
EEA-SSE.3. Identify what is unknown.
Ex. John has three cats and some dogs. Do we know the number of dogs
John has?
Ex. Allen ate some apples. Do we know how many he ate?
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Grade Level/ Course: Algebra 1 Unit 1
Standard with code: A.CED.2 Create equations in two or more variables to represent relationships between quantities, graph equations on a coordinate axes with labels and scales.
Domain: Create Equations* (*Modeling
Domain)
Cluster: Create equations that describe numbers or relationships
Quarter 1:
Identify the quantities in a mathematical problem or
real-world situation that should be represented by
distinct variables and describe what quantities the
variables represent.
Quarter 2:
Graph one or more created equation on a coordinate
axes with appropriate labels and scales.
Create at least two equations in two or more
variables to represent relationships between
quantities.
Quarter 3:
Quarter 4:
Graph one or more created equation on a
coordinate axes with appropriate labels and
scales.
Create at least two equations in two or more
variables to represent relationships between
quantities.
Justify which quantities in a mathematical problem
or real-world situation are dependent and
independent of one another and which operations
represent those relationships.
Determine appropriate units for the labels and scale
of a graph depicting the relationship between
equations created in two or more variables.
Notes from Appendix A: The targets are limited to
linear and exponential equations, and, in the case of
exponential equations, limited to situations requiring
evaluation of exponential functions at integer inputs
Make sense of
problems and
persevere in
solving them.
Reason abstractly
and quantitatively.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
Use appropriate
tools strategically.
Attend to
precision.
Look for and make
use of structure.
Look for and express
regularity in
repeated reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a contextual situation
expressing a relationship
between quantities with two or
more variables,
Model the relationship with
equations and graph the
relationship on coordinate axes
with labels and scales.
(Please Note: This standard must
be taught in conjunction with the
preceding standard).
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
Level IV Students will:
EEA-CED.2-4. Solve two-step inequalities with a variable.
Ex. If I buy two movie tickets for $5 each and two drinks at $4 each, will $15
be enough money?
Ex. I walked to the store to buy a book. I gave the cashier $10. She said, “You
need twice this amount.” How much is the book?
Ex. I went to the store to buy two items that cost x dollars each plus a $5
membership fee. The total cost is more than $25. How much must each item
cost? 2x + 5 > 25.
When a particular two variable Write equations in two variables that
equation accurately models the accurately model contextual
situation presented in a
situations,
contextual problem.
Graph equations involving two
variables on coordinate axes with
appropriate scales and labels.
There are relationships among
features of a contextual problem, a
created mathematical model for that
problem, and a graph of that
relationship.
Level III Students will:
EEA-CED.2-4. Solve one-step inequalities.
Ex. Sally wants to buy a shirt that costs $15. She has $10. How much more
money does she need?
Ex. Mike has six apples. Two of his friends are joining him for snack. Mike
wants to share his apples with his friends. Does he have enough to give each
friend two apples?
Level II Students will:
EEA-CED.2-4. Verify the solution to an inequality with one variable.
Ex. You have $10 and buy socks that cost $2. Will you get change?
Ex. I walk to the store and buy a book. If I give the cashier $10 and she says I
do not have enough money, is the book more or less than $10?
Ex. You have $1 and your breakfast costs $2. Do you need more money?
Level I Students will:
EEA-CED.2-4. Identify quantities that are greater than or less than a given
quantity.
Ex. Using a number line indicate greater than or less than a given number.
Ex. Mike has five oranges and Mary has two oranges. Who has more oranges?
Ex. Sarah has $50 and Cindy has $30. Who has more money?
Ex. Is five more or less than three?
Ex. If Sue has baseball cards and Tim has five, who has the most/fewest
baseball cards?
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Grade Level/ Course (HS): Algebra 1 Unit 1
Standard with code: A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable
options in a modeling context.
For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
Domain: Creating Equations*
Cluster: Create equations that describe numbers or relationships
Quarter 1:
Recognize when a modeling context
involves constraints.
Quarter 2:
Quarter 3:
Quarter 4:
Determine when a problem should be represented
by equations, systems of equations and/ or
inequalities.
Determine when a problem should be
represented by equations, inequalities,
systems of equations and/ or inequalities.
Represent constraints by equations or
inequalities, and by systems of equations and/or
inequalities.
Interpret solutions as viable or nonviable
options in a modeling context.
Represent constraints by equations or
inequalities..
Represent constraints by equations , and by
systems of equations
From Appendix A: Limit targets to linear
equations and inequalities.
Make sense of
problems and
persevere in solving
them.
Reason abstractly Construct viable
Model with
and quantitatively. arguments and
mathematics.
critique the
reasoning of others.
Use appropriate
tools
strategically.
Attend to
precision.
Look for and make
use of structure.
Look for and express
regularity in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a contextual situation
involving constraints,
Write equations or inequalities
or a system of equations or
inequalities that model the
situation and justify each part of
the model in terms of the
context,
Solve the equation, inequalities
or systems and interpret the
solution in the original context
including discarding solutions to
the mathematical model that
cannot fit the real world situation
(e.g., distance cannot be
negative),
Solve a system by graphing the
system on the same coordinate
grid and determine the point(s)
or region that satisfies all
members of the system,
Determine the point(s) of the
region satisfying all members of
the system that maximizes or
minimizes the variable of
interest in the case of a system of
inequalities.
Vocabulary
Knowledge
Skills
Constraint
Students know:
Students understand/are able to:
When a particular system of
two variable equations or
inequalities accurately models
the situation presented in a
contextual problem,
Which points in the solution of
a system of linear inequalities
need to be tested to maximize
or minimize the variable of
interest.
Instructional Achievement Level Descriptors
Level IV Students will:
EEA-CED.2-4. Solve two-step inequalities with a variable.
Ex. If I buy two movie tickets for $5 each and two drinks at $4 each, will $15
Graph equations and inequalities
be enough money?
involving two variables on
Ex. I walked to the store to buy a book. I gave the cashier $10. She said, “You
coordinate axes,
need twice this amount.” How much is the book?
Identify the region that satisfies both Ex. I went to the store to buy two items that cost x dollars each plus a $5
membership fee. The total cost is more than $25. How much must each item
inequalities in a system,
cost? 2x + 5 > 25.
Identify the point(s) that maximizes
or minimizes the variable of interest
in a system of inequalities,
Level III Students will:
EEA-CED.2-4. Solve one-step inequalities.
Ex. Sally wants to buy a shirt that costs $15. She has $10. How much more
money does she need?
Test a mathematical model using
Ex. Mike has six apples. Two of his friends are joining him for snack. Mike
equations, inequalities, or a system
against the constraints in the context wants to share his apples with his friends. Does he have enough to give each
friend two apples?
and interpret the solution in this
context.
Level II Students will:
EEA-CED.2-4. Verify the solution to an inequality with one variable.
A symbolic representation of
Ex. You have $10 and buy socks that cost $2. Will you get change?
relevant features of a real world
Ex. I walk to the store and buy a book. If I give the cashier $10 and she says I
problem can provide for resolution
do not have enough money, is the book more or less than $10?
of the problem and interpretation of
Ex. You have $1 and your breakfast costs $2. Do you need more money?
the situation and solution,
Representing a physical situation
with a mathematical model requires
consideration of the accuracy and
limitations of the model.
Level I Students will:
EEA-CED.2-4. Identify quantities that are greater than or less than a given
quantity.
Ex. Using a number line indicate greater than or less than a given number.
Ex. Mike has five oranges and Mary has two oranges. Who has more oranges?
Ex. Sarah has $50 and Cindy has $30. Who has more money?
Ex. Is five more or less than three?
Ex. If Sue has baseball cards and Tim has five, who has the most/fewest
baseball cards?
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Grade Level/ Course: Algebra 1 Unit 1
Standard with code: 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.
Domain: Creating Equations*
(*Modeling Domain)
Cluster: Create equations that describe numbers or relationships
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Use appropriate
Attend to precision.
tools strategically.
Look for and make
use of structure.
Define a “quantity of interest” to mean any
numerical or algebraic quantity (e.g.
, in which 2 is the quantity of interest
showing that d must be even,
and
showing that
.
Rearrange formulas to highlight a quantity of
interest, using the same reasoning as in
solving equations. (e.g. π * r2 can be rewritten as (π *r)*r which makes the form of
this expression resemble b*h.)
From Appendix A: Limit A.CED.4 to
formulas which are linear in the variable of
interest.
Make sense of
Reason abstractly
problems and
and quantitatively.
persevere in solving
them.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
Look for and express
regularity in repeated
reasoning.
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Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Rearrange formulas which arise
in contextual situations to isolate
variables that are of interest for
particular problems. For example,
if the electric company charges
for power by the formula COST =
0.03 KWH + 15, a consumer may
wish to determine how many
kilowatt hours they may use to
keep the cost under particular
amounts, by considering KWH<
(COST - 15)/0.03 which would
yield to keep the monthly cost
under $75, they need to use less
than 2000 KWH.
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
Properties of equality and
inequality (Tables 4 and 5).
Accurately rearrange equations or
inequalities to produce equivalent
forms for use in resolving situations
of interest.
Level IV Students will:
EEA-CED.2-4. Solve two-step inequalities with a variable.
Ex. If I buy two movie tickets for $5 each and two drinks at $4 each, will $15
be enough money?
Ex. I walked to the store to buy a book. I gave the cashier $10. She said, “You
need twice this amount.” How much is the book?
Ex. I went to the store to buy two items that cost x dollars each plus a $5
membership fee. The total cost is more than $25. How much must each item
cost? 2x + 5 > 25.
The structure of mathematics allows
for the procedures used in working
with equations to also be valid when
rearranging formulas,
The isolated variable in a formula is
not always the unknown and
rearranging the formula allows for
sense-making in problem solving.
Level III Students will:
EEA-CED.2-4. Solve one-step inequalities.
Ex. Sally wants to buy a shirt that costs $15. She has $10. How much more
money does she need?
Ex. Mike has six apples. Two of his friends are joining him for snack. Mike
wants to share his apples with his friends. Does he have enough to give each
friend two apples?
Level II Students will:
EEA-CED.2-4. Verify the solution to an inequality with one variable.
Ex. You have $10 and buy socks that cost $2. Will you get change?
Ex. I walk to the store and buy a book. If I give the cashier $10 and she says I
do not have enough money, is the book more or less than $10?
Ex. You have $1 and your breakfast costs $2. Do you need more money?
Level I Students will:
EEA-CED.2-4. Identify quantities that are greater than or less than a given
quantity.
Ex. Using a number line indicate greater than or less than a given number.
Ex. Mike has five oranges and Mary has two oranges. Who has more oranges?
Ex. Sarah has $50 and Cindy has $30. Who has more money?
Ex. Is five more or less than three?
Ex. If Sue has baseball cards and Tim has five, who has the most/fewest
baseball cards?
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Version 3 2015-2016
Grade Level/ Course: Algebra 1 Unit 1
Standard with code: 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.
Domain: Reasoning with equations and
inequalities
Cluster: Understand solving equations as a process of reasoning and explain the reasoning.
Quarter 1:
Quarter 2:
Know that solving an equation means that the
equation remains balanced during each step.
Quarter 3:
Quarter 4:
Know that solving an equation means that the
equation remains balanced during each step.
Recall the properties of equality.
Recall the properties of equality.
Explain why, when solving equations, it is assumed
that the original equation is equal.
Determine if an equation has a solution.
Choose an appropriate method for solving the
equation.
Justify solution(s) to equations by explaining each
step in solving a simple equation using the properties
of equality, beginning with the assumption that the
original equation is equal.
Construct a mathematically viable argument
justifying a given, or self-generated, solution
method.
From Appendix A: Students should focus on and
master A.REI.1 for linear equations and be able to
extend and apply their reasoning to other types of
equations in future courses.
Make sense of
Reason abstractly
problems and
and quantitatively.
persevere in solving
them.
Construct viable
arguments and
critique the reasoning
of others.
Model with
mathematics.
Use appropriate
tools strategically.
Attend to
precision.
Look for and make
use of structure.
Look for and express
regularity in
repeated reasoning.
30
Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Inverse operations
Students know:
Students understand/are able to:
EEA-REI.1-2. N/A
Justify each step in the solution
of equations by communicating
their understandings of inverse
operations as well as other
operation properties
(commutative, associative,
identity, distributive) and
properties of equality.
Equality
Rules for producing
equivalent equations (Table
4),
Accurately rearrange equations to
produce equivalent forms for use in
resolving situations of interest,
Properties of addition and
multiplication (Table 3 ).
Communicate reasoning behind each
step conducted in producing each
equivalent form.
The structure of mathematics present in
the properties of the operations can be
used to maintain equality while
rearranging equations, as well as to
justify steps in the finding of solutions
of equations.
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Version 3 2015-2016
Grade Level/ Course: Algebra 1 Unit 1
Standard with code: A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Domain: Reasoning with Equations and Cluster: Solve equations and inequalities in one variable
Inequalities
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Recall properties of equality
Solve multi-step equations in one variable
Solve multi-step inequalities in one variable
Solve equations with coefficients represented
by letters.
Determine the effect that rational coefficients
have on the inequality symbol and use this to
find the solution set.
Solve equations and inequalities with
coefficients represented by letters.
Make sense of
Reason abstractly
problems and
and quantitatively.
persevere in solving
them.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
Use appropriate
tools strategically.
Attend to precision.
Look for and make
use of structure.
Look for and
express regularity in
repeated reasoning.
32
Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Solve linear equations and
inequalities in one variable
including being able to isolate
variables when coefficients are
letters.
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
EEA-REI.3-4. N/A (See EEA-ECED.1-2.)
Properties of equality and
inequality (Tables 4 and 5).
Accurately use the properties of
equality and inequality (Tables 4
and 5) to produce equivalent
expressions which lead to solutions.
The structure of mathematics allows
for the procedures used in working
with equations and inequalities to
also be valid when coefficients are
letters,
The solutions arrived at though
algebraic manipulations should
make the original equation true.
33
Version 3 2015-2016
Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code: 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
1/3
1/3 3
(1/3)3
1/3 3
notation for radicals in terms of rational exponents. For example, we define 5
to be the cube root of 5 because we want (5 ) = 5
to hold, so (5 ) must equal 5.
Domain: The Real Number System
Cluster: Extend the properties of exponents to rational exponents
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Define radical notation as a convention used to
represent rational exponents.
Explain the properties of operations of rational
exponents as an extension of the properties of
integer exponents.
Explain how radical notation, rational exponents,
and properties of integer exponents relate to one
another.
Make sense of
problems and
persevere in solving
them.
Reason abstractly Construct viable
and quantitatively. arguments and
critique the reasoning
of others.
Model with
mathematics.
Use appropriate
tools strategically.
Attend to
precision.
Note from Appendix A: In implementing
the standards in curriculum, these
standards should occur before discussing
exponential functions with continuous
domains.
Look for and make
Look for and express
use of structure.
regularity in repeated
reasoning.
34
Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Vocabulary
Knowledge
Skills
Exponent
Students know:
Students understand that/are able to:
Use the repeated reasoning from Root
prior knowledge of properties of
exponents to progress from, for
example, (23*23*23) = 29 to the
extension what three equal factors
multiplied together yields 21,
Connect this result to the
definition of cube root and extend
their understanding to other roots.
Techniques for applying the
properties of exponents.
Instructional Achievement Level Descriptors
Level IV Students will:
EEN-RN.1. Illustrate concept of remainders using objects and numerical
Correctly perform the manipulations of representations.
Ex. Divide 15 objects into two groups of six and one group of three. Show
exponents which apply the properties
representation and objects in numerical representation (e.g., 15/6 = 2 r 3).
of exponents,
Ex. A group of six students sits down to have a snack. You have 25 cookies. How many
cookies does each student get? Are there any leftover? (e.g., Write number sentence
Use mathematical reasoning and prior 25/6 = 4 r 1).
Ex. If a pack of gum costs $0.49 and there are five sticks per pack, how much does each
knowledge of the meaning of integer
stick cost? Use real objects (gum and coins) to show division (e.g., 49/5 = 9 r 4).
exponents to explain notation for
radicals with rational exponents.
The properties of exponents are true
regardless of the type of numbers
being used.
The properties of exponents are true
regardless of the type of numbers
being used.
Level III Students will:
EEN-RN.1. Solve division problems with remainders using concrete objects.
Ex. Divide 13 into equal groups (two groups of six with a remainder of one, three
groups of four with a remainder of one, one group of 13, four groups of three with a
remainder of one, six groups of two with a remainder of one, 13 groups of one).
Ex. A group of six students sits down to have a snack. You have 15 cookies. How many
cookies does each student get? Are there any leftover?
Ex. A student has five quarters and wants to buy a soda that costs $1.00. How much
money is left over?
Ex. A class of seven students earns $20 doing a service project. How much does each
student receive? (Solve using money, calculator, etc.)
Level II Students will:
EEN-RN.1. Identify the difference between equal and not equal groups.
Ex. Using drawings or groups of cubes, determine if the groups are equal or not equal.
Ex. When passing out 10 pencils to nine people, do you have one for each person? Are
there some left over?
Ex. Do 10 pennies = $0.10?
Ex. Are two nickels equal to $0.11?
Ex. Given two clocks, one shows 20 minutes after the hour and another shows 30
minutes after the hour. Which clock shows the later time?
Level I Students will:
EEN-RN.1. Recognize that a whole can be divided into parts.
Ex. Use models to represent quantities as parts of a whole.
Ex. Given two sets of objects with one set divided into smaller groups, point to the
quantities that have been divided when prompted.
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Version 3 2015-2016
Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code: N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Domain: The Real Number System
Cluster: Extend the properties of exponents to rational exponents
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Using the properties of exponents, rewrite a
radical expression as an expression with a rational
exponent.
Using the properties of exponents, rewrite an
expression with a rational exponent as a radical
expression.
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and quantitatively.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate
tools strategically.
Attend to
precision.
Notes from Appendix A: In implementing
the standards in curriculum, these
standards should occur before discussing
exponential functions with continuous
domains.
Look for and make Look for and express
use of structure.
regularity in repeated
reasoning.
36
Version 3 2015-2016
Evidence of Student
Vocabulary
Attainment/Assessmen
t
Students:
Rational exponent
Given expressions
involving radicals and
rational exponents,
Use
mathematical/logical
reasoning to
demonstrate that various
forms of radicals and
roots actually represent
the same quantity, that is
(23/2)3 = 29/2 = √29 =
16√2.
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
EEN-RN.2. N/A
Properties of exponents,
Use mathematical reasoning to justify
the equality of various forms of radical
expressions,
The meaning of algebraic symbols
such as radicals and rational
exponents.
Correctly perform the manipulations of
exponents which apply properties of
exponents.
The properties of exponents are true
regardless of the type of numbers
being used.
The properties of exponents are true
regardless of the type of numbers
being used.
37
Version 3 2015-2016
Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code: 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.
Domain: Reasoning with Equations and
Inequalities
Cluster: Solve system of equations
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Recognize and use properties of equality to
maintain equivalent systems of equations.
Justify that replacing one equation in a twoequation system with the sum of that equation
and a multiple of the other will yield the same
solutions as the original system.
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and quantitatively.
Construct viable
Model with
arguments and critique mathematics.
the reasoning of
others.
Use appropriate
Attend to
tools strategically. precision.
Look for and make
use of structure.
Look for and express
regularity in repeated
reasoning.
38
Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Given a system of two equations
in the form
Ax + By = C
Dx + Ey = F
Use the properties of equality
(Table 4) (where A, B, C, D, E,
and F are real numbers) to show
that the same solution is produced
by replacing one equation by the
sum of that equation and a
multiple of the other,
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
System of two
equations in two
variables
Students know:
Students understand/are able to:
EEA-REI.5. N/A
Appropriate use of properties Accurately perform the operations of
of addition and multiplication multiplication and addition, and
(Table 3) and equality (Table techniques for manipulating equations.
4).
When the properties of operations and
equality are applied to systems of
equations, the resulting equations have
the same solution as the original.
For example, the system 2x + 3y
= 18
4x + 2y = 20
is equivalent to 0x - 4y = -16
(multiply first equation by -2 add
to the second equation).
39
Version 3 2015-2016
Grade Level/ Course: Algebra 1 Unit 2
Standard with code: A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Domain: Reasoning with Equations and
Inequalities
Cluster: Solve system of equations
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Solve systems of linear equations by any
method.
Justify the method used to solve systems of
linear equations exactly and approximately
focusing on pairs of linear equations in two
variables.
Make sense of
problems and
persevere in solving
them.
Reason abstractly Construct viable
and quantitatively. arguments and
critique the
reasoning of others.
Model with
mathematics.
Use appropriate
Attend to precision.
tools strategically.
Notes from Appendix A: 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);
connect to GPE.5 when it is taught in
Geometry, which requires students to
prove the slope criteria for parallel
lines.
Look for and make
Look for and
use of structure.
express regularity in
repeated reasoning.
40
Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Given systems of linear
equations,
Choose an appropriate method
for solving (e.g., substitution,
elimination, and graphing),
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
EEA-REI.6-7. N/A (See EEA-REI.10-12.)
Appropriate use of properties of
addition and multiplication (Table 3)
and equality (Table 4),
Accurately perform the operations of
multiplication and addition, and
techniques for manipulating
equations,
Techniques for producing and
interpreting graphs of linear equations, Graph linear equations precisely,
Solve and justify solutions,
Provide reasonable
approximations when
appropriate on a graph.
The conditions under which a system
of linear equations has 0, 1, or infinite
solutions.
Use estimation to find approximate
solutions on a graph.
The solution of a linear system is the
set of all ordered pairs that satisfy
both equations,
Solving by graphing often leads to
approximate solutions,
A system of linear equations will
have 0, 1, or infinite solutions.
41
Version 3 2015-2016
Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code: 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).
Domain: Reasoning with Equations and
Inequalities
Cluster: Represent and solve equations and inequalities graphically
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Recognize that the graphical representation
of an equation in two variables is a curve,
which may be a straight line.
Explain why each point on a curve is a
solution to its equation.
Make sense of
problems and
persevere in solving
them.
Reason abstractly and
quantitatively.
