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Algebra 2 Learning Progression (v2)

Page history last edited by Christine Koerner 12 months ago

Welcome to the new progression for Algebra 2. This progression builds upon Math Framework Project Phase 1 work (see Progression v1 here), taking many of the best features and building in an Overarching Question, Essential Questions, and Big Ideas for each unit. This new model takes the work of bundling standards to the next level by grouping together grade level concepts under Big Ideas. The Big Ideas are designed to represent the critical mathematics of this grade level in a manner we believe to be more coherent and productive as a guide for planning instruction, assessment, and intervention. Big Ideas are not a replacement for the objectives. 

 

The use of an asterisk (*) indicates an objective 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

Overarching Question

Essential Questions

Big Ideas

Full Objectives

Unit 0:

Math with a Growth Mindset

How does improving student attitudes toward math affect their learning?

 

 

  1. Math is about learning not performing.
  2. Math is about making sense.
  3. Math is filled with conjectures, creativity, and uncertainty.
  4. Mistakes are beautiful things.

 

  1. Students will understand the importance of a growth mindset (e.g., that math is not about talent or natural ability but is about thoughtful practice) and what it means to talk and listen. Students will also understand that class is where students practice thinking and doing math.
  2. Students will learn the value of taking time to think about math and listen to how others make sense of their work to arrive at a common understanding.
  3. Students will build the habits of using precise language, practicing, and sharing their thoughts. 

Unit 1:

Family of Functions


Timing

 ~5 weeks


Objectives

(Click on each objective for the objective analysis) 

 

Big Idea 1

A2.A.2.2 

(except rational) 

A2.F.2.1

 

Big Idea 2

A2.F.1.1

A2.F.1.3

(only identify characteristics)

A2.F.1.4

(only identify characteristics)

A2.F.1.5

(only identify characteristics)

A2.F.1.6

(only identify characteristics)

A2.F.1.7

(only identify characteristics)

 

Big Idea 3

A2.F.2.2

A2.F.2.3

 

Big Idea 4

A2.F.1.1

A2.F.1.2

(except parent graphs)

 

 

What makes each family of functions unique?

  1. How can arithmetic operations on functions be represented by a graph and related to an algebraic process?

  2. What are the similarities and differences between functions and within a family of functions?

  3. What families of functions also have inverses that are functions?

  4. How are functions used to represent real-world problems?

 

  1. Big Idea 1: Operations on functions can be represented with both algebraic and graphical models.
  2. Big Idea 2: Families of functions share similar behaviors, characteristics and properties.
  3. Big Idea 3: Two functions are inverses if the function f maps x to y and the inverse of f maps y to x.
  4. Big Idea 4: Functions within the same family are made by applying transformations onto the parent function. 

 

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

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

 

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.1.3 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.1.4 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.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.1.6 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.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.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 f is the range of the inverse function f-1, and the range of the function f is the domain of the inverse function f-1.

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.1.2 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.

 

Unit 2:

Quadratics


Timing

~6 weeks


Objectives

(Click on each objective for the objective analysis)

 

Big Idea 1

A2.A.1.1

A2.A.2.1

(except cubes and grouping)

 

Big Idea 2

A2.A.1.1

A2.N.1.1

A2.N.1.2

 

Big Idea 3

A2.A.2.3

A2.F.1.1

A2.F.1.2

(only quadratic)

A2.F.1.3

A2.D.1.2 

(only quadratic) 

 

 

 

 

 

 

 

Where are quadratic functions used in the world?

  1. What are the distinguishing characteristics of a quadratic?

  2. How can we identify key features of a quadratic?

  3. What mathematical and real world value do quadratic functions possess?

  4. How are quadratic functions used to solve real world problems?

  5. How are complex numbers related to real numbers?

 
  1. Big Idea 1: A quadratic function has degree 2.
  2. Big Idea 2: The graph and equation of a quadratic function can yield information about its roots.
  3. Big Idea 3: Equivalent representations of a function highlights different properties.  

 

 

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.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.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.N.1.1 By defining imaginary numbers, students can now apply the square root function to any real number and obtain a real or imaginary solution.They can solve in for any real number n.

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

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.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.1.2 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.1.3 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.D.1.2 Collect data and use scatter plots 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 3

Polynomials and Rational Functions

Note: This can be broken into Unit 3A Polynomials and 3B Rational and Radical Functions


Timing

~7 weeks


Objectives

 

Big Idea 1

A2.A.1.4

A2.A.2.1

A2.F.1.1

 

Big Idea 2

A2.N.1.2

A2.A.1.4

A2.A.2.1

 

Big Idea 3

A2.A.2.2

A2.F.1.5

 

Big Idea 4

A2.A.1.3

A2.A.2.2

A2.F.1.6

 

Big Idea 5

A2.N.1.4

A2.A.1.5

A2.A.2.4

A2.F.1.7

 

How are real situations modeled with polynomial functions?

