All lessons

Course

9th Grade Math: Algebra 1

Solve equations and inequalities, graph linear and exponential functions, factor polynomials, and solve quadratics.

Work through the lessons in order, from expressions to independent problem-solving. By the end, you can solve linear equations, rearrange formulas, and solve compound and absolute value inequalities. You can use function notation, write linear equations in several forms, and model with arithmetic sequences. You can solve systems of equations and inequalities and graph piecewise functions. You can use rational exponents, model exponential growth and decay, and compare linear and exponential models. You can add, multiply, and factor polynomials, graph quadratic functions, and solve quadratic equations by factoring, completing the square, and the quadratic formula. You can summarize data with standard deviation, two-way tables, and residuals.

How to use these lessons

  • Start at the lesson marked Do this. Go on when you can do the line under the title by yourself.
  • Play the Games first. Then take the Quizzes. Eight of ten correct passes. The Worksheets are on paper. The answers are at the bottom.
  • Each lesson builds on the one before it. If a lesson is hard, go back one.
  • Write every step. Check each solution by putting it back into the original equation. A graphing calculator or a free graphing app helps you see functions, but sketch the graph by hand first.

Each lesson has games, quizzes, and worksheets.

0 of 41

Expressions and Equations

  1. 1Expressions and PropertiesSimplify expressions using the properties of real numbers.
    • Use the commutative, associative, and distributive properties.
    • Name the terms, factors, and coefficients of an expression.
    • Interpret parts of an expression in context, such as P(1 + r)^n.
  2. 2Solve Linear EquationsSolve multi-step linear equations and justify each step.
    • Solve equations with fractions and variables on both sides.
    • Explain each step with a property.
    • Recognize equations with no solution or all real numbers.
  3. 3Literal Equations and FormulasRearrange a formula to solve for a different variable.
    • Solve A = 1/2 bh for h.
    • Solve d = rt for t.
    • Solve 2x + 3y = 6 for y.
  4. 4Units and ModelingUse units to set up and check a model.
    • Convert units with dimensional analysis.
    • Choose a sensible level of accuracy.
    • Write an equation for a real situation.

Inequalities

  1. 5Solve Linear InequalitiesSolve and graph multi-step linear inequalities.
    • Flip the sign when multiplying or dividing by a negative.
    • Graph the solution on a number line.
    • Write the solution in interval notation.
  2. 6Compound InequalitiesSolve and graph “and” and “or” inequalities.
    • Solve −3 < 2x + 1 ≤ 7.
    • Solve x < −2 or x > 5.
    • Graph the intersection or the union.
  3. 7Absolute Value Equations and InequalitiesSolve equations and inequalities with absolute value.
    • Solve |x − 3| = 5 as two equations.
    • Solve |x + 1| < 4 as an “and” inequality.
    • Know when there is no solution.

Linear Functions

  1. 8Function Notation, Domain, and RangeUse f(x) notation and find the domain and range of a function.
    • Evaluate f(3) and solve f(x) = 7.
    • Find the domain and range from a graph.
    • Choose a sensible domain for a real situation.
  2. 9Rate of ChangeFind and explain the average rate of change over an interval.
    • Find the rate of change from a table, graph, or equation.
    • Know a linear function has a constant rate of change.
    • Explain the units of the rate, such as dollars per hour.
  3. 10Forms of Linear EquationsWrite linear equations in slope-intercept, point-slope, and standard form.
    • Write y − y₁ = m(x − x₁) from a point and a slope.
    • Convert between forms.
    • Find the intercepts from standard form.
  4. 11Parallel and Perpendicular LinesWrite equations of parallel and perpendicular lines.
    • Know parallel lines have equal slopes.
    • Know perpendicular slopes multiply to −1.
    • Write a line through a point parallel or perpendicular to another.
  5. 12Arithmetic SequencesWrite arithmetic sequences as linear functions.
    • Find the common difference.
    • Write the explicit rule aₙ = a₁ + (n − 1)d.
    • Write a recursive rule.

Systems of Equations and Inequalities

  1. 13Solve Systems of EquationsSolve a system of linear equations by graphing, substitution, or elimination.
    • Choose the most efficient method.
    • Explain why elimination works.
    • Identify systems with no solution or infinitely many.
  2. 14Systems of Linear InequalitiesGraph a system of linear inequalities and find its solution region.
    • Graph y ≤ 2x + 1 with a solid line and shading.
    • Find where the shaded regions overlap.
    • Test a point to check.
  3. 15Model with SystemsWrite and solve systems for real-world problems.
    • Solve mixture and cost-comparison problems.
    • Use constraints to find possible solutions.
    • Explain the solution in context.

Absolute Value and Piecewise Functions

  1. 16Absolute Value FunctionsGraph absolute value functions and their transformations.
    • Graph y = |x| as a V shape.
    • Graph y = a|x − h| + k.
    • Find the vertex and the direction it opens.
  2. 17Piecewise and Step FunctionsGraph and evaluate piecewise and step functions.
    • Evaluate a piecewise function at a given input.
    • Graph each piece on its own domain.
    • Model parking fees with a step function.

