**What Algebra Standards & Skills should you master for the Praxis 5161 Exam?**

Since the Praxis 5161 Exam Math Credential enables one teach High School Math, it's an extremely profitable exercise to scrutinize the Math content standards applicable for High School Math teachers in your state. These are expectations for that every current and prospective Math teacher ought to be familiar with.

The following content standards apply for Algebra, in general.

Candidates shall be able to

Candidates shall be able to

Candidates shall be able to

Candidates shall be able to

All that said, these are essential skills that I have distilled from the standards above!

For

For

For

For

For

For

For

For

For

The following content standards apply for Algebra, in general.

__ALGEBRA I__**:**Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences. In addition, algebraic skills and concepts are developed and used in a wide variety of problem-solving situations.Candidates shall be able to

- identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable:
- use properties of numbers to demonstrate whether assertions are true or false.
- understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.
- solve equations and inequalities involving absolute values.
- simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12.
- solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.
- graph a linear equation and compute the x- and y- intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).
- verify that a point lies on a line, given an equation of the line. are able to derive linear equations by using the point-slope formula.
- understand the concepts of parallel lines and perpendicular lines and how those slopes are related. are able to find the equation of a line perpendicular to a given line that passes through a given point.
- solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.
- add, subtract, multiply, and divide monomials and polynomials. solve multi step problems, including word problems, by using these techniques.
- apply basic factoring techniques to second-and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.
- simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.
- add, subtract, multiply, and divide rational expressions and functions. solve both computationally and conceptually challenging problems by using these techniques.
- solve a quadratic equation by factoring or completing the square.
- apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.
- understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions.
- determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.
- determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion.
- know the quadratic formula and are familiar with its proof by completing the square.
- use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.
- graph quadratic functions and know that their roots are the x- intercepts.
- use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points.
- apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.
- use and know simple aspects of a logical argument:
- explain the difference between inductive and deductive reasoning and identify and provide examples of each.
- identify the hypothesis and conclusion in logical deduction. 24.3 use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.
- use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements
- use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.
- judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step.
- determine whether the statement is true sometimes, always, or never, Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities.

__ALGEBRA II__**:**This discipline complements and expands the mathematical content and concepts of algebra I and geometry. who master algebra II will gain experience with algebraic solutions of problems in various content areas, including the solution of systems of quadratic equations, logarithmic and exponential functions, the binomial theorem, and the complex number system.Candidates shall be able to

- solve equations and inequalities involving absolute value.
- solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices.
- perform operations on polynomials, including long division.
- factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes.
- demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane.
- add, subtract, multiply, and divide complex numbers.
- add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions, including those with negative exponents in the denominator.
- solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. apply these techniques in solving word problems. They also solve quadratic equations in the complex number system.
- demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x – b)2 + c.
- graph quadratic functions and determine the maxima, minima, and zeros of the function.
- prove simple laws of logarithms.
- understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
- judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step.
- know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay.
- use the definition of logarithms to translate between logarithms in any base.
- understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.
- determine whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential functions is sometimes true, always true, or never true.
- demonstrate and explain how the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it.
- use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola, and can then graph the equation, given a quadratic equation of the form ax2 + by2 + cx + dy + e = 0, can.
- use fundamental counting principles to compute combinations and permutations.
- use combinations and permutations to compute probabilities.
- know the binomial theorem and use it to expand binomial expressions that are raised to positive integer powers.
- apply the method of mathematical induction to prove general statements about the positive integers.
- find the general term and the sums of arithmetic series and of both finite and infinite geometric series.
- derive the summation formulas for arithmetic series and for both finite and infinite geometric series.
- solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions.
- use properties from number systems to justify steps in combining and simplifying functions.

