Integrated Math III

126 core topics + 62 prerequisite topics taught as needed · approximately 47 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

Functions & Algebra II · 24 topics

Function Notation & Evaluation [E]	Reading f(x) notation and plugging in inputs.
Domain & Range [E]	Which inputs a function accepts, and which outputs it can produce.
Function Composition [M]	Feeding one function's output into another: f(g(x)).
Inverse Functions [M]	The function that undoes f: f⁻¹(b) is the input that f sends to b.
Transformations of Functions [M]	How f(x − h) + k slides a graph around the plane.
Piecewise Functions [M]	Functions defined by different rules on different intervals.
Absolute Value Equations [M]	|x − a| = b splits into two linear equations.
Systems by Elimination [H]	Adding or subtracting equations to cancel a variable.
Nonlinear Systems [H]	Where a line meets a parabola: set the two formulas equal.
Complex Numbers [M]	The imaginary unit i = √(−1) and numbers of the form a + bi.
Operations with Complex Numbers [M]	Multiplying complex numbers with FOIL and i² = −1.
Quadratics with Complex Roots [H]	When the discriminant is negative, the roots come in a conjugate pair a ± bi.
Polynomial Division [H]	Dividing a polynomial by (x − a) with long or synthetic division.
Remainder & Factor Theorems [M]	The remainder when p(x) is divided by (x − a) is simply p(a).
Zeros of Polynomials [H]	Finding all the roots of a cubic by factoring it down.
End Behavior of Polynomials [E]	Far from the origin, only the leading term matters.
Simplifying Rational Expressions [M]	Factor top and bottom, then cancel the common factor.
Operations on Rational Expressions [H]	Multiplying and dividing algebraic fractions.
Rational Equations [H]	Clearing denominators to solve equations with x below the line.
Logarithms [M]	log_b(x) asks: to what power must b be raised to get x?
Properties of Logarithms [M]	Logs turn products into sums, quotients into differences, powers into multiples.
Exponential & Log Equations [M]	Matching bases and rewriting between exponential and log form.
Arithmetic Sequences [M]	Sequences that grow by a constant difference each step.
Geometric Sequences [M]	Sequences that grow by a constant ratio each step.

Trigonometry · 15 topics

Right-Triangle Trigonometry [M]	SOH-CAH-TOA: the three trig ratios of an acute angle in a right triangle.
Solving for Sides with Trig [M]	Using a known angle and one side to find another side.
Degrees & Radians [E]	Two ways to measure the same angle: 180° equals π radians.
The Unit Circle [M]	Exact sine, cosine, and tangent values at the special angles.
Trig of Any Angle [H]	Reference angles plus quadrant signs extend trig beyond 90°.
Graphs of Sine & Cosine [M]	Reading amplitude, period, and midline from y = a sin(bx) + c.
Phase Shifts & Other Trig Graphs [M]	Horizontal (phase) shifts of trig graphs, and the period of tangent.
The Pythagorean Identity [M]	sin²θ + cos²θ = 1 links sine and cosine of the same angle.
Basic Trig Identities [M]	Quotient, reciprocal, and even-odd identities.
Sum & Difference Formulas [H]	Expanding sin(A ± B) and cos(A ± B) to reach non-special angles.
Double-Angle Formulas [H]	sin 2x = 2 sin x cos x and cos 2x = 1 − 2 sin²x.
Trig Equations [H]	Isolating a trig function and reading solutions off the unit circle.
Law of Sines [H]	In any triangle, each side over the sine of its opposite angle is constant.
Law of Cosines [H]	c² = a² + b² − 2ab cos C generalizes the Pythagorean theorem.
Inverse Trig Functions [M]	arcsin, arccos, and arctan undo the trig functions on restricted ranges.

