Algebra foundations
Multi-Step Linear Equations
Use distribution and inverse operations, then substitute each solution into the original equation.
Download Multi-Step Linear Equations PDF + answersGrades 9 through 12
High-school math worksheets
Choose practice for algebra, functions, geometry, trigonometry, statistics, or calculus. Check the example to see whether it matches the method used in your course. You can rehearse one procedure or revisit a prerequisite without treating this collection as a complete course.
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Each PDF includes a worksheet and matching answer key. Choose the skill and level that fit the learner; these are individual practice sets, not a complete course.
Download Multi-Step Linear Equations PDF + answers
Connect equations
Systems by Substitution
Substitute one expression into the other equation. Check both coordinates in both original equations.
Download Systems by Substitution PDF + answers
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Grades 9-12 math topics
92 level-focused worksheet skills lead this collection, with 25 prerequisite and review skills listed separately. Choose a topic to see its suggested setup in the builder.
Solve equations and use functions
36 skills- Multi-Step EquationsSolve linear equations requiring distribution and more than one inverse operation.
- Variables on Both SidesSolve linear equations with variable terms on both sides.
- Literal EquationsRearrange formulas to isolate a requested variable.
- Absolute-Value EquationsSolve absolute-value equations and identify their two solutions.
- Function NotationEvaluate functions and interpret function notation.
- Domain & RangeFind domain and range from relations, tables, mappings, formulas, and graphs.
- Systems by SubstitutionSolve two-variable systems using substitution.
- Systems by EliminationSolve two-variable systems using elimination.
- Systems Word ProblemsTranslate contexts into systems and solve them.
- Systems of InequalitiesTest whether ordered pairs satisfy both inequalities in a system.
- Interpreting Parabola FeaturesIdentify a parabola's minimum value and where it occurs from its vertex.
- Average Rate of ChangeCalculate average rate of change over an interval.
- Comparing FunctionsCompare rates of change across point and equation representations.
- Function TransformationsDescribe vertical stretches, reflections, and translations of the quadratic parent function.
- Piecewise FunctionsEvaluate piecewise-defined functions at boundary and non-boundary inputs.
- Function CompositionEvaluate compositions of two functions at given input values.
- Inverse FunctionsFind inverse rules for one-to-one linear functions.
- Linear vs. Exponential ModelsDistinguish constant differences from constant ratios.
- Exponential Growth and DecayCalculate repeated percent growth or decay over a given number of periods.
- Compound InterestCalculate compound interest using standard formulas.
- Logarithm PropertiesExpand logarithmic expressions using the power and quotient properties.
- Polynomial OperationsAdd, subtract, multiply, divide, and evaluate polynomials.
- Polynomial Long DivisionDivide polynomials by exact linear factors and write each quotient.
- Synthetic DivisionUse synthetic division with a linear divisor.
- Remainder and Factor TheoremsConnect polynomial evaluation, remainders, and factors.
- Polynomial Zeros and MultiplicityRead zeros and multiplicities from factored polynomials.
- Radical ExpressionsSimplify, operate on, rationalize, and solve radical expressions.
- Radical EquationsSolve radical equations and check for extraneous solutions.
- Rational ExpressionsSimplify, operate on, and solve rational expressions.
- Rational EquationsSolve equations containing rational expressions and state excluded values.
- Quadratic FactoringFactor monic quadratic expressions with integer roots.
- Completing the SquareRewrite quadratics in completed-square form.
- Quadratic FormulaSolve quadratic equations using the quadratic formula.
- Discriminant ClassificationUse the discriminant to classify quadratic roots.
- Quadratic Solutions & GraphsSolve quadratics and connect roots, vertices, discriminants, and graphs.
- Linear-Quadratic SystemsFind intersections of a line and a quadratic curve.
Measure shapes and triangles
10 skills- Triangle CongruenceIdentify valid triangle congruence criteria.
- Triangle SimilarityUse similarity criteria and proportional sides.
- Parallel-Line Angle RelationshipsApply corresponding, alternate, and same-side angle relationships.
- Coordinate Geometry ProofsUse slope, midpoint, and distance to verify geometric properties.
- Parallel and Perpendicular LinesWrite equations of parallel and perpendicular lines.
- Segment PartitionFind points that partition directed segments in given ratios.
- Circle TheoremsApply relationships among central angles, inscribed angles, chords, and tangents.
- Arc Length and Sector AreaCalculate arc lengths and sector areas using central angles.
- Geometry ProofsComplete concise proofs about lines, triangles, perpendicular bisectors, and parallelograms.
- Geometric ConstructionsIdentify and complete standard compass-and-straightedge construction sequences.
Understand data and chance
14 skills- Standard DeviationCalculate and interpret population standard deviation.
- Distribution Shape and OutliersDescribe distribution shape and effects of outliers.
- Normal Distributions and Z-ScoresCompute z-scores and use normal-distribution percentages.
- Correlation CoefficientInterpret the sign and strength of correlation coefficients.
- ResidualsCalculate and interpret residuals from model predictions.