__MATHEMATICAL ANALYSIS__**:**This discipline combines many of the trigonometric, geometric, and algebraic techniques needed to prepare for the study of calculus and strengthens their conceptual understanding of problems and mathematical reasoning in solving problems. These standards take a functional point of view toward those topics. The most significant new concept is that of limits. Mathematical analysis is often combined with a course in trigonometry or perhaps with one in linear algebra to make a year-long precalculus course.Candidates shall be able to

- apply, polar coordinates and vectors in the plane. In particular, they can translate between polar and rectangular coordinates and can interpret polar coordinates and vectors graphically.
- perform the arithmetic of complex numbers. They can use the trigonometric form of complex numbers and understand that a function of a complex variable can be viewed as a function of two real variables. They know the proof of DeMoivre's theorem.
- give proofs of various formulas by using the technique of mathematical induction.
- apply, the fundamental theorem of algebra.
- are familiar with conic sections, both analytically and geometrically
- take a quadratic equation in two variables; put it in standard form by completing the square and using rotations and translations, if necessary; determine what type of conic section the equation represents; and determine its geometric components (foci, asymptotes, and so forth).
- take a geometric description of a conic section - for example, the locus of points whose sum of its distances from (1, 0) and (-1, 0) is 6 - and derive a quadratic equation representing it.
- find the roots and poles of a rational function and can graph the function and locate its asymptotes.
- demonstrate an understanding of functions and equations defined parametrically and can graph them.
- are familiar with the notion of the limit of a sequence and the limit of a function as the independent variable approaches a number or infinity. They determine whether certain sequences converge or diverge.

__LINEAR ALGEBRA__**:**The general goal in this discipline is for to learn the techniques of matrix manipulation so that they can solve systems of linear equations in any number of variables. Linear algebra is most often combined with another subject, such as trigonometry, mathematical analysis, or precalculus.Candidates shall be able to

- solve linear equations in any number of variables by using Gauss-Jordan elimination.
- interpret linear systems as coefficient matrices and the Gauss-Jordan method as row operations on the coefficient matrix
- reduce rectangular matrices to row echelon form.
- perform addition on matrices and vectors.
- perform matrix multiplication and multiply vectors by matrices and by scalars.
- demonstrate an understanding that linear systems are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions.
- demonstrate an understanding of the geometric interpretation of vectors and vector addition (by means of parallelograms) in the plane and in three-dimensional space.
- interpret geometrically the solution sets of systems of equations. For example, the solution set of a single linear equation in two variables is interpreted as a line in the plane, and the solution set of a two-by-two system is interpreted as the intersection of a pair of lines in the plane.
- demonstrate an understanding of the notion of the inverse to a square matrix and apply that concept to solve systems of linear equations.
- compute the determinants of 2 x 2 and 3 x 3 matrices and are familiar with their geometric interpretations as the area and volume of the parallelepipeds spanned by the images under the matrices of the standard basis vectors in two-dimensional and three-dimensional spaces.
- know that a square matrix is invertible if, and only if, its determinant is nonzero. They can compute the inverse to 2 x 2 and 3 x 3 matrices using row reduction methods or Cramer's rule.
- compute the scalar (dot) product of two vectors in n-dimensional space and know that perpendicular vectors have zero dot product.

All that said, these are essential skills that I have distilled from the standards above!

** you must**__ESSENTIAL SKILLS,__- Know distance formula.
- Midpoint formula
- Know symmetricity of graphs with respect to x-axis, y-axis, origin, y = x, line y = -x.
- Know the points slope form and slope intercept form of equation of lines.
- Know the condition for parallel and perpendicular lines using slopes.
- Prove the theorem on parallel and perpendicular lines using slopes.
- Know the definition of a function and relation.
- Know the Vertical Line Test for determining if a set of points belongs to a function.
- Be able to determine the domain and range of a function algebraically
*and*graphically. - Be able to factorize expressions using (a ± b)2, (a ± b)3, a2 – b2 , a3 ± b3
- Know how to complete the square of a quadratic expression.
- Determine if a function is even/odd.
- Be able to rationalize the denominator to simplify complex (
**i)**and irrational (√**)**expressions

