Integrated Math II (Honors)

500 core topics + 107 prerequisite topics taught as needed · approximately 186 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. Mixed checks follow every 6 lessons; each unit ends with a 12-item quiz, and course-wide assessments appear at 25%, 50%, 75%, and 100% mastery. Prerequisite gaps below the course are detected and taught rather than skipped, so completion certifies the whole tower, not just the top.

Core curriculum

Quadratics & Polynomials · 13 topics

Adding & Subtracting Polynomials [E]	Combining polynomials by collecting like terms.
Multiplying Binomials (FOIL) [M]	Expanding products of binomials.
Factoring Out the GCF [M]	Undoing the distributive property.
Factoring Trinomials [M]	Reversing FOIL: finding two numbers that multiply to c and add to b.
Special Factoring Patterns [M]	Difference of squares and perfect-square trinomials.
Solving Quadratics by Factoring [M]	Zero-product property: if a·b = 0 then a = 0 or b = 0.
Solving x² = k [M]	Taking square roots of both sides - remembering ±.
Completing the Square [M]	Turning any quadratic into a perfect square plus a constant.
The Quadratic Formula [H]	x = (−b ± √(b² − 4ac)) / 2a solves any quadratic.
The Discriminant [M]	b² − 4ac tells you how many real solutions exist.
Vertex of a Parabola [M]	The turning point at x = −b/2a.
Graphs of Quadratics [M]	Intercepts and symmetry of a parabola.
Quadratic Models [H]	Projectile motion and other parabolic models.

Radicals & Exponentials · 10 topics

Product Rule for Exponents [E]	Multiplying powers of the same base adds the exponents.
Quotient & Power Rules [E]	Dividing powers subtracts exponents; a power of a power multiplies them.
Zero & Negative Exponents [M]	Anything (nonzero) to the 0 power is 1; a negative exponent flips to a reciprocal.
Scientific Notation [M]	Writing very large or very small numbers as c × 10ⁿ.
Simplifying Radicals [M]	Pulling perfect-square factors out of a square root.
Operations with Radicals [M]	Adding like radicals and multiplying square roots.
Rational Exponents [M]	Fractional exponents are roots: x^(p/q) is the q-th root of x, raised to the p.
Radical Equations [H]	Isolate the radical, then square both sides.
Exponential Growth & Decay [M]	Quantities that multiply by the same factor each time step: y = a·bᵗ.
Compound Interest [H]	Money growing exponentially: A = P(1 + r)ᵗ.

Geometry · 14 topics

Angle Relationships [E]	Vertical, complementary, and supplementary angle pairs.
Parallel Lines & Transversals [E]	Angle pairs formed when a transversal crosses parallel lines.
Triangle Angle Sum [E]	The three angles of a triangle always add to 180°.
The Pythagorean Theorem [M]	In a right triangle, a² + b² = c².
Distance & Midpoint [M]	Measuring segments in the coordinate plane.
Similar Triangles [M]	Same shape, different size: corresponding sides are proportional.
Perimeter & Area [E]	Measuring around and inside basic shapes.
Circles: Area & Circumference [M]	C = 2πr and A = πr².
Composite Areas [H]	Adding and subtracting simple shapes to measure a complicated one.
Volume: Prisms & Cylinders [M]	Volume = base area × height.
Volume: Cones, Pyramids & Spheres [M]	Pointed solids hold one third of the matching prism; spheres use 4/3 πr³.
Surface Area [M]	The total area of all the faces of a solid.
Special Right Triangles [H]	The 45-45-90 and 30-60-90 side ratios.
Arc Length & Sector Area [H]	A central angle takes the same fraction of the circumference and the area.

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.

Geometry: Congruence, Triangles & Circles · 12 topics

The Triangle Inequality [M]	Any two sides together must outreach the third.
Isosceles & Equilateral Triangles [M]	Equal sides sit opposite equal angles.
The Exterior Angle Theorem [M]	An exterior angle equals the two far-away interior angles combined.
Congruence Criteria (SSS · SAS · ASA · AAS) [M]	Which marked parts force two triangles to match exactly.
Using Congruence (CPCTC) [M]	Once triangles are congruent, every matching part is equal.
The Midsegment Theorem [M]	The segment joining two midpoints is half the far side.
Parallelogram Properties [M]	Opposite sides equal, consecutive angles supplementary, diagonals bisect.
Special Quadrilaterals [M]	Rhombus, rectangle, square, trapezoid - by their defining properties.
The Inscribed Angle Theorem [M]	An inscribed angle is half the central angle on the same arc.
Tangent Lines to Circles [M]	A tangent meets its radius at a right angle.
Intersecting Chords [M]	Crossing chords cut each other into equal products.
Coordinate Geometry Proofs [M]	Prove geometric facts with slopes, distances, and midpoints.