Algebra II: Statistics & Probability · 12 topics

The Empirical Rule: One Deviation [E]	About 68% of normal data lies within one standard deviation.
Two and Three Deviations [M]	95% within two deviations, 99.7% within three.
Tails: Above and Below [M]	Split what's left over evenly between the two tails.
Percents Between Bounds [M]	Stack the 34 / 13.5 / 2.35 bands to cover any interval.
z-Scores [M]	How many standard deviations from the mean.
Comparing Scores with z [M]	The larger z-score is the rarer, stronger performance.
Permutations [M]	Ordered arrangements: multiply the shrinking choices.
Combinations [M]	Unordered selections: divide out the reorderings.
Order or Not? [M]	Medals and PINs care about order; committees don't.
Independent Events [M]	Independent ANDs multiply.
Without Replacement [M]	The second draw's probabilities shift after the first.
Expected Value [H]	The long-run average: weigh each outcome by its probability.

Algebra II: Rational Expressions & Equations · 13 topics

Excluded Values & Domain [E]	Factor the denominator to find every input a rational expression forbids.
Simplifying Quadratic over Quadratic [M]	Factor both trinomials, cancel the shared factor, and read off what's left.
Opposite Factors: the −1 Trick [M]	a − x and x − a are negatives of each other, so they cancel to −1.
Multiplying Rational Expressions [M]	Factor every trinomial first, then cancel across the multiplication.
Dividing Rational Expressions [M]	Flip the second fraction, then factor and cancel like a multiplication.
Adding with Unlike Polynomial Denominators [H]	Build the common denominator (x + p)(x + q) and combine the numerators.
Subtracting Rational Expressions [H]	Distribute the minus sign through the entire second numerator.
Rational Equations by Cross-Multiplying [M]	One fraction equals another: cross-multiply and solve the linear equation.
Rational Equations That Turn Quadratic [H]	Clearing an x from the denominator leaves a factorable quadratic.
Extraneous Solutions [M]	A candidate that zeroes an original denominator must be thrown out.
Direct Variation [E]	y = kx: find the constant from one data point, then predict any other.
Inverse Variation [M]	y = k/x: the product xy stays constant, so one point predicts the rest.
Combined Work-Rate Problems [H]	Add the jobs-per-hour rates — 1/a + 1/b = 1/t — and solve for the time.

Algebra II: Sequences & Series · 13 topics

Computing Terms from a Recursive Rule [E]	A recursive rule builds each term from the ones before it — step by step.
Explicit vs Recursive Definitions [M]	The same sequence can be described step-by-step or by a direct formula.
From Recursive to Explicit [M]	Convert the step rule to a direct formula, then jump straight to term n.
Counting Terms of a Sequence [E]	Solve a_n = a1 + (n − 1)d for n: divide the total climb by the step size.
The Arithmetic Series Formula [M]	Sum an arithmetic series as count times the average of the two ends.
Solving Inside the Series Formula [M]	Run S = n(a1 + an)/2 backwards to recover n, the first, or the last term.
Finite Geometric Series [M]	Sum a geometric series with Sₙ = a(rⁿ − 1)/(r − 1) — one power, no term list.
Evaluating Sigma Notation [M]	Read the limits, substitute each index value, and add the results.
Writing a Series in Sigma Notation [M]	Find the kth-term formula, then set the limits so the ends match.
From Sum Formula Back to Terms [M]	Subtract consecutive partial sums to recover a single term.
Infinite Geometric Series [M]	When |r| < 1 the whole endless series adds to exactly a/(1 − r).
Repeating Decimals as Fractions [M]	A repeating decimal is an infinite geometric series in disguise.
Modeling Savings & Loans with Sequences [H]	Deposits, payments, and interest are sequences — model them step by step.

Algebra II: Logarithms in Depth · 12 topics

Exponential ↔ Logarithmic Form [E]	Every logarithm is an exponent: b^k = x and log_b(x) = k say the same thing.
Evaluating Logarithms Exactly [E]	Read log_b(x) as a question: to what power must b be raised to give x?
Change of Base [M]	log_b(x) = log_c(x)/log_c(b) — rewrite over any convenient common base.
Expanding with the Log Laws [M]	Products become sums, quotients differences, powers coefficients.
Condensing into a Single Logarithm [M]	Run the log laws backward: sums into products, coefficients into powers.
The Power Law in Detail [M]	log_b(x^k) = k·log_b(x) — even when the exponent is a root.
Solving Exponential Equations with Logs [M]	Take a logarithm of both sides — or match a common base — to free the exponent.
Solving Logarithmic Equations [M]	Rewrite in exponential form, or combine logs first, then solve for x.
Application: the Richter Scale [M]	Each whole step up in magnitude is a tenfold jump in amplitude.
Application: the pH Scale [M]	pH = −log₁₀[H⁺]: a lower pH means an exponentially higher acid concentration.
Application: the Decibel Scale [M]	Loudness in dB is ten times the base-10 log of the intensity ratio.
Application: Doubling Time [M]	Repeated doubling is a logarithm base 2: n doublings multiply by 2^n.

