Download Algebra 2 Curriculum - Poudre School District

Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Algebra 2 Standards Aligned With the Algebra 2 PARCC Assessment
Performance Based Assessment (PBA/MYA) and End Of Year Assessment (EOY)
Cluster
Standard
PBA/MYA
Unit 1: Inferences and Conclusions from Data
HS.S.ID.A
Summarize, represent, and interpret data on a single
count of measurement variable.
HS.S.IC.A
Understand and evaluate random processes underlying
statistical experiments.
HS.S.IC.B
Make inferences and justify conclusions from sample
surveys, experiments, and observational studies.
HS.S.MD.B
Use probability to evaluate outcomes of decisions.
EOY
HS.S.ID.A.4
X
X
HS.S.IC.A.2
X
X
HS.S.IC.A.1
HS.S.IC.B.3
HS.S.IC.B.4
HS.S.IC.B.5
HS.S.IC.B.6
HS.S.MD.B.6
HS.S.MD.B.7
X
X
X
X
X
X
Page | 1
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Algebra 2 Standards Aligned With the Algebra 2 PARCC Assessment
Performance Based Assessment (PBA/MYA) and End Of Year Assessment (EOY)
Cluster
Standard
PBA/MYA
Unit 2: Polynomial, Rational, and Radical Relationships
HS.A.SSE.A
Interpret the structure of expressions.
HS.A.SSE.B
Write expressions in equivalent forms to solve
problems.
HS.N.CN.A
Perform arithmetic operations with complex numbers.
HS.N.CN.C
Use complex numbers in polynomial identities and
equations.
HS.F.IF.C
Analyze functions using different representations.
HS.A.APR.A
Performa arithmetic operations on polynomials.
HS.A.APR.B
Understand the relationship between zeros and factors
of polynomials.
HS.A.APR.C
Use polynomial identities to solve problems.
HS.A.APR.D
Review rational expressions.
HS.A.SSE.A.1
EOY
HS.A.SSE.A.2
X
X
HS.N.CN.A.1
X
X
HS.A.SSE.B.4
HS.N.CN.A.2
HS.N.CN.C.7
HS.N.CN.C.8
HS.N.CN.C.9
HS.F.IF.C.7
X
X
X
X
X
HS.A.APR.A.1
X
HS.A.APR.B.3
X
HS.A.APR.B.2
HS.A.APR.C.4
HS.A.A.PR.C.5
HS.A.APR.D.6
HS.A.APR.D.7
HS.A.REI.A
Understand solving equations as a process of reasoning HS.A.REI.A.2
and explain the reasoning.
HS.A.REI.D
Represent and solve equations and inequalities
HS.A.REI.D.11
graphically.
Unit 3: Trigonometric Functions
HS.F.TF.A
HS.F.TF.A.1
Extend the domain of trigonometric functions using the
HS.F.TF.A.2
unit circle.
HS.F.TF.B
Model period phenomena with trigonometric
HS.F.TF.B.5
functions.
HS.F.TF.C
HS.F.TF.C.8
Prove and apply trigonometric identities.
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Page | 2
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Algebra 2 Standards Aligned With the Algebra 2 PARCC Assessment
Performance Based Assessment (PBA/MYA) and End Of Year Assessment (EOY)
Cluster
Standard
Unit 4: Modeling with Functions
HS.A.CED.A
Create equations that describe numbers or
relationships.
HS.F.BF.A
Build a function that models a relationship between
two quantities.
HS.F.IF.B
Interpret functions that arise in applications in terms
of a context.
HS.F.IF.C
Analyze functions using different representations.
HS.F.BF.B
Build new functions from existing functions.
HS.F.LE.A
Construct and compare linear, quadratic, and
exponential models and solve problems.
PBA/MYA
HS.A.CED.A.1
X
HS.A.CED.A.2
HS.A.CED.A.3
HS.A.CED.A.4
HS.F.BF.A.1
HS.F.IF.B.4
HS.F.IF.B.5
HS.F.IF.B.6
HS.F.IF.C.7
X
HS.F.IF.C.8
X
HS.F.BF.B.4
X
HS.F.IF.C.9
HS.F.BF.B.3
HS.F.LE.A.4
EOY
X
X
X
X
X
X
X
X
Page | 3
Curriculum Guide 2014-2015
High School Algebra 2
Unit Overview
Poudre School District
Building on their work with linear, quadratic, and exponential functions, students extend their
repertoire of functions to include polynomial, rational, and radical functions. Students work closely
with the expressions that define the functions, and continue to expand and hone their abilities to
model situations and to solve equations, including solving quadratic equations over the set of
complex numbers and solving exponential equations using the properties of logarithms. The
Mathematical Practice Standards apply throughout each course and, together with the content
standards, prescribe that students experience mathematics as a coherent, useful, and logical subject
that makes use of their ability to make sense of problem situations. The critical areas for this
course, organized into four units are as follows:
Unit 1: Inferences and Conclusions from Data
In this unit, students see how the visual displays and summary statistics they learned in earlier
grades relate to different types of data and to probability distributions. They identify different ways
of collecting data – including sample surveys, experiments, and simulations – and the role that
randomness and careful design play in the conclusions that can be drawn.
Students will be able to…
• Summarize, represent, and interpret data on a single count of measurement variable.
HS.S.ID.A.4
• Understand and evaluate random processes underlying statistical experiments.
HS.S.IC.A.1, HS.S.IC.A.2
• Make inferences and justify conclusions from sample surveys, experiments, and observational
studies.
HS.S.IC.B.3, HS.S.IC.B.4, HS.S.IC.B.5, HS.S.IC.B.6
• Use probability to evaluate outcomes of decisions.
HS.S.MD.B.6, HS.S.MD.B.7
Page | 4
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 2: Polynomial, Rational, and Radical Relationships
This unit develops the structural similarities between the system of polynomials and the system of
integers. Students draw on analogies between polynomial arithmetic and base-ten computation,
focusing on properties of operations, particularly the distributive property. Students connect
multiplication of polynomials with multiplication of multi-digit integers, and division of
polynomials with long division of integers. Students identify zeros of polynomials, including
complex zeros of quadratic polynomials, and make connections between zeros of polynomials and
solutions of polynomial equations. The unit culminates with the fundamental theorem of algebra.
Rational numbers extend the arithmetic of integers by allowing division by all numbers except 0.
Similarly, rational expressions extend the arithmetic of polynomials by allowing division by all
polynomials except the zero polynomial. A central theme of this unit is that the arithmetic of
rational expressions is governed by the same rules as the arithmetic of rational numbers.
Students will be able to…
• Interpret the structure of expressions.
HS.A.SSE.A.1, HS.A.SSE.A.2
• Perform arithmetic operations with complex numbers.
HS.N.CN.A.1, HS.N.CN.A.2
• Use complex numbers in polynomial identities and equations.
HS.N.CN.C.7, HS.N.CN.C.8, HS.N.CN.C.9
• Analyze functions using different representations.
HS.F.IF.C.7
• Perform arithmetic operations on polynomials.
HS.A.APR.A.1
• Understand the relationship between zeros and factors of polynomials.
HS.A.APR.B.2, HS.A.APR.B.3
• Use polynomial identities to solve problems.
HS.A.APR.C.4, HS.A.APR.C.5
• Review rational expressions.
HS.A.APR.D.6, HS.A.APR.D.7
• Understand solving equations as a process of reasoning and explain the reasoning.
HS.A.REI.A.2
• Write expressions in equivalent forms to solve problems.
HS.A.SSE.B.4
• Represent and solve equations and inequalities graphically.
HS.A.REI.D.11
Unit 3: Trigonometric Functions
Building on their previous work with functions, and on their work with trigonometric ratios and
circles in Geometry, students now use the coordinate plane to extend trigonometry to model
periodic phenomena.
Students will be able to…
• Extend the domain of trigonometric functions using the unit circle.
HS.F.TF.A.1, HS.F.TF.A.2
• Model periodic phenomena with trigonometric functions.
HS.F.TF.B.5
• Prove and apply trigonometric identities.
HS.F.TF.C.8
Page | 5
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 4: Modeling with Functions
In this unit students synthesize and generalize what they have learned about a variety of function
families. They extend their work with exponential functions to include solving exponential
equations with logarithms. They explore the effects of transformations on graphs of diverse
functions, including functions arising in an application, in order to abstract the general principle
that transformations on a graph always have the same effect regardless of the type of the
underlying function. They identify appropriate types of functions to model a situation, they adjust
parameters to improve the model, and they compare models by analyzing appropriateness of fit
and making judgments about the domain over which a model is a good fit. The description of
modeling as “the process of choosing and using mathematics and statistics to analyze empirical
situations, to understand them better, and to make decisions” is at the heart of this unit. The narrative
discussion and diagram of the modeling cycle should be considered when knowledge of functions,
statistics, and geometry is applied in a modeling context.
Students will be able to…
• Create equations that describe numbers or relationships.
HS.A.CED.A.1, HS.A.CED.A.2, HS.A.CED.A.3, HS.A.CED.A.4
• Build a function that models a relationship between two quantities.
HS.F.BF.A.1
• Interpret functions that arise in applications in terms of a context.
HS.F.IF.B.4, HS.F.IF.B.5, HS.F.IF.B.6
• Analyze functions using different representations.
HS.F.IF.C.7, HS.F.IF.C.8, HS.F.IF.C.9
• Build new functions from existing functions.
HS.F.BF.B.3, HS.F.BF.B.4
• Construct and compare linear, quadratic, and exponential models and solve problems.
HS.F.LE.A.4
Page | 6
Curriculum Guide 2014-2015
High School Algebra 2
Unit 1: Inferences and Conclusions from Data
Poudre School District
In this unit, students see how the visual displays and summary statistics they learned in earlier
grades relate to different types of data and to probability distributions. They identify different ways
of collecting data – including sample surveys, experiments, and simulations – and the role that
randomness and careful design play in the conclusions that can be drawn.