Geometry: Right-Triangle Trigonometry · 13 topics

Hypotenuse, Opposite & Adjacent [E]	Name the three sides of a right triangle relative to a chosen angle.
The Sine Ratio [E]	Sine is opposite over hypotenuse - read it straight off the triangle.
The Cosine Ratio [E]	Cosine is adjacent over hypotenuse - the leg that touches the angle.
The Tangent Ratio [E]	Tangent is opposite over adjacent - the only ratio with no hypotenuse.
Choosing the Right Ratio [M]	Match the two sides in play to sine, cosine, or tangent.
Pythagoras, Then the Ratio [M]	When only two sides are given, the Pythagorean theorem supplies the third.
Finding a Side from a Given Ratio [M]	Multiply the known side by the given ratio to reach the unknown one.
When the Unknown Is on the Bottom [M]	Divide by the ratio when the unknown side sits in the denominator.
Exact Values: the 45-45-90 Triangle [M]	Half a square: legs equal, hypotenuse √2 times a leg.
Exact Values: 30° and 60° [M]	Half an equilateral triangle gives every 30° and 60° value exactly.
Sides of the 30-60-90 Triangle [M]	Short leg x, long leg x√3, hypotenuse 2x - always in that pattern.
Angle of Elevation Problems [H]	Ground distance, height, and line of sight form a right triangle.
Cofunctions: sin x = cos (90 − x) [M]	Complementary angles trade sine and cosine - same triangle, other corner.

Geometry: Transformations & Symmetry · 12 topics

Translations by a Vector [E]	Slide every point the same amount: add the vector to the coordinates.
Finding the Translation [E]	Image minus preimage recovers the vector; subtract it to go back.
Reflections over the Axes [E]	The mirror line's own coordinate stays; the other flips sign.
Reflections over y = x [E]	Over y = x the coordinates swap; over y = −x they swap and negate.
Rotations of 90° about the Origin [M]	Quarter turns swap the coordinates and flip one sign.
Rotations of 180° and 270° [M]	A half turn negates both coordinates; 270° is a quarter turn the other way.
Identifying a Transformation [M]	Read the coordinate rule off a preimage-image pair.
Composing Transformations [M]	Apply the first rule, then feed its output into the second.
Dilations with Fractional Scale Factors [M]	Multiply every coordinate by the scale factor - even when it is a fraction.
Finding the Scale Factor [M]	Scale factor = image measurement divided by original measurement.
Rotational Symmetry [E]	Order n means n matching positions per full turn - every 360/n degrees.
Congruent or Similar? [M]	Rigid motions keep congruence; any leftover dilation only keeps similarity.

Geometry: Solids, Cross-Sections & Modeling · 10 topics

Volume of a Cylinder [M]	Base circle area times height.
Volume of a Cone [M]	One-third of the matching cylinder.
Volume of a Sphere [M]	Four-thirds pi r cubed.
Composite Solids [H]	Add the volumes of the parts.
Surface Area of a Cylinder [M]	Two end circles plus the wrapped-around side.
Scaling: Area vs Volume [M]	Lengths ×k, area ×k², volume ×k³.
Solids of Revolution [E]	Spin a flat shape to sweep out a solid.
Density: Mass, Volume & Modeling [M]	Mass equals density times volume.
Cavalieri's Principle [M]	Same-area slices at every level mean equal volume.
Modeling with Prism Volume [M]	Length times width times height for a box.

Geometry: Constructions & Loci · 13 topics

Copying a Segment [E]	A compass transfers a length exactly, so a copied segment matches the original.
Copying an Angle [E]	Transferring an angle's arc and chord reproduces its measure exactly.
Bisecting a Segment [E]	Equal arcs from both endpoints locate the midpoint and the perpendicular bisector.
Bisecting an Angle [M]	The angle bisector cuts an angle into two congruent halves.
The Perpendicular Bisector Property [M]	A point is equidistant from two endpoints exactly when it lies on their perpendicular bisector.
Constructing a Perpendicular [M]	Dropping or raising a perpendicular is a perpendicular-bisector construction that yields right angles.
Constructing a Parallel [M]	Copying a transversal's angle makes equal corresponding angles, forcing the lines parallel.
Identifying a Construction [E]	Read a sequence of compass-and-straightedge steps and name the construction.
Why Constructions Work [M]	The validity of each construction rests on congruent triangles and equidistance.
Inscribing a Regular Hexagon [M]	Stepping the radius around a circle marks six points - a regular hexagon of side equal to the radius.
Inscribing an Equilateral Triangle [M]	Joining every other of the six hexagon points gives an inscribed equilateral triangle.
Loci: Sets of Equidistant Points [M]	A locus is the full set of points meeting a distance condition - a bisector, a circle, or parallels.
Points of Concurrency [M]	The centroid cuts each median 2:1, and the circumcenter is equidistant from all three vertices.

Geometry: Coordinate Geometry in Depth · 13 topics

The Midpoint of a Segment [E]	Average the x's and average the y's to land exactly in the middle.
Finding the Other Endpoint [M]	Reverse the midpoint formula: the midpoint is halfway, so double and back off.
Distance Between Two Points [M]	The distance is the hypotenuse of the run-and-rise right triangle.
Perimeter of a Polygon on the Grid [M]	Walk the vertices in order, measure every side, and add the lengths.
Area of an Axis-Aligned Rectangle [M]	Width times height, where each dimension is a coordinate difference.
Area of a Triangle from Its Vertices [M]	One side as base, the perpendicular distance to the opposite vertex as height.
Partitioning a Segment in a Ratio [M]	The dividing point sits m/(m+n) of the way from A toward B.
A Fraction of the Way Along [M]	Add k times the whole displacement to the starting point.
A Line Parallel to a Given Line [M]	Parallel lines share a slope; solve for the new intercept from the point.
A Line Perpendicular to a Given Line [M]	Flip and negate the slope, then fit the intercept to the point.
Is the Triangle Right? (Slopes) [M]	Two sides meet at a right angle exactly when their slopes multiply to −1.
Right, Acute or Obtuse (Pythagoras' Converse) [M]	Compare the longest side squared with the sum of the other two squares.
Completing a Parallelogram [H]	A quadrilateral is a parallelogram exactly when its diagonals share a midpoint.