**For**you must:__QUADRATIC FUNCTIONS__,- Be able to derive the quadratic formula
- Derive the sum of roots of a quadratic equation is –b/a and the product of roots is c/a.
- Know the relationship between the roots of a quadratic equation and its
*graph*based on the discriminant, D = b2 – 4ac, being positive, negative or equal to zero - Be able to convert a quadratic function in standard form to parabola form by completing the square.
- Determine the vertex, axis of symmetry, roots of a quadratic function/ parabola.
- Determine if a parabola opens up or down, and based on that, know if the function has a maximum or minimum.
- Be able to solve variations of quadratic equations using substitution: for example, ax4 + bx2 + c = 0, √(ax4 + bx2 + c) = 0, ax2 + b/x2 + c = 0, √(ax + b) + √(cx + d) = 0, etc. and find the maximum/ minimum of such functions.
- Be able to derive the coordinates of the vertex of a quadratic function in standard form.
- Model word problems related to quadratic functions and determine maximum/ minimum values.
- Determine roots/ x intercepts through quadratic formula/ completing the square.
- Know properties of complex numbers: addition, subtraction, multiplication, and division.
- Be able to represent the sum, difference, product and quotient of complex numbers in
*a + b***i**form. - Know that irrational and complex roots of functions occur as conjugates.
- Be able to graph and interpret inequalities relating to quadratic functions: know for which intervals the graph lies above/ below the x-axis

For

__POLYNOMIAL FUNCTIONS__**,**you must:- Be able to determine the number of turning points for a function.
- Determine the end behavior of the function.
- Determine by visual inspection if a graph represents a certain polynomial function.
- Know characteristics of even/ odd functions.
- Know shapes of the most important
**BASE GRAPHS:**

- f(x) = √x
- f(x) = x2
- f(x) = xn, for n = odd/ even
- f(x) = 1/x
- f(x) = 1/xn, for n = odd/ even
- f(x) = 3√x
- f(x) = n√x, for n = odd/ even
- f(x) = |x|

- Be able to transform the above base graphs for the following cases:

- Given f(x), graphing f(x + c) or f(x – c)
- Given f(x), graphing f(x) + c or f(x) – c
- Given f(x), graphing c·f(x), for c > 1 and c < 1
- Given f(x), graphing f(-x) and –f(x)

- Be able to graph polynomial functions in factored form

- X-intercepts
- Y-intercepts
- End behavior of function
- Intervals where graph is above/ below the x-axis by using a sign table.

- Be able to divide two polynomial functions using long division
- Be able to divide a polynomial and a linear (binomial) expression using Synthetic Division, and determine the Quotient and Remainder
- Know the proofs of Remainder Theorem and Factor Theorem
- Use the Factor Theorem and Remainder Theorem to determine if a monomial is a factor of the polynomial function.
- Be able to prove the Rational Roots Theorem.
- Apply the Rational Roots Theorem to determine the possible roots of a polynomial.
- Apply the Descartes Rule of Signs to determine the number of positive, negative and imaginary roots of a polynomial.
- Determine the polynomial function given its roots and their multiplicity
- Be able to calculate the roots, real and imaginary, of a polynomial.
- Be able to graph and interpret inequalities relating to polynomial functions: know for which intervals the graph lies above/ below the x-axis.

For

**, you must:**__RATIONAL FUNCTIONS, f(x) = P(x)/ Q(x)__- Be able to graph various forms of rational functions by determining its:

- x intercepts (solve: numerator = 0)
- y intercepts (substitute: x = 0 into the function)
- vertical asymptotes (solve: denominator = 0)
- horizontal asymptotes (imagine x to be a large number, M, and simplify.)
- slant asymptotes, if any, if the numerator has degree 1 more than the denominator

For

**you must**__INVERSE OF FUNCTIONS__- Be able to determine the inverse of functions (swap x and y in the original function and solve for y)
- Know properties of inverse functions
- Know the behavior of the inverse function, when given the graph of the original function (ie. the inverse function and the original function are symmetrical about the line y =x).
- Know the relationship of the domain and range of inverse functions and the original function.
- Know the horizontal line test for determining if a function has an inverse.
- Know what a one-to-one function is.