  1. What are the characteristics of polynomial functions?

  2. When change occurs in a polynomial equation, how is the graph affected?

  3. How do polynomials, rational and radical functions represent real-world scenarios?

  4. How are complex numbers related to real numbers?

  5. Why are solutions considered “extraneous”?

 
  1. Big Idea 1: The behavior, properties, and characteristics of a polynomial function can be determined by its degree.
  2. Big Idea 2: The graph and equation of a polynomial function yields information about its roots.
  3. Big Idea 3: Basic arithmetic operations can be performed on polynomials similar to the way they are performed on numbers.
  4. Big Idea 4: Rational functions can be described by the division of two polynomial functions.
  5. Big Idea 5: Radical expressions can be written as expressions with rational exponents, and both are inverse operations of integral exponents. 

 

 

 

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

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

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

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.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.

 

Unit 4: Exponential and Logarithmic Functions


Timing

~ 6 weeks


Objectives

 

Big Idea 1

A2.A.1.2

A2.F.1.4

A2.D.1.2

 

Big Idea 2

A2.A.1.7

A2.F.1.4

A2.F.2.4

 

Big Idea 3

A2.A.1.2

A2.A1.6

A2.A.1.8

A2.F.2.4

 

Big Idea 4

A2.A.1.6

A2.A.1.7

A2.D.1.2

 

Big Idea 5 

A2.A.1.7

A2.D.1.2

How can exponential and logarithmic functions be used to represent real world scenarios?

  1. What are the distinguishing characteristics of an exponential function and logarithmic function?

  2. How can we identify key features of exponential and logarithmic functions?

  3. What mathematical and real-world value do exponential and logarithmic functions possess?

  4. How are exponential and logarithmic functions used to solve real-world problems?

  5. How are sequences and series related to functions?

 
  1. Big Idea 1: Exponential functions are defined as f(x)=ax.
  2. Big Idea 2: Exponential functions share similar characteristics with power functions.
  3. Big Idea 3: Logarithms and exponents can be used to solve mathematical and real-world problems.
  4. Big Idea 4: Sequences  can be described as functions.
  5. Big Idea 5: Sums of series can be represented by exponential and logarithmic functions. 

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.D.1.2 Collect data and use scatter plots 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. 

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.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.

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

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.F.2.4 Apply the inverse relationship between exponential and logarithmic functions to convert from one form to another.

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

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.1.2 Collect data and use scatter plots 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. 

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.1.2 Collect data and use scatter plots 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 5:

Statistics and Data


Timing

1 week


Objectives 

 

Big Idea 1:

A2.D.1.1

A2.D.1.3

A2.D.2.1

A2.D.2.2

 

 

 

How can data and statistics be used to describe and interpret real-world situations?

1. How can data be used to display information?

2. How can data be used to skew information?

3. How are the mean and standard deviation related to a normal distribution?

 

 
1. Big Idea 1:Data can represent and statistics can interpret real-world situations. 

 

 

 

Unit 6:

Systems of Equations


Timing

~4 weeks


Objectives

 

Big Idea 1

A2.A.1.8

A2.A.1.9

 

Big Idea 2 

A2.N.1.3

A2.F.1.8

A2.D.1.2

 

Big Idea 3

A2.A.1.8 

 

Big Idea 4

A2.F.1.8 

 

How can real-world situations be modeled and solved by systems of equations?

 

 

 

  1. How can we apply our knowledge to model real-world situations?

  2. How can we represent and analyze our solutions to problems?

  3. How can we represent the same thing in multiple different ways?

 
  1. Big Idea 1: Two-variable systems can be solved using various methods.
  2. Big Idea 2: Real world data can be represented, organized and solved using systems of equations.
  3. Big Idea 3: Methods for solving real-world or mathematical two variable systems of equations can be extended to three variable systems of equations.
  4. Big Idea 4: Piecewise functions can be used to model mathematical and real-world situations. 

 

  

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.

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 scatter plots 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. 

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.F.1.8Graph 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.

Culminating Unit


Timing

~2 weeks

How can students show evidence of understanding?  

Algebra 2 Showcase:

Shark Tank: Sink or Swim 

 

 
Distance Learning Resources/
Supplemental Activities
How can students develop and show evidence of understanding?     Multiple objectives are covered in this material. These math tasks are designed to enhance current curriculum and support distance learning.

 


Introduction to the OKMath Framework

Algebra 2 Grade Introduction

 

 

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