Embed Size (px)
Citation preview
Study Resources for the Praxis® Mathematics: Content Knowledge Test (5161)The links below allow you to connect content topics
on this Praxis® test directly to free Khan Academy study resources.
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
I. Number and Quality, Algebra, Functions, and Calculus
Lesson
A. Number and Quality
1. Understands the properties of exponents
a. Understands the role of phonics and word analysis in literacy development
Algebra I / Exponent properties reviewAlgebra 2 / Rational exponents
b. Demonstrate an understanding of the properties of exponential expressions
Algebra I / Exponent properties reviewAlgebra 2 / Properties of exponents (rational exponents)
c. Use the properties of exponents to rewrite expressions that have radicals or rational exponents
Algebra I / RadicalsAlgebra I / Simplifying square rootsAlgebra 2 / Evaluating exponents & radicalsAlgebra 2 / Equivalent forms of exponential expressions
2. Understands the properties of rational and irrational numbers, and the interactions between those sets of numbers
Algebra I / Irrational numbersAlgebra I / Proofs concerning irrational numbers
a. Recognize that the sum or product of two rational numbers is rational
Algebra I / Sums and products of rational and irrational numbers
b. Recognize that the sum of a rational number and an irrational number is irrational
Algebra I / Sums and products of rational and irrational numbers
c. Recognize that the product of a nonzero rational number and an irrational number is irrational
Algebra I / Sums and products of rational and irrational numbers
d. Recognize that the sum or product of two Algebra I / Sums and products of rational and irrational numbersCopyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 1 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
irrational numbers can be rational or irrational3. Understands how to solve problems by
reasoning quantitatively (e.g., dimensional analysis, reasonableness of solutions)a. Use units as a way to understand problems
and to guide the solution of multistep problems
Algebra I / Rate conversion
b. Choose and interpret units consistently in formulas
Algebra I / Appropriate units
c. Choose and interpret the scale and the origin in graphs and data displays
d. Recognize the reasonableness of results within the context of a given problem
Algebra I / Appropriate unitsAlgebra I / Word problems with multiple units
4. Understands the structure of the natural, integer, rational, real, and complex number systems and how the basic operations (+,−,×,∧÷) on numbers in these systems are performeda. Solve problems using addition, subtraction,
multiplication, and division of rational, irrational, and complex numbers
Algebra I / Substitution and evaluating expressionsAlgebra I / Division by zeroAlgebra 2 / The imaginary unit iAlgebra 2 / Complex numbers introductionAlgebra 2 / Adding and subtracting complex numbersAlgebra 2 / Multiplying complex numbersPrecalculus / What are the imaginary numbers?Precalculus / What are the complex numbers?Precalculus / Adding and subtracting complex numbersPrecalculus / Multiplying complex numbersPrecalculus / Complex conjugates and dividing complex numbersPrecalculus / Identities with complex numbers
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 2 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
b. Apply the order of operations Pre-algebra / Order of operations
c. Given operations on a number system, determine whether the properties (e.g., commutative, associative, distributive) hold
d. Compare, classify, and order real numbers
e. Simplify and approximate radicals Algebra I / Simplifying square rootsAlgebra 2 / The imaginary unit i
f. Find conjugates of complex numbers Precalculus / Complex conjugates and dividing complex numbers
g. Demonstrate an understanding of the properties of counting numbers (e.g., prime, composite, prime factorization, even, odd, factors, multiples)
5. Understands how to work with complex numbers when solving polynomial equations and rewriting polynomial expressionsa. Solve quadratic equations with real
coefficients that have complex solutionsAlgebra I / The quadratic formulaAlgebra 2 / Quadratic equations with complex solutions
b. Extend polynomial identities to the complex
numbers (e.g.,
Algebra I / Special products of binomialsAlgebra 2 / Special products of polynomialsAlgebra 2 / Polynomial identitiesPrecalculus / Identities with complex numbers
c. Verify the fundamental theorem of algebra for quadratic polynomials
Algebra I / The quadratic formulaAlgebra 2 / Quadratic equations with complex solutions
6. Understands how to perform operations on matrices and how to use matrices in applicationsa. Use matrices to represent and manipulate
dataPrecalculus / Introduction to matricesPrecalculus / Model real-world situations with matrices
b. Multiply matrices by scalars to produce new Precalculus / Multiplying matrices by scalars
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 3 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
matrices Precalculus / Properties of matrix addition & scalar multiplicationc. Add, subtract, and multiply matrices of
appropriate dimensionsPrecalculus / Adding and subtracting matricesPrecalculus / Properties of matrix addition & scalar multiplicationPrecalculus / Multiplying matrices by matrices
d. Understand that matrix multiplication for square matrices is not a commutative operation but still satisfies the associative and distributive properties
Precalculus / Properties of matrix multiplication
e. Understand the role played by zero, and identity matrices in matrix addition and multiplication
Precalculus / Properties of matrix addition & scalar multiplicationPrecalculus / Properties of matrix multiplication
f. Understand that the determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse
Precalculus / The determinant of a 2x2 matrixPrecalculus / Introduction to matrix inversesPrecalculus / Finding the inverse of a matrix using its determinantPrecalculus / Practice finding the inverses of 2x2 matrices
g. Work with matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area
Precalculus / Matrices as transformations
7. Understands how to solve problems involving ratios, proportions, averages, percents, and metric and traditional unit conversionsa. Apply the concept of a ratio and use ratio
language and notation to describe a relationship between two quantities
Pre-algebra / Intro to ratios
b. Compute unit rates. Pre-algebra / Intro to rates
c. Use ratio reasoning to convert rates Pre-algebra / Visualize ratiosPre-algebra / Ratio application
d. Solve problems involving scale factors
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 4 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
e. Recognize and represent proportional and inversely proportional relationships between two quantities
Pre-algebra / Constant of proportionality Pre-algebra / Identifying proportional relationshipsPre-algebra / Writing & solving proportionsAlgebra (all content) / Direct and inverse variation
f. Use proportional relationships to solve multistep ratio, average, and percent problems
Pre-algebra / Intro to percentsPre-algebra / Percent, fraction, decimal conversionsPre-algebra / Percent problemsPre-algebra / Percent word problems
g. Solve measurement and estimation problems involving time, length, temperature, volume, and mass in both the U.S. customary system and the metric system, where appropriate
Algebra I / Rate conversionAlgebra I / Appropriate unitsAlgebra I / Word problems with multiple units
h. Convert units within the metric and customary systems
Algebra I / Rate conversionAlgebra I / Appropriate unitsAlgebra I / Word problems with multiple units
8. Knows how to analyze both precision and accuracy in measurement situationsa. Choose a level of accuracy appropriate to
limitations on measurement when reporting quantities.
