High School — Algebra

Standards for Mathematical Practice

Seeing Structure In Expressions

Arithmetic With Polynomials And Rational Expressions

Creating Equations

Reasoning With Equations And Inequalities

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Interpret the structure of expressions.

Write expressions in equivalent forms to solve problems.

Perform arithmetic operations on polynomials.

Understand the relationship between zeros and factors of polynomials.

Use polynomial identities to solve problems.

Rewrite rational expressions.

Create equations that describe numbers or relationships.

Understand solving equations as a process of reasoning and explain the reasoning.

Solve equations and inequalities in one variable.

Solve systems of equations.

Represent and solve equations and inequalities graphically.

Interpret expressions that represent a quantity in terms of its context.

Use the structure of an expression to identify ways to rewrite it.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

Understand that polynomials form a system analogous to the integers, namely, that they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Understand and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x − a is p(a). In particular, p(a) = 0 if and only if (x – a) is a factor of p(x).

Identify zeros of polynomials, when factoring is reasonable, and use the zeros to construct a rough graph of the function defined by the polynomial.

Prove polynomial identities and use them to describe numerical relationships.

Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers.

Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Create equations and inequalities in one variable and use them to solve problems. Include equations and inequalities arising from linear, quadratic, simple rational, and exponential functions.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Solve quadratic equations in one variable.

Verify that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Solve systems of linear equations algebraically and graphically.

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Represent a system of linear equations as a single matrix equation in a vector variable.

Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Explain why the x-coordinates of the points where the graphs of the equation y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, making tables of values, or finding successive approximations.

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.

Interpret parts of an expression, such as terms, factors, and coefficients.

Interpret complicated expressions by viewing one or more of their parts as a single entity.

Factor a quadratic expression to reveal the zeros of the function it defines.

Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Use the properties of exponents to transform expressions for exponential functions. For example, 8t can be written as 2³t.

Focus on polynomial expressions that simplify to forms that are linear or quadratic.

Extend to polynomial expressions beyond those expressions that simplify to forms that are linear or quadratic.

Focus on applying linear and simple exponential expressions.

Focus on applying simple quadratic expressions.

Extend to include more complicated function situations with the option to solve with technology.

Focus on applying linear and simple exponential expressions.

Focus on applying simple quadratic expressions.

Extend to include more complicated function situations with the option to graph with technology.

While functions will often be linear, exponential, or quadratic, the types of problems should draw from more complicated situations.

Focus on formulas in which the variable of interest is linear or square.

Focus on formulas in which the variable of interest is linear.

Focus on formulas in which the variable of interest is linear or square.

While functions will often be linear, exponential, or quadratic, the types of problems should draw from more complicated situations.

Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x − p)² = q that has the same solutions.

Solve quadratic equations as appropriate to the initial form of the equation by inspection, e.g., for x² = 49; taking square roots; completing the square; applying the quadratic formula; or utilizing the Zero-Product Property after factoring.

Derive the quadratic formula using the method of completing the square.

Limit to pairs of linear equations in two variables.

Extend to include solving systems of linear equations in three variables, but only algebraically.