Notes from Appendix A: 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.
Construct viable
Model with
arguments and
mathematics.
critique the
reasoning of
others.
Use appropriate
Attend to precision. Look for and make
tools strategically.
use of structure.
Look for and express
regularity in repeated
reasoning.
42
Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Given an equation in two
variables,
Verify that any ordered
pair that makes the
equation true is a point on
the graph,
Show that there are an
infinite number of ordered
pairs that satisfy the
equation.
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Curve (which could be a
line)
Students know:
Students understand/are able to:
Appropriate methods to find
ordered pairs that satisfy an
equation,
Accurately find ordered pairs that
satisfy the equation,
Level IV Students will:
EEA-REI.10. Make a prediction using the graph of an equation with two
variables that form a line when plotted using the trend of the line.
Ex. Given the graph of a linear function based on real-world situations (e.g.,
How much money do I earn (y) if I work a given number of hours (x) at $5
dollars per hour; (y = 5 x hours), use this information to make predictions
(e.g., If you work six hours, how much will you make?).
Ex. Given the graph of a linear function based on cost per pizza and the
number of pizzas bought [e.g., If pizza is $5, then the total cost (y) = 5 x the
number bought (x)], use this information to make predictions.
Techniques to graph the
collection of ordered pairs to
form a curve.
Accurately graph the ordered pairs
and form a curve.
An equation in two variables has an
infinite number of solutions (ordered
pairs that make the equation true), and
those solutions can be represented by Level III Students will:
EEA-REI.10. Determine the two pieces of information that are plotted on a
the graph of a curve.
graph of an equation with two variables that form a line when plotted.
Ex. Follow the line on the graph to tell the two pieces of information in each
point (total cost and Items bought).
Ex. Given the graph of a linear function based on cost per pizza and the
number of pizzas bought (e.g., number of pizzas bought and total price),
follow the line on the graph to tell the two pieces of information at a given
point.
Level II Students will:
A-REI.10. Use a graph of two variables to find the answer to a real-world
problem.
Ex. Locate objects using a map with pictorial cues using two coordinates to
find one position on a simple map.
Ex. Gain basic information from a graph (total cost of two items).
Level I Students will:
A-REI.10. Identify major parts of a graph.
Ex. Point to the numbers that tell me how many items I bought.
Ex. Point to the numbers that tell me how much the total cost is.
Ex. Trace the line with your finger – show where the line would go if it
continued.
43
Version 3 2015-2016
Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code: 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.* (Modeling standard)
Domain: Reasoning with Equations and
Inequalities
Cluster: Represent and solve equations and inequalities graphically
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Recognize that f(x) = g(x) means that there may
be particular inputs of f and g for which the
outputs of f and g are equal.
Explain why the x-coordinates of the points
where the graph of the equations y=f(x) and
y=g(x) intersect are the solutions of the
equations f(x) = g(x) . (Include cases where
f(x) and/or g(x) are linear and exponential
equations)
Recognize and use function notation to
represent equations
Recognize and use function notation to
represent linear and exponential equations
Recognize that if (x1, y1) and (x2, y2)
share the same location in the coordinate
plane that x1 = x2 and y1 = y2.
Approximate/find the solution(s) using an
appropriate method for example, 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
and exponential equations).
Notes from Appendix A: For
A.REI.11, focus on cases where f(x)
and g(x) are linear or exponential.
Make sense of
problems and
persevere in solving
them.
Reason abstractly and
quantitatively.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
Approximate/find the solution(s) using an
appropriate method for example, 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 equations).
Approximate/find the solution(s) using an
appropriate method for example, 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 equations)
Use appropriate
Attend to precision.
tools strategically.
Approximate/find the solution(s) using
an appropriate method for example,
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 and
exponential equations)
Look for and make
use of structure.
Look for and
express regularity in
repeated reasoning.
44
Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students: Given two functions
(linear, polynomial, rational,
absolute value, exponential, and
logarithmic) that intersect (e.g.,
y= 3x and y= 2x), - Graph each
function and identify the
intersection point(s), - Explain
solutions for f(x) = g(x) as the
x-coordinate of the points of
intersection of the graphs, and
explain solution paths (e.g., the
values that make 3x = 2x true,
are the x-coordinate
intersection points of y=3x and
y=2x, - Use technology, tables,
and successive approximations
to produce the graphs, as well
as to determine the
approximation of solutions.
Vocabulary
Knowledge
Skills
Functions
Students know:
Students understand/are able to:
Successive
approximations
Defining characteristics of
linear, polynomial, rational,
absolute value, exponential,
and logarithmic graphs,
Linear functions
Polynomial functions
Rational functions
Absolute value
functions
Exponential functions
Logarithmic functions
Methods to use technology,
tables, and successive
approximations to produce
graphs and tables for linear,
polynomial, rational, absolute
value, exponential, and
logarithmic functions.
Instructional Achievement Level Descriptors
Level IV Students will:
EEA-REI.10. Make a prediction using the graph of an equation with two
Determine a solution or solutions of a variables that form a line when plotted using the trend of the line.
Ex. Given the graph of a linear function based on real-world situations (e.g.,
system of two functions,
How much money do I earn (y) if I work a given number of hours (x) at $5
Accurately use technology to produce dollars per hour; (y = 5 x hours), use this information to make predictions (e.g.,
If you work six hours, how much will you make?).
graphs and tables for linear,
polynomial, rational, absolute value, Ex. Given the graph of a linear function based on cost per pizza and the number
of pizzas bought [e.g., If pizza is $5, then the total cost (y) = 5 x the number
exponential, and logarithmic
bought (x)], use this information to make predictions.
functions,
Accurately use technology to
approximate solutions on graphs.
When two functions are equal, the x
coordinate(s) of the intersection of
those functions is the value that
produces the same output (y-value)
for both functions,
Technology is useful to quickly and
accurately determine solutions and
produce graphs of functions.
Level III Students will:
EEA-REI.10. Determine the two pieces of information that are plotted on a
graph of an equation with two variables that form a line when plotted.
Ex. Follow the line on the graph to tell the two pieces of information in each
point (total cost and Items bought).
Ex. Given the graph of a linear function based on cost per pizza and the number
of pizzas bought (e.g., number of pizzas bought and total price), follow the line
on the graph to tell the two pieces of information at a given point.
Level II Students will:
A-REI.10. Use a graph of two variables to find the answer to a real-world
problem.
Ex. Locate objects using a map with pictorial cues using two coordinates to find
one position on a simple map.
Ex. Gain basic information from a graph (total cost of two items).
Level I Students will:
A-REI.10. Identify major parts of a graph.
Ex. Point to the numbers that tell me how many items I bought.
Ex. Point to the numbers that tell me how much the total cost is.
Ex. Trace the line with your finger – show where the line would go if it
continued.
45
Version 3 2015-2016
Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code: A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the
solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
Domain: Reasoning with Equations and
Inequalities
Cluster: Represent and solve equations and inequalities graphically
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Identify characteristics of a linear inequality
and system of linear inequalities, such as:
boundary line (where appropriate),
shading, and determining appropriate test
points to perform tests to find a solution set.
Explain the meaning of the intersection of the
shaded regions in a system of linear
inequalities.
Graph a line, or boundary line, and shade the
appropriate region for a two variable linear
inequality.
Make sense of
problems and
persevere in solving
them.
Reason abstractly Construct viable
Model with
and quantitatively. arguments and
mathematics.
critique the
reasoning of others.
Graph a system of linear inequalities and
shade the appropriate overlapping region for a
system of linear inequalities.
Use appropriate
Attend to precision.
tools strategically.
Look for and make
use of structure.
Look for and
express regularity in
repeated reasoning.
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Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Given a linear inequality in two
variables or a system of linear
inequalities,
Graph solutions and solution sets
using the appropriate notation
(dotted or solid line).
Vocabulary
Knowledge
Skills
Half-planes
Students know:
Students understand/are able to:
System of linear
inequalities
When to include and exclude
the boundary of linear
inequalities,
Techniques to graph the
boundaries of linear
inequalities,
Methods to find solution
regions of a linear inequality
and systems of linear
inequalities.
Instructional Achievement Level Descriptors
Level IV Students will:
EEA-REI.10. Make a prediction using the graph of an equation with two
Accurately graph a linear inequality variables that form a line when plotted using the trend of the line.
Ex. Given the graph of a linear function based on real-world situations (e.g.,
and identify values that make the
How much money do I earn (y) if I work a given number of hours (x) at $5
inequality true (solutions),
dollars per hour; (y = 5 x hours), use this information to make predictions (e.g.,
If you work six hours, how much will you make?).
Find the intersection of multiple
linear inequalities to solve a system. Ex. Given the graph of a linear function based on cost per pizza and the
number of pizzas bought [e.g., If pizza is $5, then the total cost (y) = 5 x the
Solutions to a linear inequality result number bought (x)], use this information to make predictions.
in the graph of a half-plane,
Level III Students will:
EEA-REI.10. Determine the two pieces of information that are plotted on a
Solutions to a system of linear
graph of an equation with two variables that form a line when plotted.
inequalities are the intersection of
Ex. Follow the line on the graph to tell the two pieces of information in each
the solutions of each inequality in
point (total cost and Items bought).
the system.
Ex. Given the graph of a linear function based on cost per pizza and the
number of pizzas bought (e.g., number of pizzas bought and total price),
follow the line on the graph to tell the two pieces of information at a given
point.
Level II Students will:
A-REI.10. Use a graph of two variables to find the answer to a real-world
problem.
Ex. Locate objects using a map with pictorial cues using two coordinates to
find one position on a simple map.
Ex. Gain basic information from a graph (total cost of two items).
Level I Students will:
A-REI.10. Identify major parts of a graph.
Ex. Point to the numbers that tell me how many items I bought.
Ex. Point to the numbers that tell me how much the total cost is.
Ex. Trace the line with your finger – show where the line would go if it
continued.
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Version 3 2015-2016
Grade Level/ Course: Algebra 1 Unit 2
Standard with code: 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).
Domain: Interpreting Functions
Cluster: Understand the concept of a function and use function notation.
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Identify the domain and range of a function.
Identify the domain and range of a function.
Determine if a relation is a function.
Determine if a relation is a function.
Determine the value of the function with
proper notation (i.e. f(x)=y, the y value is
the value of the function at a particular
value of x)
Determine the value of the function with proper
notation (i.e. f(x)=y, the y value is the value of
the function at a particular value of x)
Evaluate functions for given values of x.
Evaluate functions for given values of x.
Make sense of
problems and
persevere in solving
them.
Note from Appendix A: 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.
Reason abstractly Construct viable
Model with
and quantitatively. arguments and
mathematics.
critique the reasoning
of others.
Use appropriate
tools strategically.
Attend to precision. Look for and make
use of structure.
Look for and express
regularity in repeated
reasoning.
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Version 3 2015-2016
Evidence of Student
Vocabulary
Attainment/Assessment
Students:
Domain
Given input/output relations
between two variables in
Range
graphical form, verbal
description, set of ordered pairs,
or algebraic model,
Distinguish between those that
are functions and non-functions.
Given a functional relationship,
Determine that exactly one
element of the range (output) is
assigned to each element of the
domain (input) by the function,
Represent the function with a
graph and with functional
notation.
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
Distinguishing characteristics
of functions,
Accurately graph functions when
given function notation,
Conventions of function
notation,
Accurately determine domain and
range values from function notation.
In graphing functions the
ordered pairs are (x,f(x)) and
the graph is y = f(x).
Functions are relationships between
two variables that have a unique
characteristic: that for each input
there exists exactly one output,
Level IV Students will:
EEF-IF.1-3. Use the concept of functions to identify how the two variables are
affected.
Ex. Given a graph showing the growth of a plant over a period of one month, identify
that, as the number of days increase, plant height increases.*
Ex. Given a graph that shows the amount of paint in can and the area painted, identify
that, as the area painted increases, the amount of paint in the can decreases.*
Ex. Tell the cost of movie tickets for five people if movies tickets are $3 per ticket.
Ex. The amount of change you get from a drink machine if each drink cost $0.65. The
amount of change you receive will be a function of how much you put into the
machine.
Level III Students will:
EEF-IF.1-3. Use the concept of function to solve problems.
Ex. Using a store scenario, one store charges students $2 more than another store for
Function notation is useful to see the the same item. Tom purchases a caramel apple for $5. What should Becky expect to
relationship between two variables
pay for an identical apple at the more expensive store?
when the unique output for each input Ex. Look at a graph to identify relationship between two variables (distance - time,
cost - product, etc.) If every item cost $1 at a store, how much would five items cost?
relation is satisfied.
Ex. Determine the total distance traveled in 20 minutes using a table if you are
traveling at a constant speed of one mile every 10 minutes.
Level II Students will:
EEF-IF.1-3. Solve problems using a table that shows basic relationships (may not
involve a true function).
Ex. Look at a weather chart to identify relationships between the day of the week and
the temperature.
Ex. Determine the number of shoes worn by four people using a graph that
incorporates picture representations.
Ex. From a given table displaying the cost of movie tickets, determine the cost of one
ticket, two tickets, and three tickets.
Ex. From a five-day weather forecast, identify the weather for Wednesday.
Level I Students will:
EEF-IF.1-3. Identify basic information located on graphs.
Ex. Tell the day of the week on a graph/point to the activity on the graph.
Ex. Identify a line on a line graph.
Ex. Identify the highest bar on a bar graph.
Ex. Recognize different types of graphs.
*Refer to the Common Core Essential Elements document for diagram.
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Version 3 2015-2016
Grade Level/ Course: Algebra 1 Unit 2
Standard with code: 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.
Domain: Interpreting Functions
Cluster: Understand the concept of a function and use function notation.
Quarter 1:
Quarter 2:
Identify mathematical relationships
and express them using function
notation.
Make sense of
problems and
persevere in
solving them.
Reason
abstractly and
quantitatively.
Note from Appendix A: 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.
Construct viable
Model with
arguments and
mathematics.
critique the
reasoning of others.
Quarter 3:
Quarter 4:
Define a reasonable domain, which depends on the Define a reasonable domain, which depends on the
context and/or mathematical situation, for a
context and/or mathematical situation, for a
function focusing on linear functions.
function focusing on linear and exponential
functions.
Evaluate functions at a given input in the domain,
focusing on linearl functions.
Evaluate functions at a given input in the domain,
focusing on linear and exponential functions.
Interpret statements that use functions in terms of
real world situations, focusing on linear functions.
Use appropriate tools
strategically.
Attend to precision.
Look for and make
use of structure.
Look for and express
regularity in repeated
reasoning.
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Version 3 2015-2016
Evidence of Student
Vocabulary
Attainment/Assessment
Students:
Function notation
Given a contextual situation that
may be represented as a
function,
Use function notation to model
the situation,
Evaluate the function to produce
outputs when given a value in
the domain,
Explain in the original context
the meaning of the output when
related to the input.
Knowledge
Skills
Students know:
Students understand/are able to:
Instructional Achievement Level Descriptors
Level IV Students will:
EEF-IF.1-3. Use the concept of functions to identify how the two variables are affected.
Ex. Given a graph showing the growth of a plant over a period of one month, identify
Distinguishing characteristics Accurately use function notation to
that, as the number of days increase, plant height increases.*
of a function,
model physical situations,
Ex. Given a graph that shows the amount of paint in can and the area painted, identify
that, as the area painted increases, the amount of paint in the can decreases.*
Conventions of function
Accurately evaluate function
Ex. Tell the cost of movie tickets for five people if movies tickets are $3 per ticket.
notation.
equations given values in the domain. Ex. The amount of change you get from a drink machine if each drink cost $0.65. The
amount of change you receive will be a function of how much you put into the machine.
Function notation is useful to see the
relationship between two variables
when the unique output for each input
relation is satisfied.
Level III Students will:
EEF-IF.1-3. Use the concept of function to solve problems.
Ex. Using a store scenario, one store charges students $2 more than another store for the
same item. Tom purchases a caramel apple for $5. What should Becky expect to pay for
an identical apple at the more expensive store?
Ex. Look at a graph to identify relationship between two variables (distance - time, cost product, etc.) If every item cost $1 at a store, how much would five items cost?
Ex. Determine the total distance traveled in 20 minutes using a table if you are traveling
at a constant speed of one mile every 10 minutes.
Level II Students will:
EEF-IF.1-3. Solve problems using a table that shows basic relationships (may not
involve a true function).
Ex. Look at a weather chart to identify relationships between the day of the week and the
temperature.
Ex. Determine the number of shoes worn by four people using a graph that incorporates
picture representations.
Ex. From a given table displaying the cost of movie tickets, determine the cost of one
ticket, two tickets, and three tickets.
Ex. From a five-day weather forecast, identify the weather for Wednesday.
Level I Students will:
EEF-IF.1-3. Identify basic information located on graphs.
Ex. Tell the day of the week on a graph/point to the activity on the graph.
Ex. Identify a line on a line graph.
Ex. Identify the highest bar on a bar graph.
Ex. Recognize different types of graphs.
*Refer to the Common Core Essential Elements document for diagram.
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Version 3 2015-2016
Grade Level/ Course: Algebra 1 Unit 2
Standard with code: 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.
Domain: Interpreting Functions
Cluster: Understand the concept of a function and use function notation.
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
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.
Notes from Appendix A: Students should
experience a variety of types of situations
modeled by functions. Detailed analysis of
any particular class of functions at this
stage is not advised. Students should apply
these concepts throughout their future
mathematics courses. Draw examples from
linear and exponential functions. In F.IF.3,
draw connection to F.BF.2, which requires
students to write arithmetic and geometric
sequences. Emphasize arithmetic and
geometric sequences as examples of linear
and exponential functions.
Make sense of
Reason abstractly
problems and
and quantitatively.
persevere in solving
them.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
Use appropriate
tools strategically.
Attend to
precision.
Look for and make use
of structure.
Look for and express
regularity in
repeated reasoning.
52
Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Given a sequence,
Vocabulary
Knowledge
Skills
Sequence
Students know:
Students understand/are able to:
of generating sequences.
consecutive.
Recursively
Generate and justify a function
that relates the number of the
term to the value of the term in
the sequence.
Domain
Instructional Achievement Level Descriptors
Level IV Students will:
EEF-IF.1-3. Use the concept of functions to identify how the two variables are
affected.
Distinguishing characteristics Relate the number of the term to the
Ex. Given a graph showing the growth of a plant over a period of one month, identify
of a function,
value of the term in a sequence and
that, as the number of days increase, plant height increases.*
express the relation in functional
Ex. Given a graph that shows the amount of paint in can and the area painted, identify
Distinguishing characteristics notation.
that, as the area painted increases, the amount of paint in the can decreases.*
of function notation,
Ex. Tell the cost of movie tickets for five people if movies tickets are $3 per ticket.
Each term in the domain of a sequence Ex. The amount of change you get from a drink machine if each drink cost $0.65. The
amount of change you receive will be a function of how much you put into the machine.
Distinguishing characteristics defined as a function is unique and
Level III Students will:
EEF-IF.1-3. Use the concept of function to solve problems.
Ex. Using a store scenario, one store charges students $2 more than another store for the
same item. Tom purchases a caramel apple for $5. What should Becky expect to pay for
an identical apple at the more expensive store?
Ex. Look at a graph to identify relationship between two variables (distance - time, cost
- product, etc.) If every item cost $1 at a store, how much would five items cost?
Ex. Determine the total distance traveled in 20 minutes using a table if you are traveling
at a constant speed of one mile every 10 minutes.
Level II Students will:
EEF-IF.1-3. Solve problems using a table that shows basic relationships (may not
involve a true function).
Ex. Look at a weather chart to identify relationships between the day of the week and
the temperature.
Ex. Determine the number of shoes worn by four people using a graph that incorporates
picture representations.
Ex. From a given table displaying the cost of movie tickets, determine the cost of one
ticket, two tickets, and three tickets.
Ex. From a five-day weather forecast, identify the weather for Wednesday.
Level I Students will:
EEF-IF.1-3. Identify basic information located on graphs.
Ex. Tell the day of the week on a graph/point to the activity on the graph.
Ex. Identify a line on a line graph.
Ex. Identify the highest bar on a bar graph.
Ex. Recognize different types of graphs.
*Refer to the Common Core Essential Elements document for diagram.
53
Version 3 2015-2016
Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code: 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.*(*Modeling standard)
Domain: Interpreting Functions
Cluster: Interpret functions that arise in applications in terms of the context.
Quarter 1:
Quarter 2:
Define and recognize the key features
in tables and graphs of linear functions:
intercepts; intervals where the function
is increasing, decreasing, positive, or
negative, and end behavior.
Define and recognize the key features in tables
and graphs of linear and exponential functions:
intercepts; intervals where the function is
increasing, decreasing, positive, or negative,
and end behavior.
Identify whether the function is linear,
given its table or graph.
Identify whether the function is linear or
exponential, given its table or graph.
Interpret key features of graphs and
tables of functions in the terms of the
contextual quantities the function
represents.
Interpret key features of graphs and tables of
functions in the terms of the contextual
quantities the function represents.
Sketch graphs showing key features of
a function that models a relationship
between two quantities from a given
verbal description of the relationship.
Make sense of
problems and
persevere in
solving them.
Quarter 3:
Quarter 4:
Sketch graphs showing key features of a
function that models a relationship between two
quantities from a given verbal description of the
relationship.
Notes from Appendix A: Focus on linear
and exponential.
Reason abstractly Construct viable
Model with
and
arguments and
mathematics.
quantitatively.
critique the
reasoning of others.
Use appropriate tools
strategically.
Attend to precision.
Look for and make
use of structure.
Look for and express
regularity in repeated
reasoning.
54
Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Given a function that models a
relationship between two
quantities,
Produce the graph and table of
the function and show the key
features (intercepts; intervals
where the function is increasing,
decreasing, positive, or negative;
relative maximums and
minimums; symmetries; end
behavior; and periodicity) that
are appropriate for the function.
Given key features from verbal
description of a relationship,
Sketch a graph with the given
key features.
