8th Grade Math

134 core topics + 4 prerequisite topics taught as needed · approximately 33 hours of instruction including spaced review

How the course runs

An adaptive diagnostic (up to 40 questions) places the student on the course's knowledge graph — topics already known are credited, and instruction begins exactly at the learning frontier. Every topic is taught with a worked-example lesson and auto-graded practice; a topic is mastered at 75%+ and then maintained through spaced reviews on an expanding schedule. A cumulative quiz follows every 6 lessons. Prerequisite gaps below the course are detected and taught rather than skipped, so completion certifies the whole tower, not just the top.

Core curriculum

Arithmetic Foundations · 8 topics

Adding & Subtracting Whole Numbers [E]	Multi-digit addition and subtraction.
Multiplication [E]	Multiplying whole numbers.
Division [E]	Dividing whole numbers.
Order of Operations [M]	Parentheses first, then multiplication/division, then addition/subtraction.
Negative Numbers: Adding & Subtracting [M]	Working with numbers below zero on the number line.
Negative Numbers: Multiplying & Dividing [M]	Sign rules for products and quotients.
Exponents [M]	Repeated multiplication in shorthand.
Square Roots [M]	Undoing a square.

Fractions · 7 topics

Equivalent Fractions [M]	Different fractions can name the same amount.
Simplifying Fractions [M]	Reducing a fraction to lowest terms.
Adding Fractions (Like Denominators) [M]	Same-denominator addition.
Adding Fractions (Unlike Denominators) [M]	Rewrite over a common denominator first.
Multiplying Fractions [M]	Multiply straight across.
Dividing Fractions [M]	Multiply by the reciprocal.
Mixed Numbers & Improper Fractions [M]	Converting between forms.

Decimals, Percents & Ratios · 5 topics

Decimal Addition & Subtraction [M]	Line up the decimal points.
Fractions ↔ Decimals [M]	Converting between the two notations.
Percent of a Number [M]	Percent means per hundred.
Percent Increase & Decrease [M]	Applying a percent change to a quantity.
Ratios & Proportions [M]	Two quantities that scale together.

Expressions & Equations · 7 topics

Evaluating Expressions [M]	Substituting a value for a variable.
Combining Like Terms [M]	Adding the coefficients of matching variable parts.
The Distributive Property [M]	Multiplying across a sum.
One-Step Equations [M]	Undoing a single operation.
Two-Step Equations [M]	Undo addition/subtraction first, then multiplication.
Multi-Step Equations [H]	Equations needing distribution or variables on both sides.
Linear Inequalities [M]	Solving with <, >, ≤, ≥.

Linear Functions · 7 topics

The Coordinate Plane [M]	Locating points with (x, y) pairs.
Slope of a Line [M]	Rise over run between two points.
Slope-Intercept Form [M]	y = mx + b describes a whole line.
Finding a Line from Points [H]	Reconstructing y = mx + b from data.
Systems of Equations (Substitution) [H]	Two equations, two unknowns.
Elimination: A First Look [H]	Add or subtract equations so one variable cancels.
Systems: Word Problems [H]	Translating two facts into two equations.

Ratios, Data & Geometry (Middle School) · 16 topics

GCF & LCM [E]	Greatest common factor and least common multiple.
Prime Factorization [E]	Every whole number is a unique product of primes.
Integer Operations [E]	Fluent four-operation arithmetic with negative numbers.
Absolute Value & Distance [E]	Absolute value is distance from zero.
Unit Rates [E]	Per-one comparisons: dollars per item, miles per hour.
Ratio Tables & Equivalent Ratios [E]	Scaling both parts of a ratio keeps it equivalent.
Solving Proportions [E]	Cross-multiply to find the missing value.
Percent Applications: Tax, Tip & Discount [E]	Real-world percents: discounts, tips, and tax.
Area: Triangles & Trapezoids [E]	Half of base times height — and its trapezoid cousin.
Area of Composite Figures [E]	Split odd shapes into rectangles and triangles.
Volume: Rectangular Prisms [E]	Length × width × height, fractional edges included.
Surface Area & Nets [E]	Unfold the box: surface area is the area of its net.
Mean, Median & Range [E]	Three ways to summarize a data set with one number.
Reading Data Displays [E]	Pulling answers out of dot plots, tables, and bar graphs.
Probability Basics [E]	Favorable outcomes over total outcomes.
Compound Probability [E]	Independent events multiply.

