Download MA41 Honors Algebra II Sequenced Units for Year at a Glance

Sequenced Units for
Arizona’s College and Career Ready Standards
MA41 Honors Algebra II
Year at a Glance
Semester 1
Unit 1: Linear Functions (10 days)
Unit 2: Quadratic Functions (10 days)
Unit 3: Quadratic Equations and Complex Numbers (14 days)
Unit 4: Polynomial Functions (20 days)
Unit 5: Rational Exponents and Radical Functions (13 days)
Unit 6: Exponential and Logarithmic Functions (17 days)
Semester 2
Unit 7: Rational Functions (12 days)
Unit 8: Sequences and Series (13 days)
Unit 9: Trigonometric Ratios and Functions (19 days)
Unit 10: Probability (16 days)
Unit 11: Data Analysis and Statistics (16 days)
2015-2016
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Algebra II extends the knowledge students have of algebraic and statistical concepts. They have investigated linear, exponential, and quadratic functions in
previous years. Algebra II further develops important mathematical ideas introduced in Algebra I by extending techniques to solve equations and students’
knowledge of functions by studying inverses and new function families: polynomial, radical, trigonometric, and rational functions. Students will also spend a
significant portion of the school year studying probability and statistics. There are some (+) standards that are included in this course because the standards
naturally support the assessed Algebra II content.
This document reflects our current thinking related to the intent of Arizona’s College and Career Ready Standards for Mathematics and assumes 160 days for
instruction, divided among 11 units. The number of days suggested for each unit assumes 45–minute class periods and is included to convey how instructional
time should be balanced across the year. The units are sequenced in a way that we believe best develops and connects the mathematical content described in the
standards; however, the order of the standards included in any unit does not imply a sequence of content within that unit. Some standards may be revisited
several times during the course; others may be only partially addressed in different units, depending on the focus of the unit. Strikethroughs in the text of the
standards are used in some cases in an attempt to convey that focus, and comments are included throughout the document to clarify and provide additional
background for each unit.
Throughout Algebra II, students should continue to develop proficiency with Arizona’s College and Career Ready Standards’ eight Standards for Mathematical Practice:
1.
Make sense of problems and persevere in solving them.
5. Use appropriate tools strategically.
2.
Reason abstractly and quantitatively.
6. Attend to precision.
3.
Construct viable arguments and critique the reasoning of others.
7. Look for and make use of structure.
4.
Model with mathematics.
8. Look for and express regularity in repeated reasoning.
These practices should become the natural way in which students come to understand and do mathematics. While, depending on the content to be understood
or on the problem to be solved, any practice might be brought to bear, some practices may prove more useful than others. Opportunities for highlighting certain
practices are indicated in different units in this document, but this highlighting should not be interpreted to mean that other practices should be neglected in
those units.
When using this document to help in planning your district's instructional program, you will also need to refer to the standards document, relevant progressions
documents for the standards, and the appropriate assessment consortium framework.
Mesa Public Schools
1
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 1: Linear Functions
Suggested number of days: 10
Unit 1 presents topics that were studied in Algebra 1. Transformations of linear, quadratic, and absolute value functions are explored. The parent functions are established and then
transformed functions are compared to the parent. Students will review modeling with linear functions involving writing linear functions from given information and fitting a line to data.
Results from performing a linear regression are compared to the model determined by hand. Students extend prior work with systems of equations to solving linear systems in three
variables. Students may well be a bit rusty with the algebra skills. The review content should be familiar to students, with new content introduced at an appropriate level. It is assumed that
students will be using graphing technology in this book. In this first unit, many fundamental calculator skills are integrated in the lessons.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Creating equations— A–CED
A-REI.C.6 Include 3x3 systems.
A. Create equations that describe numbers or relationships
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with
labels and scales.
3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as
viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints
on combinations of different foods.
Reasoning with Equations and Inequalities—A–REI
C. Solve systems of equations
6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two
variables.
Interpreting Functions — F–IF
C. Analyze functions using different representations
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.