b. Calculate or estimate absolute and relative error in the numerical answer to a problem
9. Understands various ways to represent and compare very large and very small numbers (e.g., scientific notation, orders of magnitude)a. Represent and compare very large and very
small numbersPre-algebra / Scientific notation
10. Understands how to both estimate and perform calculations on very large and very small quantities
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 5 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
a. Use orders of magnitude to estimate very large and very small numbers
b. Perform calculations on numbers in scientific notation
Pre-algebra / Computing with scientific notation
B. Algebra
1. Understands how to write algebraic expressions in equivalent formsa. Use the structure of an expression to identify
ways to rewrite itAlgebra I / Combining like termsAlgebra I / Introduction to equivalent expressionsAlgebra I / Factoring quadratics with difference of squares Algebra 2 / Factoring using structureAlgebra 2 / Factoring monomialsAlgebra 2 / Factoring higher degree polynomials
b. Understand how to rewrite quadratic expressions for specific purposes (e.g., factoring/finding zeros, completing the square/finding maxima or minima).
Algebra I / Factoring quadratics introAlgebra I / Factoring quadratics by groupingAlgebra I / Factoring quadratics with difference of squaresAlgebra I / Factoring quadratics with perfect squaresAlgebra I / Strategy in factoring quadraticsAlgebra I / Vertex formAlgebra I / Completing the squareAlgebra I / Quadratic standard formAlgebra I / Features & forms of quadratic functions
c. Use the properties of exponents to rewrite expressions for exponential functions.
Algebra I / Exponent properties review
2. Understands how to perform arithmetic operations on polynomialsa. Add, subtract, and multiply polynomials Algebra I / Combining like terms
Algebra I / Multiplying monomials by polynomialsAlgebra I / Multiplying binomialsAlgebra 2 / Intro to polynomials
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 6 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
Algebra 2 / Adding and subtracting polynomialsAlgebra 2 / Multiplying monomials by polynomialsAlgebra 2 / Multiplying binomials by polynomialsPrecalculus / Adding and subtracting polynomials: two variables
3. Understands the relationship between zeros of polynomial functions (including their graphical representation) and factors of the related polynomial expressionsa. Know and apply the remainder theorem: for a
polynomial and a number a, the remainder on division by is , so
if and only if is a factor of .
Algebra 2 / Polynomial Remainder TheoremPrecalculus / Polynomial division
b. Use factorization to identify zeros of polynomials
Algebra I / Introduction to factoringAlgebra 2 / Dividing quadratics by linear factorsAlgebra 2 / Dividing polynomials by linear factorsAlgebra 2 / Zeros of polynomials
c. Use zeros of a polynomial to construct a rough graph of the function defined by the polynomial.
Algebra I / Solving and graphing with factored formAlgebra 2 / Zeros of polynomialsAlgebra 2 / Positive and negative intervals of polynomialsAlgebra 2 / End behavior of polynomialsAlgebra 2 / Putting it all togetherPrecalculus / The fundamental theorem of algebra
4. Understands how to use polynomial identities (e.g., difference of squares, sum and difference of cubes) to solve problems.a. Apply the binomial theorem for the expansion
of in powers of x and y for a positive integer n.
Precalculus / Adding and subtracting polynomials: two variablesPrecalculus / The binomial theorem
5. Understands how to rewrite rational expressions Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 7 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
and perform arithmetic operations on rational expressionsa. Rewrite simple rational expressions in
different formsAlgebra 2 / Cancelling common factors
b. Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.
Algebra 2 / Taking common factors Algebra 2 / Dividing quadratics by linear factorsAlgebra 2 / Dividing polynomials by linear factors
c. Add, subtract, multiply, and divide rational expressions
Algebra 2 / Dividing polynomials by xPrecalculus / Polynomial divisionAlgebra 2 / Multiplying and dividing rational expressionsAlgebra 2 / Adding and subtracting rational expressions introAlgebra 2 / Adding and subtracting rational expressions (factored)Algebra 2 / Adding and subtracting rational expressions (not factored)
6. Understands how to create equations and inequalities that describe relationshipsa. Create equations and inequalities in one
variable and use them to solve problems and graph solutions on the number line
Algebra I / Linear equations with variables on both sidesAlgebra I / Linear equations with parenthesesAlgebra I / Linear equations with unknown coefficientsAlgebra I / Multi-step inequalitiesAlgebra 2 / Modeling with equations and inequalities
b. Create equations and inequalities in two or more variables to represent relationships between quantities, solve problems, and graph them on the coordinate plane with labels and scales
Algebra I / Two-variable linear equations introAlgebra I / Graphing two-variable inequalitiesAlgebra I / Slope
c. Represent constraints by equations, inequalities, or systems of equations and/or inequalities, and interpret solutions as viable
Algebra I / Systems of equations word problemsAlgebra I / Modeling with linear inequalitiesAlgebra 2 / Modeling with equations and inequalities
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 8 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
or nonviable options in a modeling contextd. Rearrange formulas to highlight a quantity of
interest (e.g., solve )Algebra 2 / Manipulating formulas
7. Understands how to justify the reasoning process used to solve equations, including analysis of potential extraneous solutions.a. Explain each step in solving a simple equation Algebra I / Linear equations with variables on both sides
Algebra I / Linear equations with parenthesesb. Solve simple rational and radical equations in
one variable, incorporating analysis of possible extraneous solutions
Algebra 2 / Rational equationsAlgebra 2 / Square-root equationsAlgebra 2 / Extraneous solutionsAlgebra 2 / Cube-root equations
8. Understands how varied techniques (e.g., graphical, algebraic) are used to solve equations and inequalities in one variablea. Solve linear equations and inequalities in one
variable, including equations with coefficients represented by letters
Algebra I / Linear equations with variables on both sidesAlgebra I / Linear equations with parenthesesAlgebra I / Linear equations with unknown coefficientsAlgebra I / Multi-step inequalitiesAlgebra I / Compound inequalitiesAlgebra I / Summary: Forms of two-variable linear equations
b. Use the method of completing the square to transform any quadratic equation in x into the
equivalent form
Algebra I / Completing the square
c. Solve equations using a variety of methods (e.g., using graphs, using the quadratic formula, or factoring)
Algebra I / Factoring quadratics introAlgebra I / Factoring quadratics by groupingAlgebra I / Factoring quadratics with difference of squaresAlgebra I / Factoring quadratics with perfect squaresAlgebra I / Strategy in factoring quadraticsAlgebra I / Solving quadratics by factoring
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 9 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
Algebra I / The quadratic formulaPrecalculus / The fundamental theorem of algebraPrecalculus / Solving equations by graphing
d. Use different methods (e.g., discriminant analysis, graphical analysis) to determine the nature of the solutions of a quadratic equation
Algebra I / Intro to parabolasAlgebra I / Solving and graphing with factored formAlgebra I / Solving by taking the square rootAlgebra I / Vertex formAlgebra I / The quadratic formulaAlgebra I / Quadratic standard formAlgebra I / Features & forms of quadratic functions
e. Write complex solutions in the form . Algebra I / The quadratic formulaAlgebra 2 / The imaginary unit i
9. Understands how varied techniques (e.g., graphical, algebraic, matrix) are used to solve systems of equations and inequalitiesa. Explain why, when solving a system of two
equations using the elimination method, replacing one or both equations with a scalar multiple produces a system with the same solutions as the solutions of the original system
Algebra I / Equivalent systems of equations and the elimination method
b. Solve a system consisting of two linear equations in two variables algebraically and graphically
Algebra I / Introduction to systems of equationsAlgebra I / Solving systems of equations with substitutionAlgebra I / Number of solutions to systems of equations
c. Solve a system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically
Algebra 2 / Quadratic systems
d. Represent a system of linear equations as a single matrix equation
Precalculus / Representing linear systems of equations with augmented matrices
e. Find the inverse of a matrix if it exists, and use Precalculus / Solving equations with inverse matrices
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 10 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
it to solve systems of linear equationsf. Explain why the x-coordinates of the
intersection points of the graphs of and are the solutions of .