Geometry: Arc Length, Sectors & Radians · 12 topics

What Fraction of the Circle? [E]	A central angle takes the fraction theta/360 of the whole circle.
Arc Length as a Piece of the Circumference [M]	Arc length is (theta/360) of the circumference 2*pi*r.
Sector Area as a Slice of the Circle [M]	Sector area is (theta/360) of the circle area pi*r^2.
From a Fraction Back to the Angle [M]	If an arc is a given fraction of the circle, the angle is that fraction of 360°.
Degrees to Radians [M]	A radian sweeps one radius of arc; multiply degrees by pi/180.
Radians to Degrees [M]	Multiply a radian measure by 180/pi to get degrees.
Arc Length with Radians: s = r*theta [M]	In radians the arc length is simply the radius times the angle.
Sector Area with Radians: A = ½r²θ [M]	With theta in radians a sector's area is one half r squared theta.
Finding the Radius from an Arc [M]	Invert the arc-length formula to recover the radius.
Inscribed & Central Angles on One Arc [M]	The central angle equals its arc; the inscribed angle is half of it.
The Tangent-Chord Angle [M]	A tangent-chord angle is half the arc it cuts off.
Central Angle from an Arc Length [M]	Compare the arc to the whole circumference to recover the angle.

Geometry: Trigonometry in Any Triangle · 12 topics

Area with Two Sides and an Angle [M]	Two sides and the angle between them give the area directly.
Area Backwards: Find a Missing Side [M]	Turn the area formula around to recover a side length.
Law of Sines: Finding a Side [M]	Each side over the sine of its opposite angle stays constant.
Law of Sines: Finding an Angle [M]	Solve the proportion for a sine, then read off the special angle.
Law of Cosines: The Third Side [H]	c² = a² + b² − 2ab cos C reaches the side the Law of Sines can't.
Law of Cosines: Finding the Angle [H]	Three sides pin down every angle through its cosine.
Classifying a Triangle by Its Sides [M]	The sign of a² + b² − c² tells acute from right from obtuse.
Choosing Sines vs. Cosines [M]	The marked parts decide which law does the job.
Multi-Step Angle of Elevation [H]	Two sight lines to the same top pin down an unknown height.
Multi-Step Angle of Depression [H]	One height, two depression angles, and the gap between the targets.
Law of Sines in the Field [M]	Surveying and navigation triangles solved with one clean proportion.
SAS Area in Context [M]	Real plots and gardens measured from two sides and their angle.

Geometry: Geometric Probability & Modeling · 12 topics

Probability on a Segment [E]	A point on a segment lands in a region with probability length over length.
Length Models in Context [M]	Waiting times and positions are segment probabilities in disguise.
Probability by Area [M]	A dart on a region lands in a shape with probability area over area.
Triangular Targets [M]	Same ratio idea, but the favorable area is half base times height.
Landing Inside an Inscribed Circle [M]	Circle area over rectangle area keeps pi symbolic - report its coefficient.
Rings and Concentric Circles [M]	When both regions are circles, pi cancels and the answer is a clean fraction.
Composite and L-Shaped Regions [M]	Find the favorable area by adding or subtracting rectangles, then divide.
Population as Area Density [M]	Population equals people-per-area times area - density is a rate over region.
Comparing Population Densities [M]	Denser means more people per square mile, so compare population over area.
Choosing the Right Units [E]	Track units through a model: area is squared, and a probability is unitless.
Designing a Region to a Constraint [M]	Work backward from a required area to the dimension that meets it.
Expected Value from Geometric Probability [H]	Weigh each payout by its area-probability and add up the pieces.

Geometry - Deep II · 25 topics

AA Similarity: Solving for a Side [H]	Two matching angle pairs force similarity; then a proportion finds any side.
Similarity Criteria (AA · SAS~ · SSS~) [M]	Which given ratios and angles force two triangles to be similar.
Congruent Triangles: Full Solves [H]	Match corresponding parts, solve for the unknown, then total the sides.
The Side-Splitter Theorem [H]	A line parallel to one side cuts the other two sides in the same ratio.
Geometric Means in Right Triangles [H]	The altitude to the hypotenuse creates three similar triangles.
Similar Figures: Lengths vs Areas [H]	Scale lengths by k and every area scales by k squared.
Elevation with 30°, 45° and 60° [H]	Special angles give exact heights - no calculator, no rounding.
Angle of Depression Problems [H]	Looking down makes the same right triangle as looking up.
Slopes, Ramps & Road Grades [H]	A slope ratio or percent grade is the tangent of the incline angle.
Tangent-Secant Power of a Point [H]	tangent squared equals external part times whole secant.
Two Secants from an External Point [H]	external times whole matches for every secant from the same point.
Cyclic Quadrilaterals [H]	Opposite angles of an inscribed quadrilateral add to 180 degrees.
The Angle Between Two Chords [H]	An inside angle is half the sum of its two intercepted arcs.
Angles Formed Outside a Circle [H]	An outside angle is half the difference of the far and near arcs.
Coordinate Proof: Classifying a Triangle [H]	Squared distances settle both the sides and the angles, exactly.
Coordinate Proof: Collinear Points [H]	Three points are collinear exactly when the slopes agree.
Coordinate Proof: Diagonals Bisect [H]	A parallelogram is proved by one shared midpoint computation.
Coordinate Proof: Trapezoid Midsegment [H]	Midpoints of the legs join into a segment averaging the two bases.
Finding the Radius from a Sector [H]	Invert the sector-area formula to recover the radius.
Area of a Circular Segment [H]	Segment equals sector minus the triangle on the chord.
Slices of a Ring [H]	A slice of an annulus takes theta/360 of the ring's area.
Density, Mass & Cost Modeling [H]	Volume feeds density feeds cost - a chain of rates.
Filling a Tank: Volume and Rate [H]	Time to fill equals the volume divided by the flow rate.
Cross-Sections of Solids [M]	What flat shape appears when a plane slices a solid.
Dimensions of a Solid of Revolution [H]	The axis side becomes the height; the swinging side becomes the radius.

