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Skills available for Texas high school math standards

Standards are in black and IXL math skills are in dark green. Hold your mouse over the name of a skill to view a sample question. Click on the name of a skill to practice that skill.

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College and Career Readiness Standards College and Career Readiness Standards
Texas Essential Knowledge and Skills (TEKS): 111.47 Statistics Texas Essential Knowledge and Skills (TEKS): 111.47 Statistics
Texas Essential Knowledge and Skills (TEKS): 111.48 Algebraic Reasoning Texas Essential Knowledge and Skills (TEKS): 111.48 Algebraic Reasoning
State of Texas Assessments of Academic Readiness (STAAR) State of Texas Assessments of Academic Readiness (STAAR)
Texas Essential Knowledge and Skills (TEKS): Algebra 1 Texas Essential Knowledge and Skills (TEKS): Algebra 1
Texas Essential Knowledge and Skills (TEKS): Algebra 2 Texas Essential Knowledge and Skills (TEKS): Algebra 2
Texas Essential Knowledge and Skills (TEKS): Algebraic Reasoning Texas Essential Knowledge and Skills (TEKS): Algebraic Reasoning
Texas Essential Knowledge and Skills (TEKS): Geometry Texas Essential Knowledge and Skills (TEKS): Geometry
Texas Essential Knowledge and Skills (TEKS): Precalculus Texas Essential Knowledge and Skills (TEKS): Precalculus
Texas College and Career Readiness Standards: Grades: 9-12 Texas College and Career Readiness Standards: Grades: 9-12

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1 Mathematical process standards

1 The student uses mathematical processes to acquire and demonstrate mathematical understanding.
- A apply mathematics to problems arising in everyday life, society, and the workplace;
- B use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;
- C select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;
- D communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;
- E create and use representations to organize, record, and communicate mathematical ideas;
- F analyze mathematical relationships to connect and communicate mathematical ideas; and
- G display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

2 Attributes of functions and their inverses

2 The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse.
- A graph the functions f(x)=√x, f(x)=1/x, f(x)=x³, f(x)= ³√x, f(x)=b to the x power, f(x)=|x|, and f(x)=logb (x) where b is 2, 10, and e, and, when applicable, analyze the key attributes such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum given an interval;
- B graph and write the inverse of a function using notation such as f-¹ (x);
- C describe and analyze the relationship between a function and its inverse (quadratic and square root, logarithmic and exponential), including the restriction(s) on domain, which will restrict its range; and
- D use the composition of two functions, including the necessary restrictions on the domain, to determine if the functions are inverses of each other.

3 Systems of equations and inequalities

3 The student applies mathematical processes to formulate systems of equations and inequalities, use a variety of methods to solve, and analyze reasonableness of solutions.
- A formulate systems of equations, including systems consisting of three linear equations in three variables and systems consisting of two equations, the first linear and the second quadratic;
- B solve systems of three linear equations in three variables by using Gaussian elimination, technology with matrices, and substitution;
  - Solve a system of equations in three variables using substitution (A2-E.12)
  - Solve a system of equations in three variables using elimination (A2-E.13)
- C solve, algebraically, systems of two equations in two variables consisting of a linear equation and a quadratic equation;
  - Solve a system of linear and quadratic equations: parabolas (A2-P.2)
- D determine the reasonableness of solutions to systems of a linear equation and a quadratic equation in two variables;
  - Solve a system of linear and quadratic equations by graphing: parabolas (A2-P.1)
- E formulate systems of at least two linear inequalities in two variables;
- F solve systems of two or more linear inequalities in two variables; and
- G determine possible solutions in the solution set of systems of two or more linear inequalities in two variables.
  - Is (x, y) a solution to the system of inequalities? (A2-F.3)

4 Quadratic and square root functions, equations, and inequalities

4 The student applies mathematical processes to understand that quadratic and square root functions, equations, and quadratic inequalities can be used to model situations, solve problems, and make predictions.
- A write the quadratic function given three specified points in the plane;
  - Write a quadratic function from three points (A2-N.13)
- B write the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening;
- C determine the effect on the graph of f(x) = √x when f(x) is replaced by af(x), f(x) + d, f(bx), and f(x - c) for specific positive and negative values of a, b, c, and d;
  - Graph square root functions (A2-S.2)
- D transform a quadratic function f(x) = ax² + bx + c to the form f(x) = a(x - h)² + k to identify the different attributes of f(x);
  - Convert equations of parabolas from general to vertex form (A2-DD.7)
  - Find properties of a parabola from equations in general form (A2-DD.8)
- E formulate quadratic and square root equations using technology given a table of data;
- F solve quadratic and square root equations;
- G identify extraneous solutions of square root equations; and
- H solve quadratic inequalities.
  - Graph solutions to quadratic inequalities (A2-O.9)
  - Solve quadratic inequalities (A2-O.10)

