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Algebra 2 Learning Progression

Page history last edited by Brenda Butz 6 years, 4 months ago

* Indicates an objective that is repeated in another unit or an objective that is partially taught in a unit and will be taught in its entirety in a later unit. The parts of the objective that will be taught in a later unit is indicated by the “strikethroughs.” Occasionally, new words are added to the objective to ensure the objective still makes sense considering the strikethroughs.

Unit

Unit Storyline

Full Objectives

Unit 1:

Functions

 


Timing

2-4 weeks


Objectives

For this bundle ONLY linear and absolute value.

A2.F.1.1 (piecewise & absolute value)

A2.F.1.8

A2.F.2.1

A2.F.2.2

A2.F.2.3

A2.D.1.2* (correlation coefficient will be covered in unit 9)

The focus of this course is the study of functions.  Students’ prior knowledge of linear and absolute value functions from Algebra 1 are an entry point to a deeper study of functions.  These concepts should only be applied to linear and absolute value functions initially, with piecewise functions introduced in this unit.  A deeper understand will be developed by creating scatter plots and identifying the line of best fit.  We will build on this foundation as the course progresses by incorporating other functions as they are introduced.

A2.F.1.1 Use algebraic, interval, and set notations to specify the domain and range of functions of various types and evaluate a function at a given point in its domain.

A2.F.2.1 Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions.

A2.F.2.2 Combine functions by composition and recognize that

g(x) = f -1(x), the inverse function of f(x) , if and only if f(g(x)) = g(f(x)) =x

A2.F.2.3 Find and graph the inverse of a function, if it exists, in real-world and mathematical situations. Know that the domain of a function is the range of the inverse function, and the range of the function is the domain of the inverse function.

A2.F.1.8 Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant. 

A2.D.1.2 Collect data and use scatterplots to analyze patterns and describe linear, exponential or quadratic relationships between two variables. Using graphing calculators or other appropriate technology, determine regression equation and correlation

Unit 2:

Quadratics


Timing

3-5 weeks


Objectives

A2.A.1.1 (real roots only)

A2.A.2.3

A2.F.1.2 (quadratic only)

A2.F.1.3

Quadratics added to:

A2.F.1.8*

A2.F.1.1*

A2.F.2.2*

A2.F.2.3*

A2.D.1.2* (correlation coefficient will be covered in unit 9)

 

 

 

 

 

 

 

Quadratics are a starting point for the study of functions in Algebra 2. The idea of quadratics is first introduced in Algebra 1. Quadratic functions are developed in depth in this course.  In this unit, students will learn to identify, graph and solve quadratic equations by various methods. Students will graph quadratic functions, with the aid of technology at times, and identify its major characteristics. These goals are important in order to further develop the idea of transformations and critical points found in other functions. The skill of solving quadratics, algebraically or graphically, will be used throughout the remainder of the course.

The concepts of quadratics are essential and students need to be involved in lessons and activities that help them analyze and understand this graph. Students’ fluency with the vocabulary of this function is a building block for the remaining functions.

 

A2.A.2.3 Recognize that a quadratic function has different equivalent representations [f(x) =ax2+bx+c, f(x) = a(x-h)2 + k, and f(x) =(x-h)(x-k) ]. Identify and use the representation that is most appropriate to solve real-world and mathematical problems.

A2.A.1.1 Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist.

A2.F.1.3 Graph a quadratic function. Identify the x- and y-intercepts, maximum or minimum value, axis of symmetry, and vertex using various methods and tools that may include a graphing calculator or appropriate technology.

A2.F.1.2 Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x+c),f(x) +c ,f(cx), and cf(x) , where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology.

*A2.F.1.1 Use algebraic, interval, and set notations to specify the domain and range of functions of various types and evaluate a function at a given point in its domain.

*A2.F.1.8 Graph piecewise functions with no more than three branches (including linear,quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant.

*A2.F.2.2 Combine functions by composition and recognize that g(x) = f1(x), the inverse function of f(x) , if and only if f(g(x)) = g(f(x)) =x

*A2.F.2.3 Find and graph the inverse of a function, if it exists, in real-world and mathematical situations. Know that the domain of a function is the range of the inverse function, and the range of the function  is the domain of the inverse function. 