Building Functions — F–BF
F-IF.B.9 Tasks may involve polynomial,
exponential, logarithmic, and trigonometric
functions.
F-BF.A.1a Include problem-solving
opportunities utilizing a real-world context.
Function types extend to include polynomial,
radical, logarithmic, simple rational, and
trigonometric. Extend to include all
exponential functions.
F-BF.B.3 Function types extend to include
polynomial, radical, logarithmic, simple rational,
and trigonometric. Extend to include all
exponential functions.
F-LE.A.2 Extend to include solving multi-step
problems by constructing linear and
exponential functions.
A. Build a function that models a relationship between two quantities
1. Write a function that describes a relationship between two quantities.★
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
B. Build new functions from existing functions
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.
Linear, Quadratic, and Exponential Models— F–LE
A. Construct and compare linear, quadratic, and exponential models and solve problems
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).
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Mesa Public Schools
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May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 1: Linear Functions
Suggested number of days: 10
Interpreting Categorical and Quantitative Data — S–ID
B. Summarize, represent, and interpret data on two categorical and quantitative variables
6. 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, quadratic, and exponential models.
S-ID.B.6a Include problem-solving opportunities
utilizing a real-world context. Extend to include
all exponential functions.
Arizona’s College and Career Ready Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Mesa Public Schools
3
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 2: Quadratic Functions
Suggested number of days: 10
Students have studied quadratic functions in Algebra 1. Their background should include factoring quadratic expressions, graphing quadratic equations written in three forms, and solving
quadratic equations using a variety of approaches. Students will extend their knowledge of quadratic functions in this unit. Transformations on quadratic functions are introduced as well as
characteristics of quadratic functions. Understanding the connection between the characteristics of a quadratic and its equation can help students apply their knowledge when working with a
real-life application.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Arithmetic with Polynomials and Rational Expression —A–APR
A-APR.B.3 Include quadratic, cubic, and quartic
polynomials and polynomials for which factors
are not provided. For example, find the zeros of
(x 2 - 1)( x 2 + 1).
B. Understand the relationship between zeros and factors of polynomials
3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the
function defined by the polynomial.
Creating equations— A–CED
A. Create equations that describe numbers or relationships
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes
with labels and scales.
Interpreting Functions — F–IF
B. Interpret functions that arise in applications in terms of the context
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.
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.
C. Analyze functions using different representations
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for
more complicated cases.
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
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.
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Mesa Public Schools
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F-IF.B.4 Include problem-solving opportunities
utilizing a real-world context. Function types
extend to include polynomial, radical,
logarithmic, simple rational, and trigonometric.
Extend to include all exponential functions. The
function types listed here are the same as those
listed in the Algebra II column for standards FIF.6, F-IF.7, and F-IF.9.
F-IF.B.6 Include problem-solving opportunities
utilizing a real-world context. Function types extend
to include polynomial, radical, logarithmic, simple
rational, and trigonometric. Extend to include all
exponential functions. The function types listed here
are the same as those listed in the Algebra II column
for standards F-IF.6, F-IF.7, and F-IF.9.
F-IF.C.7 Include problem-solving opportunities
utilizing a real-world context. Function types
extend to include polynomial, radical,
logarithmic, simple rational, and trigonometric.
Extend to include all exponential functions. The
function types listed here are the same as those listed
in the Algebra II column for standards F-IF.6, F-IF.7,
and F-IF.9.
F-IF.C.9 Include problem-solving opportunities
utilizing a real-world context. Function types
extend to include polynomial, radical,
logarithmic, simple rational, and trigonometric.
Extend to include all exponential functions.
continued on next page
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 2: Quadratic Functions
Suggested number of days: 10
Building Functions — F–BF
F-BF.A.1a Include problem-solving opportunities
utilizing a real-world context. Function types extend
to include polynomial, radical, logarithmic, simple
rational, and trigonometric. Extend to include all
exponential functions.
A. Build a function that models a relationship between two quantities
1. Write a function that describes a relationship between two quantities.★
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
B. Build new functions from existing functions
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.