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Function
Students know:
Students understand/are able to:
Key features
Key features of function
graphs (i.e., intercepts;
intervals where the function is
increasing, decreasing,
positive, or negative; relative
maximums and minimums;
symmetries; end behavior; and
periodicity),
Accurately graph any relationship,
Level IV Students will:
EEF-IF.4-6. Evaluate key features of a graph (e.g. increasing, decreasing,
constant.).
Ex. Determine parts of graph illustrating an increase or decrease in speed.
Ex. Using a graph illustrating change in temperature over a day, indicate times
when the temperature increased, decreased, or stayed the same.
Methods of modeling
relationships with a graph or
table.
Interpret key features of a graph.
The relationship between two
variables determines the key features Level III Students will:
that need to be used when interpreting EEF-IF.4-6. Interpret rate of change (e.g. higher/lower, faster/slower).
Ex. Compare two graphs with different slopes to determine faster/slower rate
and producing the graph.
Ex. Compare a bus schedule with two buses, look and determine if one bus runs
more frequently than the next bus on the route.
Level II Students will:
EEF-IF.4-6. Graph a simple linear equation represented by a table of values.
Ex. Match the graph to its corresponding story.
Ex. Plot the points from a table of values less than 10.
Level I Students will:
EEF-IF.4-6. Read a table.
Ex. From a given table, find information.
Ex. Read a bus schedule.
Ex. Given a daily schedule, determine the time of lunch during the school day.
55
Version 3 2015-2016
Grade Level/ Course: Algebra 1 Unit 2
Standard with code: 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.*(*Modeling standard)
Domain: Interpreting Functions
Cluster: Interpret functions that arise in applications in terms of a context
Quarter 1:
Quarter 2:
Given the graph or a verbal/written
description of a function, identify and
describe the domain of the function.
Given the graph or a verbal/written
description of a function, identify and
describe the domain of the function.
Identify an appropriate domain based on
the unit, quantity, and type of function it
describes.
Identify an appropriate domain based on the
unit, quantity, and type of function it
describes.
Relate the domain of the function to its
graph and, where applicable, to the
quantitative relationship it describes.
Relate the domain of the function to its
graph and, where applicable, to the
quantitative relationship it describes.
Explain why a domain is appropriate for a
given situation.
Explain why a domain is appropriate for a
given situation.
Notes from Appendix A: For F.IF.4 and 5,
focus on linear and exponential functions
in Algebra 1 unit 2.
Notes from Appendix A: For F.IF.4
and 5, focus on linear and
exponential functions in Algebra 1
unit 2.
Construct viable
Model with
arguments and
mathematics.
critique the
reasoning of
others.
Make sense of
problems and
persevere in solving
them.
Reason
abstractly and
quantitatively.
Quarter 3:
Use appropriate tools
strategically.
Quarter 4:
Attend to precision.
Look for and make
use of structure.
Look for and
express regularity
in repeated
reasoning.
56
Version 3 2015-2016
Evidence of Student
Vocabulary
Attainment/Assessment
Students:
Function
Given a contextual situation that
is functional,
Model the situation with a graph
and construct the graph based on
the parameters given in the
domain of the context.
Knowledge
Skills
Students know:
Students understand/are able to:
Instructional Achievement Level Descriptors
Level IV Students will:
EEF-IF.4-6. Evaluate key features of a graph (e.g. increasing, decreasing,
Interpret the domain from the context, constant.).
Techniques for graphing
Ex. Determine parts of graph illustrating an increase or decrease in speed.
functions,
Produce a graph of a function based on Ex. Using a graph illustrating change in temperature over a day, indicate
times when the temperature increased, decreased, or stayed the same.
Techniques for determining the the context given.
domain of a function from its
Level III Students will:
context.
Different contexts produce different
EEF-IF.4-6. Interpret rate of change (e.g. higher/lower, faster/slower).
domains and graphs,
Ex. Compare two graphs with different slopes to determine faster/slower rate
Ex. Compare a bus schedule with two buses, look and determine if one bus
Function notation in itself may
produce graph points which should not runs more frequently than the next bus on the route.
be in the graph as the domain is limited
Level II Students will:
by the context.
EEF-IF.4-6. Graph a simple linear equation represented by a table of values.
Ex. Match the graph to its corresponding story.
Ex. Plot the points from a table of values less than 10.
Level I Students will:
EEF-IF.4-6. Read a table.
Ex. From a given table, find information.
Ex. Read a bus schedule.
Ex. Given a daily schedule, determine the time of lunch during the school
day.
57
Version 3 2015-2016
Grade Level/ Course (high school): Algebra 1 Unit 2
Standard with code: 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.*(Modeling standard)
Domain: Interpreting Functions
Cluster: Interpret functions that arise in applications in terms of a context
Quarter 1:
Quarter 2:
Quarter 3:
Recognize slope as an average rate of change.
Calculate the average rate of change of a
function (presented symbolically or as a
table) over a specified interval.
Make sense of
problems and
persevere in solving
them.
Reason abstractly and
quantitatively.
Quarter 4:
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 linear
or exponential graph.
Notes from Appendix A: Focus on linear
functions and exponential functions
whose domain is a subset of the integers.
Interpret the average rate of change of a
function (presented symbolically or as a
table) over a specified interval.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
Use appropriate
tools strategically.
Attend to
precision.
Look for and make
use of structure.
Look for and
express regularity
in repeated
reasoning.
58
Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Given an interval on a graph or
table,
Calculate the average rate of
change within the interval.
Given a graph of contextual
situation,
Estimate the rate of change
between intervals that are
appropriate for the summary of
the context.
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Average rate of
change
Students know:
Students understand/are able to:
Techniques for graphing,
Calculate rate of change over an
interval on a table or graph,
Level IV Students will:
EEF-IF.4-6. Evaluate key features of a graph (e.g. increasing, decreasing,
constant.).
Ex. Determine parts of graph illustrating an increase or decrease in speed.
Ex. Using a graph illustrating change in temperature over a day, indicate times
when the temperature increased, decreased, or stayed the same.
Techniques for finding a rate of
change over an interval on a
Estimate a rate of change over an
table or graph,
interval on a graph.
Level III Students will:
Techniques for estimating a rate The average provides information on EEF-IF.4-6. Interpret rate of change (e.g. higher/lower, faster/slower).
Ex. Compare two graphs with different slopes to determine faster/slower rate
of change over an interval on a the overall changes within an
Ex. Compare a bus schedule with two buses, look and determine if one bus runs
graph.
interval, not the details within the
interval (an average of the endpoints more frequently than the next bus on the route.
of an interval does not tell you the
Level II Students will:
significant features within the
EEF-IF.4-6. Graph a simple linear equation represented by a table of values.
interval).
Ex. Match the graph to its corresponding story.
Ex. Plot the points from a table of values less than 10.
Level I Students will:
EEF-IF.4-6. Read a table.
Ex. From a given table, find information.
Ex. Read a bus schedule.
Ex. Given a daily schedule, determine the time of lunch during the school day.
59
Version 3 2015-2016
Grade Level/ Course: Algebra 1 Unit 2
Standard with code: F.IF.7e Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated
cases.*(Modeling standard)
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and
amplitude.
Domain: Interpreting Functions
Cluster: Analyze functions using different representations
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Determine the differences between simple and
complicated linear functions and know when the
use of technology is appropriate.
Graph exponential functions, by hand in simple
cases or using technology for more complicated
cases, and show intercepts and end behavior.
Determine the differences between simple and
complicated linear and exponential functions and
know when the use of technology is appropriate.
Make sense of
problems and
persevere in solving
them.
Reason abstractly Construct viable
and quantitatively. arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate tools Attend to precision.
strategically.
Note from Appendix A: 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.
Look for and make
Look for and express
use of structure.
regularity in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a symbolic representation of a
function (including linear, quadratic, square
root, cube root, piecewise-defined functions,
polynomial, exponential, logarithmic,
trigonometric, and (+) rational),
Produce an accurate graph (by hand in
simple cases and by technology in more
complicated cases) and justify that the graph
is an alternate representation of the symbolic
function,
Identify key features of the graph and
connect these graphical features to the
symbolic function, specifically for special
functions:
quadratic or linear (intercepts, maxima, and
minima),
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
EEF-IF.7. N/A (See EEF-IF.1-3)
Techniques for graphing,
Identify the type of function from the
symbolic representation,
Key features of graphs of
functions.
Manipulate expressions to reveal
important features for identification in
the function,
Accurately graph any relationship.
Key features are different depending
on the function,
Identifying key features of functions
aid in graphing and interpreting the
function.
square root, cube root, and piecewise-defined
functions, including step functions and
absolute value functions (descriptive features
such as the values that are in the range of the
function and those that are not),
polynomial (zeros when suitable
factorizations are available, end behavior),
(+) rational (zeros and asymptotes when
suitable factorizations are available, end
behavior),
exponential and logarithmic (intercepts and
end behavior),
trigonometric functions (period, midline, and
amplitude).
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Grade Level/ Course: Algebra 1 Unit 2
Standard with code: 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.
Domain: Interpreting Functions
Cluster: Analyze functions using different representations
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Identify types of functions based on verbal ,
numerical, algebraic, and graphical descriptions
and state key properties (e.g. intercepts, growth
rates, average rates of change, and end behaviors)
Use a variety of function representations
(algebraically, graphically, numerically in
tables, or by verbal descriptions) to compare
and contrast properties of two functions.
Differentiate between exponential and linear
functions using a variety of descriptors
(graphically, verbally, numerically, and
algebraically)
Use appropriate tools
strategically.
Look for and make
use of structure.
Use a variety of function representations
(algebraically, graphically, numerically in tables, or
by verbal descriptions) to compare and contrast
properties of two functions.
Make sense of
problems and
persevere in solving
them.
Note from Appendix A: Focus on linear and
exponential functions. Include comparisons of two
functions presented algebraically. For example,
compare the growth of two linear functions, or two
exponential functions such as y=3n and y=1002.
Reason abstractly Construct viable
Model with
and
arguments and
mathematics.
quantitatively.
critique the reasoning
of others.
Attend to
precision.
Look for and
express regularity in
repeated reasoning.
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Evidence of Student
Vocabulary
Attainment/Assessment
Students:
Given two functions represented
differently (algebraically,
graphically, numerically in
tables, or by verbal
descriptions),
Use key features to compare the
functions,
Explain and justify the
similarities and differences of
the functions.
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
EEF-IF.9. N/A
Techniques to find key
features of functions when
presented in different ways,
Accurately determine which key
features are most appropriate for
comparing functions,
Techniques to convert a
function to a different form
(algebraically, graphically,
numerically in tables, or by
verbal descriptions).
Manipulate functions algebraically to
reveal key functions,
Convert a function to a different form
(algebraically, graphically, numerically
in tables, or by verbal descriptions) for
the purpose of comparing it to another
function.
Functions can be written in different
but equivalent ways (algebraically,
graphically, numerically in tables, or
by verbal descriptions),
Different representations of functions
may aid in comparing key features of
the functions.
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Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code: F.BF.1a Write a function that describes a relationship between two quantities.*(Modeling standard) a. Determine an explicit expression, a recursive process, or
steps for calculation from a context.
Domain: Building Functions
Cluster: Build a function that models a relationship between two quantities
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Define “explicit function” and “recursive
process”.
Write a function that describes a relationship
between two quantities by determining an
explicit expression, a recursive process, or steps
for calculation from a context.
Make sense of
Reason abstractly
problems and
and quantitatively.
persevere in solving
them.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate tools Attend to precision.
strategically.
Note from Appendix A: Limit to F.BF.1a to
linear and exponential functions.
Look for and make
Look for and express
use of structure.
regularity in
repeated reasoning.
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Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Given a contextual situation
containing two quantities,
Vocabulary
Knowledge
Skills
Explicit expression
Students know:
Students understand/are able to:
Recursive process
Express a relationship between Decaying exponential
the quantities through an explicit
expression using function
notation, recursive process, or
steps for calculation,
Explain and justify how the
expression or process models the
relationship between the given
quantities,
Create a new function by using
standard function types and
arithmetic operations to combine
the original functions to model
the relationship of the given
quantities, (+) standards not
covered.
Instructional Achievement Level Descriptors
Level IV Students will:
EEF.BF.1. Complete the appropriate graphical representation (first quadrant)
Techniques for expressing
Accurately develop a model that shows given a situation involving constant rate of change.
Ex. Given this scenario and a graphical representation with missing
functional relationships
the functional relationship between
information: If I mow one lawn and I make $25 and if I mow three lawns and I
(explicit expression, a recursive two quantities,
make $75, how much will I make if I mow two lawns?
process, or steps for
Ex. Given this scenario and a graphical representation with missing
calculation) between two
Accurately create a new function
quantities,
through arithmetic operations of other information: If hamburgers are four for $1 and I buy four, it will cost $1; if I
buy 12, it will cost $3 – complete the graph for eight hamburgers.
functions,
Techniques to combine
functions using arithmetic
Present an argument to show how the Level III Students will:
EEF-BF.1. Select the appropriate graphical representation (first quadrant)
operations.
function models the relationship
given a situation involving constant rate of change.
between the quantities.
Ex. Given this scenario and two completed graphs, show me the graph that
shows the following: If I mow one lawn, I make $25; if I mow two lawns, I
Relationships can be modeled by
will make $50; and if I mow three lawns I will make $75.
several methods (e.g., explicit
Ex. Given this scenario and two completed graphs, show me the graph that
expression or recursive process),
depicts that there are two cookies for every student.
Arithmetic combinations of functions
may be used to improve the fit of a
model.
Level II Students will:
EEF-BF.1. Select the appropriate graphical representation (first quadrant)
given a situation involving constant rate of change where the difference is very
clear.
Ex. Every dog has one bone. Pick the graph that would represent this concept
when given the following graphs.*
Level I Students will:
EEF-BF.1. Identify the terms in a sequence.
Ex. Identify an ABABABABAB pattern out of two different pattern sets of
colored blocks using black (B) and white (W) and one set is
BWBWBWBWBW and the other pattern set is BBWBBWBBWBBW.
Ex. Place two pencils in front of each student in the classroom.
*Refer to the Common Core Essential Elements document for diagram.
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Version 3 2015-2016
Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code: F.BF.1b Write a function that describes a relationship between two quantities.*(Modeling standard)
b. Combine standard function types using arithmetic
operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
Domain: Building Functions
Cluster: Build a function that models a relationship between two quantities
Quarter 1:
Quarter 2:
Combine two functions using the operations
of addition, subtraction, multiplication, and
division.
Quarter 3:
Quarter 4:
Evaluate the domain of the combined
function.
Given a real-world situation or
mathematical problem:
 build standard functions to represent
relevant relationships/ quantities,
 determine which arithmetic
operation should be performed to
build the appropriate combined
function, and
 relate the combined function to the
context of the problem
Make sense of
problems and
persevere in
solving them.
Note from Appendix A: Limit to linear and
exponential functions.
Reason abstractly Construct viable
Model with
and quantitatively. arguments and
mathematics.
critique the
reasoning of others.
Use appropriate tools
strategically.
Attend to precision.
Look for and make
use of structure.
Look for and
express regularity
in repeated
reasoning.
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Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Given a contextual situation
containing two quantities,
Vocabulary
Knowledge
Skills
Explicit expression
Students know:
Students understand/are able to:
Recursive process
Express a relationship between Decaying exponential
the quantities through an explicit
expression using function
notation, recursive process, or
steps for calculation,
Explain and justify how the
expression or process models the
relationship between the given
quantities,
Create a new function by using
standard function types and
arithmetic operations to combine
the original functions to model
the relationship of the given
quantities, (+) standards not
covered.
Instructional Achievement Level Descriptors
Level IV Students will:
EEF.BF.1. Complete the appropriate graphical representation (first quadrant)
Techniques for expressing
Accurately develop a model that shows given a situation involving constant rate of change.
Ex. Given this scenario and a graphical representation with missing
functional relationships
the functional relationship between
information: If I mow one lawn and I make $25 and if I mow three lawns and I
(explicit expression, a
two quantities,
make $75, how much will I make if I mow two lawns?
recursive process, or steps for
Ex. Given this scenario and a graphical representation with missing
calculation) between two
Accurately create a new function
quantities,
through arithmetic operations of other information: If hamburgers are four for $1 and I buy four, it will cost $1; if I
buy 12, it will cost $3 – complete the graph for eight hamburgers.
functions,
Techniques to combine
functions using arithmetic
Present an argument to show how the Level III Students will:
EEF-BF.1. Select the appropriate graphical representation (first quadrant)
operations.
function models the relationship
given a situation involving constant rate of change.
between the quantities.
Ex. Given this scenario and two completed graphs, show me the graph that
shows the following: If I mow one lawn, I make $25; if I mow two lawns, I
Relationships can be modeled by
will make $50; and if I mow three lawns I will make $75.
several methods (e.g., explicit
Ex. Given this scenario and two completed graphs, show me the graph that
expression or recursive process),
depicts that there are two cookies for every student.
Arithmetic combinations of functions
may be used to improve the fit of a
model.
Level II Students will:
EEF-BF.1. Select the appropriate graphical representation (first quadrant)
given a situation involving constant rate of change where the difference is very
clear.
Ex. Every dog has one bone. Pick the graph that would represent this concept
when given the following graphs.*
Level I Students will:
EEF-BF.1. Identify the terms in a sequence.
Ex. Identify an ABABABABAB pattern out of two different pattern sets of
colored blocks using black (B) and white (W) and one set is
BWBWBWBWBW and the other pattern set is BBWBBWBBWBBW.
Ex. Place two pencils in front of each student in the classroom.
*Refer to the Common Core Essential Elements document for diagram.
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Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code: 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.*(*Modeling standard)
Domain: Building Functions
Cluster: Build a function that models a relationship between two quantities
Quarter 1:
Quarter 2:
Identify arithmetic and geometric patterns in given
sequences.
Quarter 3:
Generate arithmetic and geometric sequences from
recursive and explicit formulas.
Generate arithmetic and geometric sequences from
explicit formulas.
Given an arithmetic or geometric sequence in recursive
form, translate into the explicit formula.
Given an arithmetic or geometric sequence translate
into the explicit formula.
Given an arithmetic or geometric sequence as an explicit
formula, translate into the recursive form.
Use given and constructed arithmetic and geometric
sequences, expressed both recursively and with explicit
formulas, to model real-life situations.
Use given and constructed arithmetic and geometric
sequences, expressed with explicit formulas, to
model real-life situations.
Determine the recursive rule given arithmetic and
geometric sequences.
Determine the explicit formula given arithmetic and
geometric sequences.
Determine the explicit formula given arithmetic and
geometric sequences.
Justify the translation between the recursive form &
explicit formula for arithmetic and geometric
sequences.
Make sense of
problems and
persevere in
solving them.
Notes from Appendix A: Connect arithmetic
sequences to linear functions and geometric
sequences to exponential functions.
Reason abstractly Construct viable
Model with
and quantitatively. arguments and
mathematics.
critique the reasoning
of others.
Quarter 4:
Identify arithmetic and geometric patterns in given
sequences.
Justify the translation between the recursive form & explicit
formula for arithmetic and geometric sequences.
Notes from Appendix A: Connect arithmetic sequences to
linear functions and geometric sequences to exponential
functions.
Use appropriate
tools strategically.
Attend to precision. Look for and make use
of structure.
Look for and express
regularity in
repeated reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a contextual situation that
is sequential (arithmetic and
geometric),
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Arithmetic and
geometric sequences
Students know:
Students understand/are able to:
Techniques to translate
between recursive and explicit
formulas for sequences
(geometric and arithmetic).
Use the properties of operations and
equality (Tables 3 and 4) and
knowledge of recursive functions to
justify that an explicit formula that
models a sequence is equivalent to a
recursive model.
Level IV Students will:
EEF-BF.2. Build an arithmetic sequence when provided a recursive rule with
decreasing terms, decimals, or fractions.
Ex. Starting at 100, subtract five each time to build a sequence.
Ex. Starting at $5.50, add/subtract $0.50 each time to build a sequence.
Recursively
Create both recursive and
Explicit formula
explicit models for the sequence,
Explain and justify the
relationship between the
recursive and explicit forms that
model the situation.
Arithmetic and geometric sequences
can be expressed with a recursive
model or explicit formula, and each
form may have benefits to aid in
understanding or interpreting the
situation.
Level III Students will:
EEF-FB.2. Build an arithmetic sequence when provided a recursive rule with
whole numbers.
Ex. Starting at four, add four each time to build a sequence (e.g., If one dog has
four legs, how many will two dogs have, three dogs, etc.).
Ex. Starting at five, add seven each time to build a sequence (e.g., If I have $5
and I earn $7 each hour – how much money will I have in four hours?).
Level II Students will:
EEF-BF.2. Identify a term in a sequence.
Ex. Given a clear sequence (2, 4, 6, 8, . . . ), identify the next number in the set.
Ex. Given the sequence 4, 2, 5, 1, 3, N, identify what is the value of N.
Level I Students will:
EEF-BF.2. Recognize a sequence.
Ex. Given two lists of numbers or a set of manipulatives, identify the sequence
in 5, 4, 3, 2, 1.
Ex. Given two lists of numbers or a set of manipulatives, identify the sequence
in 2, 4, 6, 8.
Ex. Given a sequence, a picture of a ball, and a fraction, student can select the
sequence.
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Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code: 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.
Domain: Building Functions
Cluster: Build new functions from existing functions
Quarter 1:
Quarter 2:
Experiment with cases and illustrate an
explanation of the effects on the graph using
technology.
Notes from Appendix A: Focus on vertical
translations of graphs of linear and exponential
functions. Relate the vertical translation of a
linear function to its y-intercept. While applying
other transformations to a linear graph is
appropriate at this level, it may be difficult for
students to identify or distinguish between the
effects of the other transformations included in this
standard.
Make sense of
Reason abstractly
problems and
and
persevere in solving quantitatively.
them.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
Quarter 3:
Given a single transformation on a function
(symbolic or graphic) identify the effect on the
graph.
Quarter 4:
Using technology, identify effects of single
transformations on graphs of functions.
Graph a given function by 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).
Describe the differences and similarities between
a parent function and the transformed function.
Find the value of k, given the graphs of a parent
function, f(x), and the transformed function: f(x)
+ k, k f(x), f(kx), or f(x + k).