Sub-Unit A:
The Normal Curve
HS.S.ID.A.4
Sub-Unit B:
Data and Populations
HS.S.IC.A.1
HS.S.IC.A.2
HS.S.IC.B.3
HS.S.IC.B.4
HS.S.IC.B.5
HS.S.IC.B.6
HS.S.MC.B.6
HS.S.MD.B.7
Page | 7
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 1: Inferences and Conclusions from Data
Sub-Unit A: The Normal Curve
August 25, 2014 – August 29, 2014
Common Core State Standards
Explanations/Examples
Resources
HS.S.ID.A: Summarize, represent, and interpret data on a single count or measurement variable.
While students may have heard of the normal distribution, it is unlikely that they will have prior experience using it to make specific estimates.
Build on students’ understanding of data distributions to help them see how the normal distribution uses area to make estimates of frequencies
(which can be expressed as probabilities). Emphasize that only some data are well described by a normal distribution.
HS.S.ID.A.4
HS.S.ID.A.4
Lessons
Use the mean and standard deviation of a data
Students may use spreadsheets, graphing calculators,
HS.S.ID.A
set to fit it to a normal distribution and to
statistical software and tables to analyze the fit
Prentice Hall
estimate population percentages. Recognize that between a data set and normal distributions and
• PH A2
there are data sets for which such a procedure is estimate areas under the curve.
o Ch 12.7
not appropriate. Use calculators, spreadsheets,
• The bar graph below gives the birth weight of a
and tables to estimate areas under the normal
The Practice of Statistics
population of 100 chimpanzees. The line shows
curve.
how the weights are normally distributed about the • Ch 2.2
mean, 3250 grams. Estimate the percent of baby
chimps weighing 3000-3999 grams.
Tasks
Activities
Practice
Assessments
Page | 8
Curriculum Guide 2014-2015
Common Core State Standards
HS.S.ID.A.4 (continued)
High School Algebra 2
Unit 1: Inferences and Conclusions from Data
Sub-Unit A: The Normal Curve
August 25, 2014 – August 29, 2014
Explanations/Examples
HS.S.ID.A.4 (continued)
•
Poudre School District
Resources
Determine which situation(s) is best modeled by a
normal distribution. Explain your reasoning.
o
o
Annual income of a household in the U.S.
Weight of babies born in one year in the U.S.
Page | 9
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 1: Inferences and Conclusions from Data
Sub-Unit B: Data and Populations
September 2, 2014 – October 3, 2014
Common Core State Standards
Explanations/Examples
Resources
HS.S.IC.A: Understand and evaluate random processes underlying statistical experiments.
For S.IC.2, include comparing theoretical and empirical results to evaluate the effectiveness of a treatment.
HS.S.IC.A.1
HS.S.IC.A.1
Lessons
Understand statistics as a process for making
HS.S.IC.A
inferences to be made about population
The Practice of Statistics
parameters based on a random sample from that
• Ch 4.1
population.
Tasks
HS.S.IC.A.2
HS.S.IC.A.2
Decide if a specified model is consistent with
Possible data-generating processes include (but are
Activities
results from a given data-generating process, e.g., not limited to): flipping coins, spinning spinners,
using simulation. For example, a model says a
rolling a number cube, and simulations using the
Practice
spinning coin will fall heads up with probability
random number generators. Students may use
0.5. Would a result of 5 tails in a row cause you to
graphing calculators, spreadsheet programs, or applets Assessments
question the model?
to conduct simulations and quickly perform large
numbers of trials.
The law of large numbers states that as the sample size
increases, the experimental probability will approach
the theoretical probability. Comparison of data from
repetitions of the same experiment is part of the model
building verification process.
•
Have multiple groups flip coins. One group flips a
coin 5 times, one group flips a coin 20 times, and
one group flips a coin 100 times. Which group’s
results will most likely approach the theoretical
probability?
Page | 10
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 1: Inferences and Conclusions from Data
Sub-Unit B: Data and Populations
September 2, 2014 – October 3, 2014
Common Core State Standards
Explanations/Examples
Resources
HS.S.IC.B: Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
In earlier grades, students are introduced to different ways of collecting data and use graphical displays and summary statistics to make
comparisons. These ideas are revisited with a focus on how they way in which data is collected determines the scope and nature of the conclusions
that can be drawn from the data. The concept of statistical significance is developed is unlikely to have occurred solely as a result of random
selection in sampling or random assignment in an experiment.
For S.IC.4 and 5, focus on the variability of results from experiments – that is, focus on statistics as a way of dealing with, not eliminating, inherent
randomness.
HS.S.IC.B.3
HS.S.IC.B.3
Lessons
Recognize the purposes of and differences among Students should be able to explain
HS.S.IC.B
sample surveys, experiments, and observational
techniques/applications for randomly selecting study
The Practice of Statistics
studies; explain how randomization relates to
subjects from a population and how those
• Ch 4.1
each.
techniques/applications differ from those used to
randomly assign existing subjects to control groups or HS.S.IC.B.4/HS.S.IC.B.5/HS.S.IC.B.6
experimental groups in a statistical experiment.
Prentice Hall
• PH A2
In statistics, an observational study draws inferences
o Ch 12.3
about the possible effect of a treatment on subjects,
Tasks
where the assignment of subjects into a treated group
HS.S.IC.B.4
Use data from a sample survey to estimate a
population mean or proportion; develop a
margin of error through the use of simulation
models for random sampling.
versus a control group is outside the control of the
investigator (for example, observing data on academic
achievement and socio-economic status to see if there
is a relationship between them). This is in contrast to
controlled experiments, such as randomized controlled
trials, where each subject is randomly assigned to a
treated group or a control group before the start of the
treatment.
HS.S.IC.B.4
Students may use computer generated simulation
models based upon sample surveys results to estimate
population statistics and margins of error.
Activities
Practice
Assessments
Page | 11
Curriculum Guide 2014-2015
Common Core State Standards
HS.S.IC.B.5
Use data from a randomized experiment to
compare two treatments; use simulations to
decide if differences between parameters are
significant.
HS.S.IC.B.6
Evaluate reports based on data.
High School Algebra 2
Unit 1: Inferences and Conclusions from Data
Sub-Unit B: Data and Populations
September 2, 2014 – October 3, 2014
Explanations/Examples
HS.S.IC.B.5
Students may use computer generated simulation
models to decide how likely it is that observed
differences in a randomized experiment are due to
chance.
Poudre School District
Resources
Treatment is a term used in the context of an
experimental design to refer to any prescribed
combination of values of explanatory variables. For
example, one wants to determine the effectiveness of
weed killer. Two equal parcels of land in a
neighborhood are treated; one with a placebo and one
with weed killer to determine whether there is a
significant difference in effectiveness in eliminating
weeds.
HS.S.IC.B.6
Explanations can include but are not limited to sample
size, biased survey sample, interval scale, unlabeled
scale, uneven scale, and outliers that distort the line-ofbest-fit. In a pictogram the symbol scale used can also
be a source of distortion.
As a strategy, collect reports published in the media
and ask students to consider the source of the data, the
design of the study, and the way the data are analyzed
and displayed.
•
A reporter used the two data sets below to
calculate the mean housing price in Arizona as
$629,000. Why is this calculation not
representative of the typical housing price in
Arizona?
Page | 12
Curriculum Guide 2014-2015
Common Core State Standards
HS.S.IC.B.5 (continued)
High School Algebra 2
Poudre School District
Unit 1: Inferences and Conclusions from Data
Sub-Unit B: Data and Populations
September 2, 2014 – October 3, 2014
Explanations/Examples
HS.S.IC.B.5 (continued)
o King River area {1.2 million, 242000, 265500,
140000, 281000, 265000, 211000}
o
Resources
Toby Ranch homes {5 million, 154000, 250000,
250000, 200000, 160000, 190000}
HS.S.MD.B: Use probability to evaluate outcomes of decisions.
Extend to more complex probability models. Include situations such as those involving quality control, or diagnostic tests that yield both false
positive and false negative results.
HS.S.MD.B.6 (+)
HS.S.MD.B.6 (+)
Lessons
Use probabilities to make fair decisions
Students may use graphing calculators or programs,
HS.S.MD.B
(e.g., drawing by lots, using a random number
spreadsheets, or computer algebra systems to model
Prentice Hall
generator).
and interpret parameters in linear, quadratic or
• PH A2
exponential functions.
o Ch 12.1
HS.S.MD.B.7 (+)
Analyze decisions and strategies using
probability concepts (e.g., product testing,
medical testing, pulling a hockey goalie at the end
of a game).
HS.S.MD.B.7 (+)
Students may use graphing calculators or programs,
spreadsheets, or computer algebra systems to model
and interpret parameters in linear, quadratic or
exponential functions.
Tasks
Activities
Practice
Assessments
Page | 13
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 2: Polynomial, Rational, and Radical Relationships
This unit develops the structural similarities between the system of polynomials and the system of
integers. Students draw on analogies between polynomial arithmetic and base-ten computation,
focusing on properties of operations, particularly the distributive property. Students connect
multiplication of polynomials with multiplication of multi-digit integers, and division of
polynomials with long division of integers. Students identify zeros of polynomials, including
complex zeros of quadratic polynomials, and make connections between zeros of polynomials and
solutions of polynomial equations. The unit culminates with the fundamental theorem of algebra.
Rational numbers extend the arithmetic of integers by allowing division by all numbers except 0.
Similarly, rational expressions extend the arithmetic of polynomials by allowing division by all
polynomials except the zero polynomial. A central theme of this unit is that the arithmetic of
rational expressions is governed by the same rules as the arithmetic of rational numbers.
