Algebra Topics Review
The following topics are essential algebraic abilities that students must have mastery of to be successful in an algebra 2 classroom. Students are expected to know and use these mechanics with fluency throughout the course as we build upon them and apply them to the key concepts of study in the course.
The following topics are essential algebraic abilities that students must have mastery of to be successful in an algebra 2 classroom. Students are expected to know and use these mechanics with fluency throughout the course as we build upon them and apply them to the key concepts of study in the course.
Multi-Step and Literal Equations |
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Distribution and FOIL |
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Factoring |
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Exponents Day 1 |
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Exponents Day 2 |
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Simplifying Radicals |
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Rationalizing the Denominator |
Properties and Attributes of Functions
This unit begins with an introduction to different forms of notation for sets of numbers. Inequality and interval are by far the most used notation, with interval being my favorite because of how much information is given in such a concise format. Features of functions include: maximums, minimums, increasing and decreasing intervals, and domain and range. All of these characteristics provide a stepping off point for the discussion of functions in later units. Additionally, the basics for transformation of functions are addressed here. Transformations are applicable to each type of function that we study and are reviewed as each comes up.
This unit begins with an introduction to different forms of notation for sets of numbers. Inequality and interval are by far the most used notation, with interval being my favorite because of how much information is given in such a concise format. Features of functions include: maximums, minimums, increasing and decreasing intervals, and domain and range. All of these characteristics provide a stepping off point for the discussion of functions in later units. Additionally, the basics for transformation of functions are addressed here. Transformations are applicable to each type of function that we study and are reviewed as each comes up.
Inequality, Interval, Set Notation |
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Domain, Range, and End Behavior |
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Characteristics of Graphs |
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Features of Functions |
Value |
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Inverse Relations |
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Parent Functions |
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Transformations |
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Transformations Day 2 |
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Review |
Quadratic Functions
Quadratics are the first and the most extensively studied function in algebra 2. The basics of parabolas are covered in algebra 1, so we extend learning to include vertex form of a parabola (both vertical and horizontal) and applications of parabolic curves.
Quadratics are the first and the most extensively studied function in algebra 2. The basics of parabolas are covered in algebra 1, so we extend learning to include vertex form of a parabola (both vertical and horizontal) and applications of parabolic curves.
Vertex Form of a Parabola |
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Vertex Form of a Parabola Day 2 |
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Vertical and Horizontal Parabolas |
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Vertical and Horizontal Parabolas Day 2 |
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Parabola Application Questions |
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Linear and Quadratic Modeling |
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Regression Modeling |
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Regression on the Calculator |
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Parabola Application Questions Set 2 |
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Review |
Solving Quadratic Functions
The second portion of quadratics involves solving quadratic functions through a variety of methods. These methods include: factoring the quadratic formula, and completing the square. Quadratic applications and solving inequalities that involve quadratics are also included in this unit. Completing the square is an element of the PAP classroom that is not covered in on-level classrooms. It represents a standard for college readiness.
The second portion of quadratics involves solving quadratic functions through a variety of methods. These methods include: factoring the quadratic formula, and completing the square. Quadratic applications and solving inequalities that involve quadratics are also included in this unit. Completing the square is an element of the PAP classroom that is not covered in on-level classrooms. It represents a standard for college readiness.
Zeros by Factoring |
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Factoring to Find Solutions |
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Quadratic Formula |
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Quadratic Formula Day 2 |
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Completing the Square |
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Completing the Square Day 2 |
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Completing the Square Day 3 |
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Quadratic Application (Calculator) |
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Quadratic Inequalities |
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Review |
Square Root, Cube Root, & Cubic Functions
After completing quadratics, the class moves toward three more parent functions: square root, cube root, and cubic functions. Special mention is made to the fact that quadratics and square roots are inverses as are cubic and cube root functions. Particular to quadratics and square roots is the idea of domain restrictions. Quadratics will only have an inverse on a certain domain because it does not pass the horizontal line test.