Recognize even and odd functions from their
graphs and from their equations.
Use appropriate tools
Attend to precision.
strategically.
Look for and make
use of structure.
Look for and
express regularity
in repeated
reasoning.
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Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Given a function in algebraic
form,
Graph the function, f(x),
conjecture how the graph of f(x)
+ k, k f(x), f(kx), and f(x + k) for
specific values of k(both positive
and negative) will change from
f(x), and test the conjectures,
Describe how the graphs of the
functions were affected (e.g.,
horizontal and vertical shifts,
horizontal and vertical stretches,
or reflections),
Use technology to explain
possible effects on the graph from
adding or multiplying the input or
output of a function by a constant
value,
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Even and odd
functions
Students know:
Students understand/are able to:
EEF-BF.3-4. N/A
Graphing techniques of
functions,
Accurately graph functions,
Check conjectures about how a
Methods of using technology to parameter change in a function
graph functions,
changes the graph and critique the
reasoning of others about such shifts,
Techniques to identify even and
odd functions both algebraically Identify shifts, stretches, or
and from a graph.
reflections between graphs,
Determine when a function is even
or odd.
Graphs of functions may be shifted,
stretched, or reflected by adding or
multiplying the input or output of a
function by a constant value,
Even and odd functions may be
identified from a graph or algebraic
form of a function.
Recognize if a function is even or
odd.
Given the graph of a function and
the graph of a translation, stretch,
or reflection of that function,
Determine the value which was
used to shift, stretch, or reflect the
graph,
Recognize if a function is even or
odd.
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Version 3 2015-2016
Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code: F.LE.1a Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal
differences over equal intervals; and that exponential functions grow by equal factors over equal intervals.
Domain: Linear, Quadratic, and
Cluster: Construct and compare linear and exponential models and solve problems
Exponential Models *(*Modeling Domain)
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Recognize that linear functions grow by equal
differences over equal intervals.
Recognize that exponential functions
grow by equal factors over equal
intervals.
Distinguish between situations that can be
modeled with linear functions and with
exponential functions to solve mathematical and
real-world problems.
Distinguish between situations that can
be modeled with linear functions and
with exponential functions to solve
mathematical and real-world problems.
Prove that linear functions grow by equal
differences over equal intervals.
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and quantitatively.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
Prove that exponential functions grow by
equal factors over equal intervals.
Use appropriate tools
strategically.
Attend to precision.
Look for and
make use of
structure.
Look for and
express regularity
in repeated
reasoning.
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Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Given a linear or exponential
function,
Create a sequence from the
functions and examine the
results to demonstrate that linear
functions grow by equal
differences, and exponential
functions grow by equal factors
over equal intervals,
Use slope-intercept form of a
linear function and the general
definition of exponential
functions to justify through
algebraic rearrangements that
linear functions grow by equal
differences, and exponential
functions grow by equal factors
over equal intervals.
Given a contextual situation
modeled by functions,
Determine if the change in the
output per unit interval is a
constant being added or
multiplied to a previous output,
and appropriately label the
function as linear, exponential,
or neither.
Vocabulary
Knowledge
Skills
Linear functions
Students know:
Students understand/are able to:
Exponential functions
Key components of linear and
exponential functions,
Properties of operations and
equality (Tables 3 and 4).
Instructional Achievement Level Descriptors
Level IV Students will:
EEF-LE.1-4. Plot points using pictures in first quadrant on a graph using
Accurately determine relationships of whole numbers and explain how y increases/decreases as x changes.
Ex. If you go to the store where every item is one dollar, students should state
data from a contextual situation to
y = x (the number of items I buy will tell me the cost). Students will then plot
determine if the situation is one in
this on the graph.
which one quantity changes at a
constant rate per unit interval relative Ex. If I get two apples for every orange I buy, students should state that y = 2x,
or for every orange I buy (x), I will get two apples (y), therefore x times two
to another (linear),
tells me the number of apples each time. Students should then plot this on the
Accurately determine relationships of graph.
data from a contextual situation to
Level III Student will:
determine if the situation is one in
EEF-LE.1-4. Model a simple linear function such as y = mx to show functions
which one quantity grows or decays
grow by equal factors over equal intervals.
by a constant percent rate per unit
Ex. Determine a simple relationship of y to x by looking at the first quadrant of
interval relative to another
a graph.
(exponential).
Ex. Identify the cost per item on a simple graph where every item in the store
cost the same amount and state the relationship between x and y.
Linear functions have a constant
Ex. Look at a graph that shows a constant ratio of boys to girls and state the
value added per unit interval, and
exponential functions have a constant relationship between x and y.
value multiplied per unit interval,
Distinguishing key features of and
categorizing functions facilitates
mathematical modeling and aids in
problem resolution.
Level II Students will:
EEF-LE.1-4. Identify a specific data point in the first quadrant and explain the
meaning behind it.
Ex. Given data points in the first quadrant, identify the named point and state
the two pieces of information that one dot provides.
Ex. When given a simple graph that shows the total cost of items purchased at
a store where every item is $1, tell the cost of four items, the cost of two items,
etc.
Level I Students will:
EEF-LE.1-4. Interpret major ideas of a graph with linear functions.
Ex. When shown two lines on a graph, tell which one is rising faster.
Ex. When shown a graph of distance driven and gas left in tank, explain that
the further one drives the less gas one has left.
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Version 3 2015-2016
Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code: F.LE.1b Distinguish between situations that can be modeled with linear functions and with exponential functions. b. Recognize situations in which one quantity
changes at a constant rate per unit interval relative to another.
Domain: Linear, Quadratic, and
Exponential Models *(*Modeling Domain)
Cluster: Construct and compare linear, quadratic, and exponential models and solve problems
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Recognize situations in which one quantity
changes at a constant rate per unit (equal
differences) interval relative to another to
solve mathematical and real-world problems.
Make sense of
problems and
persevere in solving
them.
Reason abstractly and
quantitatively.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate tools
strategically.
Attend to precision.
Look for and
make use of
structure.
Look for and
express regularity
in repeated
reasoning.
74
Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Given a linear or exponential
function,
Create a sequence from the
functions and examine the
results to demonstrate that linear
functions grow by equal
differences, and exponential
functions grow by equal factors
over equal intervals,
Use slope-intercept form of a
linear function and the general
definition of exponential
functions to justify through
algebraic rearrangements that
linear functions grow by equal
differences, and exponential
functions grow by equal factors
over equal intervals.
Given a contextual situation
modeled by functions,
Determine if the change in the
output per unit interval is a
constant being added or
multiplied to a previous output,
and appropriately label the
function as linear, exponential,
or neither.
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Linear functions
Students know:
Students understand/are able to:
Exponential functions
Key components of linear and Accurately determine relationships of
exponential functions,
data from a contextual situation to
determine if the situation is one in
Properties of operations and which one quantity changes at a
constant rate per unit interval relative
equality (Tables 3 and 4).
to another (linear),
Level IV Students will:
EEF-LE.1-4. Plot points using pictures in first quadrant on a graph using
whole numbers and explain how y increases/decreases as x changes.
Ex. If you go to the store where every item is one dollar, students should state
y = x (the number of items I buy will tell me the cost). Students will then plot
this on the graph.
Ex. If I get two apples for every orange I buy, students should state that y = 2x,
or for every orange I buy (x), I will get two apples (y), therefore x times two
tells me the number of apples each time. Students should then plot this on the
graph.
Accurately determine relationships of
data from a contextual situation to
determine if the situation is one in
which one quantity grows or decays by
a constant percent rate per unit interval
relative to another (exponential).
Level III Student will:
EEF-LE.1-4. Model a simple linear function such as y = mx to show functions
grow by equal factors over equal intervals.
Ex. Determine a simple relationship of y to x by looking at the first quadrant of
a graph.
Linear functions have a constant value Ex. Identify the cost per item on a simple graph where every item in the store
cost the same amount and state the relationship between x and y.
added per unit interval, and
exponential functions have a constant Ex. Look at a graph that shows a constant ratio of boys to girls and state the
relationship between x and y.
value multiplied per unit interval,
Distinguishing key features of and
categorizing functions facilitates
mathematical modeling and aids in
problem resolution.
Level II Students will:
EEF-LE.1-4. Identify a specific data point in the first quadrant and explain the
meaning behind it.
Ex. Given data points in the first quadrant, identify the named point and state
the two pieces of information that one dot provides.
Ex. When given a simple graph that shows the total cost of items purchased at
a store where every item is $1, tell the cost of four items, the cost of two items,
etc.
Level I Students will:
EEF-LE.1-4. Interpret major ideas of a graph with linear functions.
Ex. When shown two lines on a graph, tell which one is rising faster.
Ex. When shown a graph of distance driven and gas left in tank, explain that
the further one drives the less gas one has left.
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Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code: F.LE.1c Distinguish between situations that can be modeled with linear functions and with exponential functions. c. Recognize situations in which a quantity
grows or decays by a constant percent rate per unit interval relative to another.
Domain: Linear, Quadratic, and Exponential
Models *(*Modeling Domain)
Cluster: Construct and compare linear, quadratic, and exponential models and solve problems
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Use appropriate tools Attend to precision.
strategically.
Look for and make
use of structure.
Recognize situations in which a quantity
grows or decays by a constant percent rate
per unit (equal factors) interval relative to
another to solve mathematical and real-world
problems.
Make sense of
problems and
persevere in solving
them.
Reason abstractly and
quantitatively.
Construct viable
Model with
arguments and
mathematics.
critique the
reasoning of others.
Look for and
express regularity
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a linear or exponential
function,
Create a sequence from the
functions and examine the results
to demonstrate that linear
functions grow by equal
differences, and exponential
functions grow by equal factors
over equal intervals,
Use slope-intercept form of a
linear function and the general
definition of exponential
functions to justify through
algebraic rearrangements that
linear functions grow by equal
differences, and exponential
functions grow by equal factors
over equal intervals.
Given a contextual situation
modeled by functions,
Determine if the change in the
output per unit interval is a
constant being added or
multiplied to a previous output,
and appropriately label the
function as linear, exponential, or
neither.
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Linear functions
Students know:
Students understand/are able to:
Exponential functions
Key components of linear and Accurately determine relationships of
exponential functions,
data from a contextual situation to
determine if the situation is one in
Properties of operations and which one quantity changes at a
constant rate per unit interval relative
equality (Tables 3 and 4).
to another (linear),
Level IV Students will:
EEF-LE.1-4. Plot points using pictures in first quadrant on a graph using whole
numbers and explain how y increases/decreases as x changes.
Ex. If you go to the store where every item is one dollar, students should state y
= x (the number of items I buy will tell me the cost). Students will then plot this
on the graph.
Ex. If I get two apples for every orange I buy, students should state that y = 2x,
or for every orange I buy (x), I will get two apples (y), therefore x times two
tells me the number of apples each time. Students should then plot this on the
graph.
Accurately determine relationships of
data from a contextual situation to
determine if the situation is one in
which one quantity grows or decays by
a constant percent rate per unit interval
relative to another (exponential).
Level III Student will:
EEF-LE.1-4. Model a simple linear function such as y = mx to show functions
grow by equal factors over equal intervals.
Ex. Determine a simple relationship of y to x by looking at the first quadrant of
a graph.
Linear functions have a constant value Ex. Identify the cost per item on a simple graph where every item in the store
cost the same amount and state the relationship between x and y.
added per unit interval, and
exponential functions have a constant Ex. Look at a graph that shows a constant ratio of boys to girls and state the
relationship between x and y.
value multiplied per unit interval,
Distinguishing key features of and
categorizing functions facilitates
mathematical modeling and aids in
problem resolution.
Level II Students will:
EEF-LE.1-4. Identify a specific data point in the first quadrant and explain the
meaning behind it.
Ex. Given data points in the first quadrant, identify the named point and state
the two pieces of information that one dot provides.
Ex. When given a simple graph that shows the total cost of items purchased at a
store where every item is $1, tell the cost of four items, the cost of two items,
etc.
Level I Students will:
EEF-LE.1-4. Interpret major ideas of a graph with linear functions.
Ex. When shown two lines on a graph, tell which one is rising faster.
Ex. When shown a graph of distance driven and gas left in tank, explain that the
further one drives the less gas one has left.
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Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code: 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).
Domain: Linear, Quadratic, and
Exponential Models *(Modeling Domain)
Cluster: Construct and compare linear, quadratic, and exponential models and solve problems
Quarter 1:
Quarter 2:
Recognize arithmetic sequences can be expressed
as linear functions.
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and quantitatively.
Quarter 3:
Quarter 4:
Recognize geometric sequences can be
expressed as exponential functions.
Construct linear functions, including arithmetic
sequences, given a graph, a description of a
relationship, or two input-output pairs (include
reading these from a table).
Construct exponential functions,
including geometric sequences, given a
graph, a description of a relationship, or
two input-output pairs (include reading
these from a table).
Determine when a graph, a description of a
relationship, or two input-output pairs (include
reading these from a table) represents a linear
function in order to solve problems
Determine when a graph, a description of
a relationship, or two input-output pairs
(include reading these from a table)
represents a linear or exponential
function in order to solve problems
Construct viable
arguments and
critique the reasoning
of others.
Model with
mathematics.
Use appropriate tools
strategically.
Attend to
precision.
Look for and
make use of
structure.
Look for and
express regularity
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a contextual situation
shown by a graph, a description
of a relationship, or two inputoutput pairs,
Create a linear or exponential
function that models the
situation,
Create arithmetic and geometric
sequences from the given
situation,
Justify the equality of the
sequences and the functions
mathematically and in terms of
the original sequence.
Vocabulary
Knowledge
Skills
Arithmetic and
geometric sequences
Students know:
Students understand/are able to:
That linear functions grow by
equal differences over equal
intervals, and that exponential
functions grow by equal
factors over equal intervals,
Properties of arithmetic and
geometric sequences.
Instructional Achievement Level Descriptors
Level IV Students will:
EEF-LE.1-4. Plot points using pictures in first quadrant on a graph using
whole numbers and explain how y increases/decreases as x changes.
Accurately recognize relationships
within data and use that relationship Ex. If you go to the store where every item is one dollar, students should state
y = x (the number of items I buy will tell me the cost). Students will then plot
to create a linear or exponential
this on the graph.
function to model the data of a
Ex. If I get two apples for every orange I buy, students should state that y = 2x,
contextual situation.
or for every orange I buy (x), I will get two apples (y), therefore x times two
Linear and exponential functions may tells me the number of apples each time. Students should then plot this on the
graph.
be used to model data that is
presented as a graph, a description of
Level III Student will:
a relationship, or two input-output
EEF-LE.1-4. Model a simple linear function such as y = mx to show functions
pairs (include reading these from a
grow by equal factors over equal intervals.
table),
Ex. Determine a simple relationship of y to x by looking at the first quadrant of
a graph.
Linear functions have a constant
Ex. Identify the cost per item on a simple graph where every item in the store
value added per unit interval, and
exponential functions have a constant cost the same amount and state the relationship between x and y.
Ex. Look at a graph that shows a constant ratio of boys to girls and state the
value multiplied per unit interval.
relationship between x and y.
Level II Students will:
EEF-LE.1-4. Identify a specific data point in the first quadrant and explain the
meaning behind it.
Ex. Given data points in the first quadrant, identify the named point and state
the two pieces of information that one dot provides.
Ex. When given a simple graph that shows the total cost of items purchased at
a store where every item is $1, tell the cost of four items, the cost of two items,
etc.
Level I Students will:
EEF-LE.1-4. Interpret major ideas of a graph with linear functions.
Ex. When shown two lines on a graph, tell which one is rising faster.
Ex. When shown a graph of distance driven and gas left in tank, explain that
the further one drives the less gas one has left.
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Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code: 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.
Domain: Linear, Quadratic, and
Exponential Models *(Modeling Domain)
Cluster: Construct and compare linear, quadratic, and exponential models and solve problems
Quarter 1:
Quarter 2:
Informally define the concept of “end
behavior”.
Quarter 3:
Quarter 4:
Compare tables and graphs of linear and
exponential functions to observe that a
quantity increasing exponentially exceeds all
others to solve mathematical and real-world
problems.
Compare tables and graphs of linear
functions to observe that a quantity
increasing exponentially exceeds all others
to solve mathematical and real-world
problems.
Note from Appendix A: Limit to
comparisons between linear and
exponential models.
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and quantitatively.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate tools
strategically.
Attend to precision.
Look for and make
use of structure.
Look for and
express regularity
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a quantity increasing
exponentially and a quantity
increasing as a polynomial
function (e.g., linearly,
quadratically),
Construct graphs and tables that
demonstrate the exponential
function will exceed the
polynomial function at some
point,
Present a convincing argument
that this must be true for all
polynomial functions.
Vocabulary
Knowledge
Skills
Increasing
exponentially
Students know:
Students understand/are able to:
Techniques to graph and
create tables for exponential
and polynomial functions.
Instructional Achievement Level Descriptors
Level IV Students will:
EEF-LE.1-4. Plot points using pictures in first quadrant on a graph using whole
numbers and explain how y increases/decreases as x changes.
Accurately create graphs and tables
Ex. If you go to the store where every item is one dollar, students should state y
for exponential and polynomial
= x (the number of items I buy will tell me the cost). Students will then plot this
functions,
on the graph.
Use the graphs and tables to present a Ex. If I get two apples for every orange I buy, students should state that y = 2x,
or for every orange I buy (x), I will get two apples (y), therefore x times two tells
convincing argument that the
me the number of apples each time. Students should then plot this on the graph.
exponential function eventually
exceeds the polynomial function.
Level III Student will:
Exponential functions grow at a faster EEF-LE.1-4. Model a simple linear function such as y = mx to show functions
grow by equal factors over equal intervals.
rate than polynomial functions after
Ex. Determine a simple relationship of y to x by looking at the first quadrant of a
some point in their domain.
graph.
Ex. Identify the cost per item on a simple graph where every item in the store
cost the same amount and state the relationship between x and y.
Ex. Look at a graph that shows a constant ratio of boys to girls and state the
relationship between x and y.
Level II Students will:
EEF-LE.1-4. Identify a specific data point in the first quadrant and explain the
meaning behind it.
Ex. Given data points in the first quadrant, identify the named point and state the
two pieces of information that one dot provides.
Ex. When given a simple graph that shows the total cost of items purchased at a
store where every item is $1, tell the cost of four items, the cost of two items,
etc.
Level I Students will:
EEF-LE.1-4. Interpret major ideas of a graph with linear functions.
Ex. When shown two lines on a graph, tell which one is rising faster.
Ex. When shown a graph of distance driven and gas left in tank, explain that the
further one drives the less gas one has left.
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Grade Level/ Course (HS): Algebra 1 Unit 2
Standard with code F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
Domain: Linear, Quadratic, and
Exponential Models *(Modeling Domain)
Cluster: Interpret expressions for functions in terms of the situation they model
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Recognize the parameters in a linear function
including: vertical and horizontal shifts, vertical
and horizontal dilations.
Recognize the parameters in a linear or
exponential function including: vertical
and horizontal shifts, vertical and
horizontal dilations.
Recognize rates of change and intercepts as
“parameters” in linear functions.
Recognize rates of change and intercepts
as “parameters” in linear or exponential
functions.
Interpret the parameters in a linear function in
terms of a context.
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and quantitatively.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
Use appropriate tools
strategically.
Attend to
precision.
Interpret the parameters in a linear or
exponential function in terms of a context.
Look for and
Look for and
make use of
express regularity
structure.
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a contextual situation that
may be modeled by a linear or
exponential function,
Create a function that models the
situation,
Define and justify the parameters
(all constants used to define the
function) in terms of the original
context.
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Parameters
Students know:
Students understand/are able to:
EEF-LE.5. N/A
Key components of linear and
exponential functions.
Communicate the meaning of defining
values (parameters and variables) in
functions used to model contextual
situations in terms of the original
context.
Sense making in mathematics requires
that meaning is attached to every value
in a mathematical expression.
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Grade Level/ Course: Algebra 1 Unit 3
Standard with code: S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots). (Statistics and Probability is a Modeling Conceptual Category.)
Domain: Interpreting Categorical and
Quantitative Data
Cluster: Summarize, represent, and interpret data on a single count or measurement variable
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Represent data with plots on the real number line
using various display types by creating dot plots,
histograms and box plots.
Make sense of
problems and
persevere in solving
them.
Reason abstractly and
quantitatively.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate tools
strategically.
Attend to precision.
Look for and
make use of
structure.
Look for and
express regularity
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given numerical data in any
form (e.g., all real numbers),
Vocabulary
Knowledge
Skills
Dot plots
Students know:
Students understand/are able to:
Organize and display the data
using plots on a real number
line, including dot plots,
histograms, and box plots.
Box plots
Histograms
Instructional Achievement Level Descriptors
Level IV Students will:
EES-ID.1-2. Collect and organize data in simple graphs and use findings to
Techniques for constructing Choose from among data display (dot draw conclusions from the data.
dot plots, histograms, and box plots, histograms, box plots) to convey Ex. Ask 10 people how many hours of TV they watch a day. Put the findings
into a graph and tell which person watches the most and least TV.
plots from a set of data.
significant features of data,
Ex. Collect data on a given topic and tell what conclusions they draw from the
data, such as most common weather in two cities, cheapest price of jeans, etc.
Accurately construct dot plots,
histograms, and box plots.
Level III Students will:
EES-ID.1-2. Given data, construct a simple graph (table, line, pie, bar, or
Sets of data can be organized and
displayed in a variety of ways each of picture) and answer questions about the data.
which provides unique perspectives of Ex. Given data about the cost of jeans at three stores, place the information on a
graph (table, line, pie, bar, or picture) and answer questions about the graph.
the data set,
Ex. Read data from a given graph showing the weather for one week and
Data displays help in conceptualizing determine how many days it was rainy.
Ex. Given data from student surveys (e.g. favorite sport, subject, book)
ideas and in solving problems.
presented on a bar or pie graph and answer questions about the findings
(most/least).
Ex. Interpret weather data (e.g. temperature changes over time) presented in a
line graph.
Level II Students will:
EES-ID.1-2. Given a graph, answer simple questions.