*A2.D.1.2 Collect data and use scatterplots to analyze patterns and describe linear, exponential or quadraticrelationships between two variables. Using graphing calculators or other appropriate technology, determine regression equation and correlation

Unit 3:

Complex Numbers

 


Timing

1-3 weeks


Objectives

A2.N.1.1

A2.N.1.2

A2.A.1.1* (include non-real roots)

 

 

Complex numbers are introduced in this course for the first time.  Many students will find them challenging and not relatable to their knowledge of numbers.  Allowing the students to see a visual representation of the complex numbers within the complex plane (where the horizontal axis is the real numbers and the vertical axis is the imaginary numbers) will allow for a deeper understanding of these numbers.  The students will also be introduced to the concept of a conjugate to do division of complex numbers.  They will use this knowledge later on when they have to rationalize the denominator.  Furthermore, their understanding of complex numbers is essential in their understanding of non-real roots within polynomial functions.

A2.N.1.1 Find the value of in for any whole number n.

A2.N.1.2 Simplify, add, subtract, multiply, and divide complex numbers.

*A2.A.1.1 Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist.

 

 

 

 

 

 

Unit 4:

Systems


Timing

1-3 weeks


Objectives

A2.A.1.8

A2.A.1.9

A2.N.1.3

 

 

 

 

Solving systems of equations is first introduced in Algebra 1 as systems of linear equations.  In this course that concept is expanded to include systems with one linear and one quadratic equation.  It is further explored with the introduction of three equations with three unknowns.  The students are introduced to a three dimensional space where intersections are still important but they are no longer restricted to one coordinate point.  The fact that equations exist in a three dimensional world makes them a plane and, as such, solutions become more complex.  Students will have to understand the meaning of no solution, one solution and infinitely many solutions within a three dimensional space.  

As we continue our study of systems we find the limits of finding solutions for systems of equations algebraically or graphically.  At this point the students are introduced to matrices as a way to represent systems.  In order to have a deeper understanding of matrices this course introduces students to basic arithmetic operations involving matrices including addition, subtraction and multiplication by a scalar.

A2.A.1.8 Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology).

A2.A.1.9 Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology.

 A2.N.1.3 Use matrices to organize and represent data. Identify the order (dimension) of a matrix, add and subtract matrices of appropriate dimensions, and multiply a matrix by a scalar to create a new matrix to solve problems

 

 

 

 

 Unit 5:

Polynomials

 

 


Timing

3-5 weeks


Objectives

A2.A.1.4

A2.A.2.1

A2.A.2.2

A2.F.1.5

Polynomials added to:

A2.F.1.1*

A2.F.2.2*

 

 

 

 

 

Students first experience with polynomials is brief in Algebra 1, only getting to see them as quadratics with a leading coefficient. In this course, students are introduced to expressions and equations whose power goes beyond the second degree. They expand their knowledge of exponents and factoring quadratics to include polynomials of different degrees.  Students will learn to perform arithmetic operations on these polynomials, a skill that will be help with rational expression later on in this course.  They will solve these polynomial equations using a variety of new methods, including but not limited factoring quadratics with a leading coefficient, factoring the difference of squares, factoring the sum and differences of cubes and factoring by grouping.  Students will also be presented with polynomials graphs and should be able to identify the main points of each graph.

During this unit, students will understand the relationship between solutions, roots, and zeros. Whether factoring, solving or graphing, students will understand what their answers mean in context to the problem.

A2.A.1.4 Solve polynomial equations with real roots using various methods and tools that may include factoring, polynomial division, synthetic division, graphing calculators or other appropriate technology.

A2.A.2.1 Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies.

A2.A.2.2 Add, subtract, multiply, divide, and simplify polynomial and rational expressions 

A2.F.1.5 Analyze the graph of a polynomial function by identifying the domain, range, intercepts, zeros, relative maxima, relative minima, and intervals of increase and decrease. 