Sub-Unit A:
Complex Numbers
HS.A.SSE.A.1
HS.N.CN.A.1
HS.N.CN.A.2
HS.A.SSE.A.2
HS.N.CN.C.8
HS.N.CN.C.7
HS.F.IF.C.7
HS.N.CN.C.9
Page | 14
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 2: Polynomial, Rational, and Radical Relationships
Sub-Unit B:
Polynomial Functions
HS.APR.A.1
HS.A.APR.B.3
HS.A.APR.B.2
HS.A.APR.C.4
HS.A.APR.C.5
Sub-Unit C:
Rational Functions
HS.A.REI.A.2
HS.A.APR.D.7
HS.A.APR.D.6
HS.A.SSE.B.4
Sub-Unit D:
Solving Systems
HS.A.REI.D.11
Page | 15
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 2: Polynomials, Rational and Radical Relationships
Sub-Unit A: Complex Numbers
October 6, 2014 – October 31, 2014
Common Core State Standards
Explanations/Examples
Resources
HS.A.SSE.A: Interpret the structure of expressions.
Extend to polynomial and rational expressions.
HS.A.SSE.A.1
HS.A.SSE.A.1
Lessons
Interpret expressions that represent a quantity in Students should understand the vocabulary for the
HS.A.SSE.A
terms of its context.
parts that make up the whole expression and be able
Prentice Hall
to identify those parts and interpret their meaning in
a. Interpret parts of an expression, such as
• PH A2
terms of a context.
terms, factors, and coefficients.
o Ch 5
b. Interpret complicated expressions by
o Ch 6.1-6.5
viewing one or more of their parts as a single
entity. For example, interpret P(1+r)n as the
Tasks
product of P and a factor not depending on P.
HS.A.SSE.A.2
HS.A.SSE.A.2
Activities
Use the structure of an expression to identify
Students should extract the greatest common factor
ways to rewrite it. For example,
(whether a constant, a variable, or a combination of
Practice
see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a
each). If the remaining expression is quadratic,
difference of squares that can be factored as
students should factor the expression further.
Assessments
(x2 – y2)(x2 + y2).
• Factor
HS.N.CN.A: Perform arithmetic operations with complex numbers.
HS.N.CN.A.1
HS.N.CN.A.1
Lessons
Know there is a complex number i such that
HS.N.CN.A
i2 = −1, and every complex number has the form
Prentice Hall
a + bi with a and b real.
• PH A2
HS.N.CN.A.2
HS.N.CN.A.2
o Ch 5.6
Use the relation i2 = –1 and the commutative,
• Simplify the following expression. Justify each
associative, and distributive properties to add,
Tasks
step using the commutative, associative and
subtract, and multiply complex numbers.
distributive properties.
Activities
Practice
Page | 16
Curriculum Guide 2014-2015
Common Core State Standards
HS.N.CN.A.2 (continued)
High School Algebra 2
Unit 2: Polynomials, Rational and Radical Relationships
Sub-Unit A: Complex Numbers
October 6, 2014 – October 31, 2014
Explanations/Examples
HS.N.CN.A.2 (continued)
Assessments
Solutions may vary; one solution follows:
Poudre School District
Resources
HS.N.CN.C: Use complex numbers in polynomial identities and equations.
Limit to polynomials with real coefficients.
HS.N.CN.C.7
HS.N.CN.C.7
Lessons
Solve quadratic equations with real coefficients
HS.N.CN.7/HS.N.CN.8
• Within which number system can x2 = – 2 be
that have complex solutions.
Prentice Hall
solved? Explain how you know.
• PH A2
• Solve x2+ 2x + 2 = 0 over the complex numbers.
o Ch 5.6-5.8
• Find all solutions of 2x2 + 5 = 2x and express them
in the form a + bi.
HS.N.CN.8
HS.N.CN.C.8
HS.N.CN.C.8
Prentice Hall
Extend polynomial identities to the complex
• PH A2
numbers. For example, rewrite x2 + 4 as
o Ch 6.4-6.5
(x + 2i)(x – 2i).
Page | 17
Curriculum Guide 2014-2015
High School Algebra 2
Unit 2: Polynomials, Rational and Radical Relationships
Sub-Unit A: Complex Numbers
October 6, 2014 – October 31, 2014
Common Core State Standards
Explanations/Examples
HS.N.CN.C.9
HS.N.CN.C.9
Know the Fundamental Theorem of Algebra;
• How many zeros does
have? Find all
show that it is true for quadratic polynomials.
the zeros and explain, orally or in written format,
your answer in terms of the Fundamental
Theorem of Algebra.
• How many complex zeros does the following
polynomial have? How do you know?
Poudre School District
Resources
Lessons (continued)
HS.N.CN.9
Prentice Hall
• PH A2
o Ch 6.6
Tasks
Activities
Practice
Assessments
HS.F.IF.C: Analyze functions using different representations.
Relate F.IF.7c to the relationship between zeros of quadratic functions and their factored forms.
HS.F.IF.C.7
HS.F.IF.C.7
Lessons
Graph functions expressed symbolically and
Key characteristics include but are not limited to
HS.F.IF.C.7
show key features of the graph, by hand in simple maxima, minima, intercepts, symmetry, end behavior, Prentice Hall
cases and using technology for more complicated and asymptotes. Students may use graphing
• PH A2
cases.
calculators or programs, spreadsheets, or computer
o Ch 5.2-5.4
c. Graph polynomial functions, identifying zeros algebra systems to graph functions.
o Ch 6.1-6.2
when suitable factorizations are available,
o p. 312
• Describe key characteristics of the graph of
and showing end behavior.
f(x) = │x – 3│ + 5.
Tasks
• Sketch the graph and identify the key
characteristics of the function described below.
Activities
Practice
Assessments
Page | 18
Curriculum Guide 2014-2015
Common Core State Standards
HS.F.IF.C.7 (continued)
High School Algebra 2
Unit 2: Polynomials, Rational and Radical Relationships
Sub-Unit A: Complex Numbers
October 6, 2014 – October 31, 2014
Explanations/Examples
HS.F.IF.C.7 (continued)
•
•
•
Poudre School District
Resources
Graph the function f(x) = 2x by creating a table of
values. Identify the key characteristics of the
graph.
Graph f(x) = 2 tan x – 1. Describe its domain,
range, intercepts, and asymptotes.
Draw the graph of f(x) = sin x and f(x) = cos x.
What are the similarities and differences between
the two graphs?
Page | 19
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 2: Polynomials, Rational and Radical Relationships
Sub-Unit B: Polynomial Functions
November 3, 2014 – December 12, 2014
Common Core State Standards
Explanations/Examples
Resources
HS.A.APR.A: Perform arithmetic operations on polynomials.
Extend beyond the quadratic polynomials found in Algebra 1.
HS.A.APR.A.1
HS.A.APR.A.1
Lessons
Understand that polynomials form a system
HS.A.APR.A.1
analogous to the integers, namely, they are
Prentice Hall
closed under the operations of addition,
• PH A2
subtraction, and multiplication; add, subtract,
o Ch 5.6
and multiply polynomials.
o Ch 6.1-6.5
Tasks
Activities
Practice
Assessments
HS.A.APR.B: Understand the relationship between zeros and factors of polynomials.
HS.A.APR.B.2
HS.A.APR.B.2
Lessons
Know and apply the Remainder Theorem: For a
The Remainder theorem says that if a polynomial p(x) is HS.A.APR.B.2
polynomial p(x) and a number a, the remainder
divided by x – a, then the remainder is the constant p(a). Prentice Hall
on division by x – a is p(a), so p(a) = 0 if and only That is,
So if p(a) = 0 then p(x)
• PH A2
if (x – a) is a factor of p(x).
o Ch 6.3
= q(x)(x-a).
•
Let
. Evaluate p(-2).
What does your answer tell you about the factors of
p(x)? [Answer: p(-2) = 0 so x+2 is a factor.]
HS.A.APR.B.3
Prentice Hall
• PH A2
o Ch 6.4
Tasks
Page | 20
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 2: Polynomials, Rational and Radical Relationships
Sub-Unit B: Polynomial Functions
November 3, 2014 – December 12, 2014
Common Core State Standards
Explanations/Examples
Resources
HS.A.APR.B.3
HS.A.APR.B.3
Activities
Identify zeros of polynomials when suitable
Graphing calculators or programs can be used to
Practice
factorizations are available, and use the zeros to generate graphs of polynomial functions.
construct a rough graph of the function defined
• Factor the expression
and
Assessments
by the polynomial.
explain how your answer can be used to solve the
equation
. Explain why the
solutions to this equation are the same as the xintercepts of the graph of the function
.
HS.A.APR.C: Use polynomial identities to solve problems.
This cluster has many possibilities for optional enrichment, such as relating the example in A.APR.4 to the solution of the system u2+v2=1,
v=t(u+1), relating the Pascal triangle property of binomial coefficients to (x+y)n+1=(x+y)(x+y)n, deriving explicit formulas for the coefficients, or
proving the binomial theorem by induction.
HS.A.APR.C.4
HS.A.APR.C.4
Lessons
Prove polynomial identities and use them to
HS.A.APR.C.5
• Use the distributive law to explain why
describe numerical relationships. For example,
Prentice Hall
x2 – y2 = (x – y)(x + y) for any two numbers x and y.
the polynomial identity
• PH A2
Derive the identity (x – y)2 = x2 – 2xy + y2 from
(x2+y2)2 = (x2– y2)2 + (2xy)2 can be used to
o Ch 6.8
(x + y)2 = x2 + 2xy + y2 by replacing y by –y.
generate Pythagorean triples.
Use an identity to explain the pattern
Tasks
22 – 1 2 = 3
Activities
32 – 22 = 5
Practice
42 – 32 = 7
52 – 42 = 9
Assessments
1)2
[Answer: (n +
whole number n.]
n2
= 2n + 1 for any
Page | 21
Curriculum Guide 2014-2015
High School Algebra 2
Unit 2: Polynomials, Rational and Radical Relationships
Sub-Unit B: Polynomial Functions
November 3, 2014 – December 12, 2014
Common Core State Standards
Explanations/Examples
HS.A.APR.C.5
HS.A.APR.C.5
Know and apply the Binomial Theorem for the
• Use Pascal’s Triangle to expand the expression
expansion of (x + y)n in powers of x and y for a
.
positive integer n, where x and y are any
• Find the middle term in the expansion of
.
numbers, with coefficients determined for
example by Pascal’s Triangle. (The Binomial
Theorem can be proved by mathematical
induction or by a combinatorial argument.)