Algebra II: Polynomial & Rational Functions · 12 topics

Synthetic Substitution [E]	Horner's method: evaluate a polynomial with only multiplies and adds.
Reading the Synthetic-Division Quotient [M]	The bottom row of synthetic division is the quotient, then the remainder.
Polynomial Long Division [M]	Dividing by a quadratic: match leading terms, multiply back, subtract, repeat.
The Remainder Theorem [M]	The remainder of P(x) ÷ (x − a) is just P(a) — no division required.
The Factor Theorem: Finding a Root [M]	x = a is a root exactly when (x − a) is a factor, i.e. when P(a) = 0.
Testing Whether (x − a) Is a Factor [M]	Compute P(a): a zero remainder means (x − a) is a factor.
The Rational Root Theorem: Listing Candidates [M]	Candidate rational roots are ±(factors of the constant)/(factors of the lead).
Finding Integer Roots with the Rational Root Theorem [M]	List the candidates, then test them to pin down the actual roots.
End Behavior from the Leading Term [E]	Degree parity sets whether the ends agree; the lead sign sets the direction.
Excluded Values of a Rational Function [M]	Every zero of the denominator is barred from the domain.
Holes versus Vertical Asymptotes [M]	A cancelling factor makes a hole; a leftover denominator factor makes an asymptote.
Horizontal Asymptotes by Degree [M]	Compare top and bottom degrees to read off the horizontal asymptote.

Algebra II: Introduction to Trigonometry · 12 topics

Sine as a Ratio [E]	sin of an angle is the opposite side over the hypotenuse.
Cosine as a Ratio [E]	cos of an angle is the adjacent side over the hypotenuse.
Tangent as a Ratio [E]	tan of an angle is the opposite side over the adjacent side.
Pythagoras, Then the Ratio [M]	When only two sides are given, the Pythagorean theorem supplies the third.
The Pythagorean Identity [M]	sin^2 + cos^2 = 1 turns one ratio into the other.
Exact Values at 45 Degrees [M]	Half a square: legs equal, hypotenuse sqrt(2) times a leg.
Exact Values at 30 and 60 Degrees [M]	Half an equilateral triangle gives every 30 and 60 degree value exactly.
Coterminal Angles [E]	Add or subtract 360 degrees to land on the same terminal ray.
Reference Angles [M]	The acute angle between the terminal ray and the x-axis.
Signs by Quadrant [M]	Which of sine and cosine are positive depends on the quadrant.
Degrees to Radians [M]	Multiply degrees by pi/180 and reduce the fraction of pi.
Values at 0, 90, 180, 270 [M]	On the axes, sine and cosine are always -1, 0, or 1.

Algebra II: Complex Numbers & Matrices · 13 topics

Adding Complex Numbers [E]	Combine real parts with real parts and imaginary parts with imaginary parts.
Subtracting Complex Numbers [E]	Distribute the minus sign, then subtract part by part.
Multiplying Complex Numbers [E]	FOIL the two binomials, then replace i² with −1 and collect parts.
Powers of i [E]	Powers of i repeat every four steps: i, −1, −i, 1.
Complex Conjugates [E]	The conjugate of a + bi keeps the real part and flips the sign of i.
The Product (a + bi)(a − bi) [E]	A number times its conjugate is always the real value a² + b².
Dividing Complex Numbers [M]	Multiply top and bottom by the denominator's conjugate to clear the i.
The Modulus of a Complex Number [M]	The modulus |a + bi| = √(a² + b²) is the point's distance from the origin.
Adding 2×2 Matrices [E]	Add two matrices of the same size entry by matching entry.
Scalar Multiplication of a Matrix [E]	Multiply a matrix by a number by scaling every entry.
Multiplying 2×2 Matrices [M]	Each product entry is a row of A dotted with a column of B.
The Determinant of a 2×2 Matrix [M]	For [[a, b], [c, d]] the determinant is ad − bc.
Solving a 2×2 System with Determinants [M]	Cramer's rule reads each variable off a ratio of determinants.