Expressing Geometric Properties with Equations — G–GPE
A. Translate between the geometric description and the equation for a conic section
2. Derive the equation of a parabola given a focus and directrix.
F-BF.B.3 Function types extend to include polynomial,
radical, logarithmic, simple rational, and trigonometric.
Extend to include all exponential functions.
S-ID.B.6a Include problem-solving opportunities
utilizing a real-world context. Extend to include all
exponential functions.
Interpreting Categorical and Quantitative Data — S–ID
B. Summarize, represent, and interpret data on two categorical and quantitative variables
6. 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, quadratic, and exponential models.
Arizona’s College and Career Ready Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Mesa Public Schools
5
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 3: Quadratic Equations and Complex Numbers
Suggested number of days: 14
The strategies for solving quadratic equations presented previously were introduced at the end of Algebra 1. The difference now is that solutions are not restricted to real numbers. The
technique of completing the square so that the Quadratic Formula can be derived is introduced. Students will use five strategies for solving quadratic equations: graphing, square rooting,
factoring, completing the square, and using the Quadratic Formula. As the number of strategies increases in the unit, students should be making informed choices as to which strategy to
use given the equation. The last two lessons extend work with solving quadratic equations to solving nonlinear systems and solving quadratic inequalities. It is important throughout the
unit to be clear with students about your expectation of the role of technology versus computational and analytical skills.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Quantities – N-Q
N-Q These standards are integrated throughout both
and Algebra I and Algebra II course. Most notably in
modeling tasks. For example, in a situation involving
periodic phenomena, the student might autonomously
decide that amplitude is a key variable in a situation,
and then choose to work with peak amplitude.
A. Reason qualitatively and use units to solve problems.
2. Define appropriate quantities for the purpose of descriptive modeling.
The Complex Number System — N–CN
A. Perform arithmetic operations with complex numbers.
1. Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.
2. Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply
complex numbers.
C. Use complex numbers in polynomial identities and equations.
7. Solve quadratic equations with real coefficients that have complex solutions.
8. Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).
9. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Seeing Structure in Expressions — A–SSE
A. Interpret the structure of expressions
2. Use the structure of an expression to identify ways to rewrite it. For example, see x 4 – y4 as (x2)2 – (y2)2, thus recognizing
it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
Creating equations— A–CED
A. Create equations that describe numbers or relationships
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-SSE.A.2 Focus on polynomial, rational, or exponential
expressions. Examples: See x 4 – y 4 as (x 2 ) 2 – (y 2 ) 2 ,
thus recognizing it as a difference of squares that can be
factored as (x 2 – y 2 )(x 2 + y 2 ). In the equation x 2 + 2x +
1 + y 2 = 9, see an opportunity to rewrite the first three
terms as (x+1)2 , thus recognizing the equation of a
circle with radius 3 and center (1, 0). See (x 2 + 4)/(x 2 +
3) as ( (x 2 +3) + 1 )/(x 2 +3), thus recognizing an
opportunity to write it as 1 + 1/(x 2 + 3).
A-CED.A.1 Extend to exponential equations with
rational or real exponents and rational functions.
Include problem-solving opportunities utilizing a realworld context.
A-REI.B.4b Include all solution cases. In the case of
equations that have roots with nonzero imaginary parts,
students write the solutions as a ± bi for real numbers a
and b.
3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret
solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional
and cost constraints on combinations of different foods.
Reasoning with Equations and Inequalities—A–REI
B. Solve equations and inequalities in one variable
4. Solve quadratic equations in one variable.
b.
Solve quadratic equations by inspection (e.g., for x2 = 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.
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May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 3: Quadratic Equations and Complex Numbers
Suggested number of days: 14
C. Solve systems of equations
7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and
graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.
A-REI.D.11 Include any of the function types mentioned
in the standard. Extend to include all exponential
functions.
D. Represent and solve equations and inequalities graphically
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.
F-IF.C.7 Include problem-solving opportunities
utilizing a real-world context. Function types
extend to include polynomial, radical,
logarithmic, simple rational, and trigonometric.