Ex. Identify the highest and lowest points on a graph (costs the most).
Ex. Tell what the simple graph represents (graph about the weather, cell phone
plans, or gas prices).
Ex. Read data from a given graph showing the weather for one week to tell how
many days was it rainy.
Level I Students will:
EES-ID.1-2. Identify any part of a simple graph.
Ex. Point to and identify part of simple graph, (such as the bar, line, title, labels
on the graph).
Ex. Point or indicate to answer, “Which is the tallest/highest bar?”
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Grade Level/ Course: Algebra 1 Unit 3
Standard with code: 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.
(Statistics and Probability is a Modeling Conceptual Category.)
Domain: Interpreting Categorical and
Quantitative Data
Cluster: Summarize, represent, and interpret data on a single count or measurement variable
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Choose the appropriate measure for center
(mean, median) and spread (interquartile range,
standard deviation) based on the shape of a data
distribution.
Use appropriate statistics for center and spread
to compare two or more data sets.
Make sense of problems Reason abstractly and
and persevere in solving quantitatively.
them.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
From Appendix A: 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.
Use appropriate tools
Attend to precision. Look for and
strategically.
make use of
structure.
Look for and
express regularity
in repeated
reasoning.
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Evidence of Student
Vocabulary
Attainment/Assessment
Students:
Center
Given two or more different data
sets,
Median
Compare the center (median,
mean) and the spread
(interquartile range, standard
deviation) of the data sets to
describe differences and
similarities of the data sets.
Knowledge
Skills
Students know:
Students understand/are able to:
Techniques to calculate the
center and spread of data sets,
Mean
Spread
Interquartile range
Standard deviation
Methods to compare data sets
based on measures of center
(median, mean) and spread
(interquartile range and
standard deviation) of the data
sets.
Instructional Achievement Level Descriptors
Level IV Students will:
EES-ID.1-2. Collect and organize data in simple graphs and use findings to
draw conclusions from the data.
Accurately find the center (median
and mean) and spread (interquartile Ex. Ask 10 people how many hours of TV they watch a day. Put the findings
range and standard deviation) of data into a graph and tell which person watches the most and least TV.
Ex. Collect data on a given topic and tell what conclusions they draw from the
sets,
data, such as most common weather in two cities, cheapest price of jeans, etc.
Present viable arguments and
critique arguments of others from the Level III Students will:
comparison of the center and spread EES-ID.1-2. Given data, construct a simple graph (table, line, pie, bar, or
picture) and answer questions about the data.
of multiple data sets.
Ex. Given data about the cost of jeans at three stores, place the information on a
Multiple data sets can be compared graph (table, line, pie, bar, or picture) and answer questions about the graph.
Ex. Read data from a given graph showing the weather for one week and
by making observations about the
determine how many days it was rainy.
center and spread of the data,
Ex. Given data from student surveys (e.g. favorite sport, subject, book)
presented on a bar or pie graph and answer questions about the findings
The center and spread of multiple
(most/least).
data sets are used to justify
Ex. Interpret weather data (e.g. temperature changes over time) presented in a
comparisons of the data.
line graph.
Level II Students will:
EES-ID.1-2. Given a graph, answer simple questions.
Ex. Identify the highest and lowest points on a graph (costs the most).
Ex. Tell what the simple graph represents (graph about the weather, cell phone
plans, or gas prices).
Ex. Read data from a given graph showing the weather for one week to tell how
many days was it rainy.
Level I Students will:
EES-ID.1-2. Identify any part of a simple graph.
Ex. Point to and identify part of simple graph, (such as the bar, line, title, labels
on the graph).
Ex. Point or indicate to answer, “Which is the tallest/highest bar?”
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Grade Level/ Course: Algebra 1 Unit 3
Standard with code: S.ID.3 Interpret differences in shape, center and spread in the context of data sets, accounting for possible effects of extreme data points (outliers).
(Statistics
and Probability is a Modeling Conceptual Category.)
Domain: Interpreting Categorical and
Quantitative Data
Cluster: Summarize, represent, and interpret data on a single count or measurement variable
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Define “the context of data sets” as meaning
the specific nature of the attributes under
investigation.
Interpret differences in shape, center and
spread in the context of data sets.
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and quantitatively.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
Describe the possible effects the presence of
outliers in a set of data can have on shape,
center, and spread in the context of the data
sets.
Use appropriate tools Attend to precision.
strategically.
Look for and make
use of structure.
Look for and
express regularity
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given multiple data sets,
Recognize and explain the
differences in shape, center, and
spread, including effects of
outliers.
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Outliers
Students know:
Students understand/are able to:
Center
Techniques to calculate the
center and spread of data sets,
Accurately identify differences in
shape, center, and spread when
comparing two or more data sets,
Level IV Students will:
EES-ID.3. Extend a graph or chart to make a prediction.
Ex. If the weatherman says there is a 60% chance of rain, should you wear a
rain coat?
Ex. Show a graph, predict which direction the line will continue and answer
predictive questions.
Ex. Using a graph, estimate a future point when the trend of the line is not
extremely clear.
Shape
Spread
Methods to compare attributes
(e.g. shape, median, mean,
Accurately identify outliers,
interquartile range, and
standard deviation) of the data Explain, with justification, why there
sets,
are differences in the shape, center,
and spread of data sets.
Methods to identify outliers.
Differences in the shape, center, and
spread of data sets can result from
various causes, including outliers and
clustering.
Level III Students will:
EES-ID.3. Indicate general trends on a graph or chart.
Ex. Which chart shows an increase? A chart with an upward slope or a chart
with a downward slope.
Ex. Which chart shows a decrease? A chart with an upward slope or a chart
with a downward slope.
Ex. Using a graph, estimate a future point when the trend of the line is clear.
Level II Students will:
EES-ID.3. Demonstrate increase and decrease over time.
Ex. Is this point more or less than this point?
Ex. Is this line (slope) increasing or decreasing?
Ex. Collect data that has a trend possibility (e.g., growing plant, collecting
money).
Ex. Ordinate piles of money, items to show increase/decrease.
Ex. When shown two graphs, determine which shows increase and which shows
decrease.
Level I Students will:
EES-ID.3. Determine categories needed on a graph.
Ex. We are charting plant growth. Should I put the length of the monkey’s tail
on the graph?
Ex. Describe sample space – Are we looking at oranges or apples?
Ex. We are counting apples. Do shoes belong on this graph?
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Grade Level/ Course: Algebra 1 Unit 3
Standard with code: 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.(Statistics and Probability is a Modeling Conceptual Category.)
Domain: Interpreting Categorical and
Quantitative Data
Cluster: Summarize, represent, and interpret data on two categorical and quantitative variables
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Recognize the differences between joint,
marginal and conditional relative
frequencies.
Calculate relative frequencies including
joint, marginal and conditional relative
frequencies.
Summarize categorical data for two
categories in two-way frequency tables.
Interpret relative frequencies in the context
of the data.
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and quantitatively.
Construct viable
arguments and
critique the reasoning
of others.
Model with mathematics.
Recognize possible associations and trends
in the data.
Use appropriate
Attend to precision. Look for and
tools strategically.
make use of
structure.
Look for and
express regularity
in repeated
reasoning.
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Version 3 2015-2016
Evidence of Student
Attainment/Assessment
Students:
Given categorical data for two
categories,
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Categorical data
Students know:
Students understand/are able to:
EES-ID.5. N/A (See EEF-IF.1. and EEA-REI.6-7)
Two-way frequency
Tables
Characteristics of a two-way
frequency table,
Accurately construct frequency tables,
Relative frequency
Methods for converting
frequency tables to relative
frequency tables,
Create two-way frequency tables,
Find relative frequencies using
ratios,
Joint frequency
Recognize and justify possible
Marginal frequency
relationships and patterns in the
data by examining the joint,
Conditional relative
marginal, and conditional relative frequency
frequencies.
Accurately construct relative
frequency tables,
Accurately find the joint, marginal,
and conditional relative frequencies,
That the sum of the frequencies
in a row or a column gives the Recognize and explain possible
marginal frequency,
associations and trends in the data.
Techniques for finding
conditional relative frequency,
Two-way frequency tables may be
used to represent categorical data,
Techniques for finding the joint Relative frequency tables show the
frequency in tables.
ratios of the categorical data in terms
of joint, marginal, and conditional
relative frequencies,
Two-way frequency or relative
frequency tables may be used to aid in
recognizing associations and trends in
the data.
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Grade Level/ Course: Algebra 1 Unit 3
Standard with code: S.ID.6a Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions
fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.
(Statistics and
Probability is a Modeling Conceptual Category.)
Domain: Interpreting Categorical and
Quantitative Data
Cluster: Summarize, represent, and interpret data on two categorical and quantitative variables
Quarter 1:
Quarter 2:
Quarter 3:
Represent data on a scatter plot (2 quantitative variables).
Quarter 4:
Fit a given function class (e.g. linear, exponential) to
data.
Using given scatter plot data represented on the
coordinate plane, informally describe how the two
quantitative variables are related.
Determine which function best models scatter plot data
represented on the coordinate plane, and describe how the
two quantitative variables are related.
Use functions fitted to data to solve problems in the
context of the data.
From Appendix A: 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.
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and
quantitatively.
Construct viable
arguments and
critique the reasoning
of others.
Model with
mathematics.
Use appropriate
tools strategically.
Attend to precision.
Look for and
make use of
structure.
Look for and
express regularity in
repeated reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a data set of two
quantitative variables,
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Quantitative variables
Students know:
Students understand/are able to:
EES-ID.6. N/A
Scatter plot
Techniques for creating a
scatter plot,
Accurately create a scatter plot of data,
Create a scatter plot (with and
without technology),
Residuals
Correctly choose a function to fit the
Techniques for fitting various scatter plot,
functions (linear, quadratic,
exponential) to data,
Make reasonable assessments on the fit
of the function to the data by
Methods for using residuals
examining residuals,
to judge the closeness of the
fit of the function to the
Accurately fit a linear function to data
original data.
when there is evidence of a linear
association.
Create a function which best fits
the data (linear, quadratic, and
exponential models),
Compare the graphs of the scatter
plot and function to see the fit to
the original data,
Fit a linear function to the data if
the scatter plot indicates a linear
association.
Functions are used to create equations
representative of ordered pairs of data,
Residuals may be examined to analyze
how well a function fits the data,
When a linear association is suggested,
a linear function can be fit to the
scatter plot to aid in modeling the
relationship.
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Grade Level/ Course: Algebra 1 Unit 3
Standard with code: S.ID.6b Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. b. Informally assess the fit of a function by
plotting and analyzing residuals.
(Statistics and Probability is a Modeling Conceptual Category.)
Domain: Interpreting Categorical and
Quantitative Data
Cluster: Summarize, represent, and interpret data on two categorical and quantitative variables.
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Represent the residuals from a function and the
data set it models numerically and graphically.
Informally assess the fit of a function by
analyzing residuals from the residual plot.
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and quantitatively.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
From Appendix A: 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. Focus on linear models,
however, this standard could also preview
quadratic functions in Unit 5 of Algebra I.
Use appropriate
Attend to precision.
tools strategically.
Look for and make
use of structure.
Look for and
express regularity
in repeated
reasoning.
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Evidence of Student
Vocabulary
Attainment/Assessment
Students:
Given a contextual situation that
yields a data set of ordered pairs
that suggests a linear relationship,
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
EES-ID.7. N/A (See EEF.IF.4-6)
Techniques for creating a
scatter plot,
Accurately create a scatter plot of data
Fit a linear function to the data,
Techniques for fitting a linear
function to a scatter plot,
Determine the slope and intercept
of that function,
Interpret the slope and intercept
of the linear function in the
context of the data.
Methods to find the slope and
intercept of a linear function.
Accurately fit linear functions to
scatter plots,
Correctly find the slope and intercept
of linear functions,
Justify and explain the relevant
connections slope and intercept of the
linear function to the data.
Linear functions are used to model
data that have a relationship that
closely resembles a linear
relationship,
The slope and intercept of a linear
function may be interpreted as the
rate of change and the zero point
(starting point).
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Grade Level/ Course: Algebra 1 Unit 3
Standard with code: S.ID.6c Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. c. Fit a linear function for a scatter plot that
suggests a linear association.
(Statistics and Probability is a Modeling Conceptual Category.)
Domain: Interpreting Categorical and
Quantitative Data
Cluster: Summarize, represent, and interpret data on two categorical and quantitative variables.
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Fit a linear function for a scatter plot that
suggests a linear association.
From Appendix A: By the end of Middle School,
students were creating scatter plots and
recognizing linear trends in data. This unit
builds upon that prior experience, providing
students with more formal means of assessing
how a model fits data.
Make sense of
problems and
persevere in solving
them.
Reason abstractly Construct viable
and quantitatively. arguments and
critique the
reasoning of others.
Model with
mathematics.
Use appropriate
tools strategically.
Attend to precision.
Look for and
make use of
structure.
Look for and
express regularity in
repeated reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a data set of two
quantitative variables,
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Quantitative variables
Students know:
Students understand/are able to:
EES-ID.6. N/A
Scatter plot
Techniques for creating a
scatter plot,
Accurately create a scatter plot of data,
Create a scatter plot (with and
without technology),
Residuals
Correctly choose a function to fit the
Techniques for fitting various scatter plot,
functions (linear, quadratic,
exponential) to data,
Make reasonable assessments on the fit
of the function to the data by
Methods for using residuals
examining residuals,
to judge the closeness of the
fit of the function to the
Accurately fit a linear function to data
original data.
when there is evidence of a linear
association.
Create a function which best fits
the data (linear, quadratic, and
exponential models),
Compare the graphs of the scatter
plot and function to see the fit to
the original data,
Fit a linear function to the data if
the scatter plot indicates a linear
association.
Functions are used to create equations
representative of ordered pairs of data,
Residuals may be examined to analyze
how well a function fits the data,
When a linear association is suggested,
a linear function can be fit to the
scatter plot to aid in modeling the
relationship.
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Grade Level/ Course: Algebra 1 Unit 3
Standard with code: S.ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit. (Statistics and Probability is a Modeling Conceptual Category.)
Domain: Interpreting Categorical and
Quantitative Data
Cluster: Interpret linear models.
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Compute (using technology) the correlation
coefficient of a linear fit.
Define the correlation coefficient.
Interpret the correlation coefficient of a linear fit
as a measure of how well the data fit the
relationship.
Make sense of
problems and
persevere in solving
them.
Reason abstractly and
quantitatively.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
From Appendix A: 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 cause-and-effect relationship arises in S.ID.9.
Use appropriate tools
Attend to precision. Look for and
strategically.
make use of
structure.
Look for and
express regularity
in repeated
reasoning.
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Evidence of Student
Vocabulary
Attainment/Assessment
Students:
Correlation coefficient
Given a contextual situation that
yields a data set of ordered pairs Linear fit
that suggests a linear
relationship,
Find the correlation coefficient
of a linear fit using technology,
Communicate the relationship of
the correlation coefficient to the
data.
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
EES-ID.8-9. N/A
The relationship among the
shape of the scatter plot, the
value of the correlation
coefficient, and the strength of
the linear relationship in the
data.
Use technology accurately to find the
correlation coefficient,
Justify and communicate conclusions
about the relationship of the data
based upon the correlation coefficient.
Correlation coefficients are used to
measure the strength of the linear
relationship in a set of data,
Technology aids in finding the
correlation coefficient of a linear fit.
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Grade Level/ Course: Algebra 1 Unit 3
Standard with code: S.ID.9 Distinguish between correlation and causation. (Statistics and Probability is a Modeling Conceptual Category.)
Domain: Interpreting Categorical and
Quantitative Data
Cluster: Interpret Linear Models
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Define positive, negative, and no correlation and
explain why correlation does not imply causation.
Define causation.
Distinguish between correlation and causation.
Make sense of
problems and
persevere in solving
them.
Reason
abstractly and
quantitatively.
Construct viable
Model with
arguments and
mathematics.
critique the
reasoning of others.
From Appendix A: 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 cause-and-effect
relationship arises in S.ID.9.
Use appropriate tools
Attend to precision.
strategically.
Look for and make
use of structure.
Look for and
express regularity
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given situations where two
variables are correlated,
Explain why the correlation does
not mean that changes in one
variable cause the changes in the
other variable.
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Correlation
Students know:
Students understand/are able to:
EES-ID.8-9. N/A
Causation
Interpretations of correlation
and causation.
Justify that two variables that show a
strong correlation, do not necessarily
show that one is the cause of the
other.
Correlation may be an indication of
causation, but may also be an
indication of influence of other
variables or coincidence.
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Grade Level/ Course (HS): Algebra 1 Unit 4
Standard with code: A.SSE.1a Interpret expressions that represent a quantity in terms of its context.* (*Modeling standard) a. Interpret parts of an expression, such as terms,
factors, and coefficients.
Domain: Seeing Structure in Expressions
Cluster: Interpret the structure of expressions
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
For expressions that represent a
contextual quantity, define and recognize
parts of an expression, such as terms,
factors, and coefficients.
For expressions that represent a
contextual quantity, interpret parts of an
expression, such as terms, factors, and
coefficients in terms of the context.
Make sense of
problems and
persevere in solving
them.
Reason abstractly and
quantitatively.
Construct viable
arguments and
critique the reasoning
of others.
Model with
mathematics.
Note from Appendix A: Extend to
quadratic and exponential expressions
Use appropriate
Attend to
tools strategically. precision.
Look for and
make use of
structure.
Look for and
express regularity
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a contextual situation and
an expression that does model it,
Vocabulary
Knowledge
Skills
Terms
Students know:
Students understand/are able to:
Factors
Connect each part of the
expression to the corresponding
piece of the situation,
Coefficients
Interpretations of parts of
algebraic expressions such as
terms, factors, and
coefficients.
Interpret parts of the expression
such as terms, factors, and
coefficients.
Instructional Achievement Level Descriptors
Level IV Students will:
EEA-SSE.1. Write or match an algebraic expression for a given word
expression involving more than one operation.
Produce mathematical expressions
Ex. Write an expression to represent the problem, six weeks minus two weeks
that model given contexts,
plus four weeks, to find the total number of weeks you are working?
Ex. How would you represent five dogs plus two cats plus one mouse to find
Provide a context that a given
mathematical expression accurately the total number of animals in a pet store?
Ex. Shown pictures representing two expressions, select the one for two
fits,
drinks, plus three slices of pizza, plus two salads if d represents drinks, s
Explain the reasoning for selecting a represents salad, and p represents pizza?
Ex. Match two dimes, three nickels, and four pennies to an expression when d
particular algebraic expression by
represents dimes, n represents nickels, and p represents pennies.
connecting the quantities in the
expression to the physical situation Ex. Match 2r + 3b + 4y with two red disks, three blue disks, and four yellow
when given colored disks.
that produced them, (e.g., the
formula for the area of a trapezoid
Level III Students will:
can be explained as the average of
the two bases multiplied by height). EEA-SSE.1. Match an algebraic expression involving one operation to
represent a given word expression with an illustration.
Ex. Match the correct algebraic expression to a picture of three boys and two
Physical situations can be
represented by algebraic expressions girls if b represents boys and g represents girls (3b + 4g) when asked, “Which
is the correct way to express three boys and two girls if b represents the
which combine numbers from the
number of boys and g represents the number of girls in the classroom?”
context, variables representing
unknown quantities, and operations Ex. Shown a picture of three hamburgers at $4 each, match an expression to
the picture given two expressions when asked, “Which is the correct way to
indicated by the context,
express the cost of three hamburgers if each hamburger is $4.00? (three
hamburgers x $4).
Different but equivalent algebraic
Ex. Shown two drinks plus three slices of pizza, match an expression to the
expressions can be formed by
picture given two expressions when asked, “Which one shows two drinks plus
approaching the context from a
three slices of pizza if d represents drinks and p represents pizza?”
different perspective.
Ex. Match two dimes and three nickels to an expression where d represents
dimes and n represents nickels.
Ex. Match the expression of 2r + 3b with two red disks and three blue disks
when given an assortment of colored disks.
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Level II Students will:
EEA-SSE.1. Identify the operation used for word expressions as indicated by
an illustration.
Ex. Nancy has 10 balloons. She gives three away to her friend. What operation
(addition or subtraction) do you use to find how many are left as indicated by
an illustration or manipulatives.
Ex. Dave has 10 cookies. His friend gives him two more cookies. What
operation (addition or subtraction) should Dave use to determine how many
cookies he has in all as indicated by an illustration or manipulatives?
Ex. Jose has three times as many baseball cards as his brother. What operation
(addition or multiplication) do you use to find how many baseball cards Jose
has as indicated by an illustration?
Ex. One box has six books in it and another box only has two. How many
books are there together?
Ex. Match words (and, more, take away, times) to (addition, subtraction,
multiplication).
Ex. Given a word problem (June has four marbles and Cho has two marbles.
How many marbles do they have all together?) Student will identify if they
should add or subtract to find the answer as indicated by an illustration.
Ex. When given a pictorial number sentence, complete an algebraic
representation of the pictures by placing/drawing in the correct sign for the
operation.
Level I Students will:
EEA-SSE.1. Recognize the symbol for an operation.
Ex. What does this mean? + means add.
Ex. What does this mean? – means subtract or take away.
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Grade Level/ Course (HS): Algebra 1 Unit 4
Standard with code: A.SSE.1b Interpret expressions that represent a quantity in terms of its context.* (Modeling standard)
b. Interpret complicated expressions by viewing one or
more of their parts as a single entity. For example, interpret
as the product of P and a factor not depending on P.
Domain: Seeing Structure in Expressions
Cluster: Interpret the structure of expressions
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
For expressions that represent a contextual
quantity, define and recognize parts of an
expression, such as terms, factors, and
coefficients.
For expressions that represent a contextual
quantity, interpret complicated expressions,
in terms of the context, by viewing one or
more of their parts as a single entity.
Make sense of
problems and
persevere in solving
them.
Reason abstractly and
quantitatively.
Construct viable
arguments and
critique the reasoning
of others.
Model with
mathematics.
Notes from Appendix A: Extend to
exponential and quadratic expressions,
extend exponents to rational exponents
focusing on those that represent square or
cube roots.