↑
4C0
↑
4C1
↑
4C2
↑
4C3
Poudre School District
Resources
↑
4C4
Page | 22
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 2: Polynomials, Rational and Radical Relationships
Sub-Unit C: Rational Functions
January 6, 2015 – January 23, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.A.APR.D: Review rational expressions.
The limitations on rational functions apply to the rational expressions in A.APR.6. A.APR.7 requires the general division algorithm for
polynomials.
HS.A.APR.D.6
HS.A.APR.D.6
Lessons
Rewrite simple rational expressions in different
The polynomial q(x) is called the quotient and the
HS.A.APR.D.6
polynomial r(x) is called the remainder. Expressing a
Prentice Hall
forms; write a(x)/b(x) in the form q(x) +
rational expression in this form allows one to see
• PH A2
r(x)/b(x), where a(x), b(x), q(x), and r(x) are
different
properties
of
the
graph,
such
as
horizontal
o Ch 9.3
polynomials with the degree of r(x) less than the
asymptotes.
degree of b(x), using inspection, long division, or,
HS.A.APR.D.7
for the more complicated examples, a computer
• Find the quotient and remainder for the rational
Prentice Hall
3
2
algebra system.
𝑥𝑥 −3𝑥𝑥 +𝑥𝑥−6
expression
and
use
them
to
write
the
2
• PH A2
𝑥𝑥 +2
o Ch 9.4-9.5
expression in a different form.
•
HS.A.APR.D.7
Understand that rational expressions form a
system analogous to the rational numbers, closed
under addition, subtraction, multiplication, and
division by a nonzero rational expression; add,
subtract, multiply, and divide rational
expressions.
2𝑥𝑥+1
Express 𝑓𝑓(𝑥𝑥) =
in a form that reveals the
𝑥𝑥−1
horizontal asymptote of its graph.
[Answer: Error! Digit expected., so the horizontal
asymptote is y = 2.]
HS.A.APR.D.7
• Use the formula for the sum of two fractions to
explain why the sum of two rational expressions is
another rational expression.
•
1
Tasks
Activities
Practice
Assessments
1
Express 2 +1 − 2 −1 in the form 𝑎𝑎(𝑥𝑥)/𝑏𝑏(𝑥𝑥), where
𝑥𝑥
𝑥𝑥
a(x) and b(x) are polynomials.
Page | 23
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 2: Polynomials, Rational and Radical Relationships
Sub-Unit C: Rational Functions
January 6, 2015 – January 23, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.A.REI.A: Understand solving equations as a process of reasoning and explain the reasoning.
Extend to simple rational and radical equations.
HS.A.REI.A.2
HS.A.REI.A.2
Lessons
Solve simple rational and radical equations in
HS.A.REI.A.2
•
one variable, and give examples showing how
Prentice Hall
extraneous solutions may arise.
• PH A2
•
o Ch 7.5
o Ch 9.6
𝑥𝑥+2
•
=2
•
𝑥𝑥+3
Tasks
Activities
Practice
Assessments
HS.A.SSE.B: Write expressions in equivalent forms to solve problems.
Consider extending A.SSE.4 to infinite geometric series in curricular implementations of this course description.
HS.A.SSE.B.4
HS.A.SSE.B.4
Lessons
Derive the formula for the sum of a finite
• In February, the Bezanson family starts saving for a HS.A.SSE.B.4
geometric series (when the common ratio is not
Prentice Hall
trip to Australia in September. The Bezanson’s
1), and use the formula to solve problems. For
expect their vacation to cost $5375. They start with • PH A2
example, calculate mortgage payments.
o Ch 11.5
$525. Each month they plan to deposit 20% more
than the previous month. Will they have enough
Tasks
money for their trip?
Activities
Practice
Page | 24
Curriculum Guide 2014-2015
Common Core State Standards
High School Algebra 2
Unit 2: Polynomials, Rational and Radical Relationships
Sub-Unit C: Rational Functions
January 6, 2015 – January 23, 2015
Explanations/Examples
Poudre School District
Assessments
Resources
Page | 25
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 2: Polynomials, Rational and Radical Relationships
Sub-Unit D: Solving Systems
January 26, 2015 – January 30, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.A.REI.D: Represent and solve equations and inequalities graphically.
Include combinations of linear, polynomial, rational, radical, absolute value, and exponential functions.
HS.A.REI.D.11
HS.A.REI.D.11
Lessons
Explain why the x-coordinates of the points where Students need to understand that numerical solution
HS.A.REI.D.11
the graphs of the equations y = f(x) and
methods (data in a table used to approximate an
Prentice Hall
y = g(x) intersect are the solutions of the equation algebraic function) and graphical solution methods
• PH A2
f(x) = g(x); find the solutions approximately, e.g.,
may produce approximate solutions, and algebraic
o Ch 3.1-3.3
using technology to graph the functions, make
solution methods produce precise solutions that can
o p. 296-297
tables of values, or find successive
be represented graphically or numerically. Students
o p. 589
approximations. Include cases where f(x) and/or
may use graphing calculators or programs to generate
g(x) are linear, polynomial, rational, absolute
tables of values, graph, or solve a variety of functions.
Tasks
value, exponential, and logarithmic functions.
• Given the following equations determine the x
value that results in an equal output for both
Activities
functions.
Practice
Assessments
Page | 26
Curriculum Guide 2014-2015
High School Algebra 2
Unit 3: Trigonometric Functions
Poudre School District
Building on their previous work with functions, and on their work with trigonometric ratios and
circles in Geometry, students now use the coordinate plane to extend trigonometry to model
periodic phenomena.
HS.F.TF.A.2
HS.F.TF.A.1
HS.F.TF.B.5
HS.F.TF.C.8
Page | 27
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 3: Trigonometric Functions
February 2, 2015 – February 20, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.F.TF.A: Extend the domain of a trigonometric function using the unit circle.
HS.F.TF.A.1
HS.F.TF.A.1
Lessons
Understand radian measure of an angle as the
HS.F.TF.A
length of the arc on the unit circle subtended by
Prentice Hall
the angle.
• PH A2
o Ch 13.2-13.3
HS.F.TF.A.2
HS.F.TF.A.2
Explain how the unit circle in the coordinate
Students may use applets and animations to explore
Tasks
plane enables the extension of trigonometric
the unit circle and trigonometric functions. Students
functions to all real numbers, interpreted as
may explain (orally or in written format) their
Activities
radian measures of angles traversed
understanding.
counterclockwise around the unit circle.
Practice
Assessments
HS.F.TF.B: Model periodic phenomena with trigonometric functions.
HS.F.TF.B.5
HS.F.TF.B.5
Lessons
Choose trigonometric functions to model
Students may use graphing calculators or programs,
HS.F.TF.B.5
periodic phenomena with specified amplitude,
spreadsheets, or computer algebra systems to model
Prentice Hall
frequency, and midline.
trigonometric functions and periodic phenomena.
• PH A2
o Ch 13.1
• The temperature of a chemical reaction oscillates
o Ch 13.4-14.6
between a low of
C and a high of
C. The
temperature is at its lowest point when t = 0 and
Tasks
completes one cycle over a six hour period.
a.
b.
Sketch the temperature, T, against the elapsed
time, t, over a 12 hour period.
Find the period, amplitude, and the midline of
the graph you drew in part a).
Activities
Practice
Assessments
Page | 28
Curriculum Guide 2014-2015
Common Core State Standards
HS.F.TF.B.5 (continued)
High School Algebra 2
Poudre School District
Unit 3: Trigonometric Functions
February 2, 2015 – February 20, 2015
Explanations/Examples
HS.F.TF.B.5 (continued)
c. Write a function to represent the relationship
between time and temperature.
d.
e.
Resources
What will the temperature of the reaction be 14
hours after it began?
At what point during a 24 hour day will the
reaction have a temperature of
C?
HS.F.TF.C: Prove and apply trigonometric identities.
An Algebra II course with an additional focus on trigonometry could include the (+) standard F.TF.9: Prove the addition and subtraction formulas
for sine, cosine, and tangent and use them to solve problems. This could be limited to acute angles in Algebra II.
HS.F.TF.C.8
HS.F.TF.C.8
Lessons
Prove the Pythagorean identity
HS.F.TF.C.8
sin2(θ) + cos2(θ) = 1 and use it to find sin(θ),
Prentice Hall
cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ)
• PH A2
and the quadrant of the angle.
o Ch 14.1
Tasks
Activities
Practice
Assessments
Page | 29
Curriculum Guide 2014-2015
High School Algebra 2
Unit 4: Modeling with Functions
Poudre School District
In this unit students synthesize and generalize what they have learned about a variety of function
families. They extend their work with exponential functions to include solving exponential
equations with logarithms. They explore the effects of transformations on graphs of diverse
functions, including functions arising in an application, in order to abstract the general principle
that transformations on a graph always have the same effect regardless of the type of the
underlying function. They identify appropriate types of functions to model a situation, they adjust
parameters to improve the model, and they compare models by analyzing appropriateness of fit
and making judgments about the domain over which a model is a good fit. The description of
modeling as “the process of choosing and using mathematics and statistics to analyze empirical
situations, to understand them better, and to make decisions” is at the heart of this unit. The narrative
discussion and diagram of the modeling cycle should be considered when knowledge of functions,
statistics, and geometry is applied in a modeling context.