After completing quadratics, the class moves toward three more parent functions: square root, cube root, and cubic functions. Special mention is made to the fact that quadratics and square roots are inverses as are cubic and cube root functions. Particular to quadratics and square roots is the idea of domain restrictions. Quadratics will only have an inverse on a certain domain because it does not pass the horizontal line test.
Fractional Exponents |
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Inverse of Quadratic and Cubic Functions |
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Transformation of Root/Cubic Functions |
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Attributes of Root/Cubic Functions |
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Horizontal Line Test and Restricted Domains |
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Writing Prompt: Inverses |
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Review |
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Extra Credit: Gini Index |
Solving Radical and Cube Root Functions
After looking at the graphs and the properties of root and cubic functions, it is time to solve the functions. Additional factoring techniques and the use of technology are two of the means used to find the zeros of these functions.
After looking at the graphs and the properties of root and cubic functions, it is time to solve the functions. Additional factoring techniques and the use of technology are two of the means used to find the zeros of these functions.
Solving Square Root Equations |
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Solving Radical Equations |
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Factoring by Grouping |
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Solving Cubic and Cube Root Equations |
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Solving Cubics/Roots by Technology |
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Review |
Fall Semester Review
Review |
Polynomials
The next big unit of study is polynomials. Basic properties of polynomials are covered at first to build a common language between students and teacher. Specific to the PAP classroom is the addition of Pascal's Triangle for finding the expansion of binomials raised to a power. The unit closes with solving polynomials using a combination of graphing calculator, synthetic division, and factoring with complex roots to provide a great synthesis of many of the concepts studied in isolation thus far in the year.
The next big unit of study is polynomials. Basic properties of polynomials are covered at first to build a common language between students and teacher. Specific to the PAP classroom is the addition of Pascal's Triangle for finding the expansion of binomials raised to a power. The unit closes with solving polynomials using a combination of graphing calculator, synthetic division, and factoring with complex roots to provide a great synthesis of many of the concepts studied in isolation thus far in the year.
Adding and Subtracting Polynomials |
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Multiplying Polynomials |
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Pascal's Triangle |
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Sum and Difference of Cubes |
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Review of Factoring |
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Long Division |
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Long Division Day 2 |
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Synthetic Division |
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Polynomial Division Application Questions |
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Solving Polynomials Day 1 |
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Solving Polynomials Day 2 |
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Review |
Rational Functions
Solving rational functions is the next focus of the class. A lot of time is spent here and for good reason. Adding, subtracting, dividing, and multiplying rational functions are time consuming processes even for students that have mastery of factoring techniques. A student that cannot factor will not be successful in this unit. Oblique (slant) asymptotes are an addition for the PAP classroom as are the "Work" problems toward the end of the unit. This unit closes with direct and indirect inverse variation relationships.
Solving rational functions is the next focus of the class. A lot of time is spent here and for good reason. Adding, subtracting, dividing, and multiplying rational functions are time consuming processes even for students that have mastery of factoring techniques. A student that cannot factor will not be successful in this unit. Oblique (slant) asymptotes are an addition for the PAP classroom as are the "Work" problems toward the end of the unit. This unit closes with direct and indirect inverse variation relationships.
Graphing Rational Functions |
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Writing Equations of Rational Functions |
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Oblique (Slant) Asymptotes |
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Multiply/Divide Rationals |
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Add/Subtract Rationals |
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Add/Subtract Rationals Day 2 |
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Solving Rational Equations |
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Rational "Work" Problems |
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Solving Rational Inequalities |
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Variation Relationships |
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Review |
Exponential Functions
Exponential functions follow next. Two projects are included in this unit: Compound Interest, and a Disease project. The disease project has students research and write about a disease. Students then take a made-up growth rate to see how long it will take to impact various population levels. The final component of the project asks students to use regression to come up with an exponential equation to model the growth/decay of infected/healthy people in a closed-environment population.