Use appropriate
Attend to precision. Look for and
tools strategically.
make use of
structure.
Look for and
express regularity
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a contextual situation and
an expression that does model it,
Vocabulary
Knowledge
Skills
Terms
Students know:
Students understand/are able to:
Factors
Connect each part of the
expression to the corresponding
piece of the situation,
Coefficients
Interpretations of parts of
algebraic expressions such as
terms, factors, and
coefficients.
Interpret parts of the expression
such as terms, factors, and
coefficients.
Instructional Achievement Level Descriptors
Level IV Students will:
EEA-SSE.1. Write or match an algebraic expression for a given word
expression involving more than one operation.
Produce mathematical expressions
Ex. Write an expression to represent the problem, six weeks minus two weeks
that model given contexts,
plus four weeks, to find the total number of weeks you are working?
Ex. How would you represent five dogs plus two cats plus one mouse to find
Provide a context that a given
the total number of animals in a pet store?
mathematical expression accurately
Ex. Shown pictures representing two expressions, select the one for two
fits,
drinks, plus three slices of pizza, plus two salads if d represents drinks, s
Explain the reasoning for selecting a represents salad, and p represents pizza?
Ex. Match two dimes, three nickels, and four pennies to an expression when d
particular algebraic expression by
represents dimes, n represents nickels, and p represents pennies.
connecting the quantities in the
Ex. Match 2r + 3b + 4y with two red disks, three blue disks, and four yellow
expression to the physical situation
that produced them, (e.g., the formula when given colored disks.
for the area of a trapezoid can be
Level III Students will:
explained as the average of the two
EEA-SSE.1. Match an algebraic expression involving one operation to
bases multiplied by height).
represent a given word expression with an illustration.
Physical situations can be represented Ex. Match the correct algebraic expression to a picture of three boys and two
girls if b represents boys and g represents girls (3b + 4g) when asked, “Which
by algebraic expressions which
is the correct way to express three boys and two girls if b represents the
combine numbers from the context,
number of boys and g represents the number of girls in the classroom?”
variables representing unknown
Ex. Shown a picture of three hamburgers at $4 each, match an expression to
quantities, and operations indicated
the picture given two expressions when asked, “Which is the correct way to
by the context,
express the cost of three hamburgers if each hamburger is $4.00? (three
hamburgers x $4).
Different but equivalent algebraic
Ex. Shown two drinks plus three slices of pizza, match an expression to the
expressions can be formed by
picture given two expressions when asked, “Which one shows two drinks plus
approaching the context from a
three slices of pizza if d represents drinks and p represents pizza?”
different perspective.
Ex. Match two dimes and three nickels to an expression where d represents
dimes and n represents nickels.
Ex. Match the expression of 2r + 3b with two red disks and three blue disks
when given an assortment of colored disks.
Level II Students will:
EEA-SSE.1. Identify the operation used for word expressions as indicated by
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Version 3 2015-2016
an illustration.
Ex. Nancy has 10 balloons. She gives three away to her friend. What operation
(addition or subtraction) do you use to find how many are left as indicated by
an illustration or manipulatives.
Ex. Dave has 10 cookies. His friend gives him two more cookies. What
operation (addition or subtraction) should Dave use to determine how many
cookies he has in all as indicated by an illustration or manipulatives?
Ex. Jose has three times as many baseball cards as his brother. What operation
(addition or multiplication) do you use to find how many baseball cards Jose
has as indicated by an illustration?
Ex. One box has six books in it and another box only has two. How many
books are there together?
Ex. Match words (and, more, take away, times) to (addition, subtraction,
multiplication).
Ex. Given a word problem (June has four marbles and Cho has two marbles.
How many marbles do they have all together?) Student will identify if they
should add or subtract to find the answer as indicated by an illustration.
Ex. When given a pictorial number sentence, complete an algebraic
representation of the pictures by placing/drawing in the correct sign for the
operation.
Level I Students will:
EEA-SSE.1. Recognize the symbol for an operation.
Ex. What does this mean? + means add.
Ex. What does this mean? – means subtract or take away.
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Grade Level/ Course: Algebra 1 Unit 4
4 4
22
22
Standard with code: F A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x – y as (x ) – (y ) , thus recognizing it as a difference of squares
2 2 2
2
that can be factored as (x – y )(x + y ).
Domain: Seeing Structure in Expressions
Cluster: Interpret the structure of expressions.
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Identify ways to rewrite expressions, such
as difference of squares, factoring out a
common monomial, regrouping, etc.
Identify various structures of expressions
(e.g. an exponential monomial multiplied
by a scalar of the same base, difference of
squares in terms other than just x)
Use the structure of an expression to
identify ways to rewrite it.
Classify expressions by structure and
develop strategies to assist in
classification.
Make sense of problems Reason abstractly and
and persevere in solving quantitatively.
them.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate
tools strategically.
Notes from Appendix A: Focus on
quadratics and exponential expressions
Attend to precision. Look for and
Look for and
make use of
express regularity
structure.
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Make sense of algebraic
expressions by identifying
structures within the expression
which allow them to rewrite it in
useful ways.
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
EEA-SSE.2. N/A
Algebraic properties
Use algebraic properties to produce
(including those in Tables 3, 4, equivalent forms of the same
and 5),
expression by recognizing underlying
mathematical structures.
When one form of an algebraic
expression is more useful than Generating simpler, but equivalent,
an equivalent form of that
algebraic expressions facilitates the
same expression.
investigation of more complex
algebraic expressions.
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Grade Level/ Course: Algebra 1 Unit 4
Standard with code: A.SSE.3a Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.*(Modeling
standard)
a. Factor a quadratic expression to reveal the zeros of the function it defines.
Domain: Seeing Structure in
Expressions
Cluster: Write expressions in equivalent forms to solve problems.
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Factor a quadratic expression to produce an
equivalent form of the original expression
Explain the connection between the factored
form of a quadratic expression and the zeros
of the function it defines.
Explain the properties of the quantity
represented by the quadratic expression.
Choose and produce an equivalent form of a
quadratic expression to reveal and explain
properties of the quantity represented by the
original expression.
Make sense of
problems and
persevere in
solving them.
Reason abstractly
and quantitatively.
Construct viable
Model with
arguments and
mathematics.
critique the
reasoning of others.
Use appropriate
tools strategically.
Attend to precision.
Notes from Appendix A: 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.
Look for and make Look for and
use of structure.
express regularity in
repeated reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Quadratic expression
Students know:
Students understand/are able to:
Make sense of algebraic
expressions by identifying
structures within the expression
which allow them to rewrite it in
useful ways that assist in the
solution of given problems,
Zeros
Techniques for generating
equivalent forms of an
algebraic expression
including factoring and
completing the square for
quadratic expressions and
using properties of exponents,
Use algebraic properties including
properties of exponents to produce
equivalent forms of the same
expression by recognizing underlying
mathematical structures,
Level IV Students will:
EEA-SSE.3. Solve one-step equations (multiplication and division of two
digits) with a variable.
Ex. Solve the equation x ÷ 6 = 2 (If I buy two cakes and they were $6 each,
how much money did I spend?).
Ex. Solve the equation $8.00 x ___ = 24 (If a ticket to the movies costs $8,
how many tickets did I buy if I spent 24 dollars?).
Ex. Solve the equation 5 x __ = 45 (If I have five rows of desks and 45 desks
total – how many desks are in each row?).
When one form of an
algebraic expression is more
useful than an equivalent
form of that same expression
to solve a given problem.
Complete the square in quadratic
expressions.
Produce the useful equivalent
forms of expressions, in
particular, factor a quadratic
expression to reveal the zeros of
the function it defines and
complete the square in a quadratic
expression to reveal the
maximum or minimum value of
the function it defines,
Justify their selection of a form
for an expression by explaining
which features of the expression
are revealed by the particular
form and how these features aid
in resolving a problem situation.
Complete the square
Factor quadratic expressions,
Level III Students will:
EEA-SSE.3. Solve simple one-step equations (multiplication and division)
with a variable.
Making connections among equivalent Ex. ___ seats ÷ 8 people = 2 cars
Ex. 2 x N = 6 (box)
expressions reveals the roles of
Ex. 2 apples x ___ people = 16 apples
important mathematical features of a
problem.
Level II Students will:
EEA-SSE.3. Solve basic equations.
Ex. 4 + 3 = ___ (If I have four cups and I get three more, I will have N cups).
Ex. Adds on objects to “make one number into another.” If I have five and I
add two, I get seven.
Ex. Use a number line to show how seven is made of many different
combinations: 5 + 2, 6 + 1, etc.
Ex. Solve picture problems: 2 balloons (picture) + 2 balloons.
Ex. If you have $10 and spend $4, what will your change be?
Ex. Given pictures of monetary value, determine how much money they have
altogether?
Ex. Given money, count how much they have.
Level I Students will:
EEA-SSE.3. Identify quantity and match to the number.
Ex. Match number of objects to correct numerals.
Ex. Count objects (e.g., up to 10) and match the numerals.
Ex. Match five $1 to the number 5.
Ex. Count three tallies and match to the number 3.
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Grade Level/ Course: Algebra 1 Unit 4
Standard with code: A.SSE.3b Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* (Modeling
standard)
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
Domain: Seeing Structure in
Expressions
Cluster: Write expressions in equivalent forms to solve problems.
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Complete the square on a quadratic expression
to produce an equivalent form of an expression.
Explain the connection between the completed
square form of a quadratic expression and the
maximum or minimum value of the function it
defines.
Explain the properties of the quantity
represented by the expression.
Choose and produce an equivalent form of a
quadratic expression to reveal and explain
properties of the quantity represented by the
original expression.
Make sense of
problems and
persevere in
solving them.
Reason
abstractly and
quantitatively.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate tools
strategically.
Attend to precision.
Notes from Appendix A: 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.
Look for and make Look for and express
use of structure.
regularity in repeated
reasoning.
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Evidence of Student
Vocabulary
Attainment/Assessment
Students:
Quadratic expression
Make sense of algebraic
expressions by
identifying structures
within the expression
which allow them to
rewrite it in useful ways
that assist in the solution
of given problems,
Produce the useful
equivalent forms of
expressions, in particular,
factor a quadratic
expression to reveal the
zeros of the function it
defines and complete the
square in a quadratic
expression to reveal the
maximum or minimum
value of the function it
defines,
Justify their selection of a
form for an expression by
explaining which features
of the expression are
revealed by the particular
form and how these
features aid in resolving a
problem situation.
Zeros
Complete the square
Knowledge
Skills
Students know:
Students understand/are able to:
When one form of an algebraic
expression is more useful than
an equivalent form of that same
expression to solve a given
problem.
Complete the square in quadratic
expressions.
Instructional Achievement Level Descriptors
Level IV Students will:
EEA-SSE.3. Solve one-step equations (multiplication and division of two
digits) with a variable.
Techniques for generating
Use algebraic properties including
Ex. Solve the equation x ÷ 6 = 2 (If I buy two cakes and they were $6 each, how
equivalent forms of an
properties of exponents to produce
much money did I spend?).
algebraic expression including equivalent forms of the same
factoring and completing the
expression by recognizing underlying Ex. Solve the equation $8.00 x ___ = 24 (If a ticket to the movies costs $8, how
many tickets did I buy if I spent 24 dollars?).
square for quadratic expressions mathematical structures,
Ex. Solve the equation 5 x __ = 45 (If I have five rows of desks and 45 desks
and using properties of
total – how many desks are in each row?).
exponents,
Factor quadratic expressions,
Making connections among
equivalent expressions reveals the
roles of important mathematical
features of a problem.
Level III Students will:
EEA-SSE.3. Solve simple one-step equations (multiplication and division) with
a variable.
Ex. ___ seats ÷ 8 people = 2 cars
Ex. 2 x N = 6 (box)
Ex. 2 apples x ___ people = 16 apples
Level II Students will:
EEA-SSE.3. Solve basic equations.
Ex. 4 + 3 = ___ (If I have four cups and I get three more, I will have N cups).
Ex. Adds on objects to “make one number into another.” If I have five and I add
two, I get seven.
Ex. Use a number line to show how seven is made of many different
combinations: 5 + 2, 6 + 1, etc.
Ex. Solve picture problems: 2 balloons (picture) + 2 balloons.
Ex. If you have $10 and spend $4, what will your change be?
Ex. Given pictures of monetary value, determine how much money they have
altogether?
Ex. Given money, count how much they have.
Level I Students will:
EEA-SSE.3. Identify quantity and match to the number.
Ex. Match number of objects to correct numerals.
Ex. Count objects (e.g., up to 10) and match the numerals.
Ex. Match five $1 to the number 5.
Ex. Count three tallies and match to the number 3.
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Grade Level/ Course: Algebra 1 Unit 4
Standard with code: A.SSE.3c Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* (*Modeling
t
1/12 12t
t
standard)
c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15 can be rewritten as (1.15
)
≈ 1.01212 to reveal
the approximate equivalent monthly interest rate if the annual rate is 15%.
Domain: Seeing Structure in
Expressions
Cluster: Write expressions in equivalent forms to solve problems.
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Use the properties of exponents to transform
simple expressions for exponential
functions.
Use the properties of exponents to transform
expressions for exponential functions.
Choose and produce an equivalent form of
an exponential expression to reveal and
explain properties of the quantity
represented by the original expression.
Make sense of
Reason abstractly
problems and
and quantitatively.
persevere in solving
them.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate tools
strategically.
Attend to precision.
Explain the properties of the quantity or
quantities represented by the transformed
exponential expression.
Look for and make Look for and
use of structure.
express regularity
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Vocabulary
Knowledge
Skills
Quadratic expression
Students know:
Students understand/are able to:
When one form of an algebraic
expression is more useful than
an equivalent form of that
same expression to solve a
given problem.
Complete the square in quadratic
expressions.
Zeros
Make sense of algebraic
expressions by identifying
structures within the expression Complete the square
which allow them to rewrite it in
useful ways that assist in the
solution of given problems,
Produce the useful equivalent
forms of expressions, in
particular, factor a quadratic
expression to reveal the zeros of
the function it defines and
complete the square in a
quadratic expression to reveal
the maximum or minimum value
of the function it defines,
Justify their selection of a form
for an expression by explaining
which features of the expression
are revealed by the particular
form and how these features aid
in resolving a problem situation.
Instructional Achievement Level Descriptors
Level IV Students will:
EEA-SSE.3. Solve one-step equations (multiplication and division of two
digits) with a variable.
Techniques for generating
Use algebraic properties including
Ex. Solve the equation x ÷ 6 = 2 (If I buy two cakes and they were $6 each,
equivalent forms of an
properties of exponents to produce
how much money did I spend?).
algebraic expression including equivalent forms of the same
factoring and completing the
expression by recognizing underlying Ex. Solve the equation $8.00 x ___ = 24 (If a ticket to the movies costs $8,
how many tickets did I buy if I spent 24 dollars?).
square for quadratic
mathematical structures,
Ex. Solve the equation 5 x __ = 45 (If I have five rows of desks and 45 desks
expressions and using
total – how many desks are in each row?).
properties of exponents,
Factor quadratic expressions,
Making connections among
equivalent expressions reveals the
roles of important mathematical
features of a problem.
Level III Students will:
EEA-SSE.3. Solve simple one-step equations (multiplication and division)
with a variable.
Ex. ___ seats ÷ 8 people = 2 cars
Ex. 2 x N = 6 (box)
Ex. 2 apples x ___ people = 16 apples
Level II Students will:
EEA-SSE.3. Solve basic equations.
Ex. 4 + 3 = ___ (If I have four cups and I get three more, I will have N cups).
Ex. Adds on objects to “make one number into another.” If I have five and I
add two, I get seven.
Ex. Use a number line to show how seven is made of many different
combinations: 5 + 2, 6 + 1, etc.
Ex. Solve picture problems: 2 balloons (picture) + 2 balloons.
Ex. If you have $10 and spend $4, what will your change be?
Ex. Given pictures of monetary value, determine how much money they have
altogether?
Ex. Given money, count how much they have.
Level I Students will:
EEA-SSE.3. Identify quantity and match to the number.
Ex. Match number of objects to correct numerals.
Ex. Count objects (e.g., up to 10) and match the numerals.
Ex. Match five $1 to the number 5.
Ex. Count three tallies and match to the number 3.
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Grade Level/ Course (HS): Algebra 1 Unit 4
Standard with code: 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.
Domain: Arithmetic with Polynomial and
Rational Expressions
Cluster: Perform arithmetic operations on polynomials
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Identify that the sum, difference, or product of
two polynomials will always be a polynomial,
which means that polynomials are closed
under the operations of addition, subtraction,
and multiplication.
Define “closure”.
Apply arithmetic operations of addition,
subtraction, and multiplication to
polynomials.
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and quantitatively.
Construct viable
Model with
arguments and
mathematics.
critique the
reasoning of others.
Use appropriate
tools strategically.
Attend to precision.
Note from Appendix A: Focus on polynomial
expressions that simplify to forms that are
linear or quadratic in a positive integer
power of x.
Look for and make
Look for and
use of structure.
express regularity
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Polynomials
Students know:
Students understand/are able to:
EEA-APR.1 N/A
Corresponding rules of arithmetic of
integers, specifically what it means
for the integers to be closed under
addition, subtraction, and
multiplication, and not under
division,
Communicate the connection between
the rules for arithmetic on integers
and the corresponding rules for
arithmetic on polynomials,
Use the repeated reasoning from Closure
prior knowledge of properties of
arithmetic on integers to
progress consistently to rules for
arithmetic on polynomials,
Accurately perform
combinations of operations on
various polynomials.
Procedures for performing addition,
subtraction, and multiplication on
polynomials.
Accurately perform combinations of
operations on various polynomials.
There is an operational connection
between the arithmetic on integers
and the arithmetic on polynomials.
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Grade Level/ Course (high school): Algebra 1 Unit 1
Standard with code: 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.
Domain: Creating Equations*
(*Modeling Domain)
Cluster: Create equations that describe numbers or relationships
Quarter 1:
Quarter 2:
Solve linear and exponential equations in one variable.
Quarter 3:
Solve linear and exponential equations in one variable.
Solve inequalities in one variable.
Solve inequalities in one variable.
Describe the relationships between the quantities in the
problem (for example, how the quantities are changing
or growing with respect to each other); express these
relationships using mathematical operations to create an
appropriate equation or inequality to solve.
Describe the relationships between the quantities in the problem
(for example, how the quantities are changing or growing with
respect to each other); express these relationships using
mathematical operations to create an appropriate equation or
inequality to solve.
Create equations (linear and exponential) and
inequalities in one variable and use them to solve
problems.
Create equations (linear and exponential) and inequalities in one
variable and use them to solve problems.
Create equations and inequalities in one variable to
model real-world situations.
Compare and contrast problems that can be solved by
different types of equations (linear & exponential).
Note from Appendix A: Limit to linear and exponential
equations, and, in the case of exponential equations,
limit to situations requiring evaluation of exponential
functions at integer inputs.
Make sense of
problems and
persevere in solving
them.
Reason abstractly Construct viable
and
arguments and
quantitatively.
critique the
reasoning of
others.
Model with
mathematics.
Quarter 4:
Create equations and inequalities in one variable to model realworld situations.
Compare and contrast problems that can be solved by different
types of equations (linear & exponential).
Note from Appendix A: Limit to linear and exponential
equations, and, in the case of exponential equations, limit to
situations requiring evaluation of exponential functions at
integer inputs.
Use appropriate tools
strategically.
Attend to precision.
Look for and
make use of
structure.
Look for and
express regularity in
repeated reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a contextual situation that
may include linear, quadratic,
exponential, or rational functional
relationships in one variable,
Model the relationship with
equations or inequalities and solve
the problem presented in the
contextual situation for the given
variable.
(Please Note: This standard must
be taught in conjunction with the
standard that follows).
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Student know:
Students understand/are able to:
When the situation presented
in a contextual problem is
most accurately modeled by a
linear, quadratic, exponential,
or rational functional
relationship.
Write equations or inequalities in one
variable that accurately model
contextual situations.
Level IV Students will:
EEA-CED.1. Solve an algebraic expression with more than one variable.
Ex. If I have two bills, one of them is a $5 and one of them is unknown. What is
the value of the unknown bill if I have $10 total?
Ex. If I have some money in my pocket and some money in the other pocket and
I still need $3 more to buy the bird that cost $10, how much money is in my
pockets?
Features of a contextual problem can
be used to create a mathematical model
Level III Students will:
for that problem.
EEA-SSE.3. Solve an algebraic expression using subtraction.
Ex. If I need $10 and I have $5, how much more money do I need?
Ex. If I have two bills, one of them is a $5 and one of them is a $1, how much
money do I need to have $10?
Level II Students will:
EEA-SSE.3. Solve simple equations with unknown/missing values (without
variables).
Ex. If I have three dogs and one runs away, how many dogs are left?
Ex. I walked to the store to buy a book. I gave the cashier $10 and she gives me
back $7. How much was the book?
Ex. If I have two pens in my backpack when I get to school and I left home with
five pens, how many pens were given away on the trip from home to school?
Ex. 5 – [__] = 2.
Ex. [__] x 2 = 8.
Level I Students will:
EEA-SSE.3. Identify what is unknown.
Ex. John has three cats and some dogs. Do we know the number of dogs John
has?
Ex. Allen ate some apples. Do we know how many he ate?
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Grade Level/ Course: Algebra 1 Unit 4
Standard: A.CED.2 Create equations in two or more variables to represent relationships between quantities, graph equations on a coordinate axes with labels and scales.
Domain: Create Equations and
describe*
Cluster: Create equations that describe numbers or relationships
Quarter 1:
Quarter 2:
Quarter 3:
Identify the quantities in a mathematical problem or
real-world situation that should be represented by
distinct variables and describe what quantities the
variables represent.
Quarter 4:
Graph one or more created equation on a coordinate
axes with appropriate labels and scales.
Create at least two equations in two or more variables to
represent relationships between quantities
Justify which quantities in a mathematical problem or
real-world situation are dependent and independent of
one another and which operations represent those
relationships.
Determine appropriate units for the labels and scale of a
graph depicting the relationship between equations
created in two or more variables.