Algebra II - Deep II · 42 topics

Coefficients in a Polynomial Product [H]	Pick out one coefficient of a product without expanding everything.
Sum & Difference of Cubes [H]	a³ ± b³ factors into a binomial times an unfactorable trinomial.
Factoring by Grouping [H]	Pair up four terms, pull a factor from each pair, then share the binomial.
Finding a Coefficient from the Remainder [H]	Run the Remainder Theorem backwards to pin down an unknown constant.
Quartics in Quadratic Form [H]	Substitute u = x² to turn a quartic into a familiar quadratic.
Sum & Product of Roots (Vieta) [H]	Read the sum and product of the roots straight off the coefficients.
Complex Fractions [H]	Clear a fraction-of-fractions by multiplying through by the inner LCD.
Clearing a Denominator [H]	Multiply both sides by the denominator, then solve the linear leftovers.
Extraneous Roots of Rational Equations [H]	Clearing denominators can invent candidates the original equation rejects.
Reciprocal-Sum Formulas [H]	Solve 1/R = 1/a + 1/b style formulas exactly, not by decimal guessing.
Joint & Combined Variation [H]	One quantity driven by two others: z = kxy or z = kx/y.
Two-Step Radical Equations [H]	Isolate the radical first, then square both sides.
Extraneous Roots from Squaring [H]	Squaring both sides can create a candidate the original equation rejects.
Equations with Rational Exponents [H]	Undo x^(m/n) by raising both sides to the reciprocal power n/m.
Powers of a Complex Number [H]	Square or cube a + bi by expanding and folding every i² into -1.
Rebuilding a Quadratic from Complex Roots [H]	Conjugate roots p ± qi give sum 2p and product p² + q².
The Modulus Is Multiplicative [H]	|zw| = |z||w|: take moduli first, then multiply plain numbers.
Multiplying Square Roots of Negatives [H]	Convert to i-form BEFORE multiplying: √(-4)·√(-9) is -6, not 6.
Log Equations by Condensing [H]	Merge the logs into one, convert to exponential form, then solve.
Exponential Equations over a Common Base [H]	Rewrite both sides as powers of one base, then equate the exponents.
The Natural Logarithm [H]	ln is the base-e log: ln(e^k) = k and e^(ln m) = m.
Exact Fractional Log Values [H]	log_4 8 = 3/2: write base and argument over one prime and divide exponents.
How Long to Grow: Doubling & Tripling Time [H]	Count how many doublings are needed, then multiply by the doubling time.
Half-Life [H]	Each half-life cuts the amount in half: after n of them, 1/2^n remains.
Exponentials & Logs Undo Each Other [M]	log_b(b^k) = k and b^(log_b m) = m, with zero computation.
The y-Coordinate of a Hole [H]	Cancel the shared factor, then plug the hole's x into what remains.
Horizontal Asymptotes, Exactly [H]	Equal degrees: y = ratio of leading coefficients. Bottom-heavy: y = 0.
Intercepts of Rational Functions [H]	x-intercepts come from the numerator; the y-intercept is f(0).
Slant Asymptotes by Division [H]	Numerator one degree heavier: divide, keep the linear quotient.
Where a Line Meets a Circle [H]	Substitute the line into the circle and solve the resulting quadratic.
Systems in Three Variables [H]	Combine three equations to peel off one variable at a time.
Sum & Product Systems [H]	Numbers with sum s and product p are the roots of t² - st + p = 0.
Finite Geometric Sums in Context [H]	Total a + ar + ... + ar^(n-1) = a(r^n - 1)/(r - 1).
Sigma Notation: Geometric Sums [H]	Read the limits carefully, then apply the geometric sum formula.
Sigma Notation: Linear Sums [H]	Split Σ(ak + b) into a·Σk + b·n and use Σk = n(n+1)/2.
Geometric Means [H]	The middle of a geometric triple is the square root of the outer product.
Composition as a Formula [H]	Build f(g(x)) as a new rule by substituting g's whole formula into f.
The Inverse as a Formula [H]	Swap x and y, then solve for y: the steps of f, undone in reverse order.
Reading an Inverse from a Table [M]	f⁻¹(b) = a means the table row where f outputs b: read it backwards.
Tracking a Point through Transformations [H]	y = a·f(x - h) + k sends (p, q) to (p + h, aq + k).
Recognizing a Transformed Parent Function [H]	Inside the function moves left-right (backwards); outside moves up-down.
Even & Odd Functions [H]	Even: f(-x) = f(x), mirror symmetry. Odd: f(-x) = -f(x).