Grade 7: Proportions, Geometry & Statistics · 10 topics

Constant of Proportionality [M]	The unit rate k in y = kx.
Graphing Proportional Relationships [E]	Straight line through the origin; slope = k.
Scale Drawings & Maps [E]	Scale is a ratio between drawing and reality.
Two-Step Inequalities [M]	Solve like an equation; flip when you multiply by a negative.
Angle Relationships [E]	Complementary (90°), supplementary (180°), vertical (equal).
Circumference & Area of Circles [M]	C = 2πr and A = πr².
Simple Interest & Percent Change [M]	I = P·r·t; balance = principal + interest.
Random Sampling & Inference [E]	Use a representative sample to estimate a whole population.
Comparing Two Populations [E]	Compare centers relative to spread.
Probability Models & Simulation [E]	Experimental probability from observed frequencies.

Grade 8: Functions, Exponents & Geometry · 14 topics

Integer Exponents & Laws [M]	Add exponents to multiply, subtract to divide, multiply to raise a power.
Scientific Notation [E]	a × 10ⁿ with 1 ≤ a < 10.
Square & Cube Roots [E]	√ undoes squaring; ∛ undoes cubing.
Rational vs Irrational Numbers [E]	Irrational numbers never end or repeat.
Functions: Inputs & Outputs [E]	One input, exactly one output.
Rate of Change & Initial Value [M]	Slope is the rate of change; the y-intercept is the initial value.
Linear vs Nonlinear Functions [E]	Linear graphs are straight lines with constant slope.
Systems of Linear Equations [M]	The solution is where the two lines intersect.
The Pythagorean Theorem [M]	a² + b² = c² for right triangles.
Distance on the Coordinate Plane [M]	Horizontal and vertical gaps are the legs of a right triangle.
Transformations [M]	Translations slide, reflections flip, rotations turn.
Dilations & Similarity [M]	Dilations scale from a center; similar figures share shape.
Volume of Cylinders, Cones & Spheres [M]	V = πr²h, ⅓πr²h, and 4⁄3·πr³.
Scatter Plots & Association [E]	Positive, negative, or no association between two variables.

Grade 7: Rational Numbers & Multi-Step Problems · 12 topics

Adding & Subtracting Signed Fractions [M]	Common denominators first, then the integer sign rules.
Adding & Subtracting Signed Decimals [M]	Balances, debts, and deposits: decimal arithmetic below zero.
Multiplying Signed Rationals [M]	Multiply the fractions, then apply the sign rules.
Dividing Signed Rationals [M]	Flip, multiply, and keep track of the sign.
Markup Then Discount [H]	Percent changes chain by multiplying, never by adding.
Discount, Tax & Tip Chains [H]	Work one percent step at a time — each acts on the previous result.
Proportional Word Problems [M]	Find the rate for one, then scale it to the amount asked.
Angle Equations: Complementary & Supplementary [M]	Complements sum to 90°, supplements to 180° — write the equation.
Angle Equations: Vertical Angles & Linear Pairs [H]	Vertical angles are equal; a linear pair sums to 180°.
Expanding with Rational Coefficients [M]	Distribute fractions and negatives to every term inside.
Factoring Linear Expressions [M]	Pull the GCF out front; distributing it back must restore the original.
Multi-Step Problems with Rationals [H]	Chain the operations one sentence at a time — fractions act on what's left.

Grade 8: Linear Modeling & Data · 12 topics

Building y = mx + b from a Story [M]	The per-unit amount is m; the one-time starting amount is b.
Using a Linear Model [M]	Substitute into y = mx + b — or solve backward for x.
Interpreting the Slope [E]	Slope answers: how much does y change for each one unit of x?
Interpreting the y-Intercept [E]	The intercept is the value of y when x = 0 — the starting amount.
A Model from Two Data Points [M]	Rate first (Δy over Δx), then walk back to x = 0.
Comparing Rates Across Forms [M]	Put both functions' rates in the same form, then compare numbers.
Comparing Linear Functions' Values [M]	Rebuild the table's equation, then evaluate both functions.
Variables on Both Sides in Context [M]	Model each side, set them equal, and collect the variable on one side.
One, None, or Infinitely Many Solutions [M]	Compare the x-coefficients first, then the constants.
Two-Way Tables [E]	Rows are one category, columns the other; totals come from adding.
Relative Frequency in a Table [M]	Divide a cell by its row total to see the pattern, not just the count.
Predicting with a Trend Line [M]	A fitted line turns a cloud of points into predictions.