Prerequisite material — taught automatically when the diagnostic finds gaps

Arithmetic Foundations · 8 topics

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

Fractions · 6 topics

Equivalent Fractions	Different fractions can name the same amount.
Simplifying Fractions	Reducing a fraction to lowest terms.
Adding Fractions (Like Denominators)	Same-denominator addition.
Adding Fractions (Unlike Denominators)	Rewrite over a common denominator first.
Multiplying Fractions	Multiply straight across.
Dividing Fractions	Multiply by the reciprocal.

Decimals, Percents & Ratios · 4 topics

Fractions ↔ Decimals	Converting between the two notations.
Percent of a Number	Percent means per hundred.
Percent Increase & Decrease	Applying a percent change to a quantity.
Ratios & Proportions	Two quantities that scale together.

Expressions & Equations · 7 topics

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

Linear Functions · 6 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.
Systems of Equations (Substitution)	Two equations, two unknowns.
Arithmetic Sequences	Add the same amount each step.
Geometric Sequences	Multiply by the same ratio each step.

Quadratics & Polynomials · 10 topics

Adding & Subtracting Polynomials	Combining polynomials by collecting like terms.
Multiplying Binomials (FOIL)	Expanding products of binomials.
Factoring Out the GCF	Undoing the distributive property.
Factoring Trinomials	Reversing FOIL: finding two numbers that multiply to c and add to b.
Special Factoring Patterns	Difference of squares and perfect-square trinomials.
Solving Quadratics by Factoring	Zero-product property: if a·b = 0 then a = 0 or b = 0.
Solving x² = k	Taking square roots of both sides — remembering ±.
Completing the Square	Turning any quadratic into a perfect square plus a constant.
The Quadratic Formula	x = (−b ± √(b² − 4ac)) / 2a solves any quadratic.
Vertex of a Parabola	The turning point at x = −b/2a.

Radicals & Exponentials · 7 topics

Product Rule for Exponents	Multiplying powers of the same base adds the exponents.
Quotient & Power Rules	Dividing powers subtracts exponents; a power of a power multiplies them.
Zero & Negative Exponents	Anything (nonzero) to the 0 power is 1; a negative exponent flips to a reciprocal.
Simplifying Radicals	Pulling perfect-square factors out of a square root.
Rational Exponents	Fractional exponents are roots: x^(p/q) is the q-th root of x, raised to the p.
Exponential Growth & Decay	Quantities that multiply by the same factor each time step: y = a·bᵗ.
Compound Interest	Money growing exponentially: A = P(1 + r)ᵗ.

Geometry · 7 topics

Angle Relationships	Vertical, complementary, and supplementary angle pairs.
Triangle Angle Sum	The three angles of a triangle always add to 180°.
The Pythagorean Theorem	In a right triangle, a² + b² = c².
Similar Triangles	Same shape, different size: corresponding sides are proportional.
Perimeter & Area	Measuring around and inside basic shapes.
Circles: Area & Circumference	C = 2πr and A = πr².
Special Right Triangles	The 45-45-90 and 30-60-90 side ratios.

Ratios, Data & Geometry (Middle School) · 4 topics

Absolute Value & Distance	Absolute value is distance from zero.
Mean, Median & Range	Three ways to summarize a data set with one number.
Probability Basics	Favorable outcomes over total outcomes.
Compound Probability	Independent events multiply.

Algebra I: Descriptive Statistics · 3 topics

Computing the Mean	The mean is the total shared out equally — and totals work backwards too.
Mean Absolute Deviation	MAD is the average distance of the data from its own mean.
Variance of a Data Set	Square each deviation from the mean, then average the squares.

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