Geometry - Triangle Relationships & Similarity (Deep) · 42 topics

The Smallest Admissible Third Side [H]	The third side must exceed the difference of the other two, so the smallest whole-number value sits just above that difference.
Which Three Lengths Build a Triangle [H]	Three lengths form a triangle exactly when the two shorter ones together exceed the longest.
Bounds on the Whole Perimeter [H]	Adding the two known sides to each end of the third-side range bounds the perimeter of the triangle.
Triangle Inequality with a Variable Side [H]	Writing all three inequalities in terms of the variable pins down the range of admissible values.
Ordering Sides by Their Opposite Angles [H]	In any triangle the longer side lies opposite the larger angle, so ordering the angles orders the sides.
Exterior Angles of an Isosceles Triangle [H]	Equal base angles turn the exterior angle at a base vertex into 90 degrees plus half the apex angle.
Two Exterior Angles at Once [H]	One exterior angle at each vertex of a triangle totals 360 degrees, which links any two of them to the third.
The Exterior Angle Inequality [H]	Because an exterior angle equals the sum of two positive remote interior angles, it strictly exceeds each of them.
The Midsegment Triangle's Perimeter [H]	Joining the three midpoints of a triangle gives a triangle whose perimeter is exactly half the original.
Area on Either Side of a Midsegment [H]	A midsegment cuts off a triangle similar at ratio 1 to 2, so it takes one quarter of the area and leaves three quarters.
Midsegments on the Coordinate Plane [H]	On a grid a midsegment is found by averaging coordinates, and its length is half the distance across the side it parallels.
Median Pieces at the Centroid [H]	The centroid cuts each median so the vertex piece is twice the other, making the whole median three halves of the vertex piece.
Recovering a Vertex from the Centroid [H]	Because the centroid is the average of the three vertices, any missing vertex is three times the centroid minus the other two.
The Length of a Median [H]	A median's length is the distance from a vertex to the average of the other two vertices.
Medians and Equal Areas [H]	A median halves a triangle's area, all three medians cut it into six equal pieces, and each vertex-centroid triangle takes one third.
Altitudes from the Area [H]	Every side of a triangle pairs with its own altitude through the same area, so one base-height pair determines all the others.
Locating the Orthocenter [H]	The orthocenter is the common point of the three altitudes, found by intersecting two perpendicular-to-a-side lines through the opposite vertices.
Where Each Triangle Center Lies [H]	The centroid and incenter always lie inside, while the circumcenter and orthocenter move outside for an obtuse triangle.
The Circumcenter of a Right Triangle [H]	In a right triangle the circumcenter is the midpoint of the hypotenuse, so the circumradius is half the hypotenuse.
The Circumcenter from Three Vertices [H]	The circumcenter is the single point equidistant from all three vertices, located by intersecting two perpendicular bisectors.
Equidistance on a Perpendicular Bisector [H]	A point on the perpendicular bisector of a segment is equidistant from its endpoints, which turns two expressions into one equation.
The Angle at the Incenter [H]	The angle subtended at the incenter by one side equals 90 degrees plus half the opposite angle.
The Inradius from Area and Semiperimeter [H]	The incircle's radius equals the triangle's area divided by its semiperimeter.
Between the Bisector and the Altitude [H]	The angle between the bisector and the altitude from one vertex equals half the difference of the other two angles.
Writing the Similarity Statement [H]	A similarity statement is correct only when matching positions in the two names hold equal angles.
Completing an SAS Similarity [H]	With the included angles equal, similarity holds exactly when the two pairs of sides around them share one ratio.
SSS Similarity and the Scale Factor [H]	When all three side pairs share one ratio the triangles are similar, and that ratio is the scale factor.
Proportions with the Unknown on Both Sides [H]	Cross multiplying a proportion from similar triangles turns a repeated unknown into a linear equation.
Corresponding Altitudes, Medians and Bisectors [H]	In similar triangles every corresponding length, including altitudes medians and angle bisectors, shares the side ratio.
Is the Segment Parallel to the Side? [H]	A segment joining two sides of a triangle is parallel to the third side exactly when it divides those sides in the same ratio.
Three Parallel Lines Cutting Two Transversals [H]	Three parallel lines cut off segments in the same ratio on every transversal that crosses them.
The Angle-Bisector Proportionality Theorem [H]	An angle bisector divides the opposite side into two pieces proportional to the two adjacent sides.
Angle Bisectors and the Whole Triangle [H]	Combining the bisector ratio with the perimeter recovers sides that no single measurement gives directly.
The Mean Proportional [H]	The geometric mean of two numbers is the square root of their product, the value that makes a proportion with itself in both middle positions.
Working Backwards from a Leg or an Altitude [H]	The three similar triangles made by the altitude let any two known lengths on the hypotenuse recover the rest.
Area and Perimeter from the Hypotenuse Split [H]	The two hypotenuse segments determine the altitude and both legs, and therefore the whole triangle's area and perimeter.
Scale Factor and Perimeter [H]	Perimeter is a length, so it scales by the scale factor itself rather than by its square.
Material and Cost Under Scaling [H]	Anything proportional to area, such as cloth paint or coating cost, scales by the square of the length ratio.
Enlargements Run Forwards and Backwards [H]	An enlargement multiplies each dimension by the scale factor and the area by its square, so an area ratio recovers the factor by a square root.
Indirect Measurement with Shadows [H]	Objects and their shadows at the same moment form similar right triangles, so height compares to shadow in a fixed ratio.
The Mirror Method [H]	A mirror on the ground reflects at equal angles, creating two similar right triangles that relate eye height to object height.
Sighting Across a Gap [H]	Two sight lines crossing at a stake create vertical angles and similar triangles, so an unreachable width follows from measurable baselines.

