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Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
In Algebra I, students have already begun their study of algebraic concepts. They have used equations, tables, and graphs to describe relationships between quantities, with a particular focus on linear, quadratic, and exponential functions and equations. The Algebra II course outlined in this scope and sequence document begins with connections back to that earlier work, efficiently reviewing algebraic and statistical concepts that students have already studied while at the same time moving students forward into new concepts. Students expand their library of functions to include polynomial functions, logarithmic functions, rational functions, and trigonometric functions. With a larger library of functions, students increase their ability to model situations, make predictions and answer questions about the situation.
This scope and sequence assumes 160 days for instruction, divided among 13 units. The units are sequenced in a way that we believe best develops and connects the mathematical content described in the Common Core State Standards for Mathematics; 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 mathematical focus of the unit. Throughout Algebra II, students should continue to develop proficiency with the Common Core's eight 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.
These practices should become the natural way in which students come to understand and to 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 of study in this sample scope and sequence, but this highlighting should not be interpreted to mean that other practices should be neglected in those units.
This scope and sequence reflects our current thinking related to the intent of the CCSS for Mathematics, but it is an evolving document. We expect to make refinements to this scope and sequence in the coming months in response to new learnings about the standards. In planning your district's instructional program, you should be prepared to have similar flexibility in implementing your district's own scope and sequence for the next 2 to 3 years, as you transition from your state's current standards to full implementation of the CCSS for Mathematics. Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
10/28/11
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Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
Unit
Standards for Mathematical Content
Quadratic functions and the complex number system F-‐IF.8.a (Use the process of factoring and completing the square in a quadratic function to 1. Make sense of problems show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a and persevere in solving context.)
them.
N-‐CN.1 (Know there is a complex number i such that i2 = –1, and every complex number has the 2. Reason abstractly and quantitatively.
form a + bi with a and b real.)
N-‐CN.2 (Use the relation i2 = –1 and the commutative, associative, and distributive properties 3. Construct viable arguments and critique the to add, subtract, and multiply complex numbers.)
reasoning of others.
N-‐CN.7 (Solve quadratic equations with real coefficients that have complex solutions.)
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N-‐CN.8 ((+) Extend polynomial identities to the complex numbers. For example, rewrite x + 4 4. Model with mathematics.
5. Use appropriate tools as (x + 2i)(x – 2i).)
strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
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Builds on students' learning in Algebra I around quadratic functions and equations and the existence of the complex number system. They understand the complex number system and perform operations on complex numbers. They solve quadratic equations with complex solutions. Building new functions
A-‐APR.1 (Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.)
A-‐SSE.1.a (Interpret parts of an expression, such as terms, factors, and
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This topic links the basic functions to which students were introduced in Algebra I to the new functions they will master in Algebra II. This topic builds on previous work with linear and quadratic functions to help students make sense of the behavior they see in the larger family of polynomial functions.The topic leads students to understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication. Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
coefficients.★)
A-‐SSE.1.b (Interpret complicated expressions by viewing one or more of their parts as a single 3. Construct viable arguments and critique the entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.★)
reasoning of others.
F-‐BF.1.b (Combine standard function types using arithmetic operations. For example, build a 4. Model with mathematics.
function that models the temperature of a cooling body by adding a constant function to a 5. Use appropriate tools decaying exponential, and relate these functions to the model.★)
strategically.
F-‐IF.6 (Calculate and interpret the average rate of change of a function (presented symbolically 6. Attend to precision.
7. Look for and make use of or as a table) over a specified interval. Estimate the rate of change from a graph.★)
structure.
8. Look for and express regularity in repeated reasoning.
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
10/28/11
Days Comments
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Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
Unit
Standards for Mathematical Content
Characteristics of polynomial functions
N-‐CN.9 ((+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic 1. Make sense of problems polynomials.)
and persevere in solving A-‐APR.2 (Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the them.
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).)
2. Reason abstractly and A-‐APR.3 (Identify zeros of polynomials when suitable factorizations are available, and use the quantitatively.
zeros to construct a rough graph of the function defined by the polynomial.)
3. Construct viable F-‐IF.4 (For a function that models a relationship between two quantities, interpret key features arguments and critique the of graphs and tables in terms of the quantities, and sketch graphs showing key features given a reasoning of others.
verbal description of the relationship. Key features include: intercepts; intervals where the 4. Model with mathematics.
function is increasing, decreasing, positive, or negative; relative maximums and minimums; 5. Use appropriate tools strategically.
symmetries; end behavior; and periodicity.★)
6. Attend to precision.
F-‐IF.5 (Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-‐hours it 7. Look for and make use of structure.
takes to assemble n engines in a factory, then the positive integers would be an appropriate 8. Look for and express domain for the function.★)
regularity in repeated F-‐IF.7.c (Graph polynomial functions, identifying zeros when suitable factorizations are reasoning.
available, and showing end behavior.★)
F-‐IF.8.a
F-‐IF.9 (Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.)
