High School — Algebra

Standards for Mathematical Practice

Seeing Structure in Expressions

Arithmetic with Polynomials and Rational Expressions

Creating Equations and Inequalities

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.

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

Apply the Remainder Theorem.

Identify zeros of polynomials when suitable factorizations are available. Use the zeros to construct a rough graph of the function defined by the polynomial.

Apply the Binomial Theorem for the expansion of (x + y)ⁿ in powers of x and y for a positive integer n.

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.

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

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

Create equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with appropriate 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.

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

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.

Solve systems of linear equations exactly and approximately, focusing on pairs of linear equations in two variables.

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.

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.

Using graphs, technology, tables, or successive approximations, show that the solution(s) to the equation f(x) = g(x) are the x-value(s) that result in the y-values of f(x) and g(x) being the same.

Graph the solutions to a linear inequality in two variables as a half-plane. 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 examining 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 produce an equivalent expression.

Use the properties of exponents to transform exponential expressions.

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.(+) Derive the quadratic formula from this form.

Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a a ± bi for real numbers a and b.