Extend to include all exponential functions. The function
types listed here are the same as those listed in the
Algebra II column for standards F-IF.6, F-IF.7, and F-IF.9.
Interpreting Functions — F–IF
C. Analyze functions using different representations
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using
technology for more complicated cases.
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
8. 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-BF.A.1a Include problem-solving opportunities
utilizing a real-world context. Function types extend to
include polynomial, radical, logarithmic, simple rational,
and trigonometric. Extend to include all exponential
functions.
Building Functions — F–BF
A. Build a function that models a relationship between two quantities
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
Arizona’s College and Career Ready Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Mesa Public Schools
7
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 4: Polynomial Functions
Suggested number of days: 20
Linear and quadratic functions are two types of polynomials, so connections to earlier work are easily made and polynomial functions are defined and graphed. The notation and vocabulary
can be overwhelming for students, though some of the vocabulary was used in Algebra 1. End behavior of even- and odd-degree polynomials is explored. Prior work with factoring is
extended to third- and fourth-degree expressions. Synthetic division is used to efficiently check for possible rational roots when rewriting polynomials in factored form in order to solve
polynomial equations. All of the work with operations on polynomials, factoring, and solving leads to the Fundamental Theorem of Algebra in this unit. This unit also deals with polynomial
functions, in particular the graphs of these functions. Earlier work with transformations is applied to polynomials.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
The Complex Number System — N–CN
A-SSE.A.2 Focus on polynomial, rational, or
exponential expressions. Examples: See x 4
– y 4 as (x 2 ) 2 – (y 2 ) 2 , thus recognizing it
as a difference of squares that can be
factored as (x 2 – y 2 )(x 2 + y 2 ). In the
equation x 2 + 2x + 1 + y 2 = 9, see an
opportunity to rewrite the first three terms
as (x+1)2 , thus recognizing the equation of
a circle with radius 3 and center (1, 0).
See (x 2 + 4)/(x 2 + 3) as ( (x 2 +3) + 1 )/(x 2
+3), thus recognizing an opportunity to
write it as 1 + 1/(x 2 + 3).
C. Use complex numbers in polynomial identities and equations.
8. Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).
9. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Seeing Structure in Expressions — A–SSE
A. Interpret the structure of expressions
2. Use the structure of an expression to identify ways to rewrite it. For example, see x 4 – y4 as (x2)2 – (y2)2, thus recognizing it as a
difference of squares that can be factored as (x2 – y2)(x2 + y2).
Arithmetic with Polynomials and Rational Expression —A–APR
A. Perform arithmetic operations on polynomials
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.
B. Understand the relationship between zeros and factors of polynomials
2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0
if and only if (x – a) is a factor of p(x).
A-APR.B.3 Include quadratic, cubic, and
quartic polynomials and polynomials for
which factors are not provided. For
example, find the zeros of (x 2 - 1)( x 2 + 1).
3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function
defined by the polynomial.
C. Use polynomial identities to solve problems
4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity
(x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.
5. Know and apply the Binomial Theorem for the expansion of (x + y) n in powers of x and y for a positive integer n, where x and y are
any numbers, with coefficients determined for example by Pascal’s Triangle. (The Binomial Theorem can be proved by mathematical
induction or by a combinatorial argument.)
D. Rewrite rational expressions
6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are
polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated
examples, a computer algebra system.
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May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 4: Polynomial Functions
Suggested number of days: 20
Creating equations— A–CED
F-IF.B.4 Include problem-solving
opportunities utilizing a real-world
context. Function types extend to include
polynomial, radical, logarithmic, simple
rational, and trigonometric. Extend to
include all exponential functions. The
function types listed here are the same as
those listed in the Algebra II column for
standards F-IF.6, F-IF.7, and F-IF.9.
A. Create equations that describe numbers or relationships
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with
labels and scales.
Interpreting Functions — F–IF
B. Interpret functions that arise in applications in terms of the context
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.