Exponents and Exponential Functions

  1. 18Exponent Properties and Rational ExponentsSimplify expressions with integer and rational exponents.
    • Use all the exponent rules together.
    • Write x^(1/2) as √x and 8^(2/3) as 4.
    • Rewrite radicals with rational exponents.
  2. 19Exponential Growth and DecayWrite and graph exponential growth and decay functions.
    • Write y = a(1 + r)^t for growth.
    • Write y = a(1 − r)^t for decay.
    • Solve compound interest and population problems.
  3. 20Geometric SequencesWrite geometric sequences as exponential functions.
    • Find the common ratio.
    • Write aₙ = a₁ × r^(n − 1).
    • Write a recursive rule.
  4. 21Linear and Exponential ModelsDecide if a situation is linear or exponential and compare their growth.
    • Check for equal differences or equal ratios.
    • Know exponential growth eventually passes linear growth.
    • Build a model from a table.

Polynomials and Factoring

  1. 22Add and Subtract PolynomialsAdd and subtract polynomials.
    • Name the degree and leading coefficient.
    • Write a polynomial in standard form.
    • Combine like terms to add and subtract.
  2. 23Multiply PolynomialsMultiply polynomials.
    • Multiply a monomial by a polynomial.
    • Multiply two binomials with an area model.
    • Multiply a binomial by a trinomial.
  3. 24Special ProductsUse patterns for squares of binomials and differences of squares.
    • Use (a + b)² = a² + 2ab + b².
    • Use (a + b)(a − b) = a² − b².
    • Multiply quickly with the patterns.
  4. 25Factor by GCF and GroupingFactor out the greatest common factor and factor by grouping.
    • Factor 6x³ + 9x² as 3x²(2x + 3).
    • Factor four terms by grouping.
    • Check by multiplying.
  5. 26Factor TrinomialsFactor trinomials of the form ax² + bx + c.
    • Factor x² + 7x + 12 = (x + 3)(x + 4).
    • Factor when a ≠ 1.
    • Recognize a trinomial that does not factor.
  6. 27Factor Special CasesFactor perfect square trinomials and differences of squares.
    • Factor x² − 49.
    • Factor 4x² + 12x + 9.
    • Factor completely in several steps.

Quadratic Functions and Equations

  1. 28Graph Quadratic FunctionsGraph a quadratic function and find its key features.
    • Find the vertex, axis of symmetry, and intercepts.
    • Use x = −b/(2a) to find the vertex.
    • Know when the parabola opens up or down.
  2. 29Vertex Form and TransformationsGraph y = a(x − h)² + k and describe the transformations.
    • Read the vertex (h, k).
    • Describe stretches, shrinks, reflections, and shifts.
    • Write a quadratic from its graph.
  3. 30Solve Quadratics by FactoringSolve a quadratic equation by factoring.
    • Use the zero product property.
    • Set the equation equal to 0 first.
    • Connect the solutions to the x-intercepts.
  4. 31Solve Quadratics with Square RootsSolve quadratic equations by taking square roots.
    • Solve x² = 25 to get x = ±5.
    • Solve (x − 3)² = 16.
    • Simplify square roots in answers.
  5. 32Completing the SquareComplete the square to solve a quadratic and find its vertex.
    • Add (b/2)² to both sides.
    • Solve x² + 6x − 7 = 0.
    • Rewrite standard form in vertex form.
  6. 33The Quadratic FormulaSolve any quadratic equation with the quadratic formula.
    • Use x = (−b ± √(b² − 4ac)) / (2a).
    • Use the discriminant to count real solutions.
    • Choose the best method for a given equation.
  7. 34Quadratic ModelsUse quadratic functions to model projectiles and areas.
    • Find the maximum height of a thrown ball.
    • Find when it hits the ground.
    • Find the dimensions that give a maximum area.
  8. 35Linear-Quadratic SystemsSolve a system with a line and a parabola.
    • Solve by substitution.
    • Know there can be 0, 1, or 2 solutions.
    • Check by graphing.

Statistics

  1. 36Data Distributions and Standard DeviationCompare data sets using shape, center, and spread.
    • Find the mean and standard deviation.
    • Choose median and IQR for skewed data.
    • Explain the effect of outliers.
  2. 37Two-Way Tables and Relative FrequencyFind joint, marginal, and conditional relative frequencies.
    • Build a two-way frequency table.
    • Find relative frequencies by row and column.
    • Describe possible associations.
  3. 38Correlation and ResidualsFit a line to data, find the correlation, and read residuals.
    • Find the line of best fit with technology.
    • Interpret the correlation coefficient r.
    • Use a residual plot to judge the fit, and know correlation is not causation.

Review, Mixed Practice, and Challenges

  1. 39Linear and Exponential ReviewSolve mixed problems with equations, inequalities, systems, and exponential models.
    • Solve equations and inequalities.
    • Write and graph linear functions and systems.
    • Build exponential models.
  2. 40Polynomial and Quadratic ReviewSolve mixed problems with polynomials, quadratics, and data.
    • Multiply and factor polynomials.
    • Solve quadratics by the best method.
    • Summarize and model data.
  3. 41Algebra 1 ChallengeComplete mixed questions covering the major units.
    • Solve problems from every unit without hints.
    • Choose the right model for a situation.
    • Find the areas that need more practice.