- Correlation vs. CausationDistinguish association from evidence of causation.
- Conditional ProbabilityCalculate and interpret conditional probability.
- Independent EventsDetermine whether events are independent using probabilities.
- Probability RulesApply addition and multiplication rules of probability.
- Permutations and CombinationsCount ordered and unordered selections.
- Expected ValueCompute expected values and compare decisions.
- Types of Statistical StudiesDistinguish surveys, observational studies, and experiments.
- Margin of ErrorInterpret confidence intervals and margins of error.
- Simulation-Based InferenceUse simulated results to judge whether an outcome is surprising.
Use exponents and complex numbers
5 skills- Exponential & Logarithmic FunctionsEvaluate, convert, and solve exponential and logarithmic forms.
- Sequences & SeriesWork with arithmetic and geometric sequences and finite series.
- Conic SectionsIdentify and graph circles, parabolas, ellipses, and hyperbolas.
- Complex ArithmeticAdd, subtract, and multiply complex numbers.
- Rational ExponentsRewrite and evaluate rational exponents and radicals.
Study change and accumulation
8 skills- Advanced Derivative RulesDifferentiate with product, quotient, chain, exponential, logarithmic, and trig rules.
- Definite IntegrationEvaluate definite integrals and basic area applications.
- Intro Differential EquationsSolve basic rate, growth, decay, and separable differential equations.
- Power Rule DerivativesDifferentiate polynomial terms with the power rule.
- Basic Power IntegralsIntegrate polynomial terms with the power rule.
- Polynomial LimitsEvaluate limits of polynomial expressions by substitution.
- Applications of DerivativesApply derivatives to velocity, tangent slopes, extrema, and optimization.
- Applications of IntegrationApply definite integrals to net change, area, average value, and volumes of revolution.
Work with matrices and vectors
4 skillsUse trigonometry and periodic functions
10 skills- Right-Triangle Trigonometric RatiosFind sine, cosine, and tangent ratios in right triangles.
- Missing Sides and AnglesUse trigonometric ratios to solve right triangles.
- Degrees and RadiansConvert angle measures between degrees and radians.
- Unit-Circle ValuesRecall exact sine and cosine values for special angles.
- Reference AnglesFind reference angles and signs by quadrant.
- Graphs of Sine and CosineIdentify key points, amplitude, midline, and period of basic trig graphs.
- Trig Function TransformationsAnalyze amplitude, frequency, phase shift, and midline.
- Inverse Trigonometric FunctionsUse inverse trigonometric functions to find angles.
- Trigonometric IdentitiesApply fundamental trigonometric identities.
- Law of Sines and CosinesSolve non-right triangles using the Laws of Sines and Cosines.
Model with quantities and units
5 skills- Dimensional AnalysisConvert compound units using dimensional analysis.
- Derived UnitsConstruct and interpret units such as density and acceleration.
- Measurement PrecisionChoose reporting precision consistent with measurement limits.
- Graph Scales and OriginsChoose and interpret graph scales without distortion.
- Density and Rate ModelingUse density and other derived rates in contextual models.
Prerequisite and review skillsUseful practice whose primary level is another grade range.25 skills
Solve equations and use functions
8 skills- Algebraic ExpressionsEvaluate, combine, distribute, and factor algebraic expressions.
- One-Step EquationsSolve one-step equations with integer solutions.
- Two-Step EquationsSolve equations of the form ax + b = c.
- Linear InequalitiesSolve one-step and two-step inequalities, including sign reversals.
- Function TablesComplete input-output tables, find missing inputs, and identify linear rules.
- Slope From Two PointsFind slope from two coordinate points.
- Evaluate Linear FunctionsEvaluate linear functions at given inputs.
- Linear SystemsSolve two-variable linear systems with integer solutions.
Measure shapes and triangles
6 skills- Coordinate Planes & Ordered PairsPlot and identify ordered pairs, quadrants, axes, and distances.
- Distances on the Coordinate PlaneFind exact diagonal distances between plotted points.
- Shapes & Perimeter on the Coordinate PlaneFind rectangle perimeter from plotted vertices.
- Coordinate Plane Word ProblemsSolve exact coordinate-distance problems in context.
- Transformations & SimilarityTranslate and reflect points and scale similar figures.
- Pythagorean TheoremFind a missing side in right triangles.
Understand data and chance
8 skills- Mean, Median, Mode & RangeFind the mean, median, mode, or range for small data sets.
- Charts & Graph ComprehensionRead bar graphs, line graphs, and circle graphs.
- Basic ProbabilityFind simple probabilities from counts.
- Two-Way Frequency TablesCalculate joint, marginal, and conditional relative frequencies.
- ScatterplotsDescribe direction, form, strength, and unusual features in scatterplots.
- Regression LinesUse and interpret least-squares regression equations.
- Compound EventsRepresent unions, intersections, and complements of events.
- Sampling Methods and BiasRecognize representative samples and common sources of bias.
Use exponents and complex numbers
3 skillsQuestions
Using these Grades 9-12 worksheets
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