*A2.F.1.1 Use algebraic, interval, and set notations to specify the domain and range of functions of various types and evaluate a function at a given point in its domain.

*A2.F.2.2 Combine functions by composition and recognize that

g(x) = f1(x), the inverse function of f(x) , if and only if f(g(x)) = g(f(x)) =x

 

 

 

 

 

Unit 6:

Radicals

 


Timing

2-4 weeks


Objectives

A2.N.1.4

A2.A.1.5

A2.A.2.4

A2.F.1.7

Radicals added to:

A2.F.1.1*

A2.F.1.2*

A2.F.2.2*

A2.F.2.3*

 

 

 

 

Radical functions are introduced in this course for the first time. Students will solve square roots equations using their prior knowledge and techniques for solving equations and simplifying radicals.  They will also learn to convert expressions between rational exponent notation and radical notation.  To fully understand radical functions, students will graph square root and cube root functions and apply the transformations that have been used throughout the course.

 

 

 

A2.N.1.4 Understand and apply the relationship of rational exponents to integer exponents and radicals to solve problems

A2.A.1.5 Solve square root equations with one variable and check for extraneous solutions.

A2.A.2.4 Rewrite expressions involving radicals and rational exponents using the properties of exponents 

A2.F.1.7 Graph a radical function (square root and cube root only) and identify the x- and y-intercepts using various methods and tools that may include a graphing calculator or other appropriate technology.

*A2.F.1.2 Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations

[f(x+c),f(x)+c ,f(cx), and cf(x)] , where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology.

*A2.F.1.1 Use algebraic, interval, and set notations to specify the domain and range of functions of various types and evaluate a function at a given point in its domain.

*A2.F.2.2 Combine functions by composition and recognize that g(x) = f1(x), the inverse function of f(x) , if and only if f(g(x)) = g(f(x)) =x

*A2.F.2.3 Find and graph the inverse of a function, if it exists, in real-world and mathematical situations. Know that the domain of a function f is the range of the inverse function f1, and the range of the function f is the domain of the inverse function f1 .

Unit 7: 

Rational

 

 

 


Timing

2-4 weeks


Objectives

A2.A.1.3

A2.F.1.6

Rational added to:

A2.A.2.2*

A2.F.1.1*

Students’ understanding of rational numbers expands in Algebra 2 to include rational expressions and equations with variables. Students learn in this unit solving a rational equation sometimes produces a solution that will not work in the given problem and therefore must learn to check their solutions for these extraneous solutions. They also explore the relationship between radicals and rational exponents and learn to convert between the two forms. Students should feel comfortable using either form to solve equations. Students will also be able to graph a rational function and identify intercepts and asymptotes, using a variety of methods including a graphing calculator.

A2.A.1.3 Solve one-variable rational equations and check for extraneous solutions.

A2.F.1.6 Graph a rational function and identify the x- and y-intercepts, vertical and horizontal asymptotes, using various methods and tools that may include a graphing calculator or other appropriate technology. (Excluding slant or oblique asymptotes and holes.) 

*A2.A.2.2 Add, subtract, multiply, divide, and simplify polynomial and rational expressions

*A2.F.1.1 Use algebraic, interval, and set notations to specify the domain and range of functions of various types and evaluate a function at a given point in its domain.

 

 

 

 

 

 

Unit 8:

Log/ Exponents

 

 

 


Timing

2-4 weeks


Objectives

A2.A.1.2

A2.A.1.6

A2.F.1.4

A2.F.2.4

Log/Exp. added to:

A2.F.1.1*

A2.F.1.2*

A2.F.1.8*

A2.F2.2*

A2.F.2.3*

A2.D.1.2* (correlation coefficient will be covered in unit 9)

 

 

 

 

 

 

Exponential and logarithmic functions are introduced in this course for the first time.  The students will discover the inverse relationship between exponential and logarithm functions.   The understanding of this relationship will allow for a deeper understanding of logarithms, their computation and properties.  The students will be involved in the discussion of real-world examples of exponential growth (compound interest), exponential decay (half-lives) and logarithms (Richter scale) which will lead to a deeper insight into their graphs and the practicality of these functions.  

