Prealgebra

103 core topics + 14 prerequisite topics taught as needed · approximately 27 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 <, >, ≤, ≥.

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 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 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.

Prerequisite material — taught automatically when the diagnostic finds gaps

Linear Functions · 4 topics

The Coordinate Plane	Locating points with (x, y) pairs.
Slope of a Line	Rise over run between two points.
Slope-Intercept Form	y = mx + b describes a whole line.
Finding a Line from Points	Reconstructing y = mx + b from data.

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 · 4 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.

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