Middle School Math Foundations
A guided, multi-course journey that builds this skill from the ground up. Work through the chapters in order, or jump to any lesson.
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Factors and Multiples: Two Views of the Same RelationshipPrime, Composite, and Fast Divisibility ChecksPrime Factorization as a Multiplicative FingerprintGreatest Common Factor as the Largest Shared GroupLeast Common Multiple as the First Shared Landing PointGCF or LCM? Choose the Structure That Fits
Fractions and Mixed Numbers on the Number LineBenchmarking Fraction Size With 0, 1/2, 1, and Whole NumbersEquivalent Fractions From Scaling and Visual ModelsSimplifying Fractions Without Changing Their ValueComparing Fractions With Unlike DenominatorsImproper Fractions and Mixed Numbers: Converting Without Changing the AmountAdding and Subtracting Fractions and Mixed Numbers Using Common UnitsMulti-Step Fraction Problems in Measurement and Real LifeFixing Fraction Misconceptions: Denominators, Size, and “Same Units”
Read, Write, Compare, and Locate Decimals Using Place Value and Number LinesRelate Decimal Places to Fractions With Denominators That Are Powers of TenAdd and Subtract Decimals by Aligning Place Values and Checking MagnitudeMultiply Decimals Through Place-Value Scaling and Partial-Product ReasoningDivide Decimals Through Sharing, Measurement, and Powers-of-Ten EquivalenceConvert Common Fractions to Decimals Using Equivalent Fractions or DivisionConvert Terminating Decimals to Fractions and SimplifyRecognize Terminating and Repeating Decimal Patterns InformallyUse Estimation to Detect Misplaced-Decimal Errors and Unreasonable Results
Interpret Ratio Language as Multiplicative ComparisonWrite and Read Ratios in Multiple Forms Without Confusing Ratios With Fractions MechanicallyRepresent Ratios Using Tape Diagrams, Double Number Lines, and Ratio TablesGenerate Equivalent Ratios by Scaling Both Quantities by the Same FactorDistinguish Ratios From Rates and Interpret the Meaning of “Per”Compute and Interpret Unit Rates, Including Rates Involving Fractions and DecimalsCompare Options Such as Prices, Speeds, Recipes, or Plans Using Unit RatesTrack Units Throughout Calculations and Use Dimensional Meaning to Detect ErrorsTranslate Among Context, Diagram, Table, and Simple Equation RepresentationsDiagnose Additive Reasoning Incorrectly Applied to Multiplicative Situations
Determine Whether a Relationship Is ProportionalFind and Interpret the Constant of Proportionality as a Unit RateRepresent Proportional Relationships With y = kxWhy Proportional Graphs Are Lines Through the OriginTranslate Among Context, Table, Graph, Diagram, and EquationPercent as “Per 100” and as a Scale FactorPercent-of, Find the Part, Find the Whole, and Find the PercentDiscounts, Markups, Tax, Tips, Fees, Commissions, and Simple InterestPercent Increase and Decrease Using the Original as the BasePercent Change vs Percentage-Point ChangeScale Drawings and Maps With Scale FactorWhy Cross-Multiplication Works (and When to Use It)
Positive and Negative Quantities in Real-World ContextsNumber Lines: Locate, Order, and Compare Signed Rational NumbersOpposites and Absolute Value: Direction and Distance From ZeroAdding and Subtracting Signed Numbers on the Number LineMultiplying, Dividing, and Computing With Signed Rational NumbersCoordinates, Quadrants, Distances, and the Meanings of the Negative Sign
Read Numerical Expressions by Identifying Operations, Groups, and Nested StructureApply Grouping Symbols and Order of Operations With UnderstandingInterpret Whole-Number Exponents as Repeated MultiplicationBases and Exponents: Notation That People Commonly Mix UpVariables as Quantities, Not Just Mystery NumbersTranslate and Evaluate Expressions by SubstitutionRewrite Expressions Using Properties, Distribution, Factoring, Like Terms, and Equivalence
Expressions, Equations, Identities, and InequalitiesChecking Solutions by SubstitutionOne-Step and Two-Step Equations Using Inverse OperationsMulti-Step Linear Equations With Distribution and Like TermsChoosing and Comparing Equation-Solving StrategiesOne Solution, No Solution, or Infinitely Many SolutionsModeling and Solving Inequalities as Solution Sets
Statistical Questions vs. Fixed-Answer QuestionsWhy Variability Matters and How to Organize DataDot Plots: Seeing a Distribution One Point at a TimeHistograms: Grouping Data Into IntervalsBox Plots, Quartiles, Median, Range, and IQRMean, Median, and Mean Absolute Deviation (MAD): Choosing the Right Summary
Populations, Samples, and What We’re Trying to LearnWhy Samples Must Represent the PopulationBiased Samples: How Conclusions Get BentRandom Samples and Informal Population EstimatesComparing Two Populations Using Samples (With Center and Spread)Probability as a Number From 0 to 1Experimental Probability From Repeated TrialsTheoretical Probability Models When Outcomes Are Equally LikelyExperimental vs. Theoretical: When Should They Match?Sample Spaces: Lists, Tables, and Tree DiagramsSimple Compound-Event Probabilities (Using Sample Spaces)Simulations for Hard-to-Count ProbabilitiesIndependence vs. Certainty: What Chance Can and Can’t PromiseThe Gambler’s Fallacy and Other Probability Traps
Angle Size, Angle Types, and Angle AdditionAngle Relationships: Complementary, Supplementary, Adjacent, and VerticalTurning Angle Diagrams Into Simple EquationsTriangle Angle Sums and Related Polygon ReasoningArea Formulas From Decomposition and RearrangementArea of Quadrilaterals and Composite FiguresCircles: π, Circumference, Area, and Composite RegionsNets, Surface Area, Volume, and Choosing the Right Measure
Translations on the Coordinate Plane: Sliding Without Changing ShapeReflections: Flips Across a Line and What They PreserveRotations and Congruence by Sequences of Rigid TransformationsDilations: Scale Factor, Center, and How Coordinates ChangeSimilarity From Transformations: Corresponding Parts and Scale-Factor Reasoning
Constant Rate of Change: From Unit Rate to SlopeSlope From a Graph: Rise, Run, and “Same Steepness”Slope From Two Points: Changes in CoordinatesSlope From Tables and Contexts: Patterns, Units, and Special CasesThe Y-Intercept and Writing y = mx + bComparing Linear Relationships and Repairing Slope Traps
Powers and Exponent Structure: Bases, Exponents, and What They MeanExponent Rules From Patterns: Product, Quotient, and Power of a PowerZero and Negative Exponents: Making the Rules Still WorkScientific Notation and Orders of MagnitudeMultiplying and Dividing in Scientific NotationRoots and Real Numbers: Perfect Powers, Approximations, and Irrationality
Squares on a Right Triangle: Seeing the Pythagorean RelationshipWhy the Pythagorean Theorem Works: An Area Rearrangement ArgumentFinding Missing Sides and Estimating Irrational LengthsRight Triangle or Not? Using the Converse in Real Situations and on the Coordinate PlaneVolume as Layers: Cylinders and ConesSpheres and Sense-Making: Comparing Volumes and Catching Unit Errors