Make sense of
problems and
persevere in solving
them.
Reason
abstractly and
quantitatively.
Construct viable
Model with
arguments and
mathematics.
critique the
reasoning of others.
Appendix A: the targets extend work on linear and
exponential equation in Unit 1 to quadratic equations.
Use appropriate tools
Attend to precision.
strategically.
Look for and make
use of structure.
Look for and
express regularity
in repeated
reasoning.
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Evidence of Student
Vocabulary
Attainment/Assessment
Students:
Given a contextual situation
expressing a relationship
between quantities with two or
more variables,
Model the relationship with
equations and graph the
relationship on coordinate axes
with labels and scales.
(Please Note: This standard
must be taught in conjunction
with the preceding standard).
Knowledge
Skills
Students know:
Students understand/are able to:
When a particular two variable
equation accurately models the
situation presented in a contextual
problem.
Instructional Achievement Level Descriptors
Level IV Students will:
EEA-CED.2-4. Solve two-step inequalities with a variable.
Ex. If I buy two movie tickets for $5 each and two drinks at $4 each, will $15
Write equations in two variables that
accurately model contextual situations, be enough money?
Ex. I walked to the store to buy a book. I gave the cashier $10. She said, “You
need twice this amount.” How much is the book?
Graph equations involving two
Ex. I went to the store to buy two items that cost x dollars each plus a $5
variables on coordinate axes with
membership fee. The total cost is more than $25. How much must each item
appropriate scales and labels.
cost? 2x + 5 > 25.
There are relationships among features
Level III Students will:
of a contextual problem, a created
mathematical model for that problem, EEA-CED.2-4. Solve one-step inequalities.
Ex. Sally wants to buy a shirt that costs $15. She has $10. How much more
and a graph of that relationship.
money does she need?
Ex. Mike has six apples. Two of his friends are joining him for snack. Mike
wants to share his apples with his friends. Does he have enough to give each
friend two apples?
Level II Students will:
EEA-CED.2-4. Verify the solution to an inequality with one variable.
Ex. You have $10 and buy socks that cost $2. Will you get change?
Ex. I walk to the store and buy a book. If I give the cashier $10 and she says I
do not have enough money, is the book more or less than $10?
Ex. You have $1 and your breakfast costs $2. Do you need more money?
Level I Students will:
EEA-CED.2-4. Identify quantities that are greater than or less than a given
quantity.
Ex. Using a number line indicate greater than or less than a given number.
Ex. Mike has five oranges and Mary has two oranges. Who has more oranges?
Ex. Sarah has $50 and Cindy has $30. Who has more money?
Ex. Is five more or less than three?
Ex. If Sue has baseball cards and Tim has five, who has the most/fewest
baseball cards?
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Grade Level/ Course: Algebra 1 Unit 4
Standard with code: 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.
Domain: Creating Equations* (*Modeling
Domain)
Cluster: Create equations that describe numbers and relationships
Quarter 1:
Quarter 2:
Quarter 3
Define a “quantity of interest” to mean any
numerical or algebraic quantity (e.g.,
Quarter 4:
in which 2 is the quantity of
interest showing that d must be even;
and
showing that
)
Rearrange formulas to highlight a quantity of
interest, using the same reasoning as in
solving equations. (e.g. π * r2 can be rewritten as (π *r)*r which makes the form of
this expression resemble b*h.)
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and quantitatively.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
From Appendix A: Extend A.CED.4 to
formulas involving squared variables.
Use appropriate
Attend to precision.
tools strategically.
Look for and make
use of structure.
Look for and
express regularity
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Rearrange formulas which arise
in contextual situations to isolate
variables that are of interest for
particular problems. For
example, if the electric company
charges for power by the
formula COST = 0.03 KWH +
15, a consumer may wish to
determine how many kilowatt
hours they may use to keep the
cost under particular amounts,
by considering KWH< (COST 15)/0.03 which would yield to
keep the monthly cost under
$75, they need to use less than
2000 KWH.
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
Properties of equality and
inequality (Tables 4 and 5).
Accurately rearrange equations or
inequalities to produce equivalent
forms for use in resolving situations
of interest.
Level IV Students will:
EEA-CED.2-4. Solve two-step inequalities with a variable.
Ex. If I buy two movie tickets for $5 each and two drinks at $4 each, will $15
be enough money?
Ex. I walked to the store to buy a book. I gave the cashier $10. She said, “You
need twice this amount.” How much is the book?
Ex. I went to the store to buy two items that cost x dollars each plus a $5
membership fee. The total cost is more than $25. How much must each item
cost? 2x + 5 > 25.
The structure of mathematics allows
for the procedures used in working
with equations to also be valid when
rearranging formulas,
The isolated variable in a formula is
not always the unknown and
rearranging the formula allows for
sense-making in problem solving.
Level III Students will:
EEA-CED.2-4. Solve one-step inequalities.
Ex. Sally wants to buy a shirt that costs $15. She has $10. How much more
money does she need?
Ex. Mike has six apples. Two of his friends are joining him for snack. Mike
wants to share his apples with his friends. Does he have enough to give each
friend two apples?
Level II Students will:
EEA-CED.2-4. Verify the solution to an inequality with one variable.
Ex. You have $10 and buy socks that cost $2. Will you get change?
Ex. I walk to the store and buy a book. If I give the cashier $10 and she says I
do not have enough money, is the book more or less than $10?
Ex. You have $1 and your breakfast costs $2. Do you need more money?
Level I Students will:
EEA-CED.2-4. Identify quantities that are greater than or less than a given
quantity.
Ex. Using a number line indicate greater than or less than a given number.
Ex. Mike has five oranges and Mary has two oranges. Who has more oranges?
Ex. Sarah has $50 and Cindy has $30. Who has more money?
Ex. Is five more or less than three?
Ex. If Sue has baseball cards and Tim has five, who has the most/fewest
baseball cards?
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Grade Level/ Course: Algebra 1 Unit 4
Standard with code: A.REI.4a Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an equation of the
2
form (x-p) =q that has the same solutions. Derive the quadratic formula from this form.
Domain: Reasoning with Equations and
Inequalities
Cluster: Solve equations and inequalities in one variable.
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
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.
Solve quadratic equations in one variable.
Derive the quadratic formula by
completing the square on a quadratic
equation in x.
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and quantitatively.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
Use appropriate
tools strategically.
Attend to
precision.
Notes from Appendix A: Students should
learn of the existence of the complex
number system, but will not solve
quadratics with complex solutions until
Algebra II.
Look for and
Look for and
make use of
express regularity
structure.
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Vocabulary
Completing the square Students know:
Solve quadratic equations where Quadratic equations
both sides of the equation have
evident square roots by
Quadratic formula
inspection,
Complex solutions
Transform quadratic equations to
a form where the square root of Inspection
each side of the equation may be
taken, including completing the
square,
Use the method of completing
the square on the equation in
standard form (ax2+bx+c=0) to
derive the quadratic formula,
Identify quadratic equations
which may be solved efficiently
by factoring, and then use
factoring to solve the equation,
Use the quadratic formula to
solve quadratic equations,
Explain when the roots are real
or complex for a given quadratic
equation, and when complex
write them as a ± bi for real
numbers a and b,
Demonstrate that a proposed
solution to a quadratic equation
is truly a solution by making the
original true.
Knowledge
Any real number has two
square roots, that is, if a is the
square root of a real number
then so is -a,
Skills
Instructional Achievement Level Descriptors
Students understand/are able to:
EEA-REI.3-4. N/A (See EEA-ECED.1-2.)
Accurately use properties of equality
(Table 4) and other algebraic
manipulations including taking square
roots of both sides of an equation,
The method for completing the Accurately complete the square on a
square,
quadratic polynomial as a strategy for
finding solutions to quadratic
equations,
Notational methods for
expressing complex numbers,
Factor quadratic polynomials as a
strategy for finding solutions to
A quadratic equation in
quadratic equations,
standard form (ax2+bx+c=0)
has real roots when b2-4ac is
greater than or equal to zero
Rewrite solutions to quadratic
and complex roots when b2equations in useful forms including a ±
4ac is less than zero.
bi and simplified radical expressions,
Make strategic choices about which
procedures (inspection, completing the
square, factoring, and quadratic
formula) to use to reach a solution to a
quadratic equation.
Solutions to a quadratic equation must
make the original equation true and
this should be verified,
When the quadratic equation is derived
from a contextual situation, proposed
solutions to the quadratic equation
should be verified within the context
given, as well as mathematically,
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Different procedures for solving
quadratic equations are necessary
under different conditions,
If ab=0, then at least one of a or b must
be zero (a=0 or b=0) and this is then
used to produce the two solutions to
the quadratic equation,
Whether the roots of a quadratic
equation are real or complex is
determined by the coefficients of the
quadratic equation in standard form
(ax2+bx+c=0).
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Grade Level/ Course: Algebra 1 Unit 4
2
Standard with code: A.REI.4b Solve quadratic equations in one variable.
b. Solve quadratic equations by inspection (e.g., for x = 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.
Domain: Reasoning with Equations and
Inequalities
Cluster: Solve equations and inequalities in one variable.
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Solve quadratic equations by inspection (e.g.,
for x2 = 49), taking square roots, completing
the square, the quadratic formula and factoring
Express complex solutions as a ± bi for real
numbers solutions as a and b.
Determine appropriate strategies (see first
knowledge target listed) to solve problems
involving quadratic equations, as appropriate
to the initial form of the equation.
Recognize when the quadratic formula gives
complex solutions.
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and quantitatively.
Construct
viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate tools
strategically.
Attend to precision.
Note from Appendix A: Students should learn
of the existence of the complex number system,
but will not solve quadratics with complex
solutions until Algebra II.
Look for and make
Look for and express
use of structure.
regularity in
repeated reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Vocabulary
Solve quadratic equations where
both sides of the equation have
evident square roots by
inspection,
Skills
Instructional Achievement Level Descriptors
Completing the square Students know:
Students understand/are able to:
EEA-REI.3-4. N/A (See EEA-ECED.1-2.)
Quadratic equations
Accurately use properties of equality
(Table 4) and other algebraic
manipulations including taking
square roots of both sides of an
equation,
Quadratic formula
Complex solutions
Transform quadratic equations to
a form where the square root of
Inspection
each side of the equation may be
taken, including completing the
square,
Use the method of completing the
square on the equation in standard
form (ax2+bx+c=0) to derive the
quadratic formula,
Identify quadratic equations
which may be solved efficiently
by factoring, and then use
factoring to solve the equation,
Use the quadratic formula to
solve quadratic equations,
Explain when the roots are real or
complex for a given quadratic
equation, and when complex
write them as a ± bi for real
numbers a and b,
Demonstrate that a proposed
solution to a quadratic equation is
truly a solution by making the
original true.
Knowledge
Any real number has two
square roots, that is, if a is the
square root of a real number
then so is -a,
The method for completing the
square,
Accurately complete the square on a
quadratic polynomial as a strategy
for finding solutions to quadratic
Notational methods for
expressing complex numbers, equations,
A quadratic equation in
standard form (ax2+bx+c=0)
has real roots when b2-4ac is
greater than or equal to zero
and complex roots when b24ac is less than zero.
Factor quadratic polynomials as a
strategy for finding solutions to
quadratic equations,
Rewrite solutions to quadratic
equations in useful forms including a
± bi and simplified radical
expressions,
Make strategic choices about which
procedures (inspection, completing
the square, factoring, and quadratic
formula) to use to reach a solution to
a quadratic equation.
Solutions to a quadratic equation
must make the original equation true
and this should be verified,
When the quadratic equation is
derived from a contextual situation,
proposed solutions to the quadratic
equation should be verified within
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the context given, as well as
mathematically,
Different procedures for solving
quadratic equations are necessary
under different conditions,
If ab=0, then at least one of a or b
must be zero (a=0 or b=0) and this is
then used to produce the two
solutions to the quadratic equation,
Whether the roots of a quadratic
equation are real or complex is
determined by the coefficients of the
quadratic equation in standard form
(ax2+bx+c=0).
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Grade Level/ Course (HS): Algebra 1 Unit 5
Standard with code: N.RN.3 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.
Domain: The Real Number System
Cluster: Use properties of rational and irrational numbers.
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Find the sums and products of rational and
irrational numbers.
Recognize that the sum of a rational number
and an irrational number is irrational.
Recognize that the product of a nonzero
rational number and an irrational number is
irrational.
Explain why rational numbers are closed
under addition or multiplication.
Make sense of
problems and
persevere in solving
them.
Reason abstractly Construct viable
and quantitatively. arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate tools
strategically.
Attend to precision.
Note from Appendix A: Connect N.RN.3 to
physical situations, e.g., finding the
perimeter of a square of area 2.
Look for and
Look for and express
make use of
regularity in
structure.
repeated reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given the definition of rational
numbers,
Show that the sum or product of
two rational numbers is rational
and explain their reasoning,
Use repeated reasoning from
examples to demonstrate that the
sum (or product) of a rational
number and an irrational number
appears to always result in an
infinite non-repeating decimal,
for example 2 + π = 2 +
3.14159... = 5.14159...
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Irrational number
Students know:
Students understand/are able to:
EEN-RN.3. N/A
The distinction between
rational and irrational
numbers.
Convert between fraction and decimal
form of rational numbers.
Reasoning from repeated examples is
not a proof but can lend credibility to
a statement when there is reasoning to
why the regularity continues.
The decimal approximation of an
irrational will never produce an exact
or infinite repeating decimal,
Making the same change to every
digit in an infinite non-repeating
decimal will not make the decimal
repeat,
Reasoning from repeated examples is
not a proof but can lend credibility to
a statement when there is reasoning to
why the regularity continues.
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Grade Level/ Course: Algebra 1 Unit 4
Standard with code: A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the
2
2
points of intersection between the line y = –3x and the circle x + y = 3.
Domain: Reasoning with Equations and
Inequalities
Cluster: Solve systems of equations
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Transform a simple system consisting of a linear
equation and a quadratic equation in 2 variables
so that a solution can be found algebraically and
graphically.
Notes from Appendix A: Include systems
consisting of one linear and one quadratic
equation. Include systems that lead to work with
fractions. For example, finding the
Make sense of
Reason abstractly
problems and
and quantitatively.
persevere in solving
them.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate
tools strategically.
Attend to precision.
Look for and make
use
of structure.
Look for and express
regularity in
repeated reasoning.
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Evidence of Student
Vocabulary
Attainment/Assessment
Students:
Given a system of a linear
equation and a quadratic equation,
Solve the system algebraically by
substitution,
Graph the linear equation and the
quadratic equation on the same
Cartesian plane, and identify the
intersection point(s),
Make sense of the existence of 0,
1, or 2 solutions to the system by
explaining the relationship of the
solutions to the graph,
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
EEA-REI.6-7. N/A (See EEA-REI.10-12.)
Appropriate use of properties
of equality (Table 4),
Accurately use properties of equality
(Table 4) to solve a system of a linear
and a quadratic equation,
Techniques to solve quadratic
equations,
Graph linear and quadratic equations
precisely and interpret the results.
The conditions under which a
linear equation and a quadratic Solutions of a system of equations is
equation have 0, 1, or 2
the set of all ordered pairs that make
solutions,
both equations true simultaneously,
Techniques for producing and
interpreting graphs of linear
and quadratic equations.
A system consisting of a linear
equation and a quadratic equation will
have 0,1, or 2 solutions.
Verify that the proposed solutions
satisfy both equations.
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Grade Level/ Course (HS): Algebra 1 Unit 5
Standard with code: 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.*(Modeling standard)
Domain: Interpreting Functions
Cluster: Interpret functions that arise in applications in terms of the context.
Quarter 1:
Quarter 2:
Quarter 3
Quarter 4:
Define and recognize the key features in tables and
graphs of linear, exponential, and quadratic
functions: intercepts; intervals where the function is
increasing, decreasing, positive, or negative, relative
maximums and minimums, symmetries, and end
behavior.
Identify whether the function is linear, exponential,
or quadratic, given its table or graph.
Interpret key features of graphs and tables of
functions in the terms of the contextual quantities the
function represents.
Sketch graphs showing key features of a function
that models a relationship between two quantities
from a given verbal description of the relationship.
Notes from Appendix A: Focus on quadratic
functions; compare with linear and exponential
functions studied in unit 2.
Make sense of
problems and
persevere in solving
them.
Reason abstractly and
quantitatively.
Construct viable
arguments and
critique the
reasoning of
others.
Model with
mathematics.
Use appropriate tools Attend to
strategically.
precision.
Look for and
make use of
structure.
Look for and
express regularity in
repeated reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a function that models a
relationship between two
quantities,
Produce the graph and table of
the function and show the key
features (intercepts; intervals
where the function is increasing,
decreasing, positive, or negative;
relative maximums and
minimums; symmetries; end
behavior; and periodicity) that
are appropriate for the function.
Given key features from verbal
description of a relationship,
Sketch a graph with the given
key features.
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Function
Students know:
Students understand/are able to:
Key features
Key features of function
graphs (i.e., intercepts;
intervals where the function is
increasing, decreasing,
positive, or negative; relative
maximums and minimums;
symmetries; end behavior; and
periodicity),
Accurately graph any relationship,
Level IV Students will:
EEF-IF.4-6. Evaluate key features of a graph (e.g. increasing, decreasing,
constant.).
Ex. Determine parts of graph illustrating an increase or decrease in speed.
Ex. Using a graph illustrating change in temperature over a day, indicate times
when the temperature increased, decreased, or stayed the same.
Methods of modeling
relationships with a graph or
table.
Interpret key features of a graph.
The relationship between two variables
Level III Students will:
determines the key features that need
EEF-IF.4-6. Interpret rate of change (e.g. higher/lower, faster/slower).
to be used when interpreting and
Ex. Compare two graphs with different slopes to determine faster/slower rate
producing the graph.
Ex. Compare a bus schedule with two buses, look and determine if one bus runs
more frequently than the next bus on the route.
Level II Students will:
EEF-IF.4-6. Graph a simple linear equation represented by a table of values.
Ex. Match the graph to its corresponding story.
Ex. Plot the points from a table of values less than 10.
Level I Students will:
EEF-IF.4-6. Read a table.
Ex. From a given table, find information.
Ex. Read a bus schedule.
Ex. Given a daily schedule, determine the time of lunch during the school day.
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Grade Level/ Course: Algebra 1 Unit 5
Standard with code: 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.*(Modeling standard)
Domain: Interpreting Functions
Cluster: Interpret functions that arise in applications in terms of a context.
Quarter 1:
Quarter 2:
Quarter 3
Quarter 4:
Given the graph or a verbal/written
description of a function, identify and
describe the domain of the function.
Identify an appropriate domain based on
the unit, quantity, and type of function it
describes.
Relate the domain of a function to its
graph and, where applicable, to the
quantitative relationship it describes.
Explain why a domain is appropriate for a
given real-world situation.
Make sense of
Reason abstractly
problems and
and quantitatively.
persevere in solving
them.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
Use appropriate tools
strategically.
Note from Appendix A: Focus on
quadratic functions; compare with linear
and exponential functions studied in Unit
2.
Attend to precision. Look for and
Look for and
make use of
express regularity
structure.
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a contextual situation
that is functional,
Model the situation with a
graph and construct the graph
based on the parameters given
in the domain of the context.
Vocabulary
Knowledge
Skills
Function
Students know:
Students understand/are able to:
Instructional Achievement Level Descriptors
Level IV Students will:
EEF-IF.4-6. Evaluate key features of a graph (e.g. increasing, decreasing,
Interpret the domain from the context, constant.).
Techniques for graphing
Ex. Determine parts of graph illustrating an increase or decrease in speed.
functions,
Produce a graph of a function based on Ex. Using a graph illustrating change in temperature over a day, indicate times
when the temperature increased, decreased, or stayed the same.
Techniques for determining the the context given.
domain of a function from its
Level III Students will:
context.
Different contexts produce different
EEF-IF.4-6. Interpret rate of change (e.g. higher/lower, faster/slower).
domains and graphs,
Ex. Compare two graphs with different slopes to determine faster/slower rate
Ex. Compare a bus schedule with two buses, look and determine if one bus
Function notation in itself may
produce graph points which should not runs more frequently than the next bus on the route.
be in the graph as the domain is limited
Level II Students will:
by the context.
EEF-IF.4-6. Graph a simple linear equation represented by a table of values.
Ex. Match the graph to its corresponding story.
Ex. Plot the points from a table of values less than 10.
Level I Students will:
EEF-IF.4-6. Read a table.
Ex. From a given table, find information.
Ex. Read a bus schedule.
Ex. Given a daily schedule, determine the time of lunch during the school day.
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Grade Level/ Course (HS): Algebra 1 Unit 5
Standard with code: 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.*(Modeling standard)
Domain: Interpreting Functions
Cluster: Interpret functions that arise in applications in terms of the context.
Quarter 1:
Quarter 2:
Quarter 3:
Quarter 4:
Recognize slope as an average rate of change.
Recognize slope as an average rate of change.
Calculate the average rate of change of a
Calculate the average rate of change of a
function (presented symbolically or as a table) function (presented symbolically or as a table)
over a specified interval.
over a specified interval.
Estimate the rate of change from a linear,
exponential, or quadratic graph.
Estimate the rate of change from a linear,
exponential, or quadratic graph.
Interpret the average rate of change of a
Interpret the average rate of change of a
function (presented symbolically or as a table) function (presented symbolically or as a table)
over a specified interval.
over a specified interval.
Make sense of
problems and
persevere in solving
them.
Reason abstractly Construct viable
Model with
and quantitatively. arguments and
mathematics.
critique the
reasoning of others.
Note from Appendix A: Focus on quadratic
functions; compare with linear and
exponential functions studied in Unit 2 of the
Traditional Algebra 1 Pathway.
Use appropriate
Attend to precision.
tools strategically.
Note from Appendix A: Focus on quadratic
functions; compare with linear and exponential
functions studied in Unit 2 of the Traditional
Algebra 1 Pathway.
Look for and make
Look for and express
use of structure.
regularity in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given an interval on a graph or
table,
Calculate the average rate of
change within the interval.
