7th Grade Math

137 core topics + 16 prerequisite topics taught as needed · approximately 36 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 · 6 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.
Linear Inequalities [M]	Solving with <, >, ≤, ≥.

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 6: Rational Numbers & the Coordinate Plane · 12 topics

Opposites in Context [E]	Every number has a mirror twin on the other side of zero.
Absolute Value in Context [E]	Absolute value answers 'how far from zero?' — order answers 'which is greater?'
Comparing & Ordering Rational Numbers [E]	Smaller means farther left on the number line — even below zero.
Rational Numbers on the Number Line [E]	Fractions and decimals claim exact spots between the integers.
Plotting in All Four Quadrants [E]	Negative coordinates open up the other three quadrants.
Reflections Across the Axes [E]	Reflecting a point just flips the sign of one coordinate.
Distances on the Coordinate Plane [M]	Same x or same y: subtract and take the absolute value.
Rectangles on the Coordinate Plane [M]	Coordinates give the side lengths; perimeter and area follow.
Dividing Fractions: Word Problems [M]	'How many of this size fit?' is a division by a fraction.
Decimal Operations in Context [E]	Money problems are decimal arithmetic with two places, always.
Dividing Decimals in Context [M]	Slide both decimal points until the divisor is whole, then divide.
Percent of a Quantity: Parts & Wholes [M]	Percent problems run in three directions: find the part, the percent, or the whole.

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 6: Equations & Inequalities in Context · 12 topics

Writing Addition & Subtraction Equations [E]	Turn a join-or-separate story into a one-step equation.
Writing Multiplication & Division Equations [E]	Equal-groups and fair-share stories become px = q or x ÷ p = q.
Solving Add & Subtract Equations in Context [E]	Undo one addition or subtraction to answer the story's question.
Solving Multiply & Divide Equations in Context [E]	Undo one multiplication or division to find the missing amount.
Is It a Solution? [E]	Substitute a value for the variable and see if both sides agree.
Money Equations [M]	One-step equations where the amounts are dollars and cents.
Evaluating Expressions in Context [E]	Substitute a number for the variable and follow the order of operations.
Dependent & Independent Variables [E]	Which quantity drives the relationship, and which one responds?
Using Two-Variable Equations [E]	Feed the independent variable into a rule to get the dependent one.
Writing Inequalities from Constraints [E]	Translate 'at least', 'more than', 'at most', 'fewer than' into symbols.
Graphing Inequalities on a Number Line [E]	Open or closed circle, shaded left or right — read and draw both ways.
Boundary Values of an Inequality [E]	Find the smallest or greatest whole number an inequality allows.

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 7: Proportional Reasoning in Depth · 13 topics

Constant of Proportionality from a Table [M]	Divide any y by its x — the ratio is the same every row, even when it is a fraction.
Constant of Proportionality from a Graph [M]	On a line through the origin, the constant of proportionality is the rise over the run to any point.
Constant of Proportionality from an Equation [M]	Solve the equation for y = kx; the coefficient of x is the constant, fractions and all.
Unit Rates as Complex Fractions [M]	A rate can be a fraction over a fraction — divide by multiplying by the reciprocal.
Complex-Fraction Rate Problems [M]	Real 'per one unit' rates when both quantities are fractions.
Redrawing at a New Scale [M]	Find the real length once, then convert it into any other drawing's scale.
Scale Drawings and Area [M]	Lengths scale by the factor; areas scale by the factor squared.
Successive Discounts [M]	Two discounts in a row multiply — they never simply add.
Commission [M]	Commission is a percent of sales; total pay may add a base salary.
Percent Error [M]	How far a measurement is off, as a percent of the true value.
Net Percent Change over Several Steps [M]	Chain percent changes by multiplying factors, then compare to the start.
Proportions with Unit Conversion [M]	Convert the units to match the rate before you scale.
Comparing Unit Rates (Best Buy) [M]	Reduce each deal to a price-per-one and compare those rates.