Geometry - Circles, Solids & Measurement (Deep) · 43 topics

Arc Addition Around a Full Circle [H]	The arcs cut by points on a circle add to exactly 360 degrees.
Inscribed Angles with Algebraic Arcs [H]	Set the inscribed angle equal to half its arc and solve for the variable.
Two Inscribed Angles on One Chord [H]	Inscribed angles standing on the same arc are congruent.
The Angle in a Semicircle [H]	An angle inscribed in a semicircle is a right angle.
Angles of an Inscribed Triangle [H]	Each angle of an inscribed triangle is half the arc opposite it.
Arcs Given as a Ratio [H]	Share 360 degrees among the ratio parts, then read off the arc or angle.
Where the Vertex Sits Decides the Rule [H]	Center, on, inside or outside the circle selects which arc rule to use.
Tangent-Chord Angles Run Backwards [H]	A tangent-chord angle is half the arc it closes off, from either side.
A Tangent and a Secant Outside a Circle [H]	The external angle is half the difference of the far and near arcs.
Cyclic Quadrilateral Angles from Arcs [H]	Each angle of an inscribed quadrilateral is half the two arcs across from it.
Tangent Length from an External Point [H]	Radius, tangent segment and center distance form a right triangle.
Congruent Tangents and Perimeters [H]	Two tangent segments drawn from one external point are congruent.
A Radius Perpendicular to a Chord [H]	The perpendicular from the center bisects a chord and builds a right triangle.
Comparing Chords and Their Distances [H]	Chords equally far from the center are congruent, and longer chords sit closer.
The Power of a Point [H]	The number d squared minus r squared measures every product through a point.
Chord Products That Need a Quadratic [H]	Intersecting-chord products become quadratic equations when a piece is unknown.
The Perimeter of a Sector [H]	A sector's perimeter is its arc length plus two radii.
The Sector That Rolls into a Cone [H]	A cone's net is a sector whose arc becomes the base circumference.
Rolling Wheels and Turning Arcs [H]	A rolling wheel advances by the arc length that touches the ground.
Writing a Circle's Equation [H]	A center and one radius or one point on the circle fix the equation.
Reading Center and Radius Back [H]	Standard form displays the center and the square of the radius.
Circles by Completing the Square [H]	Completing the square converts general form to center-radius form.
A Circle from the Ends of a Diameter [H]	The center is the diameter's midpoint and the radius is half its length.
Inside, On, or Outside a Circle [H]	Compare the squared distance from the center with the squared radius.
Circles Tangent to the Axes [H]	Tangency to a line means the distance from the center equals the radius.
Pyramid Volume, Run Backwards [H]	One third base times height solves for whichever measurement is missing.
Slant Height, Radius and Height [H]	A cone's radius, height and slant height form a right triangle.
Surface Area of a Cone [H]	Lateral area is pi r l and the base adds pi r squared.
Surface Area of a Square Pyramid [H]	Four triangles on a square base: s squared plus two s times the slant height.
Sphere Surface Area and Volume Together [H]	Four pi r squared and four thirds pi r cubed share the same radius.
Surface Area of a Triangular Prism [H]	Two triangular ends plus the perimeter of the base times the length.
Surface Area of a Composite Solid [H]	Add only the faces that remain exposed after the pieces are joined.
A Missing Dimension from a Volume [H]	Solve the volume formula for the radius or height instead of the volume.
Volume of a Frustum [H]	A frustum is a solid with its top cut off, measured by subtraction or by one formula.
Which Solid Holds the Most [H]	Compare volumes by their pi coefficients, not by how large the solids look.
Troughs: Prisms Lying on Their Side [H]	A trough's volume is its cross-sectional area times its length.
The Area of a Cross-Section [H]	Find the slice's dimensions first, then apply the flat-shape area formula.
Slicing a Cube [H]	A cube's cross-sections range from squares and triangles to a regular hexagon.
From a Ratio Back to the Scale Factor [H]	Square root an area ratio and cube root a volume ratio to recover lengths.
Scaling in Context: Paint, Mass and Capacity [H]	Coverage scales with the square of length and mass or capacity with the cube.
Density with Cylinders, Balls and Pipes [H]	Mass equals density times volume, even when the volume carries a factor of pi.
Packing, Filling and Counting Loads [H]	Volume division counts loads, but packing solid pieces counts along each edge.
Coating Cost from Surface Area [H]	Cost follows the area actually covered, rounded up to whole units of product.

Geometry - Coordinate Proof, Transformations & Right Triangle Trigonometry (Deep) · 42 topics