Sub-Unit A:
Writing Functions
HS.A.CED.A.1
HS.A.CED.A.2
HS.A.CED.A.4
HS.F.IF.B.5
HS.F.BF.A.1
Curriculum Guide 2014-2015
High School Algebra 2
Unit 4: Modeling with Functions
Poudre School District
Sub-Unit B:
Interpreting Functions
HS.F.IF.B.6
HS.F.IF.C.8
HS.F.IF.B.4
HS.F.IF.C.7
HS.F.IF.C.9
HS.F.BF.B.3
HS.A.CED.A.3
Sub-Unit C:
Inverse Functions
HS.F.BF.B.4
HS.F.LE.A.4
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 4: Modeling with Functions
Sub-Unit A: Writing Functions
February 23, 2015 – March 13, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.A.CED.A: Create equations that describe numbers or relationships.
For A.CED.1, use all available types of functions to create such equations, including root functions, but constrain to simple cases. While functions
used in A.CED.2, 3, and 4, will often be linear, exponential, or quadratic the types of problems should draw from more complex situations than
those addressed in Algebra 1. For example, finding the equation of a line through a given point perpendicular to another line allows one to find the
distance from a point to a line. Note that the examples given for A.cED.4 applies to earlier instances of this standard, not to the current course.
HS.A.CED.A.1
HS.A.CED.A.1
Lessons
Create equations and inequalities in one variable Equations can represent real world and mathematical
HS.A.CED.A.2
and use them to solve problems. Include
problems. Include equations and inequalities that arise Prentice Hall
equations arising from linear and quadratic
when comparing the values of two different functions,
• PH A2
functions, and simple rational and exponential
such as one describing linear growth and one
o Ch 2.2-2.7
functions.
describing exponential growth.
o Ch 5.1
o Ch 5.3
• Given that the following trapezoid has area 54
o Ch 5.5
cm2, set up an equation to find the length of the
o Ch 6.1
base, and solve the equation.
•
HS.A.CED.A.2
Create equations in two or more variables to
represent relationships between quantities;
graph equations on coordinate axes with labels
and scales.
Lava coming from the eruption of a volcano
follows a parabolic path. The height h in feet of a
piece of lava t seconds after it is ejected from the
volcano is given by ℎ(𝑡𝑡) = −𝑡𝑡 2 + 16𝑡𝑡 + 936.
After how many seconds does the lava reach its
maximum height of 1000 feet?
Tasks
Activities
Practice
Assessments
HS.A.CED.A.2
Page | 32
Curriculum Guide 2014-2015
Common Core State Standards
HS.A.CED.A.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.
High School Algebra 2
Unit 4: Modeling with Functions
Sub-Unit A: Writing Functions
February 23, 2015 – March 13, 2015
Explanations/Examples
HS.A.CED.A.4
• The Pythagorean Theorem expresses the relation
between the legs a and b of a right triangle and its
hypotenuse c with the equation a2 + b2 = c2.
o Why might the theorem need to be solved for
c?
o Solve the equation for c and write a problem
situation where this form of the equation
might be useful.
o
•
Solve
Poudre School District
Resources
for radius r.
Motion can be described by the formula below,
where t = time elapsed, u=initial velocity, a =
acceleration, and s = distance traveled
s = ut+½at2
o Why might the equation need to be rewritten in
terms of a?
o Rewrite the equation in terms of a.
HS.F.BF.A: Build a function that models a relationship between two quantities.
Develop models for more complex or sophisticated situations than in previous courses.
HS.F.BF.A.1
HS.G.BF.A.1
Lessons
Write a function that describes a relationship
Students will analyze a given problem to determine the HS.F.BF.A
between two quantities.
function expressed by identifying patterns in the
Prentice Hall
function’s rate of change. They will specify intervals of
b. Combine standard function types using
• PH A2
increase, decrease, constancy, and, if possible, relate
arithmetic operations. For example, build a
o Ch 2.2-2.7
them to the function’s description in words or
function that models the temperature of a
o Ch 5.1
graphically. Students may use graphing calculators or
cooling body by adding a constant function to
o Ch 5.3
programs, spreadsheets, or computer algebra systems
a decaying exponential, and relate these
o Ch 5.5
to model functions.
functions to the model.
o Ch 6.1
Page | 33
Curriculum Guide 2014-2015
Common Core State Standards
HS.F.BF.A.1 (continued)
High School Algebra 2
Unit 4: Modeling with Functions
Sub-Unit A: Writing Functions
February 23, 2015 – March 13, 2015
Explanations/Examples
HS.F.BF.A.1 (continued)
• You buy a $10,000 car with an annual interest rate
of 6 percent compounded annually and make
monthly payments of $250. Express the amount
remaining to be paid off as a function of the
number of months, using a recursion equation.
•
•
A cup of coffee is initially at a temperature of 93º
F. The difference between its temperature and the
room temperature of 68º F decreases by 9% each
minute. Write a function describing the
temperature of the coffee as a function of time.
Poudre School District
Tasks
Resources
Activities
Practice
Assessments
The radius of a circular oil slick after t hours is
given in feet by 𝑟𝑟 = 10𝑡𝑡 2 − 0.5𝑡𝑡, for 0 ≤ t ≤ 10.
Find the area of the oil slick as a function of time.
HS.F.IF.B: Build new functions from existing functions.
Use transformations of functions to find models as students consider increasingly more complex situations.
HS.F.IF.B.5
HS.F.IF.B.5
Lessons
Relate the domain of a function to its graph and,
Students may explain orally, or in written format, the
HS.F.IF.B.5
where applicable, to the quantitative relationship existing relationships.
Prentice Hall
it describes. For example, if the function h(n) gives
• PH A2
the number of person-hours it takes to assemble n
o Ch 2.1
engines in a factory, then the positive integers
would be an appropriate domain for the function.
Tasks
Activities
Practice
Assessments
Page | 34
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 4: Modeling with Functions
Sub-Unit B: Interpreting Functions
March 23, 2014 – April 24, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.F.IF.B: Interpret functions that arise in applications in terms of a context.
Emphasize the selection of a model function based on behavior of data and context.
HS.F.IF.B.4
HS.F.IF.B.4
Lessons
For a function that models a relationship
Students may be given graphs to interpret or produce
HS.F.IF.B.4
between two quantities, interpret key features of graphs given an expression or table for the function, by Prentice Hall
graphs and tables in terms of the quantities, and
hand or using technology.
• PH A2
sketch graphs showing key features given a
o Ch 2.2
• A rocket is launched from 180 feet above the
verbal description of the relationship. Key
o Ch 2.5
ground at time t = 0. The function that models this
features include: intercepts; intervals where the
o Ch 5.1
situation is given by h = – 16t2 + 96t + 180, where
function is increasing, decreasing, positive, or
o Ch 6.1
t is measured in seconds and h is height above the
negative; relative maximums and minimums;
o Ch 8.1
ground measured in feet.
symmetries; end behavior; and periodicity.
o Ch 9.3
o What is a reasonable domain restriction for t
o Ch 13.1
in this context?
o
o
o
o
o
•
Determine the height of the rocket two
seconds after it was launched.
Determine the maximum height obtained by
the rocket.
Determine the time when the rocket is 100
feet above the ground.
Tasks
Activities
Practice
Assessments
Determine the time at which the rocket hits
the ground.
How would you refine your answer to the first
question based on your response to the
second and fifth questions?
Compare the graphs of y = 3x2 and y = 3x3.
Page | 35
Curriculum Guide 2014-2015
Common Core State Standards
HS.F.IF.B.4 (continued)
High School Algebra 2
Unit 4: Modeling with Functions
Sub-Unit B: Interpreting Functions
March 23, 2014 – April 24, 2015
Explanations/Examples
HS.F.IF.B.4 (continued)
•
•
•
HS.F.IF.B.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.
Poudre School District
Resources
. Find the domain of R(x). Also
Let
find the range, zeros, and asymptotes of R(x).
Let
. Graph the function and
identify end behavior and any intervals of
constancy, increase, and decrease.
It started raining lightly at 5am, then the rainfall
became heavier at 7am. By 10am the storm was
over, with a total rainfall of 3 inches. It didn’t rain
for the rest of the day. Sketch a possible graph for
the number of inches of rain as a function of time,
from midnight to midday.
HS.F.IF.B.6
The average rate of change of a function y = f(x) over
𝛥𝛥𝛥𝛥
𝑓𝑓(𝑏𝑏)−𝑓𝑓(𝑎𝑎)
an interval [a,b] is
= 𝑏𝑏−𝑎𝑎
𝛥𝛥𝛥𝛥
In addition to finding average rates of change from
functions given symbolically, graphically, or in a table,
Students may collect data from experiments or
simulations (ex. falling ball, velocity of a car, etc.) and
find average rates of change for the function modeling
the situation.
Page | 36
Curriculum Guide 2014-2015
Common Core State Standards
HS.F.IF.B.6 (continued)
High School Algebra 2
Unit 4: Modeling with Functions
Sub-Unit B: Interpreting Functions
March 23, 2014 – April 24, 2015
Explanations/Examples
HS.F.IF.B.6 (continued)
• Use the following table to find the average rate of
change of g over the intervals [-2, -1] and [0,2]:
x
g(x)
-2 2
-1 -1
0
-4
2
-10
• The table below shows the elapsed time when two
different cars pass a 10, 20, 30, 40 and 50 meter
mark on a test track.
o
o
Poudre School District
Resources
For car 1, what is the average velocity (change
in distance divided by change in time)
between the 0 and 10 meter mark? Between
the 0 and 50 meter mark? Between the 20 and
30 meter mark? Analyze the data to describe
the motion of car 1.
How does the velocity of car 1 compare to that
of car 2?
Car 1 Car 2
d
t
t
10 4.472 1.742
20 6.325 2.899
30 7.746 3.831
40 8.944 4.633
50 10
5.348
Page | 37
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 4: Modeling with Functions
Sub-Unit B: Interpreting Functions
March 23, 2014 – April 24, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.F.IF.C: Analyze functions using different representation.