Precalculus / Solving equations by graphing
g. Find the solutions of approximately (e.g., use technology to graph the functions, make tables of values, find successive approximations). Include cases where and/or are linear, polynomial, rational, absolute value, exponential, or logarithmic functions
Precalculus / Solving equations by graphing
h. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes
Algebra I / Checking solutions of two-variable inequalitiesAlgebra I / Graphing two-variable inequalities
10. Understands the properties of number systems under various operationsa. Given operations on algebraic expressions,
determine whether the properties hold (e.g., commutative, associative, distributive)
11. Understands the concept of rate of change of nonlinear functions
Algebra I / Average rate of changeAlgebra I / Average rate of change word problemsAlgebra 2 / Average rate of change of polynomials
a. Calculate and interpret the average rate of change of a function presented symbolically, numerically, or graphically over a specified interval
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 11 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
12. Understands the concepts of intercept(s) of a line and slope as a rate of changea. Calculate and interpret the intercepts of a line Algebra I / x-intercepts and y-intercepts
Algebra I / Applying intercepts and slopeb. Calculate and interpret the slope of a line
presented symbolically, numerically, or graphically
Algebra I / SlopeAlgebra I / Horizontal & vertical linesAlgebra I / Applying intercepts and slopeAlgebra I / Intro to slope-intercept formAlgebra I / Graphing slope-intercept equationsAlgebra I / Writing slope-intercept equationsAlgebra I / Point-slope formAlgebra I / Standard formAlgebra I / Summary: Forms of two-variable linear equations
c. Estimate the rate of change of a linear function from a graph
Algebra I / Applying intercepts and slope
13. Understands how to find the zero(s) of functions
a. Uses a variety of techniques to find and analyze the zero(s) (real and complex) of functions
Algebra 2 / Zeros of polynomialsAlgebra 2 / Solving equations by graphingPrecalculus / Solving equations by graphing
C. Functions
1. Understands the function concept and the use of function notationa. Recognize that functions are sets of ordered
pairsAlgebra I / Recognizing functions
b. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range
Algebra I / Inputs and outputs of a function
c. Use function notation, evaluate functions, and Algebra I / Evaluating functionsCopyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 12 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
interpret statements that use function notation in terms of a context
Algebra I / Functions and equationsAlgebra I / Interpreting function notation
d. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers
2. Understands how to find the domain and range of a function and a relationa. Identify the domain and range of a function or
relationAlgebra I / Introduction to the domain and range of a function
b. Determine the domain of a function from a function rule (e.g., , graph, set of ordered pairs, or table
Algebra I / Determining the domain of a function
3. Understands how function behavior is analyzed using different representations. (e.g., graphs, mappings, tables).a. For a function that models a relationship
between two quantities, interpret key features of graphs and tables (e.g., increasing/decreasing, maximum/minimum, periodicity) in terms of the quantities
Algebra I / Maximum and minimum pointsAlgebra I / Intervals where a function is positive, negative, increasing, or decreasingAlgebra 2 / Interpreting features of functions
b. Given a verbal description of a relation, sketch graphs that show key features of that relation
Algebra I / Interpreting features of graphs
c. Graph functions (i.e., radical, piecewise, absolute value, polynomial, rational, logarithmic, trigonometric) expressed symbolically and identify key features of the graph
Algebra I / Graphs of absolute value functionsAlgebra I / Piecewise functionsAlgebra I / Graphs of exponential growthAlgebra I / Quadratic standard formAlgebra I / Features & forms of quadratic functionsAlgebra I / Transforming quadratic functionsAlgebra 2 / Putting it all togetherAlgebra 2 / Graphs of square and cube root functions
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 13 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
Algebra 2 / Graphs of exponential functionsAlgebra 2 / Graphs of logarithmic functionsAlgebra 2 / Graphs of sin(x), cos(x), and tan(x)Algebra 2 / Graphing sinusoidal functionsAlgebra 2 / End behavior of rational functionsAlgebra 2 / Discontinuities of rational functionsAlgebra 2 / Graphs of rational functions
d. Write a function that is defined by an expression in different but equivalent forms to reveal different properties of the function (e.g., zeros, extreme values, symmetry of the graph)
Algebra I / Solving and graphing with factored formAlgebra I / Vertex form
e. Interpret the behavior of exponential functions (e.g., growth, decay)
Algebra I / Exponential growth & decay
f. Understand how to determine if a function is odd, even, or neither and any resulting symmetries
Algebra 2 / Symmetry of functions
4. Understands how functions and relations are used to model relationships between quantities.a. Write a function that relates two quantities Algebra I / Exponential vs. linear models
Algebra 2 / Solving exponential modelsAlgebra 2 / Modeling with rational functions
b. Determine an explicit expression or a recursive process that builds a function from a context
Algebra I / Exponential vs. linear modelsAlgebra 2 / Solving exponential models
5. Understands how new functions are obtained from existing functions (e.g., compositions, transformations, inverses)
a. Describe how the graph of is related to the graph of , where ,
Algebra I / Transforming quadratic functionsAlgebra 2 / Shifting functions
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 14 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
, , or for specific values of k (both positive and negative), and find the value of k given the graphs
Algebra 2 / Reflecting functionsAlgebra 2 / Scaling functionsAlgebra 2 / Putting it all together
b. Determine if a function has an inverse and write an expression for the inverse
Algebra I / Intro to inverse functionsPrecalculus / Invertible functions
c. Verify by composition if one function is the inverse of another
Algebra I / Intro to inverse functionsPrecalculus / Verifying inverse functions by composition
d. Given that a function f has an inverse, find values of the inverse function from a graph or a table of f
Algebra I / Intro to inverse functionsPrecalculus / Invertible functions
e. Given a noninvertible function, determine a largest possible domain of the function that produces an invertible function
Precalculus / Invertible functions
f. Understand the inverse relationship between exponential and logarithmic functions and use this relationship to solve problems
Algebra 2 / Introduction to logarithmsAlgebra 2 / The constant e and the natural logarithmPrecalculus / Invertible functions
g. Combine standard function types using arithmetic operations
Algebra 2 / Modeling with function combination
h. Perform domain analysis on functions resulting from arithmetic operations
i. Compose functions algebraically, numerically, and graphically
Precalculus / Composing functionsPrecalculus / Modeling with composite functions
j. Perform domain analysis on functions resulting from compositions
6. Understands differences between linear, quadratic, and exponential models, including how their equations are created and used to solve problems.a. Understand that linear functions grow by Algebra I / Exponential vs. linear growth
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 15 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another
Algebra I / Exponential vs. linear growth
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another
Algebra I / Exponential expressionsAlgebra 2 / Interpreting the rate of change of exponential modelsAlgebra 2 / Advanced interpretation of exponential models
d. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two ordered pairs (include reading these from a table).