Classifying a Quadrilateral from Its Vertices [H]	Slopes decide which sides are parallel or perpendicular, and squared distances decide which sides are congruent.
Diagonals of a Coordinate Quadrilateral [H]	A diagonal is just the distance between two opposite vertices, so the distance formula measures and compares diagonals.
Area of a Quadrilateral from Its Vertices [H]	The coordinate area formula pairs each vertex with the next and halves the alternating sum of the cross products.
Side Lengths in Simplest Radical Form [H]	A coordinate length is a square root, and pulling out perfect-square factors leaves it exact instead of rounded.
Completing an Isosceles Trapezoid [H]	An isosceles trapezoid is symmetric about the perpendicular bisector of its bases, so the two top vertices are inset equally from the ends.
A Point Equidistant from Two Others [H]	Setting the two squared distances equal turns the equidistance condition into a linear equation.
Completing a Rectangle or a Square [H]	Opposite sides of a rectangle are equal vectors, and a square's next side is that vector turned a quarter turn.
The Equation of a Perpendicular Bisector [H]	The perpendicular bisector passes through the midpoint with slope the opposite reciprocal of the segment's slope.
Parallel and Perpendicular Lines from Standard Form [H]	Keeping the coefficients of a standard-form line gives a parallel line, while swapping them and changing one sign gives a perpendicular one.
Where a Perpendicular Meets a Line [H]	The foot of a perpendicular is the intersection of the given line with the perpendicular line through the outside point.
Distance from a Point to a Line [H]	The distance from a point to a line is measured along the perpendicular, and the standard-form coefficients compute it directly.
Recovering the Ratio from the Point [H]	Comparing the run from A to P with the run from A to B recovers the fraction and hence the ratio of the two pieces.
Finding an Endpoint from a Partition Ratio [H]	If P cuts AB in the ratio m to n, then the step from A to P is m/(m+n) of the whole step, which recovers either endpoint.
Reflections over x = a and y = b [H]	A mirror line halfway between a point and its image gives the rule x maps to 2a minus x for a vertical mirror.
Finding the Line of Reflection [H]	The mirror line is the perpendicular bisector of the segment joining any point to its image.
Two Reflections over Parallel Lines [H]	Reflecting in two parallel mirrors is a translation perpendicular to them through twice the distance between them.
Two Reflections over Intersecting Lines [H]	Reflecting in two intersecting mirrors is a rotation about their intersection through twice the angle between them.
Glide Reflections [H]	A glide reflection is a translation along a line followed by a reflection in that same line, and the two steps commute.
Rotating about a Point Other than the Origin [H]	Shift the center to the origin, apply the origin rotation rule, then shift back.
A Sequence That Maps One Figure onto Another [H]	Two congruent figures are related by a sequence of rigid motions, and testing every vertex decides which sequence works.
Is the Rule a Rigid Motion? [H]	A rigid motion preserves every distance, a similarity scales all distances by one factor, and anything else distorts the figure.
What a Transformation Preserves [H]	Rigid motions preserve length and angle, dilations preserve angle and slope but not length, and reflections reverse orientation.
Counting Lines of Symmetry [H]	A line of symmetry is a mirror that maps the figure exactly onto itself, and each candidate must send every vertex to a vertex.
The Equation of a Line of Symmetry [H]	An axis of symmetry passes through the midpoints of the segments joining mirror-image vertices.
Rotations That Carry a Regular Polygon onto Itself [H]	A regular n-gon returns to itself exactly at the multiples of 360/n degrees, and it has n lines of symmetry as well.
Dilations Centered Away from the Origin [H]	A dilation multiplies the vector from the center to the point by the scale factor, so the center behaves like a temporary origin.
How a Dilation Changes Slope and Length [H]	A dilation leaves the slope of every segment unchanged while multiplying every length by the scale factor.
Recovering the Center of a Dilation [H]	Because a point, the center and the image are collinear, the center is the point that solves the image equals center plus k times point minus center.
The Image of a Line under a Dilation [H]	A dilation sends a line to a parallel line, so only its intercept moves and the slope is untouched.
Similarity Transformations Mapping One Figure onto Another [H]	Two similar figures are related by a dilation followed by a rigid motion, and the ratio of corresponding lengths gives the scale factor.
From One Trig Ratio to Another [H]	One ratio fixes two sides of a right triangle, and the Pythagorean theorem supplies the third so any other ratio follows.
Recovering the Angle from an Exact Ratio [H]	The three special ratios run backwards: a ratio of 1/2, sqrt(2)/2 or sqrt(3) identifies the angle without a calculator.
Base Angles, Heights and Legs [H]	Dropping perpendiculars from the shorter base turns a trapezoid into a rectangle flanked by two right triangles.
30-60-90 Triangles: Exact Sides and Areas [H]	In a 30-60-90 triangle the sides run short leg, short leg times sqrt(3), and twice the short leg, which makes every area exact.
45-45-90 Triangles: Squares and Diagonals [H]	A square's diagonal is its side times sqrt(2), so the diagonal determines the side, the perimeter and the area exactly.
Equilateral Triangles via the 30-60-90 Split [H]	An altitude splits an equilateral triangle into two 30-60-90 triangles, giving altitude s times sqrt(3) over 2 and area s squared times sqrt(3) over 4.
Apothem and Area of a Regular Hexagon [H]	A regular hexagon splits into six equilateral triangles, so its apothem is half the side times sqrt(3) and its area is six of those triangles.
Elevation Measured from Eye Level [H]	The right triangle of a sighting starts at eye level, so the observer's eye height must be added back to the computed rise.
Two Sightings from One Height [H]	Two depression angles from the same height give two horizontal distances, and their difference is the gap between the targets.
Slope and the Angle a Line Makes [H]	The slope of a line equals the tangent of the angle it makes with the positive x-axis, which links steepness to angle measure.
How the Ratios Behave as the Angle Grows [H]	As an acute angle grows the sine increases toward 1, the cosine decreases toward 0, and the tangent increases without bound.
Finding an Angle of Depression from Measurements [H]	A height and a horizontal distance determine the tangent of the depression angle, and the special ratios name that angle exactly.