For

__COMPOSITE FUNCTIONS__**f(x) = h(g(x)); h(x)/g(x); h(x)·g(x), etc.**you must- Be able to determine the domain and range.
- Find values of composite functions for given values of x (x
*= a*…)

For

**you must**__LOGARITHMIC AND EXPONENTIAL FUNCTIONS__- Know the general form (algebraically and graphically) of logarithmic (ie. log x) and exponential (ie. ax or ex) functions.
- Know the relationship between logarithmic (log10 x or log ex) and exponential (axor ex) functions, and transform exponential to logarithmic functions and
*vice versa.* - Be able to calculate the inverse of logarithmic and exponential functions.
- Graph logarithmic and exponential functions using elementary transformations
- Be able to determine the domain and range of transformed logarithmic and exponential functions.
- Know the properties of logarithms and exponents, and apply properties to simplify expressions
- Solve equations and inequalities with logs and exponents using definitions and properties of logs and exponents
- Be able to determine extraneous solutions to equations/inequalities
- Know the formula for compound interest for the case of continuous compounding
- Be familiar with exponential growth and decay situations, and related doubling time and half-life problems

For

**you must:**__VECTORS__- Be able to determine the magnitude and direction of a vector
- Know the representation of vectors in standard (a
**i +**b**j)**and component form - Know basic vector operations
- Be able to determine unit vectors in a given direction
- Be able to calculate the direction angles of vectors
- Apply elementary vector properties to solve real-world problems
- Be able to calculate the angle between 2 vectors using dot product of vectors, and determine if 2 vectors are parallel or perpendicular

For

__SYSTEMS OF EQUATIONS__**,**you must:- Be able to solve systems of equations (lines, circles, ellipses, parabolas, hyperbolas) simultaneously to determine points of intersection.
- Solve systems of inequalities and be able to graph the shaded region representing all possible values of (x, y).
- Be able to determine the maximum and minimum values of an objective function of a linear programming problem/ situation.
- Be able to solve word problems involving a linear programming situation: choose appropriate variables, determine constraints and objective function, plot lines representing constraints, determine corner points for shaded polygon and calculate maximum/ minimum values of the objective function.
- Be able to solve systems of linear equations by finding the reduced echelon form of a matrix by performing row transformations
- Be able to classify systems of linear equations as consistent/ inconsistent/ possessing infinite solutions.

For

__MATRICES and DETERMINANTS__**,**you must:- Know basic properties of matrices
- Know the criteria for multiplying matrices
- Be able to multiply matrices
- Know properties of matrix multiplication.
- Be able to find the inverse of a matrix using determinants.
- Be able to calculate the inverse of a matrix by reducing it to echelon form.
- Be able to solve systems of linear equations by calculating the inverse of a matrix.
- Be able to find the value of a 2 X 2 and 3 X 3 determinants.
- Know properties of determinants pertaining to row and column transformations.
- Know Cramer’s Rule for solving linear equations using determinants.

For

**you must:**__SEQUENCES, SERIES, PERMUTATIONS AND COMBINATIONS__- Be able to determine the general term of an arithmetic/ geometric series using the common difference/ ratio.
- Be able to calculate the sum of n terms of an arithmetic/ geometric series.
- Be able to calculate the sum of an infinite geometric series.
- Know the principle of Mathematical Induction.
- Be able to apply the Principle of Mathematic Induction to prove elementary propositions/ conjectures.
- Know the (
*r*+ 1)th term in the expansion of (a + b)n using the binomial theorem. - Expand binomial expressions using the binomial theorem and calculate specific terms.
- Be familiar with Pascal’s Triangle.
- Know the fundamental counting principle.
- Be able to calculate the number of different permutations & combinations of
*r*elements that can be obtained from a set of*n*elements.

**Comments?**Email me at**innovationguy@gmail.com**