Focus on applications and how key features relate to characteristics of a situation, making selection of a particular type of function model
appropriate.
HS.F.IF.C.7
HS.F.IF.C.7
Lessons
Graph functions expressed symbolically and
Key characteristics include but are not limited to
HS.F.IF.C.7
show key features of the graph, by hand in simple maxima, minima, intercepts, symmetry, end behavior,
Prentice Hall
cases and using technology for more complicated and asymptotes. Students may use graphing
• PH A2
cases.
calculators or programs, spreadsheets, or computer
o Ch 2.2
b. Graph square root, cube root, and piecewise- algebra systems to graph functions.
o Ch 2.5
defined functions, including step functions
o p. 71
• Describe key characteristics of the graph of
and absolute value functions.
o Ch 5.2-5.4
f(x) = │x – 3│ + 5.
o Ch 6.1-6.2
e. Graph exponential and logarithmic functions,
• Sketch the graph and identify the key
o p. 312
showing intercepts and end behavior, and
characteristics of the function described below.
o Ch 7.8
trigonometric functions, showing period,
o Ch 8.1
midline, and amplitude.
o Ch 9.3
Tasks
Activities
Practice
Assessments
•
Graph the function f(x) = 2x by creating a table of
values. Identify the key characteristics of the
graph.
Page | 38
Curriculum Guide 2014-2015
Common Core State Standards
HS.F.IF.C.7 (continued)
HS.F.IF.C.8
Write a function defined by an expression in
different but equivalent forms to reveal and
explain different properties of the function.
HS.F.IF.C.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.
High School Algebra 2
Unit 4: Modeling with Functions
Sub-Unit B: Interpreting Functions
March 23, 2014 – April 24, 2015
Explanations/Examples
HS.F.IF.C.7 (continued)
•
Graph f(x) = 2 tan x – 1. Describe its domain,
range, intercepts, and asymptotes.
•
Draw the graph of f(x) = sin x and f(x) = cos x.
What are the similarities and differences
between the two graphs?
Poudre School District
Resources
HS.F.IF.C.8
HS.F.IF.C.9
•
Examine the functions below. Which function
has the larger maximum? How do you know?
Page | 39
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 4: Modeling with Functions
Sub-Unit B: Interpreting Functions
March 23, 2014 – April 24, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.F.BF.B: Build new functions from existing functions.
Use transformation of functions to find models as students consider increasingly more complex situations. For F.BF.3, note the effect of multiple
transformations on a single graph and the common effect of each transformation across function types.
HS.F.BF.B.3
HS.F.BF.B.3
Lessons
Identify the effect on the graph of replacing f(x)
Students will apply transformations to functions and
Prentice Hall
by f(x) + k, k f(x), f(kx), and f(x + k) for specific
recognize functions as even and odd. Students may use
• PH A2
values of k (both positive and negative); find the
graphing calculators or programs, spreadsheets, or
o Ch 2.6
value of k given the graphs. Experiment with
computer algebra systems to graph functions.
o Ch 5.3
cases and illustrate an explanation of the effects
o Ch 7.8
• Is f(x) = x3 - 3x2 + 2x + 1 even, odd, or neither?
on the graph using technology. Include
o Ch 8.2
Explain your answer orally or in written format.
recognizing even and odd functions from their
o Ch 9.3
• Compare the shape and position of the graphs of
graphs and algebraic expressions for them.
o Ch 13.7
and
, and explain the
differences in terms of the algebraic expressions
Tasks
for the functions.
Activities
Practice
Assessments
•
Describe effect of varying the parameters a, h, and
k have on the shape and position of the graph of
f(x) = a(x-h)2 + k.
Page | 40
Curriculum Guide 2014-2015
Common Core State Standards
HS.F.BF.B.3 (continued)
High School Algebra 2
Unit 4: Modeling with Functions
Sub-Unit B: Interpreting Functions
March 23, 2014 – April 24, 2015
Explanations/Examples
HS.F.BF.B.3 (continued)
• Compare the shape and position of the graphs of
to
, and explain the
differences, orally or in written format, in terms of
the algebraic expressions for the functions.
•
•
Poudre School District
Resources
Describe the effect of varying the parameters a, h,
and k on the shape and position of the graph f(x) =
ab(x + h) + k., orally or in written format. What effect
do values between 0 and 1 have? What effect do
negative values have?
Compare the shape and position of the graphs of y
= sin x to y = 2 sin x.
Page | 41
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 4: Modeling with Functions
Sub-Unit B: Interpreting Functions
March 23, 2014 – April 24, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.A.CED.A: Create equations that describe numbers or relationships.
While functions used in A.CED.2, 3, and 4 will often be linear, exponential, or quadratic, the types of problems should draw from more complex
situations that those addressed in Algebra 1. For example, finding the equation of a line through a given point perpendicular to another line
allows one to find the distance from a point to a line.
HS.A.CED.A.3
HS.A.CED.A.3
Lessons
Represent constraints by equations or
HS.A.CED.A.3
• A club is selling hats and jackets as a fundraiser.
inequalities, and by systems of equations and/or
Prentice Hall
Their budget is $1500 and they want to order at
inequalities, and interpret solutions as viable or
• PH A2
least 250 items. They must buy at least as many
non-viable options in a modeling context. For
o Ch 3.4
hats as they buy jackets. Each hat costs $5 and
example, represent inequalities describing
each jacket costs $8.
nutritional and cost constraints on combinations
o Write a system of inequalities to represent the Tasks
of different foods.
situation.
Activities
o Graph the inequalities.
o If the club buys 150 hats and 100 jackets, will
Practice
the conditions be satisfied?
o What is the maximum number of jackets they
Assessments
can buy and still meet the conditions?
Page | 42
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 4: Modeling with Functions
Sub-Unit C: Inverse Functions
April 27, 2015 – May 15, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.F.BF.B: Build new functions from existing functions.
Use transformations of functions to find models as students consider increasingly more complex situations.
Extend F.BF.4a to simple rational, simple radical, and simple exponential functions; connect F.BF.4a to F.LE.4.
HS.F.BF.B.4
HS.F.BF.B.4
Lessons
Find inverse functions.
Students may use graphing calculators or programs,
HS.F.IF.B
spreadsheets, or computer algebra systems to model
Prentice Hall
a. Solve an equation of the form f(x) = c for a
functions.
• PH A2
simple function f that has an inverse and
o Ch 7.7
write an expression for the inverse. For
• For the function h(x) = (x – 2)3, defined on the
3
example, f(x) =2 x or f(x) = (x+1)/(x-1) for
domain of all real numbers, find the inverse
Tasks
x ≠ 1.
function if it exists or explain why it doesn’t exist.
•
•
Graph h(x) and h-1(x) and explain how they relate
to each other graphically.
Find a domain for f(x) = 3x2 + 12x - 8 on which it
has an inverse. Explain why it is necessary to
restrict the domain of the function.
Activities
Practice
Assessments
Page | 43
Curriculum Guide 2014-2015
High School Algebra 2
Poudre School District
Unit 4: Modeling with Functions
Sub-Unit C: Inverse Functions
April 27, 2015 – May 15, 2015
Common Core State Standards
Explanations/Examples
Resources
HS.F.LE.A: Construct and compare linear, quadratic, and exponential models and solve problems.
Consider extending this unit to include the relationship between properties of logarithms and properties of exponents, such as the connection
between the properties of exponents and the basic logarithm property that 𝐥𝐥𝐥𝐥𝐥𝐥 𝒙𝒙𝒙𝒙 = 𝐥𝐥𝐥𝐥𝐥𝐥 𝒙𝒙 + 𝐥𝐥𝐥𝐥𝐥𝐥 𝒚𝒚.
HS.F.LE.A.4
HS.F.LE.A.4
Lessons
For exponential models, express as a logarithm
Students may use graphing calculators or programs,
HS.F.LE.A.4
ct
the solution to ab = d where a, c, and d are
spreadsheets, or computer algebra systems to analyze
Prentice Hall
numbers and the base b is 2, 10, or e; evaluate the exponential models and evaluate logarithms.
• PH A2
logarithm using technology.
o Ch 8.3-8.6
• Solve 200 e0.04t = 450 for t.
Solution:
We first isolate the exponential part by dividing
both sides of the equation by 200.
e0.04t = 2.25
Now we take the natural logarithm of both sides.
ln e0.04t = ln 2.25
Tasks
Activities
Practice
Assessments
The left hand side simplifies to 0.04t, by
logarithmic identity 1.
0.04t = ln 2.25
Lastly, divide both sides by 0.04.
t = ln (2.25) / 0.04
t
20.3
Page | 44
Performance Level Descriptors – Algebra II
Equivalent
Expressions
HS.N.RN.A.2
A.Int.1
HS.A.REI.A.2
HS.A.SSE.A.2-3
HS.A.SSE.A.2-6
HS.A.SSE.B.3c-2
Interpreting
Functions
HS.A.APR.A.2
HS.A.REI.D.11-2
HS.F.IF.B.4-2
Algebra II: Sub-Claim A
The student solves problems involving the Major Content for the grade/course with connections to the
Standards for Mathematical Practice.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
Uses mathematical
properties and structure of
polynomial, exponential,
rational and radical
expressions to create
equivalent expressions that
aid in solving mathematical
and contextual problems
with three or more steps
required.
Rewrites exponential
expressions to reveal
quantities of interest that
may be useful.
Uses mathematical
properties and relationships
to reveal key features of
polynomial, exponential,
rational, trigonometric and
logarithmic functions, using
them to sketch graphs and
identify characteristics of the
relationship between two
quantities, and applying the
remainder theorem where
appropriate.
Identifies how changing the
Uses mathematical
properties and structure of
polynomial, exponential,
rational and radical
expressions to create
equivalent expressions that
aid in solving mathematical
and contextual problems
with two steps required.
Rewrites exponential
expressions to reveal
quantities of interest that
may be useful.