5 Exponential and logarithmic functions and equations

5 The student applies mathematical processes to understand that exponential and logarithmic functions can be used to model situations and solve problems.
- A determine the effects on the key attributes on the graphs of f(x) = b to the x power and f(x) = logb (x) where b is 2, 10, and e when f(x) is replaced by af(x), f(x) + d, and f(x - c) for specific positive and negative real values of a, c, and d;
  - Match exponential functions and graphs (A2-Z.5)
- B formulate exponential and logarithmic equations that model real-world situations, including exponential relationships written in recursive notation;
- C rewrite exponential equations as their corresponding logarithmic equations and logarithmic equations as their corresponding exponential equations;
- D solve exponential equations of the form y = ab to the x power where a is a nonzero real number and b is greater than zero and not equal to one and single logarithmic equations having real solutions; and
- E determine the reasonableness of a solution to a logarithmic equation.

6 Cubic, cube root, absolute value and rational functions, equations, and inequalities

6 The student applies mathematical processes to understand that cubic, cube root, absolute value and rational functions, equations, and inequalities can be used to model situations, solve problems, and make predictions.
- A analyze the effect on the graphs of f(x) = x³ and f(x) = ³√x when f(x) is replaced by af(x), f(bx), f(x - c), and f(x) + d for specific positive and negative real values of a, b, c, and d;
  - Graph cube root functions (A2-S.3)
- B solve cube root equations that have real roots;
- C analyze the effect on the graphs of f(x) = |x| when f(x) is replaced by af(x), f(bx), f(x-c), and f(x) + d for specific positive and negative real values of a, b, c, and d;
  - Graph an absolute value function (A2-H.3)
  - Transformations of absolute value functions (A2-H.4)
- D formulate absolute value linear equations;
  - Write absolute value equations from graphs (A2-G.3)
- E solve absolute value linear equations;
  - Solve absolute value equations (A2-G.1)
  - Graph solutions to absolute value equations (A2-G.2)
- F solve absolute value linear inequalities;
  - Solve absolute value inequalities (A2-G.4)
  - Graph solutions to absolute value inequalities (A2-G.5)
- G analyze the effect on the graphs of f(x) = 1/x when f(x) is replaced by af(x), f(bx), f(x-c), and f(x) + d for specific positive and negative real values of a, b, c, and d;
- H formulate rational equations that model real-world situations;
- I solve rational equations that have real solutions;
  - Solve rational equations (A2-X.8)
- J determine the reasonableness of a solution to a rational equation;
  - Solve rational equations (A2-X.8)
- K determine the asymptotic restrictions on the domain of a rational function and represent domain and range using interval notation, inequalities, and set notation; and
  - Rational functions: asymptotes and excluded values (A2-X.1)
- L formulate and solve equations involving inverse variation.
  - Write and solve inverse variation equations (A2-Y.2)

7 Number and algebraic methods

7 The student applies mathematical processes to simplify and perform operations on expressions and to solve equations.
- A add, subtract, and multiply complex numbers;
- B add, subtract, and multiply polynomials;
  - Add and subtract polynomials (A2-L.2)
  - Multiply polynomials (A2-L.3)
- C determine the quotient of a polynomial of degree three and of degree four when divided by a polynomial of degree one and of degree two;
  - Divide polynomials using synthetic division (A2-L.5)
  - Divide polynomials using long division (A2)
- D determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods;
  - Factor polynomials (A2-M.9)
- E determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum and difference of two cubes and factoring by grouping;
- F determine the sum, difference, product, and quotient of rational expressions with integral exponents of degree one and of degree two;
- G rewrite radical expressions that contain variables to equivalent forms;
- H solve equations involving rational exponents; and
  - Solve equations with rational exponents (A2-T.8)
- I write the domain and range of a function in interval notation, inequalities, and set notation.

8 Data

8 The student applies mathematical processes to analyze data, select appropriate models, write corresponding functions, and make predictions.
- A analyze data to select the appropriate model from among linear, quadratic, and exponential models;
  - Identify linear, quadratic, and exponential functions from graphs (A2)
  - Identify linear, quadratic, and exponential functions from tables (A2)
- B use regression methods available through technology to write a linear function, a quadratic function, and an exponential function from a given set of data; and
- C predict and make decisions and critical judgments from a given set of data using linear, quadratic, and exponential models.

Texas

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1 Mathematical process standards

1 The student uses mathematical processes to acquire and demonstrate mathematical understanding.

A apply mathematics to problems arising in everyday life, society, and the workplace;

B use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

C select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

D communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;

E create and use representations to organize, record, and communicate mathematical ideas;

F analyze mathematical relationships to connect and communicate mathematical ideas; and

G display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

2 Attributes of functions and their inverses

2 The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse.

B graph and write the inverse of a function using notation such as f-¹ (x);

C describe and analyze the relationship between a function and its inverse (quadratic and square root, logarithmic and exponential), including the restriction(s) on domain, which will restrict its range; and

D use the composition of two functions, including the necessary restrictions on the domain, to determine if the functions are inverses of each other.