Grade 7: Expressions & Equations Mastery · 13 topics

Expand, Then Combine Like Terms [M]	Distribute each factor, then gather the x-terms and the constants.
Collecting Like Terms [M]	Reorder freely, then add coefficients and add constants.
Factoring Out the GCF [M]	Pull the greatest common factor in front; the numbers left share no factor.
Identifying Equivalent Expressions [M]	Two expressions are equivalent when they agree for every value of x.
Writing Equivalent Expressions [M]	Describe a situation two ways, then simplify to a single equivalent form.
Two-Step Equation Word Problems [M]	Translate the story into ax + b = c, then undo the two operations.
Two-Step Equations with Money [M]	Subtract the flat fee, then divide by the count — answer to the cent.
Two-Step Equations: Geometry & Number Puzzles [M]	Perimeter formulas and number riddles both become ax + b = c.
Two-Step Inequality Word Problems [M]	“At most” and “at least” become ≤ and ≥; solve, then read off the count.
Graphing an Inequality's Solution [M]	Closed dot for ≤ or ≥, open dot for < or >; shade toward the solutions.
Rewriting to Reveal a Relationship [M]	A single combined expression can show a relationship the pieces hide.
Properties of Operations [M]	Name the property — distributive, commutative, associative, or identity.
Interpreting a Factored Form [M]	The factor pulled out front tells you what the value is a multiple of.

Grade 7: Statistics & Probability in Depth · 13 topics

Sample Proportion as a Fraction [E]	The share of a random sample with a trait, written as a reduced fraction.
Estimating a Population Count [M]	Scale the sample proportion up to a whole population total.
A Sample Mean Estimates a Total [M]	Population total ≈ sample mean × population size.
Mean Absolute Deviation [M]	Average distance of the data from its mean — a measure of spread.
Comparing Two Means [M]	Compute each group's mean, then the gap between them.
Comparing Spread with MAD [M]	The data set with the larger MAD is the more variable one.
Difference as a Multiple of the MAD [M]	Measure the gap between two populations in MAD-sized steps.
Theoretical Probability of One Event [M]	Favorable outcomes over total equally likely outcomes, reduced.
Experimental Probability from Data [M]	Observed frequency of an outcome over the total number of trials.
Theoretical vs. Experimental [M]	Expected count = theoretical probability × trials; compare to what happened.
Compound Events by Organized List [M]	List every equally likely combined outcome, then count the favorable ones.
The Counting Principle [M]	Multiply the number of choices at each stage; multiply probabilities of independent events.
Estimating Probability from a Simulation [M]	A simulation's relative frequency estimates the true probability.

Prerequisite material — taught automatically when the diagnostic finds gaps

Linear Functions · 1 topics

The Coordinate Plane

Locating points with (x, y) pairs.

Quadratics & Polynomials · 1 topics

Factoring Out the GCF

Undoing the distributive property.

Geometry · 1 topics

Perimeter & Area

Measuring around and inside basic shapes.

Elementary Math Skills · 6 topics

Place Value & Comparing	Each digit's position multiplies its value by ten.
Multi-Digit Multiplication	Split by place value, multiply each part, add.
Long Division	Divide, multiply, subtract, bring down — repeat.
Perimeter & Area of Rectangles	Perimeter walks around; area covers.
Reading Graphs & Tables	Read the values first; the math is the easy part.
Multi-Step Word Problems	One sentence at a time: what do I know, what changes, what's asked?

Grade 5: Decimals, Volume & the Coordinate Plane · 5 topics

Powers of 10 & Patterns	Each power of ten shifts every digit one place.
Multiplying Decimals	Multiply as whole numbers, then place the decimal.
Dividing Decimals	Divide, keeping the decimal point lined up.
Coordinate Plane (First Quadrant)	(x, y): right then up from the origin.
Order of Operations with Grouping	Parentheses first, then multiply/divide, then add/subtract.

Grade 5: Patterns, Expressions & Data · 2 topics

Writing Expressions from Words	Turn a phrase into symbols — parentheses show what happens first.
Rules & Input-Output Tables	Apply a two-step rule forward — or undo it backward.

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