Algebra I / Exponential functions from tables & graphsAlgebra 2 / Constructing exponential models according to rate of changeAlgebra 2 / Advanced interpretation of exponential models
e. Observe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
Algebra I / Exponential vs. linear growth over time
f. Express the solution to an exponential equation with base b as a logarithm (e.g.,
, )
Algebra 2 / Introduction to logarithmsAlgebra 2 / The constant e and the natural logarithmAlgebra 2 / Properties of logarithmsAlgebra 2 / The change of base formula for logarithmsAlgebra 2 / Solving exponential equations with logarithms
g. Use technology to evaluate logarithms that have any base
Algebra 2 / Introduction to logarithmsAlgebra 2 / The constant e and the natural logarithmAlgebra 2 / Properties of logarithmsAlgebra 2 / The change of base formula for logarithms
h. Interpret the parameters in a linear or exponential function in terms of a context
(e.g., )
Algebra I / Exponential vs. linear modelsAlgebra 2 / Solving exponential models
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 16 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
i. Use quantities that are inversely related to model phenomena
7. Understands how to construct the unit circle and how to use it to find values of trigonometric functions for all angle measures in their domains.a. Understand radian measure (e.g., 1 radian is
the measure of a central angle that subtends an arc with length equal to the length of the radius
High school geometry / Introduction to radiansAlgebra 2 / Rotation-by-radians-and-quadrants
b. Understand how the domains of trigonometric functions can be extended beyond 0 to 2π using the unit circle
c. Use special triangles (i.e., 30-60-90, 45-45-90) to determine geometrically the values of sine,
cosine, and tangent for , , and .
Algebra 2 / Trigonometric values of special angles
d. Use reference angles to find the values of trigonometric functions at angles outside the
interval 0 to .e. Use the unit circle to explain symmetry and
periodicity of trigonometric functions8. Understands how periodic phenomena are
modeled using trigonometric functionsa. Choose trigonometric functions to model
periodic phenomena with specified amplitude, frequency, and midline
Algebra 2 / Amplitude, midline and periodAlgebra 2 / Transforming sinusoidal graphsAlgebra 2 / Sinusoidal modelsPrecalculus / Sinusoidal models
b. Understand how to restrict the domain of a Precalculus / Invertible functionsCopyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 17 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
trigonometric function so that its inverse can be constructed
Precalculus / Inverse trigonometric functions
c. Use inverse functions to solve trigonometric equations that arise in modeling contexts, and interpret them in terms of the context
Precalculus / Inverse trigonometric functions
9. Understands the application of trigonometric identities (e.g., Pythagorean, double angle, half angle, sum of angles, difference of angles).a. Use Pythagorean identities (e.g.,
)Algebra 2 / The Pythagorean identity
b. Use trigonometric identities to rewrite expressions and solve equations
Precalculus / Trigonometric identitiesPrecalculus / Angle addition identitiesPrecalculus / Using trigonometric identities
c. Understand trigonometric identities in the context of equivalent graphs of trigonometric
functions (e.g., and are equivalent graphs)
Algebra 2 / Transforming sinusoidal graphsAlgebra 2 / Graphing sinusoidal functions
d. Prove Pythagorean identities (e.g., .
Algebra 2 / The Pythagorean identity
10. Knows how to interpret representations of functions of two variables (e.g., three-dimensional graphs, tables).a. Interpret representations of functions of two
variables11. Understands how to solve equations (e.g.,
trigonometric, logarithmic, exponential).a. Solve trigonometric, logarithmic, and
exponential equationsAlgebra 2 / Solving exponential equations using properties of exponentsAlgebra 2 / Solving exponential equations with logarithms
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 18 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
Algebra 2 / Solving equations by graphingPrecalculus / Sinusoidal equations
D. Calculus
1. Understands the meaning of a limit of a function and how to calculate limits of functions, determine when the limit does not exist, and solve problems using the properties of limits.
a. Graphically analyze the limit of as x approaches a fixed value from both left and right
Differential calculus / Limits introDifferential calculus / Estimating limits from graphsDifferential calculus / Limits at infinity
b. Solve limit problems (e.g., a constant times a function, the sum of two functions, the product and quotient of two functions) using properties of limits, where all limits of the individual functions exist at the value that x is approaching
Differential calculus / Properties of limitsDifferential calculus / Limits by direct substitutionDifferential calculus / Limits using algebraic manipulationDifferential calculus / Strategy in finding limitsDifferential calculus / L’Hôpital’s ruleDifferential calculus / L’Hôpital’s rule: composite exponential functions
c. Analyze one-sided limits for various functions to see whether or not the limit exists
Differential calculus / Estimating limits from graphsDifferential calculus / Estimating limits from tables
d. Recognize limits that do not exist, such as
and .
