All lessons

Course

8th Grade Math

Solve linear equations and systems, understand functions, use the Pythagorean theorem, and apply transformations.

Work through the lessons in order, from real numbers to independent problem-solving. By the end, you can tell rational from irrational numbers, estimate square roots, and use exponent rules and scientific notation. You can solve linear equations with variables on both sides and know when there is one, no, or infinitely many solutions. You can find slope, graph y = mx + b, and compare linear and nonlinear functions. You can solve systems of two equations by graphing, substitution, and elimination. You can translate, reflect, rotate, and dilate figures, use angle facts and the Pythagorean theorem, find the volume of cylinders, cones, and spheres, and fit a line to a scatter plot.

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.
  • Use graph paper for slope, systems, and transformations, and tracing paper to check rotations and reflections. A calculator helps with square roots and scientific notation. Check each solution by putting it back in.

Each lesson has games, quizzes, and worksheets.

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Real Numbers

  1. 1Rational and Irrational NumbersTell rational numbers from irrational numbers.
    • Know rational numbers can be written as fractions.
    • Know √2 and π are irrational.
    • Sort numbers into the real number system.
  2. 2Repeating Decimals to FractionsChange a repeating decimal into a fraction.
    • Change 0.444… to 4/9.
    • Change 0.272727… to 27/99 = 3/11.
    • Explain the method with an equation.
  3. 3Estimate Irrational NumbersEstimate an irrational number and place it on a number line.
    • Show √2 is between 1.4 and 1.5.
    • Place √20 between 4 and 5.
    • Compare irrational numbers.
  4. 4Square Roots and Cube RootsSolve x² = p and x³ = p.
    • Know the perfect squares to 225 and cubes to 1,000.
    • Solve x² = 49 to get x = ±7.
    • Solve x³ = 64.

Exponents and Scientific Notation

  1. 5Exponent RulesMultiply, divide, and raise powers with the same base.
    • Use a^m × a^n = a^(m+n).
    • Use a^m ÷ a^n = a^(m−n).
    • Use (a^m)^n = a^(mn).
  2. 6Zero and Negative ExponentsEvaluate expressions with zero and negative exponents.
    • Explain why a⁰ = 1.
    • Write 3⁻² as 1/9.
    • Simplify 3² × 3⁻⁵ = 1/27.
  3. 7Scientific NotationWrite very large and very small numbers in scientific notation.
    • Write 3,400,000 as 3.4 × 10⁶.
    • Write 0.00072 as 7.2 × 10⁻⁴.
    • Compare numbers in scientific notation.
  4. 8Operations with Scientific NotationAdd, subtract, multiply, and divide in scientific notation.
    • Multiply (3 × 10⁴)(2 × 10⁵).
    • Divide and rewrite the answer in scientific notation.
    • Choose sensible units, such as millimeters or kilometers.

Linear Equations

  1. 9Variables on Both SidesSolve equations with the variable on both sides.
    • Solve 5x + 3 = 2x + 18.
    • Move the variable terms to one side.
    • Check by substituting the answer.
  2. 10Multi-Step EquationsSolve equations that need the distributive property and like terms.
    • Solve 3(x − 4) + 2 = 2x − 5.
    • Solve equations with fractions and decimals.
    • Write and solve an equation for a word problem.
  3. 11One, None, or Infinitely Many SolutionsTell if an equation has one solution, no solution, or infinitely many.
    • Get x = a for one solution.
    • Get a = b with a ≠ b for no solution.
    • Get a = a for infinitely many.

Slope and Linear Relationships

  1. 12Compare Proportional RelationshipsCompare two proportional relationships shown in different ways.
    • Read the unit rate from a graph as the slope.
    • Compare a table with an equation.
    • Decide which is faster or cheaper.
  2. 13SlopeFind the slope of a line from a graph or two points.
    • Use rise over run.
    • Use similar triangles to show slope is the same anywhere on a line.
    • Know positive, negative, zero, and undefined slopes.
  3. 14Slope-Intercept FormWrite the equation of a line as y = mx + b.
    • Find m and b from a graph.
    • Write the equation from a slope and a point.
    • Write the equation from two points.
  4. 15Graph Linear EquationsGraph a line from its equation.
    • Start at b, then use the slope.
    • Graph from a table of values.
    • Graph horizontal and vertical lines.

Functions

  1. 16What Is a FunctionTell if a relation is a function.
    • Know each input has exactly one output.
    • Check tables, mappings, and graphs.
    • Use the vertical line test.
  2. 17Compare FunctionsCompare two functions shown in different ways.
    • Compare rates of change.
    • Compare starting values.
    • Compare a table with an equation or a graph.
  3. 18Linear and Nonlinear FunctionsTell linear functions from nonlinear ones.
    • Check for a constant rate of change in a table.
    • Know y = x² is not linear.
    • Match equations to graphs.
  4. 19Model with Linear FunctionsWrite a linear function for a real situation.
    • Find the rate of change and the starting value.
    • Write a phone-plan cost as C = 10 + 0.5m.
    • Explain what the slope and intercept mean.
  5. 20Describe GraphsDescribe how a graph increases, decreases, or stays the same.
    • Describe a graph of a trip.
    • Sketch a graph from a story.
    • Find where it is linear and where it is not.