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
Standards for Mathematical Practice
10/28/11
Days Comments
15
Students learn how polynomials model some behaviors with varying rates of change, and they see how the degree of the polynomial relates to the number of real zeros and the number of local extreme values of the polynomial function. Students then apply this knowledge to choose appropriate models for situations based on how quantities in the situation vary, with particular emphasis on short term and end behavior. Students will also interpret parts of an expression, such as terms, factors, and coefficients as well as interpret complicated expressions by viewing one or more of their parts as a single entity.
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Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
Unit
Standards for Mathematical Content
Applying polynomials
A-‐SSE.2 (Use the structure of an expression to identify ways to rewrite it. For example, see x4 – 1. Make sense of problems y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 and persevere in solving them.
+ y2).)
2. Reason abstractly and A-‐APR.3
quantitatively.
A-‐APR.4 (Prove polynomial identities and use them to describe numerical relationships. For 3. Construct viable 2
2
2
2 2
2
example, the polynomial identity (x + y )2 = (x – y ) + (2xy) can be used to generate arguments and critique the Pythagorean triples.)
reasoning of others.
A-‐APR.5 ((+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x 4. Model with mathematics.
and y for a positive integer n, where x and y are any numbers, with coefficients determined for 5. Use appropriate tools 1 1
example by Pascal’s Triangle. ) The binomial Theorem can be proved by mathematical induction or by a strategically.
combinatorial argument. 6. Attend to precision.
A-‐REI.11 (Explain why the x-‐coordinates of the points where the graphs of the equations y = 7. Look for and make use of f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions structure.
approximately, e.g., using technology to graph the functions, make tables of values, or find 8. Look for and express successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, regularity in repeated absolute value, exponential, and logarithmic functions.★)
reasoning.
A-‐CED.1 (Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and Standards for Mathematical Practice
Days Comments
10
Students solve equations that arise from situations that can be modeled using polynomial functions. Students also explore the structure of polynomial expressions and investigate and apply polynomial identities. exponential functions.★)
A-‐CED.2 (Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★)
A-‐CED.3 (Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-‐ viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.★)
F-‐IF.5
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
10/28/11
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Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
Unit
Standards for Mathematical Content
Modeling with rational functions
A-‐APR.6 (Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) 1. Make sense of problems + 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 and persevere in solving degree of b(x), using inspection, long division, or, for the more complicated examples, a them.
computer algebra system.)
2. Reason abstractly and A-‐APR.7 ((+) Understand that rational expressions form a system analogous to the rational quantitatively.
numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational 3. Construct viable expression; add, subtract, multiply, and divide rational expressions.)
arguments and critique the A-‐REI.2 (Solve simple rational and radical equations in one variable, and give examples showing reasoning of others.
how extraneous solutions may arise.)
4. Model with mathematics.
A-‐REI.11; F-‐IF.4; F-‐IF.5; A-‐CED.1;A-‐CED.2; A-‐CED.3
5. Use appropriate tools A-‐CED.4 (Rearrange formulas to highlight a quantity of interest, using the same reasoning as in strategically.
solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.)
6. Attend to precision.
F-‐BF.4.a (Solve an equation of the form f(x) = c for a simple function f that has an inverse and 7. Look for and make use of structure.
write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1.)
8. Look for and express F-‐BF.1.b A-‐SSE.4 (Derive the formula for the sum of a finite geometric series (when the common ratio is regularity in repeated reasoning.
not 1), and use the formula to solve problems. For example, calculate mortgage payments.★)
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Students use rational functions to model and investigate situations, and solve equations arising from those situations. Through this work, they learn about the general characteristics and behavior of rational functions and create and understand graphs of rational functions. They also develop the formula for the sum of a finite geometric series and connect that formula to the family of rational functions.
Modeling with radical functions
A-‐REI.2; A-‐REI.11; A-‐CED.1; A-‐CED.2; A-‐CED.3; A-‐CED.4; F-‐IF.4; F-‐IF.5; F-‐IF.6
F-‐IF.7.b (Graph square root, cube root, and piecewise-‐defined functions, including step 15
This topic builds on students' understanding of simple polynomial functions and inverses to develop simple radical functions. Students investigate the characteristics of these functions and solve equations that arise from situations that can be modeled by these functions.