C. Analyze functions using different representations
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more
complicated cases.
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
Building Functions — F–BF
A. Build a function that models a relationship between two quantities
1. Write a function that describes a relationship between two quantities.★
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
B. Build new functions from existing functions
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-IF.C.7 Include problem-solving
Opportunities utilizing a real-world context.
Function types extend to include polynomial,
radical, logarithmic, simple rational, and
trigonometric.
Extend to include all exponential functions.
The function types listed here are the same
as those listed in the Algebra II column for
standards F-IF.6, F-IF.7, and F-IF.9.
F-BF.A.1a Include problem-solving
opportunities utilizing a real-world context.
Function types extend to include
polynomial, radical, logarithmic, simple
rational, and trigonometric. Extend to
include all exponential functions.
F-BF.B.3 Function types extend to include
polynomial, radical, logarithmic, simple
rational, and trigonometric. Extend to
include all exponential functions.
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 5: Rational Exponents and Radical Functions
Suggested number of days: 13
This unit introduces radicals and nth roots and how these may be written as rational exponents. A connection is made to the properties of exponents studied in Algebra 1, noting that now
exponents can be rational numbers and are no longer restricted to being nonzero integers. Radical expressions, also written in rational exponent form, are represented as functions and
are graphed. This leads to a look at the difference between even- and odd-degree functions and what the domains are for each function type. Even and odd functions are defined. The
graphs of radical functions are used to help students think about solutions of radical equations and inequalities. Certainly, one goal is for students to recognize that solving radical
equations is an extension of solving other types of functions. The difference, however, is that sometimes extraneous solutions are introduced when solving radical equations, so it is
necessary to check apparent solutions. Inverse functions are presented—finding the inverse of linear, simple polynomial, and radical functions, and noting that the graphs of inverse
functions are reflections in the line y = x.
Arizona’s College and Career Ready Standards for Mathematical Content
The Real Number System — N–RN
A. Extend the properties of exponents to rational exponents.
1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer
exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define
51/3 to be the cube root of 5 because we want (51/3)3= 5(1/3)3 to hold, so (51/3)3 must equal 5.
2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Creating equations— A–CED
A. Create equations that describe numbers or relationships
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.A.1 Extend to simple rational and radical
equations.
F-IF.C.7 Include problem-solving opportunities
utilizing a real-world context. Function types
extend to include polynomial, radical,
logarithmic, simple rational, and trigonometric.
Extend to include all exponential functions. The function
types listed here are the same as those listed in the
Algebra II column for standards F-IF.6, F-IF.7, and F-IF.9.
Reasoning with Equations and Inequalities—A–REI
A. Understand solving equations as a process of reasoning and explain the reasoning
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.
2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may
arise.
Interpreting Functions — F–IF
C. Analyze functions using different representations
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using
technology for more complicated cases.
b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
Building Functions — F–BF
A. Build a function that models a relationship between two quantities
1. Write a function that describes a relationship between two quantities.
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.
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Mesa Public Schools
10
continued on next page
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 5: Rational Exponents and Radical Functions
Suggested number of days: 13
B. Build new functions from existing functions
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.
F-BF.B.3 Function types extend to include polynomial,
radical, logarithmic, simple rational, and trigonometric.
Extend to include all exponential functions.
4. Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the
inverse. For example, f(x) =2x3 or f(x) = (x+1)/(x–1) for x≠1.