Differential calculus / Estimating limits from graphsDifferential calculus / Estimating limits from tables
2. Understands the derivative of a function as a limit, as the slope of a line tangent to a curve, and as a rate of change. a. Construct a function graph for a given function
and a given point , and explain what happens to the succession of slopes of secant lines connecting to as x approaches a, from both the right side and the
Differential calculus / Average vs. instantaneous rate of changeDifferential calculus / Secant lines
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 19 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
left sideb. State the limit definition of the derivative, and
use it to find the derivative of a given function at a given value of x and to find the derivative function
Differential calculus / Derivative definitionDifferential calculus / Estimating derivatives
3. Understands how to show that a particular function is continuous. a. Apply the three steps (i.e., exists,
exists, and ) that are part of the definition of what it means for a function to be continuous at to verify whether a given function is continuous at a given point
Differential calculus / Types of discontinuitiesDifferential calculus / Continuity at a pointDifferential calculus / Continuity over an intervalDifferential calculus / Removing discontinuities
4. Knows the relationship between continuity and differentiabilitya. Give examples of functions that are
continuous at but not differentiable at , and explain why
Differential calculus / Differentiability
5. Understands how to approximate derivatives and integrals numericallya. Given a table of values, use the slope of a
secant line to approximate a derivative.Differential calculus / Secant linesDifferential calculus / Estimating derivatives
b. Use the midpoint rule, trapezoid rule, or other Riemann sums to find numerical approximations for integrals
6. Understands how and when to use standard differentiation and integration techniquesa. Use standard differentiation techniques Differential calculus / Power rule
Differential calculus / Derivative rules: constant, sum, difference, and
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 20 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
constant multipleDifferential calculus / Combining the power rule with other derivative rulesDifferential calculus / Derivatives of cos(x), sin(x), 𝑒 ˣ, and ln(x) Differential calculus / Product ruleDifferential calculus / Quotient ruleDifferential calculus / Derivatives of tan(x), cot(x), sec(x), and csc(x)Differential calculus / Chain ruleDifferential calculus / More chain rule practiceDifferential calculus / Implicit differentiationDifferential calculus / Implicit differentiation (advanced examples)Differential calculus / Differentiating inverse functionsDifferential calculus / Derivatives of inverse trigonometric functionsDifferential calculus / Strategy in differentiating functionsDifferential calculus / Differentiation using multiple rulesDifferential calculus / Disguised derivativesDifferential calculus / Logarithmic differentiation
b. Use standard integration techniques Integral calculus / Properties of definite integralsIntegral calculus / Reverse power ruleIntegral calculus / Indefinite integrals of common functionsIntegral calculus / Definite integrals of common functionsIntegral calculus / Integrating with u-substitutionIntegral calculus / Integrating using long division and completing the squareIntegral calculus / Integrating using trigonometric identitiesIntegral calculus / Trigonometric substitutionIntegral calculus / Integration by partsIntegral calculus / Integrating using linear partial fractionsIntegral calculus / Improper integrals
c. Understand the relationship between position, Differential calculus / Straight-line motion
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 21 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
velocity, and acceleration functions of a particle in motion
Integral calculus / Straight-line motion
7. Understands how to analyze the behavior of a function (e.g., extrema, concavity, symmetrya. Use the first and second derivatives to analyze
the graph of a functionDifferential calculus / Second derivativesDifferential calculus / Intervals on which a function is increasing or decreasingDifferential calculus / Relative (local) extremaDifferential calculus / Absolute (global) extremaDifferential calculus / Concavity and inflection points introDifferential calculus / Analyzing concavity and inflection pointsDifferential calculus / Second derivative testDifferential calculus / Sketching curvesDifferential calculus / Connecting f, f', and f''
8. Understands how to apply derivatives to solve problems (e.g., related rates, optimization)a. Apply derivatives to solve problems Differential calculus / Introduction to related rates
Differential calculus / Solving related rates problemsDifferential calculus / Approximation with local linearityDifferential calculus / Solving optimization problemsDifferential calculus / Analyzing implicit relations
9. Understands the foundational theorems of calculus (e.g., fundamental theorems of calculus, mean value theorem, intermediate value theorem)a. Solve problems using the foundational
theorems of calculusDifferential calculus / Intermediate value theoremDifferential calculus / Mean value theoremDifferential calculus / Extreme value theorem and critical points
b. Understand the relationship between differentiation and integration, including the
Integral calculus / Fundamental theorem of calculus and accumulation functions
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 22 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
role of the fundamental theorems of calculus Integral calculus / Interpreting the behavior of accumulation functionsIntegral calculus / Fundamental theorem of calculus and definite integrals
c. Match graphs of functions with graphs of their derivatives or accumulations
d. Understand how to use differentiation and integration of a function to express rates of change and total change
Differential calculus / Meaning of the derivative in contextDifferential calculus / Non-motion applications of derivatives
e. Understand and calculate the average value of a function over an interval (i.e., mean value theorem of integrals).
Integral calculus / Average value of a function
10. Understands integration as a limit of Riemann sums
Integral calculus / Approximation with Riemann sumsIntegral calculus / Summation notation reviewIntegral calculus / Riemann sums in summation notationIntegral calculus / Defining integrals with Riemann sums
a. Calculate a definite integral using a limit of Riemann sums.