3 Systems of equations and inequalities

3 The student applies mathematical processes to formulate systems of equations and inequalities, use a variety of methods to solve, and analyze reasonableness of solutions.

A formulate systems of equations, including systems consisting of three linear equations in three variables and systems consisting of two equations, the first linear and the second quadratic;

B solve systems of three linear equations in three variables by using Gaussian elimination, technology with matrices, and substitution;

C solve, algebraically, systems of two equations in two variables consisting of a linear equation and a quadratic equation;

D determine the reasonableness of solutions to systems of a linear equation and a quadratic equation in two variables;

E formulate systems of at least two linear inequalities in two variables;

F solve systems of two or more linear inequalities in two variables; and

G determine possible solutions in the solution set of systems of two or more linear inequalities in two variables.

4 Quadratic and square root functions, equations, and inequalities

4 The student applies mathematical processes to understand that quadratic and square root functions, equations, and quadratic inequalities can be used to model situations, solve problems, and make predictions.

A write the quadratic function given three specified points in the plane;

B write the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening;

C determine the effect on the graph of f(x) = √x when f(x) is replaced by af(x), f(x) + d, f(bx), and f(x - c) for specific positive and negative values of a, b, c, and d;

D transform a quadratic function f(x) = ax² + bx + c to the form f(x) = a(x - h)² + k to identify the different attributes of f(x);

E formulate quadratic and square root equations using technology given a table of data;

F solve quadratic and square root equations;

G identify extraneous solutions of square root equations; and

H solve quadratic inequalities.

5 Exponential and logarithmic functions and equations

5 The student applies mathematical processes to understand that exponential and logarithmic functions can be used to model situations and solve problems.

A determine the effects on the key attributes on the graphs of f(x) = b to the x power and f(x) = logb (x) where b is 2, 10, and e when f(x) is replaced by af(x), f(x) + d, and f(x - c) for specific positive and negative real values of a, c, and d;

B formulate exponential and logarithmic equations that model real-world situations, including exponential relationships written in recursive notation;

C rewrite exponential equations as their corresponding logarithmic equations and logarithmic equations as their corresponding exponential equations;

D solve exponential equations of the form y = ab to the x power where a is a nonzero real number and b is greater than zero and not equal to one and single logarithmic equations having real solutions; and

E determine the reasonableness of a solution to a logarithmic equation.

6 Cubic, cube root, absolute value and rational functions, equations, and inequalities

6 The student applies mathematical processes to understand that cubic, cube root, absolute value and rational functions, equations, and inequalities can be used to model situations, solve problems, and make predictions.

A analyze the effect on the graphs of f(x) = x³ and f(x) = ³√x when f(x) is replaced by af(x), f(bx), f(x - c), and f(x) + d for specific positive and negative real values of a, b, c, and d;

B solve cube root equations that have real roots;

C analyze the effect on the graphs of f(x) = |x| when f(x) is replaced by af(x), f(bx), f(x-c), and f(x) + d for specific positive and negative real values of a, b, c, and d;

D formulate absolute value linear equations;

E solve absolute value linear equations;

F solve absolute value linear inequalities;

G analyze the effect on the graphs of f(x) = 1/x when f(x) is replaced by af(x), f(bx), f(x-c), and f(x) + d for specific positive and negative real values of a, b, c, and d;

H formulate rational equations that model real-world situations;

I solve rational equations that have real solutions;

J determine the reasonableness of a solution to a rational equation;

K determine the asymptotic restrictions on the domain of a rational function and represent domain and range using interval notation, inequalities, and set notation; and

L formulate and solve equations involving inverse variation.

7 Number and algebraic methods

7 The student applies mathematical processes to simplify and perform operations on expressions and to solve equations.

A add, subtract, and multiply complex numbers;

B add, subtract, and multiply polynomials;

C determine the quotient of a polynomial of degree three and of degree four when divided by a polynomial of degree one and of degree two;

D determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods;

E determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum and difference of two cubes and factoring by grouping;

F determine the sum, difference, product, and quotient of rational expressions with integral exponents of degree one and of degree two;

G rewrite radical expressions that contain variables to equivalent forms;

H solve equations involving rational exponents; and

I write the domain and range of a function in interval notation, inequalities, and set notation.

8 Data

8 The student applies mathematical processes to analyze data, select appropriate models, write corresponding functions, and make predictions.

A analyze data to select the appropriate model from among linear, quadratic, and exponential models;

B use regression methods available through technology to write a linear function, a quadratic function, and an exponential function from a given set of data; and

C predict and make decisions and critical judgments from a given set of data using linear, quadratic, and exponential models.