Mesa Public Schools
11
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 6: Exponential and Logarithmic Functions
Suggested number of days: 17
This unit presents two new types of functions, exponential and logarithmic. Students should have some prior experience with exponential functions from Algebra 1, particularly with
growth and decay models. The natural base e, an irrational number, is introduced and students write and graph exponential functions for base e and other bases. Compound interest and
continuous compounding are two of the many applications explored. The logarithmic function, which is the inverse of the exponential function, is introduced, and the connection to
properties of exponents is made. The unit also looks at solving exponential and logarithmic equations using different approaches: analytical, numerical, and graphical. The unit uses
mathematical modeling - given a set of data, an exponential or logarithmic equation is fit to the data.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Seeing Structure in Expressions — A–SSE
A. Interpret the structure of expressions
2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing
it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
A-SSE.A.2 Focus on polynomial, rational, or exponential
expressions. Examples: See x 4 – y 4 as (x 2 ) 2 – (y 2 ) 2 ,
thus recognizing it as a difference of squares that can be
factored as (x 2 – y 2 )(x 2 + y 2 ). In the equation x 2 + 2x +
1 + y 2 = 9, see an opportunity to rewrite the first three
terms as (x+1)2 , thus recognizing the equation of a
circle with radius 3 and center (1, 0). See (x 2 + 4)/(x 2 +
3) as ( (x 2 +3) + 1 )/(x 2 +3), thus recognizing an
opportunity to write it as 1 + 1/(x 2 + 3).
B. Write expressions in equivalent forms to solve problems
3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by
the expression.
c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t
can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate
is 15%.
Creating equations— A–CED
A. Create equations that describe numbers or relationships
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate
axes with labels and scales.
Reasoning with Equations and Inequalities—A–REI
A. Understand solving equations as a process of reasoning and explain the reasoning
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.
Interpreting Functions — F–IF
C. Analyze functions using different representations
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using
technology for more complicated cases.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions,
showing period, midline, and amplitude.
8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of
the function.
b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate
of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing
exponential growth or decay.
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Mesa Public Schools
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A-SSE.B.3c Include problem-solving opportunities
utilizing a real-world context. As described in the
standard, there is an interplay between the
mathematical structure of the expression and the
structure of the situation such that choosing and
producing an equivalent form of the expression reveals
something about the situation. Extend to include
expressions with real number exponents.
A-REI.A.1 Extend to simple rational and radical
equations.
F-IF.C.7 Include problem-solving opportunities
utilizing a real-world context. Function types
extend to include polynomial, radical,
logarithmic, simple rational, and trigonometric.
Extend to include all exponential functions. The function
types listed here are the same as those listed in the
Algebra II column for standards F-IF.6, F-IF.7, and F-IF.9.
continued on next page
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 6: Exponential and Logarithmic Functions
Suggested number of days: 17
Building Functions — F–BF
A. Build a function that models a relationship between two quantities
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
F-BF.A.1a Include problem-solving opportunities
utilizing a real-world context. Function types extend to
include polynomial, radical, logarithmic, simple rational,
and trigonometric. Extend to include all exponential
functions.
B. Build new functions from existing functions
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.
4. Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the
inverse. For example, f(x) =2x3 or f(x) = (x+1)/(x–1) for x≠1.
Linear, Quadratic, and Exponential Models— F–LE
A. Construct and compare linear, quadratic, and exponential models and solve problems
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-BF.B.3 Function types extend to include polynomial,
radical, logarithmic, simple rational, and trigonometric.
Extend to include all exponential functions.
F-LE.A.2 Extend to include solving multi-step problems
by constructing linear and exponential functions.
F-LE.B.5 Include problem-solving opportunities utilizing
a real-world context. Extend to include all exponential
functions.
4. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2,
10, or e; evaluate the logarithm using technology.
B. Interpret expressions for functions in terms of the situation they model
5. Interpret the parameters in a linear or exponential function in terms of a context.
Arizona’s College and Career Ready Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Mesa Public Schools
12
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 7: Rational Functions
Suggested number of days: 12
Unit 7 introduces rational functions, a new type of function for students to work with. The simplest of rational functions, inverse variation, is introduced. The inverse variation function is
distinguished from the direct variation function, and it also provides the introduction to rational functions and their graphs. Students learn to identify the horizontal and vertical
asymptotes by inspecting the equations. Simple transformations of rational functions are also performed. Connections are made to operations with fractions, and symbolic manipulation
skills are necessary to perform the operations.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Arithmetic with Polynomials and Rational Expression —A–APR
A-CED.A.1 Extend to exponential equations with
rational or real exponents and rational functions.