Uses mathematical
properties and relationships
to reveal key features of
polynomial, exponential,
rational, trigonometric and
logarithmic functions, using
them to sketch graphs and
identify characteristics of
the relationship between
two quantities, and
applying the remainder
theorem where
appropriate.
Uses mathematical
properties and structure of
polynomial, exponential and
rational expressions to
create equivalent
expressions.
Uses provided
mathematical properties and
structure of polynomial and
exponential expressions to
create equivalent
expressions.
Rewrites exponential
expressions to reveal
quantities of interest that
may be useful.
Interprets key features of
graphs and tables, and uses
mathematical properties
and relationships to reveal
key features of polynomial,
exponential and rational
functions, using them to
sketch graphs.
Uses provided
mathematical properties
and relationships to reveal
key features of polynomial
and exponential functions,
using them to sketch
graphs.
Page | 45
Performance Level Descriptors – Algebra II
Algebra II: Sub-Claim A
The student solves problems involving the Major Content for the grade/course with connections to the
Standards for Mathematical Practice.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
parameters of functions
impacts key features of
graphs.
Rate of Change
HS.FIF.B.6-2
HS.F.IF.B.6-7
Calculates and interprets the
average rate of change of
polynomial, exponential,
logarithmic or trigonometric
functions (presented
symbolically or as a table)
over a specified interval, and
estimates the rate of change
from a graph.
Compares rates of change
associated with different
intervals.
Building
Functions
HS.A.SSE.B.4-2
F-Int.3
HS.F.BF.A.1b-1
HS.F.BF.A.2
Builds functions that model
mathematical and contextual
situations, including those
requiring multiple
trigonometric functions,
sequences and combinations
of these and other functions,
and uses the models to solve,
interpret and generalize
about problems.
Calculates and interprets
the average rate of change
of polynomial, exponential,
logarithmic or
trigonometric functions
(presented symbolically or
as a table) over a specified
interval, and estimates the
rate of change from a
graph.
Calculates the average rate
of change of polynomial and
exponential functions
(presented symbolically or
as a table) over a specified
interval, and estimates the
rate of change from a
graph.
Calculates the average rate
of change of polynomial and
exponential functions
(presented symbolically or
as a table) over a specified
interval.
Builds functions that model
mathematical and
contextual situations,
including those requiring
trigonometric functions,
sequences and
combinations of these and
other functions, and uses
the models to solve,
interpret and generalize
about problems.
Builds functions that model
mathematical and
contextual situations,
including those requiring
trigonometric functions,
sequences and
combinations of these and
other functions, and uses
the models to solve and
interpret problems.
Builds functions that model
mathematical and
contextual situations,
limited to those requiring
arithmetic and geometric
sequences, and uses the
models to solve and
interpret problems.
Page | 46
Performance Level Descriptors – Algebra II
Statistics &
Probability
HS.S.IC.B.3-1
Algebra II: Sub-Claim A
The student solves problems involving the Major Content for the grade/course with connections to the
Standards for Mathematical Practice.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
Determines why a sample
survey, experiment or
observational study is most
appropriate.
Determines why a sample
survey, experiment or
observational study is most
appropriate.
Given an inappropriate choice
of a sample survey,
experiment or observational
study, determines how to
change the scenario to make
the choice appropriate.
Given an inappropriate
choice of a sample survey,
experiment or
observational study,
identifies and supports the
appropriate choice.
Determines whether a
sample survey, experiment
or observational study is
most appropriate.
Identifies whether a given
scenario represents a
sample survey, experiment
or observational study.
Page | 47
Performance Level Descriptors – Algebra II
Interpreting
Functions
HS.F.IF.C.7c
HS.F.IF.C.7e-1
HS.F.IF.C.7e-2
HS.F.IF.C.8b
HS.F.IF.C.9-2
F-Int.1-2
Algebra II: Sub-Claim B
The student solves problems involving the Additional and Supporting Content for the grade/course with
connections to the Standards for Mathematical Practice.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
Given multiple functions in
different forms
(algebraically, graphically,
numerically and by verbal
description), writes multiple
equivalent versions of the
functions, and identifies and
compares key features.
Given multiple functions in
different forms
(algebraically, graphically,
numerically and by verbal
description), writes multiple
equivalent versions of the
functions, and identifies and
compares key features.
Graphs polynomial,
exponential, trigonometric
and logarithmic functions,
showing key features.
Graphs exponential,
polynomial and
trigonometric functions,
showing key features.
Given functions represented
algebraically, graphically,
numerically and by verbal
description, writes multiple
equivalent versions of the
functions and identifies key
features.
Given functions represented
algebraically, graphically,
numerically and by verbal
description, writes
equivalent versions of the
functions, and identifies key
features.
Graphs exponential and
polynomial functions,
showing key features.
Graphs polynomial
functions, showing key
features.
Uses commutative and
associative properties to
perform operations with
complex numbers.
Determines how the
changes of a parameter in
functions impact their
other representations.
Equivalent
Expressions
HS.N.CN.A.1
HS.N.CN.A.2
HS.A.APR.D.6
Uses commutative,
associative and distributive
properties to perform
operations with complex
numbers.
Uses commutative,
associative and distributive
properties to perform
operations with complex
numbers.
Uses commutative,
associative and distributive
properties to perform
operations with complex
numbers.
Rewrites simple rational
expressions using inspection
or long division, and
determines how different
Rewrites simple rational
expressions using inspection
or long division.
Rewrites simple rational
expressions using
inspection.
Page | 48
Performance Level Descriptors – Algebra II
Algebra II: Sub-Claim B
The student solves problems involving the Additional and Supporting Content for the grade/course with
connections to the Standards for Mathematical Practice.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
forms of an expression
reveal useful information.
Function
Transformations
HS.F.BF.B.3-2
HS.F.BF.B.3-3
HS.F.BF.B.3-5
Trigonometry
HS.F.TF.A.1
HS.F.TF.C.8-2
Given a context that infers
particular transformations,
identifies the effects on
graphs of polynomial,
exponential, logarithmic
and trigonometric
functions, and determines if
the resulting function is
even or odd.
Identifies the effects of
multiple transformations on
graphs of polynomial,
exponential, logarithmic
and trigonometric
functions, and determines if
the resulting function is
even or odd.
Identifies the effects of a
single transformation on
graphs of polynomial,
exponential, logarithmic
and trigonometric functions
– including f(x)+k, kf(x),
f(kx), and f(x+k) – and
determines if the resulting
function is even or odd.
Identifies the effects of a
single transformation on
graphs of polynomial,
exponential, logarithmic
and trigonometric functions
– limited to f(x)+k and kf(x)
– and determines if the
resulting function is even or
odd.
Given a trigonometric value
and quadrant for an angle,
utilizes the structure and
relationships of
trigonometry, including
relationships in the unit
circle, to identify other
trigonometric values for
that angle, and describes
the relationship between
the radian measure and the
subtended arc in the circle
in contextual situations.
Given a trigonometric value
and quadrant for an angle,
utilizes the structure and
relationships of
trigonometry, including
relationships in the unit
circle, to identify other
trigonometric values for
that angle, and describes
the relationship between
the radian measure and the
subtended arc in the circle.
Given a trigonometric value
and quadrant for an angle,
utilizes the structure and
relationships of
trigonometry, including
relationships in the unit
circle, to identify other
trigonometric values for
that angle.
Given a trigonometric value
and quadrant for an angle,
utilizes the structure and
relationships of
trigonometry to identify
other trigonometric values
for that angle.
Page | 49
Performance Level Descriptors – Algebra II
Solving Equations
and Systems
HS.N.CN.C.7
HS.A.REI.B.4b-2
HS.A.REI.C.6-2
HS.A.REI.C.7
F-Int.3
F-BF.Int.2
HS.F.LE.A.2-3
HS-Int.3-3
Algebra II: Sub-Claim B
The student solves problems involving the Additional and Supporting Content for the grade/course with
connections to the Standards for Mathematical Practice.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
Solves multi-step contextual
word problems to find
similarities and differences
between solution
approaches involving linear,
exponential, quadratic (with
real or complex solutions)
and trigonometric
equations and systems of
equations, using inverses
where appropriate.
HS.S.ID.A.4
HS.S.ID.B.6a-1
HS.S.ID.B.6a-2
Solves problems involving
linear, exponential,
quadratic (with real or
complex solutions) and
trigonometric equations
and systems of equations,
using inverses where
appropriate.
Solves problems involving
linear, exponential and
quadratic (with real
solutions) equations and
systems of equations, using
inverses where appropriate.
Constructs linear and
exponential function
models in multi-step
contextual problems with
mathematical prompting.
Constructs linear and
exponential function
models in multi-step
contextual problems.
Constructs linear and
exponential function
models in multi-step
contextual problems with
mathematical prompting.
Uses the means and standard
deviations of data sets to fit
them to normal distributions.
Uses the means and standard
deviations of data sets to fit
them to normal distributions.
Uses the means and standard
deviations of data sets to fit
them to normal distributions.
Uses the means and standard
deviations of data sets to fit
them to normal distributions.
Fits exponential or
trigonometric functions to
data in order to solve multistep contextual problems.
Fits exponential and
trigonometric functions to
data in order to solve multistep contextual problems.
Fits exponential functions to
data in order to solve multistep contextual problems.
Uses fitted exponential
functions to solve multi-step
contextual problems.
Constructs linear and
exponential function
models in multi-step
contextual problems.
Data – Univariate
and Bivariate
Solves multi-step
contextual word problems
involving linear,
exponential, quadratic (with
real or complex solutions)
and trigonometric
equations and systems of
equations, using inverses
where appropriate.
Identifies when these
procedures are not
appropriate.