Asymptotes are a new topic in this course.  Students will investigate how asymptotes apply to the graphs of exponential and logarithmic functions. This concept will be further developed in calculus.

Students will perform transformations of these parent graphs and determine their domain and range, intercepts and asymptotes.  Furthermore, students will apply the inverse relationship of exponents and logarithms and use the properties of logarithms as part of the procedure to solve exponential and logarithmic equations.

 

 

A2.A.1.6 Solve common and natural logarithmic equations using the properties of logarithms.

A2.A.1.2 Represent real-world or mathematical problems using exponential equations, such as compound interest, depreciation, and population growth, and solve these equations graphically (including graphing calculator or other appropriate technology) or algebraically. 

A2.F.1.4 Graph exponential and logarithmic functions. Identify asymptotes and x- and y-intercepts using various methods and tools that may include graphing calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically.

A2.F.2.4 Apply the inverse relationship between exponential and logarithmic functions to convert from one form to another function  is the range of the inverse function , and the range of the function  is the domain of the inverse function . 

*A2.F.1.1 Use algebraic, interval, and set notations to specify the domain and range of functions of various types and evaluate a function at a given point in its domain.

*A2.F.1.8 Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant.

**A2.F.1.2 Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations

[f(x+c),f(x) +c ,f(cx), and cf(x) , where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology.

*A2.F.2.2 Combine functions by composition and recognize that g(x) = f1(x), the inverse function of f(x) , if and only if f(g(x)) = g(f(x)) =x .

*A2.F.2.3 Find and graph the inverse of a function, if it exists, in real-world and mathematical situations. Know that the domain of a function f is the range of the inverse function f1, and the range of the function f is the domain of the inverse function f1 .

*A2.D.1.2 Collect data and use scatterplots to analyze patterns and describe linear, exponential or quadratic relationships between two variables. Using graphing calculators or other appropriate technology, determine regression equation and correlation coefficients; use regression equations to make predictions and correlation coefficients to assess the reliability of those predictions.

Unit 9:

Stats/ Data

 


Timing

2-4 weeks


Objectives

A2.A.1.7

A2.D.1.1

A2.D.1.2

A2.D.1.3

A2.D.2.1

A2.D.2.2

 

 

 

 

 

This unit builds on the students understanding of arithmetic and geometric series/sequences.   The students will analyze world problems and decide the most appropriate formula to use to find a solution.   This will lead to the discussion of discrete and continuous graphs.  A better understanding of graphs will come when students gathers data, set up a scatter plot and find the correlation coefficient.  The correlation coefficient will be used to decide how accurate any future prediction from a line of best fit will be.  

The student will go deeper into the study of graphs when they begin the analysis of statistical data.  They will identify the source and design of a study and how a study can be distorted.  They will also detect misleading uses of data and understand that correlation does not imply causation.  By the end of this unit the students will have a deep understanding of graphs and how they work in the real world.

A2.A.1.7 Solve real-world and mathematical problems that can be modeled using arithmetic or finite geometric sequences or series given the nth terms and sum formulas. Graphing calculators or other appropriate technology may be used. 

A2.D.1.1 Use the mean and standard deviation of a data set to fit it to a normal distribution (bell-shaped curve).

A2.D.1.2 Collect data and use scatterplots to analyze patterns and describe linear, exponential or quadratic relationships between two variables. Using graphing calculators or other appropriate technology, determine regression equation and correlation

A2.D.1.3 Based upon a real-world context, recognize whether a discrete or continuous graphical representation is appropriate and then create the graph.

A2.D.2.1 Evaluate reports based on data published in the media by identifying the source of the data, the design of the study, and the way the data are analyzed and displayed. Given spreadsheets, tables, or graphs, recognize and analyze distortions in data displays. Show how graphs and data can be distorted to support different points of view.

A2.D.2.2 Identify and explain misleading uses of data. Recognize when arguments based on data confuse correlation and causation.

 

 

 


OKMath Framework Introduction

Algebra 2 Grade Introduction

 

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