Systems of Equations

  1. 21Solve Systems by GraphingSolve a system of two linear equations by graphing.
    • Graph both lines.
    • Find the point where they cross.
    • Know parallel lines have no solution and the same line has infinitely many.
  2. 22Solve Systems by SubstitutionSolve a system by substitution.
    • Solve one equation for a variable.
    • Substitute into the other equation.
    • Check the solution in both equations.
  3. 23Solve Systems by EliminationSolve a system by adding or subtracting the equations.
    • Add equations to cancel a variable.
    • Multiply first when needed.
    • Choose the best method for a system.
  4. 24Systems Word ProblemsWrite and solve a system for a word problem.
    • Define two variables.
    • Write two equations from the story.
    • Explain the solution in the story.

Transformations

  1. 25TranslationsSlide a figure and describe the move with coordinates.
    • Translate a figure on a grid.
    • Write the rule (x, y) → (x + 3, y − 2).
    • Know lengths and angles stay the same.
  2. 26ReflectionsReflect a figure across an axis.
    • Reflect across the x-axis: (x, y) → (x, −y).
    • Reflect across the y-axis: (x, y) → (−x, y).
    • Find the line of reflection.
  3. 27RotationsRotate a figure 90°, 180°, or 270° around the origin.
    • Use tracing paper to rotate.
    • Use (x, y) → (−y, x) for 90° counterclockwise.
    • Describe a rotation by center, angle, and direction.
  4. 28DilationsEnlarge or shrink a figure by a scale factor.
    • Use (x, y) → (kx, ky).
    • Know angles stay the same and lengths scale.
    • Find the scale factor from two figures.
  5. 29Congruence and SimilarityUse transformations to show figures are congruent or similar.
    • Know congruent figures match by rigid moves.
    • Know similar figures also need a dilation.
    • Describe a sequence of moves that maps one figure to another.

Angles and the Pythagorean Theorem

  1. 30Angles of TrianglesUse the angle sum and exterior angles of a triangle.
    • Show the angles of a triangle add to 180°.
    • Know an exterior angle equals the two far interior angles.
    • Find unknown angles with equations.
  2. 31Parallel Lines and TransversalsFind angles made when a line crosses parallel lines.
    • Name corresponding, alternate interior, and alternate exterior angles.
    • Know which pairs are equal and which add to 180°.
    • Find all eight angles from one.
  3. 32Angle-Angle SimilarityShow triangles are similar using two angles.
    • Know two matching angles make triangles similar.
    • Find a missing side with a proportion.
    • Use shadows to find the height of a tree.
  4. 33The Pythagorean TheoremUse a² + b² = c² to find a missing side of a right triangle.
    • Explain a proof with squares.
    • Find the hypotenuse.
    • Find a missing leg.
  5. 34Pythagorean Theorem ApplicationsUse the Pythagorean theorem in real problems and in 3D.
    • Find the length of a ladder against a wall.
    • Find the diagonal of a box.
    • Use the converse to test for a right triangle.
  6. 35Distance Between Two PointsFind the distance between two points on the coordinate plane.
    • Draw a right triangle between the points.
    • Use the Pythagorean theorem.
    • Find the perimeter of a figure on a grid.

Volume

  1. 36Volume of CylindersFind the volume of a cylinder.
    • Use V = πr²h.
    • Find a missing height or radius.
    • Solve real-world problems.
  2. 37Volume of ConesFind the volume of a cone.
    • See that a cone holds 1/3 of a cylinder.
    • Use V = 1/3 πr²h.
    • Solve real-world problems.
  3. 38Volume of SpheresFind the volume of a sphere.
    • Use V = 4/3 πr³.
    • Find the volume of a hemisphere.
    • Compare a sphere, cone, and cylinder of the same size.

Bivariate Data

  1. 39Scatter PlotsMake a scatter plot and describe the pattern.
    • Plot paired data.
    • Describe positive, negative, or no association.
    • Find clusters and outliers.
  2. 40Lines of FitDraw a line of fit and use its equation to predict.
    • Draw a line close to most points.
    • Write its equation.
    • Explain what the slope and intercept mean.
  3. 41Two-Way TablesMake and read a two-way table.
    • Organize survey data into rows and columns.
    • Find relative frequencies.
    • Decide if two variables are associated.

Review, Mixed Practice, and Challenges

  1. 42Equations and Functions ReviewSolve mixed problems with exponents, equations, functions, and systems.
    • Use exponent rules and scientific notation.
    • Solve linear equations and systems.
    • Write and compare linear functions.
  2. 43Geometry and Data ReviewSolve mixed problems with transformations, angles, the Pythagorean theorem, volume, and data.
    • Describe transformations.
    • Use angle facts and the Pythagorean theorem.
    • Find volume and fit a line to data.
  3. 44Eighth Grade Math ChallengeComplete mixed questions covering the major units.
    • Choose a strategy without being told.
    • Solve multi-step problems.
    • Find the areas that need more practice.