11. Understands how to use integration to compute area, volume, distance, or other accumulation processes
Integral calculus / Accumulations of change introductionIntegral calculus / Fundamental theorem of calculus and accumulation functionsIntegral calculus / Interpreting the behavior of accumulation functionsIntegral calculus / Non-motion applications of integralsIntegral calculus / Area: vertical area between curvesIntegral calculus / Area: horizontal area between curvesIntegral calculus / Area: curves that intersect at more than two pointsIntegral calculus / Volume: squares and rectangles cross sectionsIntegral calculus / Volume: triangles and semicircles cross sectionsIntegral calculus / Volume: disc method (revolving around x- and y-axes)Integral calculus / Volume: disc method (revolving around other axes)Integral calculus / Volume: washer method (revolving around x- and y-
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 23 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
axes)Integral calculus / Volume: washer method (revolving around other axes)
a. Use integration techniques to compute area, volume, distance, or other accumulation processes
12. Knows how to determine the limits of sequences, if they exist.a. Determine the limits of sequences when they
exist13. Is familiar with simple infinite series Algebra 2 / Geometric series
Precalculus / Geometric seriesPrecalculus / Geometric series (with summation notation)Precalculus / Arithmetic seriesIntegral calculus / Convergent and divergent infinite seriesIntegral calculus / Infinite geometric seriesIntegral calculus / nth-term testIntegral calculus / Integral testIntegral calculus / Harmonic series and p-seriesIntegral calculus / Comparison testsIntegral calculus / Alternating series testIntegral calculus / Ratio testIntegral calculus / Absolute and conditional convergence
a. Determine if simple infinite series converge or diverge
Algebra 2 / Geometric seriesPrecalculus / Geometric seriesPrecalculus / Geometric series (with summation notation)Precalculus / Arithmetic seriesIntegral calculus / Convergent and divergent infinite series
b. Find the sum of a simple infinite series if it exists
Algebra 2 / Geometric seriesPrecalculus / Geometric seriesPrecalculus / Geometric series (with summation notation)
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 24 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
Precalculus / Arithmetic seriesIntegral calculus / Convergent and divergent infinite series
c. Find the partial sum of a simple infinite series Integral calculus / Approximation with Riemann sumsIntegral calculus / Summation notation reviewIntegral calculus / Riemann sums in summation notationIntegral calculus / Defining integrals with Riemann sums
II. Geometry, Probability and Statistics, and Discrete Mathematics
Lesson
A. Geometry
1. Understands transformations in a plane
a. Know precise definitions of angle, circle, line segment, perpendicular lines, and parallel lines
High school geometry / Intro to Euclidean geometryHigh school geometry / Circle basics
b. Represent transformations in the plane High school geometry / Introduction to rigid transformations
c. Describe transformations as functions that take points in the plane as inputs, and give other points as outputs
High school geometry / Introduction to rigid transformations
d. Recognize whether a transformation preserves distance and angle measure
High school geometry / TranslationsHigh school geometry / RotationsHigh school geometry / Reflections
e. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that map it onto itself
High school geometry / Rigid transformations overviewHigh school geometry / Symmetry
f. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments
High school geometry / Properties & definitions of transformations
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 25 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
g. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure
High school geometry / Rigid transformations overview
h. Specify a sequence of transformations that will map a given figure onto another figure
High school geometry / Rigid transformations overviewHigh school geometry / Properties & definitions of transformations
2. Understands how to prove geometric theorems such as those about lines and angles, triangles, and parallelogramsa. Prove theorems about lines and angles High school geometry / Angles
b. Prove theorems about triangles High school geometry / Theorems concerning triangle propertiesHigh school geometry / Proofs of general theorems that use triangle congruenceHigh school geometry / Angle bisector theoremHigh school geometry / Pythagorean theorem proofsHigh school geometry / Polygons
c. Prove theorems about parallelograms High school geometry / Theorems concerning quadrilateral propertiesHigh school geometry / Polygons
3. Understands how geometric constructions are made with a variety of tools and methodsa. Recognize formal geometric constructions High school geometry / Constructing bisectors of lines & angles
b. Explain how formal geometric constructions are made (e.g., an equilateral triangle, a square, a regular hexagon inscribed in a circle).
High school geometry / Constructing bisectors of lines & anglesHigh school geometry / Inscribed shapes problem solvingHigh school geometry / Constructing regular polygons inscribed in circlesHigh school geometry / Constructing circumcircles & incircles
4. Understands congruence and similarity in terms of transformationsa. Use geometric descriptions of rigid motions to
transform figures and to predict the effect of a given rigid motion on a given figure
High school geometry / Rigid transformations overview
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 26 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
b. Verify the properties of dilations given by a center and a scale factor
High school geometry / Dilations
c. Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent
High school geometry / Transformations & congruence
d. Given two figures, use the definition of similarity in terms of dilations to decide if the figures are similar
High school geometry / Definitions of similarity
e. Explain how the criteria for triangle congruence (e.g., ASA, SAS, HL) follow from the definition of congruence in terms of rigid motions
High school geometry / Triangle congruence
f. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar
High school geometry / Introduction to triangle similarityHigh school geometry / Solving similar triangles
g. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures
High school geometry / Theorems concerning triangle propertiesHigh school geometry / Working with trianglesHigh school geometry / Proofs of general theorems that use triangle congruenceHigh school geometry / Solving problems with similar & congruent trianglesHigh school geometry / Solving modeling problems with similar & congruent triangles
5. Understands how trigonometric ratios are defined in right trianglesa. Understand that by similarity, side ratios in
right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
High school geometry / Introduction to the trigonometric ratiosHigh school geometry / Trigonometric ratios & similarityAlgebra 2 / Unit circle introduction
b. Explain and use the relationship between the High school geometry / Introduction to the trigonometric ratios
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 27 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
sine and cosine of complementary angles High school geometry / Sine & cosine of complementary anglesc. Use trigonometric ratios and the Pythagorean
theorem to solve right triangles in applied problems
High school geometry / Pythagorean theoremHigh school geometry / Solving for a side in a right triangle using the trigonometric ratiosHigh school geometry / Solving for an angle in a right triangle using the trigonometric ratiosHigh school geometry / Modeling with right triangles
6. Understands how trigonometry is applied to general triangles
a. Derive the formula for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side and use it to solve problems
High school geometry / Law of sinesHigh school geometry / Solving general triangles
b. Apply the law of sines and the law of cosines to find unknown measurements in triangles.
High school geometry / Law of sinesHigh school geometry / Law of cosinesHigh school geometry / Solving general triangles
7. Understands and applies theorems about circles.
a. Identify and describe relationships among inscribed angles, radii, and chords
High school geometry / Arc measureHigh school geometry / Inscribed shapes problem solving
b. Prove properties of angles for a quadrilateral inscribed in a circle
High school geometry / Arc length (from degrees)High school geometry / Arc length (from radians)High school geometry / Inscribed anglesHigh school geometry / Inscribed shapes problem solving
c. Construct a tangent line from a point outside a given circle to the circle.