Page | 50
Performance Level Descriptors – Algebra II
Inference
HS.S.IC.A.2
S-IC.Int.1
Probability
S-CP.Int.1
Algebra II: Sub-Claim B
The student solves problems involving the Additional and Supporting Content for the grade/course with
connections to the Standards for Mathematical Practice.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
Uses sample data to make,
justify and critique
inferences and conclusions
about the corresponding
population.
Uses sample data to make
inferences and justify
conclusions about the
corresponding population.
Decides if specified models
are consistent with results
from given data-generating
processes.
Decides if specified models
are consistent with results
from given data-generating
processes.
Recognizes, determines and
uses conditional probability
and independence in multistep contextual problems,
using appropriate set
language and appropriate
representations, including
two-way frequency tables.
Recognizes, determines and
uses conditional probability
and independence in multistep contextual problems,
using appropriate set
language and appropriate
representations, including
two-way frequency tables.
Applies the Addition Rule of
probability and interprets
answers in context.
Applies the Addition Rule
of probability.
Uses sample data to make
inferences about the
corresponding population.
Identifies when sample data
can be used to make
inferences about the
corresponding population.
Recognizes, determines and
uses conditional probability
and independence in
contextual problems, using
appropriate set language
and appropriate
representations, including
two-way frequency tables.
Recognizes and determines
conditional probability and
independence in contextual
problems.
Page | 51
Performance Level Descriptors – Algebra II
Reasoning
HS.C.3.1
HS.C.3.2
HS.C.4.1
HS.C.5.4
HS.C.5.11
HS.C.6.2
HS.C.6.4
HS.C.7.1
HS.C.8.2
HS.C.8.3
HS.C.9.2
HS.C.11.1
HS.C.12.2
HS.C.16.3
HS.C.17.2
HS.C.17.3
HS.C.17.4
HS.C.17.5
HS.C.18.4
HS.C.CCR
Algebra II: Sub-Claim C
The student expresses grade/course-level appropriate mathematical reasoning by constructing viable
arguments, critiquing the reasoning of others and/or attending to precision when making mathematical
statements.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
Clearly constructs and
communicates a complete
response based on:
Clearly constructs and
communicates a complete
response based on:
Constructs and
communicates a response
based on:
•
•
•
•
•
•
•
•
a response to a given
equation or system of
equations
a chain of reasoning to
justify or refute
algebraic, function or
number system
propositions or
conjectures
a response based on
data
a response based on the
graph of an equation in
two variables, the
principle that a graph is
a solution set or the
relationship between
zeros and factors of
polynomials
a response based on
trigonometric functions
and the unit circle
a response based on
•
•
•
•
•
a response to a given
equation or system of
equations
a chain of reasoning to
justify or refute
algebraic, function or
number system
propositions or
conjectures,
a response based on
data
a response based on the
graph of an equation in
two variables, the
principle that a graph is
a solution set or the
relationship between
zeros and factors of
polynomials
a response based on
trigonometric functions
and the unit circle
a response based on
•
•
•
•
•
Constructs and
communicates an
incomplete response based
on:
a response to a given • a response to a given
equation or system of
equation or system of
equations
equations
a chain of reasoning to • a chain of reasoning to
justify or refute
justify or refute
algebraic, function or
algebraic, function or
number system
number system
propositions
or
propositions or
conjectures
conjectures
• a response based on
a response based on
data
data
a response based on the • a response based on the
graph of an equation in
graph of an equation in
two variables, the
two variables, the
principle that a graph is
principle that a graph is
a solution set or the
a solution set or the
relationship between
relationship between
zeros and factors of
zeros and factors of
polynomials
polynomials
• a response based on
a response based on
trigonometric functions
trigonometric functions
and the unit circle
and the unit circle
•
a response based on
a response based on
Page | 52
Performance Level Descriptors – Algebra II
Algebra II: Sub-Claim C
The student expresses grade/course-level appropriate mathematical reasoning by constructing viable
arguments, critiquing the reasoning of others and/or attending to precision when making mathematical
statements.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
•
•
•
•
•
•
transformations of
functions
OR
a response based on
properties of exponents
by:
using a logical approach
based on a conjecture
and/or stated
assumptions, utilizing
mathematical
connections (when
appropriate)
providing an efficient
and logical progression
of steps or chain of
reasoning with
appropriate justification
performing precise
calculations
using correct gradelevel vocabulary,
symbols and labels
providing a justification
of a conclusion
•
•
•
•
•
•
transformations of
functions
OR
a response based on
properties of exponents
by:
using a logical approach
based on a conjecture
and/or stated
assumptions, utilizing
mathematical
connections (when
appropriate)
providing a logical
progression of steps or
chain of reasoning with
appropriate
justification
performing precise
calculations
using correct gradelevel vocabulary,
symbols and labels
providing a justification
of a conclusion
•
•
•
•
•
•
transformations of
functions
OR
a response based on
properties of exponents
by:
using a logical approach
based on a conjecture
and/or stated
assumptions
providing a logical, but
incomplete, progression
of steps or chain of
reasoning
performing minor
calculation errors
using some grade-level
vocabulary, symbols
and labels
providing a partial
justification of a
conclusion based on
own calculations
•
•
•
•
•
•
transformations of
functions
OR
a response based on
properties of exponents
by :
using an approach
based on a conjecture
and/or stated or faulty
assumptions
providing an incomplete
or illogical progression
of steps or chain of
reasoning
making an intrusive
calculation error
using limited gradelevel vocabulary,
symbols and labels
providing a partial
justification of a
conclusion based on
own calculations
Page | 53
Performance Level Descriptors – Algebra II
Algebra II: Sub-Claim C
The student expresses grade/course-level appropriate mathematical reasoning by constructing viable
arguments, critiquing the reasoning of others and/or attending to precision when making mathematical
statements.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
• evaluating, interpreting • evaluating the validity
• determining whether
•
an argument or
conclusion is
generalizable
evaluating, interpreting
and critiquing the
validity and efficiency of
others’ responses,
approaches and
reasoning – utilizing
mathematical
connections (when
appropriate) – and
providing a counterexample where
applicable
and critiquing the
validity of others’
responses, approaches
and reasoning –
utilizing mathematical
connections (when
appropriate)
of others’ approaches
and conclusions
Page | 54
Performance Level Descriptors – Algebra II
Algebra II: Sub-Claim D
The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying
knowledge and skills articulated in the standards for the current grade/course (or for more complex problems,
knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the
Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning
abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure
and/or looking for and expressing regularity in repeated reasoning.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
Modeling
HS.D.2-3
HS.D.2-4
HS.D.2-7
HS.D.2-10
HS.D.2-12
HS.D.2-13
HS.D.3-5
HS.D.3-6
HS.D.CCR
Devises a plan to apply
mathematics in solving
problems arising in
everyday life, society and
the workplace by:
Devises a plan to apply
mathematics in solving
problems arising in
everyday life, society and
the workplace by:
•
•
•
•
•
using stated
assumptions and
approximations to
simplify a real-world
situation
mapping relationships
between important
quantities
selecting appropriate
tools to create the
appropriate model
analyzing relationships
mathematically
between important
quantities (either given
or created) to draw
conclusions
•
•
•
•
using stated
assumptions and
approximations to
simplify a real-world
situation
mapping relationships
between important
quantities
selecting appropriate
tools to create the
appropriate model
analyzing relationships
mathematically
between important
quantities (either given
or created) to draw
conclusions
interpreting
Devises a plan to apply
mathematics in solving
problems arising in
everyday life, society and
the workplace by:
•
•
•
•
•
using stated
assumptions and
approximations to
simplify a real-world
situation
illustrating
relationships between
important quantities
using provided tools to
create appropriate but
inaccurate model
analyzing relationships
mathematically
between important
given quantities to
draw conclusions
interpreting
mathematical results in
Devises a plan to apply
mathematics in solving
problems arising in
everyday life, society and
the workplace by:
•
•
•
•
•
•
using stated
assumptions and
approximations to
simplify a real-world
situation
identifying important
given quantities
using provided tools to
create inaccurate model
analyzing relationships
mathematically to draw
conclusions
writing an expression,
equation or function to
describe a situation
using securely held
content incompletely
reporting a conclusion,
Page | 55
Performance Level Descriptors – Algebra II
Algebra II: Sub-Claim D
The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying
knowledge and skills articulated in the standards for the current grade/course (or for more complex problems,
knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the
Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning
abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure
and/or looking for and expressing regularity in repeated reasoning.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
mathematical results in
a simplified context
with some inaccuracy
• interpreting
the
context
of
the
within the reporting
•
reflecting
on
whether
mathematical results in
situation
the results make sense • indiscriminately using
the context of the
data from a data source
• reflecting on whether
situation
• modifying the model if
the results make sense
it has not served its
• reflecting on whether
• improving the model if
purpose
the results make sense
it has not served its
• writing an expression,
• improving the model if
•
•
•
•
•
it has not served its
purpose
writing a complete, clear
and correct expression,
equation or function to
describe a situation
analyzing and/or
creating constraints,
relationships and goals
justifying and
defending models
which lead to a
conclusion
using geometry to solve
design problems
using securely held
•
•
•
•
purpose
writing a complete,
clear and correct
expression, equation or
function to describe a
situation
using geometry to solve
design problems
using securely held
content, briefly, but
accurately reporting the
conclusion
identifying and using
relevant data from a
data source
•
•
•
•
equation or function to
describe a situation
using geometry to solve
design problems
using securely held
content, incompletely
reporting a conclusion
selecting and using
some relevant data
from a data source
making an evaluation or
recommendation
Page | 56
Performance Level Descriptors – Algebra II
Algebra II: Sub-Claim D
The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying
knowledge and skills articulated in the standards for the current grade/course (or for more complex problems,
knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the
Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning
abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure
and/or looking for and expressing regularity in repeated reasoning.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
content, accurately
• making an appropriate
•
•
reporting and justifying
the conclusion
identifying and using
relevant data from a
data source
making an appropriate
evaluation or
recommendation
evaluation or
recommendation
Page | 57