Standards for Mathematical Practice
functions and absolute value functions.★)
F-‐IF.9
F-‐BF.3 (Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.)
F-‐BF.4.a
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
10/28/11
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.
Days Comments
5
Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
Unit
Standards for Mathematical Content
Standards for Mathematical Practice
Days Comments
Exponential and F-‐LE.4 (For exponential models, express as a logarithm the solution to ab =d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. )
logarithmic F-‐IF.7.e (Graph exponential and logarithmic functions, showing intercepts and end behavior, models
and trigonometric functions, showing period, midline, and amplitude. )
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.
10
This topic builds on students' understanding of exponential functions and inverses to develop logarithmic functions. Students investigate the characteristics of these functions and solve equations that arise from situations that can be modeled by these functions.
Symmetry and F-‐BF.3
transformations in functions
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.
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In this topic, students generalize the basic function transformations. They also explore symmetry of the graphs of functions and learn to recognize odd and even functions from their graphs and symbolic representations. Finally, they begin to investigate situations that exhibit periodic behavior and how those situations motivate the need to extend their understanding of transformations to function models of the form f(kx). ct
★
★
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
10/28/11
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Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
Unit
Standards for Mathematical Content
Modeling with trigonometric functions
F-‐TF.1 (Understand radian measure of an angle as the length of the arc on the unit circle 1. Make sense of problems subtended by the angle.)
and persevere in solving F-‐TF.2 (Explain how the unit circle in the coordinate plane enables the extension of them.
trigonometric functions to all real numbers, interpreted as radian measures of angles traversed 2. Reason abstractly and counterclockwise around the unit circle.)
quantitatively.
F-‐TF.5 (Choose trigonometric functions to model periodic phenomena with specified 3. Construct viable arguments and critique the amplitude, frequency, and midline.★)
2
2
F-‐TF.8 (Prove the Pythagorean identity sin (θ) + cos (θ) = 1 and use it to find sin(θ), cos(θ), or reasoning of others.
4. Model with mathematics.
tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.)
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.
15
Choosing a function model
F-‐IF.4; F-‐IF.5; F-‐IF.6; F-‐IF.9; F-‐BF.1.b; F-‐BF.3; A-‐CED.3
15
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
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.
10/28/11
Days Comments
In this topic, students define three trigonometric functions: y = sin α, y = cos α and y = tan α based on the unit circle. Students learn to use transformations to model periodic situations. They also develop the Pythagorean identity from the unit circle and derive related identities. In this topic, students draw upon their knowledge of different parent functions, including linear, polynomial, power, exponential, logarithmic and trigonometric functions, to choose the appropriate model for a given situation. Students write and solve equations, inequalities, and systems of equations and inequalities to answer questions that arise from these situations. 7
Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
Unit
Standards for Mathematical Content
Standards for Mathematical Practice
Days Comments
The design of statistical studies
S-‐IC.1 (Understand statistics as a process for making inferences about population parameters based on a random sample from that population.)
S-‐IC.3 (Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.)
S-‐IC.6 (Evaluate reports based on data.)
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.
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Drawing correct conclusions from data is highly dependent on how the data are collected. In particular, "cause and effect" conclusions can only arise from properly conducted experiments, in which the researcher actively imposes a treatment. Students learn the purposes of and differences among, surveys, experiments, and observational studies and explain how randomization is used in each case.
S.-‐ID.4 (Use the mean and standard deviation of a data set to fit it to a normal distribution and 1. Make sense of problems 10
Normal to estimate population percentages. Recognize that there are data sets for which such a and persevere in solving distribution as a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under them.
2. Reason abstractly and model for data the normal curve.)
Students investigate the normal distribution and estimate percentages based on the normal curve. Students also learn to evaluate whether the normal distribution is an appropriate model for a set of data.
S-‐IC.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?)
S-‐IC.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.)
S-‐MD.6 ((+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).)
S-‐MD.7 ((+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).)
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
10/28/11
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.
8
Sample Scope and Sequence for Algebra II for the Common Core State Standards for Mathematics
Unit
Standards for Mathematical Content
Drawing conclusions from data S-‐IC.1; S-‐IC.4
1. Make sense of problems S-‐IC.5 (Use data from a randomized experiment to compare two treatments; use simulations to and persevere in solving decide if differences between parameters are significant.)
them.
S-‐IC.6
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.
Copyright © 2011, The Charles A. Dana Center
at The University of Texas at Austin
Standards for Mathematical Practice
10/28/11
Days Comments
15
In this topic students use simulation as a tool to estimate population parameters and margins of error, and to compare two treatments.
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