High school geometry / Properties of tangentsHigh school geometry / Constructing a line tangent to a circle
8. Understands arc length and area measurements of sectors of circlesa. Derive and use the fact that the length of the High school geometry / Arc measure
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 28 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
arc intercepted by a central angle is proportional to the circumference
b. Derive and use the formula for the area of a sector
High school geometry / Sectors
9. Knows how to translate between a geometric description (e.g., focus, asymptotes, directrix) and an equation for a conic sectiona. Derive and use the equation of a circle of
given center and radiusHigh school geometry / Standard equation of a circlePrecalculus / The features of a circlePrecalculus / Standard equation of a circle
b. Complete the square to find the center and radius of a circle given by an equation in standard form
High school geometry / Expanded equation of a circlePrecalculus / Expanded equation of a circlePrecalculus / Identifying conic sections from their equations
c. Derive the equation of a parabola given a focus and directrix
Precalculus / Introduction to conic sectionsPrecalculus / Focus and directrix of a parabolaPrecalculus / Identifying conic sections from their equations
d. Derive and use the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from a point on the curve to the foci is constant
Precalculus / Introduction to conic sectionsPrecalculus / Center and radii of an ellipsePrecalculus / Foci of an ellipsePrecalculus / Introduction to hyperbolasPrecalculus / Foci of a hyperbolaPrecalculus / Hyperbolas not centered at the originPrecalculus / Identifying conic sections from their equations
10. Understands how to use coordinate geometry to algebraically prove simple geometric theoremsa. Use coordinates to prove simple geometric
theorems algebraicallyHigh school geometry / Distance and midpoints
b. Prove the slope criteria for parallel and perpendicular lines, and use parallel and perpendicular lines to solve geometric
High school geometry / Parallel & perpendicular lines on the coordinate planeHigh school geometry / Equations of parallel & perpendicular lines
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 29 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
problemsc. Find the point on a directed line segment
between two given points that partitions the segment in a given ratio
High school geometry / Dividing line segments
d. Use coordinates to compute perimeters of polygons and areas of triangles and quadrilaterals
High school geometry / Problem solving with distance on the coordinate plane
11. Understands how perimeter, area, surface area, and volume formulas are used to solve problemsa. Give an informal argument for the formulas
for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone
High school geometry / AreaHigh school geometry / Solid geometry intro
b. Use the perimeter and area of geometric shapes to solve problems
High school geometry / Area
c. Use the surface area and volume of prisms, cylinders, pyramids, cones, and spheres to solve problems
High school geometry / Area
12. Knows how to visualize relationships (e.g., cross section, nets, rotations) between two-dimensional and three-dimensional objectsa. Identify the shapes of two-dimensional cross
sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects
High school geometry / 2D vs. 3D objects
b. Use two-dimensional representations of three-dimensional objects to visualize and solve problems.
High school geometry / Area
13. Knows how to apply geometric concepts in real-world situationsa. Use geometric shapes, their measures, and
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 30 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
their properties to describe objectsb. Apply concepts of density based on area and
volume in modeling situationsHigh school geometry / Density
c. Apply geometric methods to solve design problems
14. Understands the properties of parallel and perpendicular lines, triangles, quadrilaterals, polygons, and circles and how they can be used in problem solvinga. Solve problems involving parallel,
perpendicular, and intersecting linesHigh school geometry / Angles
b. Apply angle relationships (e.g., supplementary, vertical, alternate interior) to solve problems
High school geometry / Angles
c. Solve problems that involve medians, midpoints, and altitudes
High school geometry / PolygonsHigh school geometry / Proofs of general theorems that use triangle congruence
d. Solve problems involving special triangles (e.g., isosceles, equilateral, right)
High school geometry / Special right trianglesHigh school geometry / PolygonsHigh school geometry / Working with triangles
e. Know geometric properties of various quadrilaterals (e.g., parallelograms, trapezoids)
High school geometry / PolygonsHigh school geometry / Theorems concerning quadrilateral properties
f. Know relationships among quadrilateral High school geometry / Polygons
g. Solve problems involving angles and diagonals High school geometry / AnglesHigh school geometry / Theorems concerning quadrilateral properties
h. Solve problems involving polygons with more than four sides
High school geometry / Polygons
B. Probability and Statistics
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 31 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
1. Understands how to summarize, represent, and interpret data collected from measurements on a single variable (e.g., box plots, dot plots, normal distributionsa. Represent data with plots on the real number
line (e.g., dot plots, histograms, and box plots).
Statistics and probability / Displaying quantitative data with graphsStatistics and probability / More on data displaysStatistics and probability / Box and whisker plotsHigh school statistics / Displays of distributionsHigh school statistics / Box and whisker plots
b. Use statistics appropriate to the shape of the data distribution to compare center (e.g., median, mean) and spread (e.g., interquartile range, standard deviation) of two or more different data sets.
Statistics and probability / Describing and comparing distributionsStatistics and probability / Measuring center in quantitative dataStatistics and probability / Interquartile range (IQR)Statistics and probability / Variance and standard deviation of a populationStatistics and probability / Other measures of spreadHigh school statistics / Summarizing center of distributions (central tendency)High school statistics / Summarizing spread of distributionsHigh school statistics / Comparing distributions
c. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of outliers
Statistics and probability / More on mean and median
d. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages, and recognize that there are data sets for which such a procedure is not appropriate
Statistics and probability / Variance and standard deviation of a sampleStatistics and probability / More on standard deviation
e. Estimate areas under the normal curve Statistics and probability / PercentilesStatistics and probability / Z-scoresStatistics and probability / Effects of linear transformations
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 32 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
Statistics and probability / Density curvesStatistics and probability / Normal distributions and the empirical ruleStatistics and probability / Normal distribution calculationsStatistics and probability / More on normal distributionsHigh school statistics / Normal distributions
2. Understands how to summarize, represent, and interpret data collected from measurements on two variables, either categorical or quantitative (e.g., scatterplots, time series)a. Summarize and interpret categorical data for
two categories in two-way frequency tables (e.g., joint, marginal, conditional relative frequencies
Statistics and probability / Two-way tablesStatistics and probability / Distributions in two-way tablesHigh school statistics / Two-way frequency tablesHigh school statistics / Two-way relative frequency tables
b. Recognize possible associations and trends in the data
Statistics and probability / Introduction to scatterplotsHigh school statistics / Creating and interpreting scatterplots
c. Represent data for two quantitative variables on a scatterplot, and describe how the variables are related.
Statistics and probability / Introduction to scatterplots
3. Understands how to create and interpret linear regression models (e.g., rate of change, intercepts, correlation coefficienta. Use technology to fit a function to data (i.e.,
linear regression)Statistics and probability / Introduction to trend linesHigh school statistics / Estimating with trend lines
b. Use functions fitted to data to solve problems in the context of the data
Statistics and probability / Introduction to trend lines
c. Assess the fit of a function by plotting and analyzing residuals
Statistics and probability / Least-squares regression equationsStatistics and probability / Assessing the fit in least-squares regressionStatistics and probability / More on regression
d. Interpret the slope and the intercept of a regression line in the context of the data.