Given a graph of contextual
situation,
Estimate the rate of change
between intervals that are
appropriate for the summary of
the context.
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Average rate of
change
Students know:
Students understand/are able to:
Techniques for graphing,
Calculate rate of change over an
interval on a table or graph,
Level IV Students will:
EEF-IF.4-6. Evaluate key features of a graph (e.g. increasing, decreasing,
constant.).
Ex. Determine parts of graph illustrating an increase or decrease in speed.
Ex. Using a graph illustrating change in temperature over a day, indicate times
when the temperature increased, decreased, or stayed the same.
Techniques for finding a rate
of change over an interval on a Estimate a rate of change over an
table or graph,
interval on a graph.
Level III Students will:
Techniques for estimating a
The average provides information on EEF-IF.4-6. Interpret rate of change (e.g. higher/lower, faster/slower).
rate of change over an interval the overall changes within an interval, Ex. Compare two graphs with different slopes to determine faster/slower rate
on a graph.
not the details within the interval (an Ex. Compare a bus schedule with two buses, look and determine if one bus runs
more frequently than the next bus on the route.
average of the endpoints of an
interval does not tell you the
Level II Students will:
significant features within the
EEF-IF.4-6. Graph a simple linear equation represented by a table of values.
interval).
Ex. Match the graph to its corresponding story.
Ex. Plot the points from a table of values less than 10.
Level I Students will:
EEF-IF.4-6. Read a table.
Ex. From a given table, find information.
Ex. Read a bus schedule.
Ex. Given a daily schedule, determine the time of lunch during the school day.
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Grade Level/ Course: Algebra 1 Unit 5
Standard with code: F.IF.7a Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated
cases.*(Modeling standard)
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
Domain: Interpreting Functions
Cluster: Analyze functions using different representations
Quarter 1:
Quarter 2:
Quarter 3
Quarter 4:
Graph linear and quadratic functions, by
hand in simple cases or using technology
for more complicated cases, and
show/label intercepts, maxima, and
minima of the graph.
Make sense of
Reason abstractly
problems and
and quantitatively.
persevere in solving
them.
Construct viable
arguments and
critique the reasoning
of others.
Model with
mathematics.
Use appropriate tools Attend to
strategically.
precision.
Determine the differences between simple
and complicated linear, exponential and
quadratic functions and know when the
use of technology is appropriate.
Look for and
Look for and
make use of
express regularity
structure.
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a symbolic representation of a function
(including linear, quadratic, square root, cube
root, piecewise-defined functions, polynomial,
exponential, logarithmic, trigonometric, and
(+) rational),
Produce an accurate graph (by hand in simple
cases and by technology in more complicated
cases) and justify that the graph is an alternate
representation of the symbolic function,
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
EEF-IF.7. N/A (See EEF-IF.1-3)
Techniques for graphing,
Identify the type of function from the
symbolic representation,
Key features of graphs of
functions.
Manipulate expressions to reveal
important features for identification in
the function,
Accurately graph any relationship.
Identify key features of the graph and connect
these graphical features to the symbolic
function, specifically for special functions:
Key features are different depending
on the function,
quadratic or linear (intercepts, maxima, and
minima),
Identifying key features of functions
aid in graphing and interpreting the
function.
square root, cube root, and piecewise-defined
functions, including step functions and
absolute value functions (descriptive features
such as the values that are in the range of the
function and those that are not),
polynomial (zeros when suitable
factorizations are available, end behavior),
(+) rational (zeros and asymptotes when
suitable factorizations are available, end
behavior),
exponential and logarithmic (intercepts and
end behavior),
trigonometric functions (period, midline, and
amplitude).
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Grade Level/ Course: Algebra 1 Unit 5
Standard with code: F.IF.7b Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated
cases.*(Modeling standard)
b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
Domain: Interpreting Functions
Cluster: Analyze functions using different representations
Quarter 1:
Quarter 2:
Quarter 4:
Graph square root, cube root, and piecewise-defined
functions, including step functions and absolute
value functions, by hand in simple cases or using
technology for more complicated cases, and
show/label key features of the graph.
Quarter 3
Determine the difference between simple and
complicated linear, quadratic, square root, cube root,
and piecewise-defined functions, including step
functions and absolute value functions and know
when the use of technology is appropriate.
Compare and contrast the domain and range of
absolute value, step and piece-wise defined functions
with linear, quadratic, and exponential.
Notes from Appendix A: Compare and contrast
absolute value, step and piece-wise defined functions
with linear, quadratic, and exponential functions.
Highlight issues of domain, range, and usefulness
when examining piece-wise defined functions.
Make sense of
problems and
persevere in solving
them.
Reason abstractly and
quantitatively.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
Use appropriate
tools strategically.
Attend to
precision.
Look for and make
use of structure.
Look for and
express regularity
in repeated
reasoning.
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Evidence of Student
Vocabulary
Attainment/Assessment
Students:
Given a symbolic representation
of a function (including linear,
quadratic, square root, cube root,
piecewise-defined functions,
polynomial, exponential,
logarithmic, trigonometric, and
(+) rational),
Produce an accurate graph (by
hand in simple cases and by
technology in more complicated
cases) and justify that the graph
is an alternate representation of
the symbolic function,
Identify key features of the
graph and connect these
graphical features to the
symbolic function, specifically
for special functions:
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
EEF-IF.7. N/A (See EEF-IF.1-3)
Techniques for graphing,
Identify the type of function from the
symbolic representation,
Key features of graphs of
functions.
Manipulate expressions to reveal
important features for identification in
the function,
Accurately graph any relationship.
Key features are different depending
on the function,
Identifying key features of functions
aid in graphing and interpreting the
function.
quadratic or linear (intercepts,
maxima, and minima),
square root, cube root, and
piecewise-defined functions,
including step functions and
absolute value functions
(descriptive features such as the
values that are in the range of the
function and those that are not),
polynomial (zeros when suitable
factorizations are available, end
behavior),
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(+) rational (zeros and
asymptotes when suitable
factorizations are available, end
behavior),
exponential and logarithmic
(intercepts and end behavior),
trigonometric functions (period,
midline, and amplitude).
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Grade Level/ Course (HS): Algebra 1 Unit 5
Standard with code: F.IF.8a Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
a. Use the process
of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
Domain: Interpreting Functions
Cluster: Analyze functions using different representations.
Quarter 1:
Quarter 2:
Quarter 3
Quarter 4:
Identify different forms of a quadratic
expression.
Write functions in equivalent forms using
the process of factoring
Identify zeros, extreme values, and
symmetry of the graph of a quadratic
function.
Interpret different but equivalent forms of
a function defined by an expression in
terms of a context
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.
Make sense of
problems and
persevere in solving
them.
Reason abstractly Construct viable
and quantitatively. arguments and critique
the reasoning of others.
Model with
mathematics.
Use appropriate tools
strategically.
Attend to
precision.
Note from Appendix A: Extend work with
quadratics to include the relationship
between coefficients and roots, and that
once roots are known, a quadratic
equation can be factored.
Look for and
Look for and
make use of
express regularity
structure.
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a contextual situation
containing a function defined by
an expression,
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Zeros
Students know:
Students understand/are able to:
EEF-IF.8. N/A
Extreme values
Techniques to factor and
complete the square,
Accurately manipulate (e.g., factoring,
completing the square) expressions
using appropriate technique(s) to
reveal key properties of a function.
Symmetry
Use algebraic properties to
rewrite the expression in a form
that makes key features of the
function easier to find,
Manipulate a quadratic function
by factoring and completing the
square to show zeros, extreme
values, and symmetry of the
graph,
Properties of exponential
Exponential growth or expressions,
decay
Algebraic properties of
equality (Table 4).
An expression may be written in
various equivalent forms,
Some forms of the expression are more
beneficial for revealing key properties
of the function.
Explain and justify the meaning
of zeros, extreme values, and
symmetry of the graph in terms of
the contextual situation,
Apply exponential properties to
expressions and explain and
justify the meaning in a
contextual situation.
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Grade Level/ Course (HS): Algebra 1 Unit 5
Standard with code: F.IF.8b Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function:
b. Use the
t
t
12t
t/10
properties of exponents to interpret expressions for exponential functions. For example: identify percent rate of change in functions such as y= (1.02) , y=(.97) , y=(1.01) , y=(1.2)
,
and classify them as representing exponential growth or decay.
Domain: Interpreting Functions
Cluster: Analyze functions using different representations.
Quarter 1:
Quarter 2:
Quarter 3
Quarter 4:
Classify the exponential function as
exponential growth or decay by
examining the base.
Use the properties of exponents to
interpret expressions for exponential
functions in a real-world context.
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and quantitatively.
Construct viable
arguments and
critique the reasoning
of others.
Model with
mathematics.
Note from Appendix A: Note this unit
extends the work begun in Unit 2 on
exponential functions with integer
exponents.
Use appropriate tools Attend to precision. Look for and
Look for and
strategically.
make use of
express regularity
structure.
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a contextual situation
containing a function defined by
an expression,
Vocabulary
Knowledge
Skills
Instructional Achievement Level Descriptors
Zeros
Students know:
Students understand/are able to:
EEF-IF.8. N/A
Extreme values
Techniques to factor and
complete the square,
Accurately manipulate (e.g., factoring,
completing the square) expressions
using appropriate technique(s) to reveal
key properties of a function.
Symmetry
Use algebraic properties to
rewrite the expression in a form
that makes key features of the
function easier to find,
Manipulate a quadratic function
by factoring and completing the
square to show zeros, extreme
values, and symmetry of the
graph,
Properties of exponential
Exponential growth or expressions,
decay
Algebraic properties of
equality (Table 4).
An expression may be written in
various equivalent forms,
Some forms of the expression are more
beneficial for revealing key properties
of the function.
Explain and justify the meaning
of zeros, extreme values, and
symmetry of the graph in terms of
the contextual situation,
Apply exponential properties to
expressions and explain and
justify the meaning in a
contextual situation.
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Grade Level/ Course: Algebra 1 Unit 5
Standard with code: 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.
Domain: Interpreting Functions
Cluster: Analyze functions using different representations
Quarter 1:
Quarter 2:
Quarter 3
Quarter 4:
Identify types of functions based on verbal ,
numerical, algebraic, and graphical descriptions and
state key properties (e.g. intercepts, maxima,
minima, growth rates, average rates of change, and
end behaviors)
Differentiate between exponential, linear, and
quadratic functions using a variety of descriptors
(graphically, verbally, numerically, and
algebraically)
Use a variety of function representations
(algebraically, graphically, numerically in tables, or
by verbal descriptions) to compare and contrast
properties of two functions
Note from Appendix A: Focus on expanding the
types of functions considered to include, linear,
exponential, and quadratic. Extend work with
quadratics to include the relationship between
coefficients and roots, and that once roots are
known, a quadratic equation can be factored.
Make sense of
Reason abstractly
problems and
and quantitatively.
persevere in solving
them.
Construct viable
arguments and
critique the reasoning
of others.
Model with
mathematics.
Use appropriate tools Attend to precision.
strategically.
Look for and make
use of structure.
Look for and
express regularity
in repeated
reasoning.
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Evidence of Student
Vocabulary
Attainment/Assessment
Students:
Given two functions represented
differently (algebraically,
graphically, numerically in
tables, or by verbal
descriptions),
Use key features to compare the
functions,
Explain and justify the
similarities and differences of
the functions.
Knowledge
Skills
Instructional Achievement Level Descriptors
Students know:
Students understand/are able to:
EEF-IF.9. N/A
Techniques to find key
features of functions when
presented in different ways,
Accurately determine which key
features are most appropriate for
comparing functions,
Techniques to convert a
function to a different form
(algebraically, graphically,
numerically in tables, or by
verbal descriptions).
Manipulate functions algebraically to
reveal key functions,
Convert a function to a different form
(algebraically, graphically,
numerically in tables, or by verbal
descriptions) for the purpose of
comparing it to another function.
Functions can be written in different
but equivalent ways (algebraically,
graphically, numerically in tables, or
by verbal descriptions),
Different representations of functions
may aid in comparing key features of
the functions.
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Grade Level/ Course (HS): Algebra 1 Unit 5
Standard with code: F.BF.1a Write a function that describes a relationship between two quantities.*(Modeling standard) a. Determine an explicit expression, a recursive process, or
steps for calculation from a context.
Domain: Building Functions
Cluster: Build a function that models a relationship between two quantities
Quarter 1:
Quarter 2:
Quarter 3
Quarter 4:
Define “explicit function” and “recursive
process”.
Write a function that describes a
relationship between two quantities by
determining an explicit expression, a
recursive process, or steps for calculation
from a context.
Make sense of
problems and
persevere in solving
them.
Reason abstractly and
quantitatively.
Construct viable
arguments and
critique the
reasoning of others.
Model with
mathematics.
Use appropriate
tools strategically.
Attend to
precision.
Note from Appendix A: Focus on situations
that exhibit a quadratic relationship. This
standard builds from Algebra 1 Unit 2.
Look for and make
Look for and
use of structure.
express regularity
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a contextual situation
containing two quantities,
Vocabulary
Knowledge
Skills
Explicit expression
Students know:
Students understand/are able to:
Recursive process
Techniques for expressing
functional relationships
(explicit expression, a
recursive process, or steps for
calculation) between two
quantities,
Express a relationship between Decaying exponential
the quantities through an explicit
expression using function
notation, recursive process, or
steps for calculation,
Explain and justify how the
expression or process models the
relationship between the given
quantities,
Create a new function by using
standard function types and
arithmetic operations to combine
the original functions to model
the relationship of the given
quantities, (+) standards not
covered.
Techniques to combine
functions using arithmetic
operations.
Instructional Achievement Level Descriptors
Level IV Students will:
EEF.BF.1. Complete the appropriate graphical representation (first quadrant)
Accurately develop a model that shows given a situation involving constant rate of change.
Ex. Given this scenario and a graphical representation with missing information:
the functional relationship between
If I mow one lawn and I make $25 and if I mow three lawns and I make $75,
two quantities,
how much will I make if I mow two lawns?
Ex. Given this scenario and a graphical representation with missing information:
Accurately create a new function
through arithmetic operations of other If hamburgers are four for $1 and I buy four, it will cost $1; if I buy 12, it will
cost $3 – complete the graph for eight hamburgers.
functions,
Present an argument to show how the
function models the relationship
between the quantities.
Relationships can be modeled by
several methods (e.g., explicit
expression or recursive process),
Arithmetic combinations of functions
may be used to improve the fit of a
model.
Level III Students will:
EEF-BF.1. Select the appropriate graphical representation (first quadrant) given
a situation involving constant rate of change.
Ex. Given this scenario and two completed graphs, show me the graph that
shows the following: If I mow one lawn, I make $25; if I mow two lawns, I will
make $50; and if I mow three lawns I will make $75.
Ex. Given this scenario and two completed graphs, show me the graph that
depicts that there are two cookies for every student.
Level II Students will:
EEF-BF.1. Select the appropriate graphical representation (first quadrant) given
a situation involving constant rate of change where the difference is very clear.
Ex. Every dog has one bone. Pick the graph that would represent this concept
when given the following graphs.*
Level I Students will:
EEF-BF.1. Identify the terms in a sequence.
Ex. Identify an ABABABABAB pattern out of two different pattern sets of
colored blocks using black (B) and white (W) and one set is BWBWBWBWBW
and the other pattern set is BBWBBWBBWBBW.
Ex. Place two pencils in front of each student in the classroom.
*Refer to the Common Core Essential Elements document for diagram.
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Grade Level/ Course (HS): Algebra 1 Unit 5
Standard with code: F.BF.1b Write a function that describes a relationship between two quantities.*(Modeling standard)
b. Combine standard function types using arithmetic
operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
Domain: Building Functions
Cluster: Build a function that models a relationship between two quantities
Quarter 1:
Quarter 2:
Quarter 3
Quarter 4:
Combine two functions using the operations
of addition, subtraction, multiplication, and
division
Evaluate the domain of the combined
function.
Given a real-world situation or
mathematical problem:
 build standard functions to represent
relevant relationships/ quantities
 determine which arithmetic
operation should be performed to
build the appropriate combined
function
 relate the combined function to the
context of the problem
Make sense of
problems and
persevere in solving
them.
Reason abstractly
and quantitatively.
Construct viable
arguments and critique
the reasoning of others.
Model with
mathematics.
Use appropriate tools Attend to
strategically.
precision.
Note from Appendix: Focus on situations
that exhibit a quadratic relationship.
Look for and make Look for and
use of structure.
express regularity
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a contextual situation
containing two quantities,
Vocabulary
Knowledge
Skills
Explicit expression
Students know:
Students understand/are able to:
Recursive process
Express a relationship between
the quantities through an explicit
expression using function
notation, recursive process, or
steps for calculation,
Decaying exponential
Techniques for expressing
functional relationships
(explicit expression, a
recursive process, or steps for
calculation) between two
quantities,
Explain and justify how the
expression or process models the
relationship between the given
quantities,
Create a new function by using
standard function types and
arithmetic operations to combine
the original functions to model
the relationship of the given
quantities, (+) standards not
covered.
Techniques to combine
functions using arithmetic
operations.
Instructional Achievement Level Descriptors
Level IV Students will:
EEF.BF.1. Complete the appropriate graphical representation (first quadrant)
given a situation involving constant rate of change.
Accurately develop a model that
Ex. Given this scenario and a graphical representation with missing
shows the functional relationship
information: If I mow one lawn and I make $25 and if I mow three lawns and I
between two quantities,
make $75, how much will I make if I mow two lawns?
Ex. Given this scenario and a graphical representation with missing
Accurately create a new function
through arithmetic operations of other information: If hamburgers are four for $1 and I buy four, it will cost $1; if I
buy 12, it will cost $3 – complete the graph for eight hamburgers.
functions,
Present an argument to show how the Level III Students will:
EEF-BF.1. Select the appropriate graphical representation (first quadrant) given
function models the relationship
a situation involving constant rate of change.
between the quantities.
Ex. Given this scenario and two completed graphs, show me the graph that
shows the following: If I mow one lawn, I make $25; if I mow two lawns, I will
Relationships can be modeled by
make $50; and if I mow three lawns I will make $75.
several methods (e.g., explicit
Ex. Given this scenario and two completed graphs, show me the graph that
expression or recursive process),
depicts that there are two cookies for every student.
Arithmetic combinations of functions
Level II Students will:
may be used to improve the fit of a
EEF-BF.1. Select the appropriate graphical representation (first quadrant) given
model.
a situation involving constant rate of change where the difference is very clear.
Ex. Every dog has one bone. Pick the graph that would represent this concept
when given the following graphs.*
Level I Students will:
EEF-BF.1. Identify the terms in a sequence.
Ex. Identify an ABABABABAB pattern out of two different pattern sets of
colored blocks using black (B) and white (W) and one set is
BWBWBWBWBW and the other pattern set is BBWBBWBBWBBW.
Ex. Place two pencils in front of each student in the classroom.
*Refer to the Common Core Essential Elements document for diagram.
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Grade Level/ Course (HS): Algebra 1 Unit 5
Standard with code: 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.
Domain: Linear, Quadratic, and
Exponential Models *(Modeling
Domain)
Cluster: Construct and compare linear, quadratic, and exponential models and solve problems
Quarter 1:
Quarter 2:
Quarter 3
Quarter 4:
Fluently compute growth rates for linear,
exponential and quadratic functions.
Compare tables and graphs of exponential
and other polynomial functions to observe
that a quantity increasing exponentially
exceeds all others to solve mathematical
and real-world problems.
Make sense of
Reason abstractly
problems and
and quantitatively.
persevere in solving
them.
Construct viable
arguments and
critique the reasoning
of others.
Model with
mathematics.
Use appropriate tools
strategically.
Attend to
precision.
Notes from Appendix A: Compare linear
and exponential growth to quadratic
growth.
Look for and make Look for and
use of structure.
express regularity
in repeated
reasoning.
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Evidence of Student
Attainment/Assessment
Students:
Given a quantity increasing
exponentially and a quantity
increasing as a polynomial
function (e.g., linearly,
quadratically),
Construct graphs and tables that
demonstrate the exponential
function will exceed the
polynomial function at some
point,
Present a convincing argument
that this must be true for all
polynomial functions.
Vocabulary
Knowledge
Skills
Increasing
exponentially
Students know:
Students understand/are able to:
Instructional Achievement Level Descriptors
Level IV Students will:
EEF-LE.1-4. Plot points using pictures in first quadrant on a graph using
whole numbers and explain how y increases/decreases as x changes.
Techniques to graph and create Accurately create graphs and tables
Ex. If you go to the store where every item is one dollar, students should state
tables for exponential and
for exponential and polynomial
y = x (the number of items I buy will tell me the cost). Students will then plot
polynomial functions.
functions,
this on the graph.
Use the graphs and tables to present a Ex. If I get two apples for every orange I buy, students should state that y = 2x,
or for every orange I buy (x), I will get two apples (y), therefore x times two
convincing argument that the
tells me the number of apples each time. Students should then plot this on the
exponential function eventually
graph.
exceeds the polynomial function.
Exponential functions grow at a faster Level III Student will:
EEF-LE.1-4. Model a simple linear function such as y = mx to show functions
rate than polynomial functions after
grow by equal factors over equal intervals.
some point in their domain.
Ex. Determine a simple relationship of y to x by looking at the first quadrant of
a graph.
Ex. Identify the cost per item on a simple graph where every item in the store
cost the same amount and state the relationship between x and y.
Ex. Look at a graph that shows a constant ratio of boys to girls and state the
relationship between x and y.
Level II Students will:
EEF-LE.1-4. Identify a specific data point in the first quadrant and explain the
meaning behind it.
Ex. Given data points in the first quadrant, identify the named point and state
the two pieces of information that one dot provides.
Ex. When given a simple graph that shows the total cost of items purchased at
a store where every item is $1, tell the cost of four items, the cost of two items,
etc.
Level I Students will:
EEF-LE.1-4. Interpret major ideas of a graph with linear functions.
Ex. When shown two lines on a graph, tell which one is rising faster.
Ex. When shown a graph of distance driven and gas left in tank, explain that
the further one drives the less gas one has left.
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