Exponential functions follow next. Two projects are included in this unit: Compound Interest, and a Disease project. The disease project has students research and write about a disease. Students then take a made-up growth rate to see how long it will take to impact various population levels. The final component of the project asks students to use regression to come up with an exponential equation to model the growth/decay of infected/healthy people in a closed-environment population.
Exponential Functions |
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Exponential Growth/Decay and Simple Interest |
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Compound and Continuous Interest |
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Mini-Project: Compound Interest |
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Exponential Modeling/Regression |
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PROJECT: Exponential Diseases |
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Review of Properties of Exponents |
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Change of Base Exponentials |
Logarithms
Logarithms follow exponential functions since they are inverses of each other. Basic properties of the functions and the graphs are developed first before the PAP classroom moves toward expanding and contracting logarithmic expressions using logarithm rules.
Logarithms follow exponential functions since they are inverses of each other. Basic properties of the functions and the graphs are developed first before the PAP classroom moves toward expanding and contracting logarithmic expressions using logarithm rules.
Graphing Logarithms |
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Properties of Logarithms |
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Solving Exponential and Log Equations |
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Log Application Problems |
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Extra Credit: Baseball Card Problem |
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Properties of Logarithms Take 2 |
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Review |
Absolute Value and Piecewise Functions
The next unit of study covers absolute value and piecewise functions. This seems like an odd pairing at first, but part of our standards require writing absolute value functions as piecewise functions. In addition to graphing and writing absolute value and piecewise functions, students are required to solve each type of equation. This presents some consternation with absolute value functions because they must be solved twice: once for the positive and once for the negative of the argument of the absolute value.
The next unit of study covers absolute value and piecewise functions. This seems like an odd pairing at first, but part of our standards require writing absolute value functions as piecewise functions. In addition to graphing and writing absolute value and piecewise functions, students are required to solve each type of equation. This presents some consternation with absolute value functions because they must be solved twice: once for the positive and once for the negative of the argument of the absolute value.
Absolute Value Functions |
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Solving Absolute Value Equations |
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Absolute Value Inequalities |
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Absolute Value Application Questions |
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Writing Absolute Value as Piecewise |
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Piecewise Functions |
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Piecewise Word Problems |
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Review |
Systems of Equations
The final major unit covers solving systems of equations. Specifically, the focus is on solving 3x3 systems; 3 equations with 3 variables. Students in PAP have the additional requirement to solve using the Gaussian or Gauss-Jordan method of elimination.
The final major unit covers solving systems of equations. Specifically, the focus is on solving 3x3 systems; 3 equations with 3 variables. Students in PAP have the additional requirement to solve using the Gaussian or Gauss-Jordan method of elimination.
Systems of 3 Linear Equations |
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Linear and Quadratic Systems |
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Solving Systems of Linear Inequalities |
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PSAT Systems Practice |
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Gaussian Elimination |
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Real-World Application of Systems |
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Review |
Matrices and Complex Numbers
Matrices are the natural follow-up unit to solving systems. Matrices are 100% new to our students, so we spend a lot of time going over the basics and the vocabulary to build knowledgeable learners. We show students how to solve all matrix problems in the calculator, but require addition, subtraction, multiplication, and finding the determinant of a 2x2 matrix be able to be done by hand. This unit closes with basic complex number properties and operations including simplifying powers of i.
Matrices are the natural follow-up unit to solving systems. Matrices are 100% new to our students, so we spend a lot of time going over the basics and the vocabulary to build knowledgeable learners. We show students how to solve all matrix problems in the calculator, but require addition, subtraction, multiplication, and finding the determinant of a 2x2 matrix be able to be done by hand. This unit closes with basic complex number properties and operations including simplifying powers of i.
Matrix Basics |
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Multiplying Matrices |
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Multiplying Matrices Application |
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Determinant of a Matrix |
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Inverse of a Matrix |
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Complex Number Basics |
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Multiplying Complex Numbers |
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Review |
5 Big Things for PreCalculus |
Spring Semester Review
Spring Semester Exam Review |