Include problem-solving opportunities utilizing a
real-world context.
D. Rewrite rational expressions
6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and
r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more
complicated examples, a computer algebra system.
7. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and
r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more
complicated examples, a computer algebra system.
Creating equations— A–CED
A-REI.A.1 Extend to simple rational and radical
equations.
F-BF.B.3 Function types extend to include polynomial,
radical, logarithmic, simple rational, and trigonometric.
Extend to include all exponential functions.
A. Create equations that describe numbers or relationships
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.
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate
axes with labels and scales.
3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions
as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost
constraints on combinations of different foods.
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.
Reasoning with Equations and Inequalities—A–REI
A. Understand solving equations as a process of reasoning and explain the reasoning
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.
2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Building Functions — F–BF
B. Build new functions from existing functions
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.
Mesa Public Schools
13
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 8: Sequences and Series
Suggested number of days: 13
This unit builds on skills from Algebra 1, where arithmetic and geometric sequences were first introduced, by extending the work students have previously done. New in this unit is the
skill of adding terms of a sequence. Partial sums and sums of infinite geometric series are explored numerically and graphically. This unit involves recursively defined functions by
reviewing knowledge of arithmetic and geometric sequences with connections to linear and exponential functions.
Seeing Structure in Expressions — A–SSE
B. Write expressions in equivalent forms to solve problems4.
4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to
solve problems. For example, calculate mortgage payments.
Interpreting Functions — F–IF
A. Understand the concept of a function and use function notation
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.
Building Functions — F–BF
A. Build a function that models a relationship between two quantities
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
Comments
F-IF.A.3 This standard is Supporting work in
Algebra II. This standard should support the Major
work in F-BF.2 for coherence.
F-BF.A.1a Include problem-solving opportunities
utilizing a real-world context. Function types extend to
include polynomial, radical, logarithmic, simple rational,
and trigonometric. Extend to include all exponential
functions.
F-LE.A.2 Extend to include solving multi-step problems
by constructing linear and exponential functions.
2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations,
and translate between the two forms.
Linear, Quadratic, and Exponential Models— F–LE
A. Construct and compare linear, quadratic, and exponential models and solve problems
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).
Mesa Public Schools
14
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 9: Trigonometric Ratios and Functions
Suggested number of days: 19
Right triangle trigonometry that students learned in geometry is reviewed. Students are introduced to radian measure, and the six trigonometric functions are defined in terms
of a unit circle and lessons focus on graphing the six trigonometric functions. The graphs of sine and cosine are developed by plotting functional values for benchmark angles,
and the concept of periodic functions is introduced. The graphs of the remaining four trigonometric functions are deduced from knowing the relationship between these
functions and sine and cosine. Knowledge of transformations is used to plot graphs beyond the parent functions. The unit introduces students to trigonometric identities and
sum and difference formulas.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Creating equations— A–CED
A. Create equations that describe numbers or relationships
2. Create equations in two or more variables to represent relationships between quantities; graph equations on
coordinate axes with labels and scales.
F-IF.C.7 Include problem-solving opportunities
utilizing a real-world context. Function types
extend to include polynomial, radical,
logarithmic, simple rational, and trigonometric.
Extend to include all exponential functions. The
function types listed here are the same as those listed in
the Algebra II column for standards F-IF.6, F-IF.7, and FIF.9.
Interpreting Functions — F–IF
C. Analyze functions using different representations
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using
technology for more complicated cases.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric
functions, showing period, midline, and amplitude.
Building Functions — F–BF
A. Build a function that models a relationship between two quantities
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
B. Build new functions from existing functions
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.
F-BF.A.1a Include problem-solving opportunities
utilizing a real-world context. Function types extend to
include polynomial, radical, logarithmic, simple
rational, and trigonometric. Extend to include all
exponential functions.
F-BF.B.3 Function types extend to include polynomial,
radical, logarithmic, simple rational, and trigonometric.
Extend to include all exponential functions.