Grade 7: Geometry Applications · 10 topics

Area of a Circle [E]	Area is π times the radius squared.
Circumference of a Circle [E]	Circumference is π times the diameter.
Semicircles [M]	Half a circle: halve the area and the circumference.
Area of Composite Figures [M]	Break the figure into pieces, then add or subtract.
Scaling and Area [M]	Double the lengths, quadruple the area.
Volume of a Triangular Prism [M]	Triangle area times the prism's length.
Surface Area of a Prism [M]	Add the areas of all six rectangular faces.
Cross-Sections of Solids [E]	The shape revealed when a solid is sliced.
Angle Relationships as Equations [M]	Complementary sum to 90°, supplementary to 180°.
Working Backward from Area [M]	Undo πr² to recover the radius.

Grade 8: Functions & Bivariate Data in Depth · 13 topics

Comparing Rates in Different Forms [M]	Pull each function's slope out of its own form before comparing.
Comparing Initial Values in Different Forms [M]	The initial value is the output at x = 0 — recover it from any form.
Comparing Outputs at a Chosen Input [M]	A steeper function is not always the larger one — evaluate both at x.
Building a Function from a Description [M]	The repeated per-unit amount is the rate; the one-time amount is b.
Building a Function from Two Points [M]	Slope from the two points first, then step back to x = 0 for b.
Interpreting a Model's Output [M]	f(input) = output: the answer carries y's units, at that one input.
Linear or Nonlinear from a Table [E]	Equal x-steps with equal y-jumps means linear; changing jumps means not.
Increasing, Decreasing, or Constant [E]	Rising left-to-right is increasing; falling is decreasing; flat is constant.
Matching a Story to a Graph's Shape [M]	Steady rate gives a line; a changing rate gives a curve.
Reading Association in a Scatter Plot [E]	Rising cloud is positive, falling is negative, shapeless is none.
Clusters & Outliers in a Scatter Plot [E]	Clusters are bunched groups; an outlier stands alone, far from the rest.
Predicting with a Line of Best Fit [M]	Substitute into the line to predict; its slope is the per-unit change.
Relative Frequencies in a Two-Way Table [M]	Divide a cell by a total — row, column, or grand — for a relative frequency.

Grade 8: Geometry, Roots & Scientific Notation · 13 topics

The Pythagorean Theorem in Applications [M]	Ladders, screens, wires, and ramps are right triangles in disguise.
The Converse: Testing for a Right Angle [M]	Compare a² + b² to c² to decide if a triangle is right, acute, or obtuse.
Coordinate Distance from Triples [M]	The horizontal and vertical gaps are legs; a triple gives the exact length.
Square Roots of Larger Perfect Squares [E]	Every perfect square has a whole-number square root — even the big ones.
Cube Roots of Perfect Cubes [M]	A cube root undoes cubing: ∛n is the number whose cube is n.
Estimating Irrational Square Roots [M]	Trap an irrational root between the perfect squares on either side.
Classifying Rational vs Irrational [E]	Spot the odd one out: which number's decimal never ends or repeats.
Multiplying in Scientific Notation [M]	Multiply the coefficients, add the exponents, then re-normalize.
Dividing in Scientific Notation [M]	Divide the coefficients, subtract the exponents.
Adding with the Same Power of Ten [M]	Matching powers of ten factor out — just add the coefficients.
Cylinder Volume as a Coefficient of π [M]	V = πr²h — report the coefficient of π and keep the answer exact.
Cone Volume as a Coefficient of π [M]	A cone is a third of its cylinder: V = ⅓πr²h.
Sphere Volume as a Coefficient of π [M]	V = 4⁄3·πr³ for a full sphere; half that for a hemisphere.

Prerequisite material — taught automatically when the diagnostic finds gaps

Elementary Math Skills · 4 topics

Place Value & Comparing	Each digit's position multiplies its value by ten.
Multi-Digit Multiplication	Split by place value, multiply each part, add.
Perimeter & Area of Rectangles	Perimeter walks around; area covers.
Reading Graphs & Tables	Read the values first; the math is the easy part.

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