Statistics and probability / Least-squares regression equations
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 33 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
e. Compute and interpret a correlation coefficient
Statistics and probability / Correlation coefficients
f. Distinguish between correlation and causation Statistics and probability / correlation-and-causalityHigh school statistics / Creating and interpreting scatterplots
4. Understands statistical processes and how to evaluate thema. Understand statistics as a process for making
inferences about population parameters based on a random sample from that population
Statistics and probability / Statistical questions
b. Decide if a specified model is consistent with results from a given data-generating process (e.g., using simulation
Statistics and probability / Experimental probabilityStatistics and probability / Randomness, probability, and simulation
5. Understands how to make inferences and justify conclusions from samples, experiments, and observational studiesa. Recognize the purposes of and differences
among samples, experiments, and observational studies, and explain how randomization relates to each
Statistics and probability / Sampling and observational studiesStatistics and probability / Sampling methodsStatistics and probability / Types of studies (experimental vs. observational)Statistics and probability / ExperimentsHigh school statistics / Samples and surveysHigh school statistics / Observational studies and experiments
b. Use data from a sample to estimate a population mean or proportion
c. Use data from a randomized experiment to compare two treatments
d. Use results of simulations to decide if differences between parameters are significant
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 34 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
e. Evaluate reports based on data
6. Understands the concepts of independence and conditional probability and how to apply these concepts to dataa. Describe events as subsets of a sample space
using characteristics of the outcomes, or as unions, intersections, or complements of other events
Precalculus / Venn diagrams and the addition ruleStatistics and probability / Basic theoretical probabilityStatistics and probability / Probability using sample spacesStatistics and probability / Basic set operationsHigh school statistics / Probability basics
b. Understand that two events, A and B, are independent if and only if
Precalculus / Compound probability of independent events using the multiplication rulePrecalculus / Dependent eventsStatistics and probability / Multiplication rule for independent eventsStatistics and probability / Multiplication rule for dependent eventsHigh school statistics / Multiplication rule for independent eventsHigh school statistics / Multiplication rule for dependent events
c. Understand the conditional probability of A
given B as , and interpret independence of A and B as saying that
and
Statistics and probability / Conditional probability and independence
d. Recognize and explain the concepts of conditional probability and independence
Statistics and probability / Conditional probability and independence
7. Understands how to compute probabilities of simple events, probabilities of compound events, and conditional probabilitiesa. Calculate probabilities of simple and
compound eventsPrecalculus / Basic probabilityStatistics and probability / Basic theoretical probability
b. Construct and interpret two-way frequency High school statistics / Conditional probabilityCopyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 35 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
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
c. Find , and interpret it in terms of a given model.
High school statistics / Conditional probability
d. Apply the addition rule, and
interpret it in terms of a given model
Precalculus / Venn diagrams and the addition ruleStatistics and probability / Addition ruleHigh school statistics / Addition rule for probability
e. Apply the general multiplication rule in a uniform probability model,
, and interpret it in terms of a given model
f. Calculate probabilities using the binomial probability distribution
Precalculus / PermutationsPrecalculus / CombinationsPrecalculus / Probability using combinatoricsStatistics and probability / Counting principle and factorialStatistics and probability / PermutationsStatistics and probability / CombinationsStatistics and probability / Combinatorics and probabilityHigh school statistics / Counting with permutationsHigh school statistics / Counting with combinationsHigh school statistics / Probability with counting, permutations, combinations
8. Knows how to make informed decisions using probabilities and expected valuesa. Define a random variable for a quantity of
interest by assigning a numerical value to each Statistics and probability / Discrete random variables
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 36 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
event in a sample space, and graph the corresponding probability distribution using the same graphical displays as for data distributions
b. Calculate the expected value of a random variable, and interpret it as the mean of the probability distribution
Statistics and probability / Discrete random variablesHigh school statistics / Expected value
c. Develop a probability distribution for a random variable, defined for a sample space in which theoretical probabilities can be calculated, and find the expected value
High school statistics / Expected value
d. Develop a probability distribution for a random variable, defined for a sample space in which probabilities are assigned empirically, and find the expected value.
High school statistics / Expected value
e. Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values
Statistics and probability / More on expected valueHigh school statistics / Expected value
f. Analyze decisions and strategies using probability concepts (e.g., fairness).
9. Understands how to use simulations to construct experimental probability distributions and to make informal inferences about theoreticala. Given the results of simulations, construct
experimental probability distributionsHigh school statistics / Simulation and randomness
b. Given the results of simulations, make informal inferences about theoretical probability distributions.
High school statistics / Simulation and randomness
10. Understands how to find probabilities involving finite sample spaces and independent trials
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 37 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
a. Use the fundamental counting principle to find probabilities involving finite sample spaces and independent trials
Statistics and probability / Counting principle and factorialStatistics and probability / PermutationsStatistics and probability / CombinationsStatistics and probability / Combinatorics and probability
C. Discrete Mathematics
1. Understands sequences (e.g., arithmetic, recursively defined, geometric).a. Write arithmetic and geometric sequences
both recursively and with an explicit formula, use them to model situations, and translate between the two forms
Algebra I / Constructing arithmetic sequencesAlgebra I / Constructing geometric sequencesPrecalculus / Arithmetic sequencesPrecalculus / Geometric sequences
b. Evaluate, extend, or algebraically represent rules that involve number patterns
Algebra I / Introduction to arithmetic sequencesAlgebra I / Introduction to geometric sequencesAlgebra I / General sequences
c. Explore patterns in order to make conjectures, predictions, or generalizations
Algebra I / Modeling with sequences
2. Is familiar with how recursion can be used to model various phenomenaa. Find values of functions defined recursively,
and understand how recursion can be used to model various phenomena
b. Convert between recursive and closed-form expressions for a function, where possible
3. Has knowledge of equivalence relations
a. Determine whether a binary relation on a set is reflexive, symmetric, or transitive
b. Determine whether a relation is an equivalence relation
4. Understands the differences between discrete Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 38 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
and continuous representations (e.g., data, functions) and how each can be used to model various phenomenaa. Understand the differences between discrete
and continuous representations (e.g., data, functions
b. Understand how discrete and continuous representations can be used to model various phenomena.
5. Understands basic terminology and symbols of logica. Understand the basic terminology of logic
b. Understand the symbols of logic
c. Use logic to evaluate the truth of statements
d. Use logic to evaluate the equivalence of statements (e.g., statement and contrapositive
6. Understands how to use counting techniques such as the multiplication principle, permutations, and combinationsa. Use counting techniques to solve problems Precalculus / Permutations
Precalculus / CombinationsStatistics and probability / Counting principle and factorialStatistics and probability / PermutationsStatistics and probability / CombinationsHigh school statistics / Counting with permutationsHigh school statistics / Counting with combinations
7. Understands basic set theory (e.g., unions,
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 39 of 40
Praxis Mathematics: Content Knowledge (5161) Content Topics
Study Resources
differences, Venn diagrams). a. Solve problems using basic set theory (i.e.,
union, intersection, complement, differenceStatistics and probability / Basic set operations
b. Use Venn diagrams to answer questions about sets.
Statistics and probability / Basic set operations
Copyright © 2020 by Educational Testing Service. All rights reserved. ETS, the ETS logo and PRAXIS are registered trademarks of Educational Testing Service (ETS). Khan Academy is a registered trademark of Khan Academy.
Page 40 of 40