Trigonometric Functions — F–TF
A. Extend the domain of trigonometric functions using the unit circle
1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real
numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
B. Model periodic phenomena with trigonometric functions
5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and
midline. 
C. Prove and apply trigonometric identities
8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ),
or tan(θ) and the quadrant of the angle.
9. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
Mesa Public Schools
15
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 10: Probability
Suggested number of days: 16
This unit on probability presumes knowledge of basic probability concepts from middle school and some data analysis techniques from Algebra 1. It begins with finding sample spaces for
a variety of classic contexts and also focuses on computing the probability of independent and dependent events, and compound events. Students construct and interpret two-way
frequency tables of data and use the entries to compute joint and marginal relative frequencies. Permutations and combinations are introduced to allow students to compute
probabilities of compound events and to solve real-life problems. The unit introduces students to probability distributions. One type of probability distribution is a binomial distribution.
Another type, a normal distribution, will be introduced in the next unit.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Arithmetic with Polynomials and Rational Expression —A–APR
C. Use polynomial identities to solve problems
5. Know and apply the Binomial Theorem for the expansion of (x + y) n in powers of x and y for a positive integer n, where x
and y are any numbers, with coefficients determined for example by Pascal’s Triangle. (The Binomial Theorem can be
proved by mathematical induction or by a combinatorial argument.)
Conditional Probability and the Rules of Probability — S–CP
A. Understand independence and conditional probability and use them to interpret data
1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes,
or as unions, intersections, or complements of other events (“or,” “and,” “not”).
2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of
their probabilities, and use this characterization to determine if they are independent.
3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying
that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given
A is the same as the probability of B.
4. Construct and interpret two–way frequency tables of data when two categories are associated with each object being
classified. Use the two–way table as a sample space to decide if events are independent and to approximate conditional
probabilities. For example, collect data from a random sample of students in your school on their favorite subject
among math, science, and English. Estimate the probability that a randomly selected student from your school will favor
science given that the student is in tenth grade. Do the same for other subjects and compare the results.
5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday
situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a
smoker if you have lung cancer.
B. Use the rules of probability to compute probabilities of compound events in a uniform probability model
6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the
answer in terms of the model.
7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
8. Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret
the answer in terms of the model.
9. Use permutations and combinations to compute probabilities of compound events and solve problems.
continued on next page
Mesa Public Schools
16
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 10: Probability
Suggested number of days: 16
Using Probability to Make Decisions – S-MD
B. Use probability to evaluate outcomes of decisions.
6. Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
7. Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey
goalie at the end of a game).
Mesa Public Schools
17
May 2015
Sequenced Units for Arizona’s College and Career Ready Standards in Mathematics
High School – MA41 Honors Algebra II
Unit 11: Data Analysis and Statistics
Suggested number of days: 16
This unit on data analysis and statistics builds upon the content in the previous unit. The overarching goal is to develop strategies for analyzing data that are collected from the result of an
experiment or survey. The normal distribution is defined and its properties are described. Students will learn to calculate z-scores and find the associated probabilities from a standard
normal table and how populations can be sampled and data collected without introducing bias. Random samples are defined. Students are introduced to experiments and observational
studies, and they analyze experimental designs. Correlation and causation are discussed.
Arizona’s College and Career Ready Standards for Mathematical Content
Comments
Interpreting Categorical and Quantitative Data — S–ID
A. Summarize, represent, and interpret data on a single count or measurement variable
4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population
percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators,
spreadsheets, and tables to estimate areas under the normal curve.
Making Inferences and Justifying Conclusions — S–IC
A. Understand and evaluate random processes underlying statistical experiments
1. Understand statistics as a process for making inferences about population parameters based on a random sample from
that population.
2. Decide if a specified model is consistent with results from a given data–generating process, e.g., using simulation. For
example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to
question the model?
B. Make inferences and justify conclusions from sample surveys, experiments, and observational studies
3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how
randomization relates to each.
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.
5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between
parameters are significant.
6. Evaluate reports based on data.
Arizona’s College and Career Ready Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Mesa Public Schools
18
May 2015