Algebra II - Polynomial & Rational Functions (Deep) · 42 topics

Degrees in a Division Statement [H]	Quotient degree is dividend minus divisor; the remainder stays below the divisor.
Dividing When Powers Are Missing [H]	Insert a zero coefficient for every absent power before dividing.
Dividing by (ax - b) [H]	Synthetic division uses the root b/a, then the quotient row is divided by a.
Rebuilding the Dividend [H]	Multiply divisor by quotient and add the remainder to recover p(x).
The Remainder Left by a Quadratic Divisor [H]	Dividing by (x - a)(x - b) leaves mx + n, fixed by the values p(a) and p(b).
The Remainder Theorem at a Fractional Root [H]	The remainder on division by (ax - b) is p(b/a), which is often a fraction.
A Coefficient Forced by a Required Factor [H]	Set p at the divisor's root equal to zero and solve for the unknown coefficient.
Two Coefficients from Two Conditions [H]	Each stated factor or value gives one equation in the unknown coefficients.
Finishing a Factorization from One Zero [H]	Divide out the known factor, then factor the smaller quotient completely.
Ruling a Value Out with the Rational Root Theorem [H]	A rational zero p/q needs p dividing the constant and q dividing the leading coefficient.
Finding a Fractional Zero [H]	When the leading coefficient is not 1, the rational zero can be a fraction p/q.
Conjugate Pairs Forced by Real Coefficients [H]	Nonreal zeros of a real polynomial arrive in conjugate pairs, so their count is even.
Building a Cubic from a Conjugate Pair [H]	A conjugate pair contributes the real quadratic factor x² - 2px + (p² + q²).
The Real Zero a Conjugate Pair Leaves Behind [H]	Divide out the quadratic from the conjugate pair, or use the sum of the zeros.
A Quartic from Two Conjugate Pairs [H]	Two conjugate pairs give two real quadratic factors whose product is the quartic.
Writing a Polynomial from Its Zeros [H]	Each zero r contributes a factor (x - r); multiply them out for standard form.
The Leading Coefficient Fixed by a Point [H]	Zeros determine the factors; one extra point determines the leading coefficient.
Radical Conjugate Zeros [H]	Rational coefficients force a + √k to be paired with a - √k.
Degree and Multiplicity [H]	The multiplicities of all zeros add up to the degree.
The Least Degree a Graph Can Have [H]	Crossings need odd multiplicity, touches need even, and nonreal zeros come in pairs.
The Leading Term of a Factored Form [H]	Multiply the leading term of each factor, raised to that factor's power.
Crossing, Touching, and Flattening [H]	Odd multiplicity crosses the axis, even multiplicity touches and turns back.
Counting the Intervals Where a Polynomial Is Positive [H]	Sign can only change at a zero of odd multiplicity, so alternation can skip.
Adding Three Rational Expressions [H]	Build one common denominator for all three terms, then combine numerators.
Multiply-Divide Chains [H]	Invert every divisor, factor everything, cancel, and only then substitute.
Least Common Denominators with Repeated Factors [H]	The LCD takes each distinct factor to its highest power anywhere.
Canceling with Sums and Differences of Cubes [H]	x³ - a³ = (x - a)(x² + ax + a²), which cancels against a linear factor.
Complex Fractions with Binomial Denominators [H]	Multiply the whole complex fraction, top and bottom, by the inner LCD.
Continued Fractions [H]	Simplify a nested fraction from the innermost level outward.
Rational Equations That Clear to a Quadratic [H]	Multiply by the LCD, collect everything on one side, and factor the quadratic.
Which Candidates Survive the Check [H]	Any candidate that zeroes an original denominator must be discarded.
Motion Problems with Wind and Current [H]	Time equals distance over rate, so wind and current problems are rational equations.
Behavior on Each Side of a Vertical Asymptote [H]	A test value just past the asymptote gives the sign, hence the direction.
Where a Graph Crosses Its Horizontal Asymptote [H]	Set f(x) equal to the asymptote's y-value and solve the resulting equation.
Repeated Factors: Hole or Asymptote? [H]	Compare the powers of the shared factor: leftovers in the denominator still make an asymptote.
Classifying the End-Behavior Asymptote [H]	Degree gap 0 gives a horizontal line, gap 1 a slant line, more than 1 neither.
Building a Rational Function from Its Features [H]	Zeros go in the numerator, asymptotes in the denominator, and the ratio sets the horizontal line.
Solving a Polynomial Inequality [H]	Mark the zeros, test one point per interval, and collect the intervals that qualify.
Inequalities with Repeated Factors [H]	A factor of even multiplicity touches zero without changing the sign.
Solving a Rational Inequality [H]	Sign changes happen at zeros of the numerator AND of the denominator.
Counting the Integers That Satisfy an Inequality [H]	Solve the inequality first, then count the integers inside the solution set.
The Extreme Integer Solution [H]	Factor first, build the sign chart, then read off the extreme integer allowed.

Algebra II - Exponential, Logarithmic & Radical Functions (Deep) · 47 topics

Radical Form and Rational Exponent Form [H]	The n-th root of x to the m equals x raised to the power m/n.
Evaluating Powers with Rational Exponents [H]	Take the root named by the denominator first, then raise to the numerator.
Combining Fractional Exponents [H]	Multiply powers by adding fractional exponents, divide by subtracting.
Roots of Variable Powers [H]	Divide each exponent by the index; pull out the largest perfect power.
Domains of Even and Odd Roots [H]	Even roots need a nonnegative radicand; odd roots accept every real number.
Rational Exponent Models [H]	Power models y = kx^(m/n) are evaluated and inverted with reciprocal powers.
Simplifying Larger Square Roots [H]	Split the radicand into the largest perfect square times the rest.
Simplifying Cube Roots [H]	Pull out perfect CUBE factors: a factor must appear three times to escape.
Adding Radicals After Simplifying [H]	Radicals combine only after each one is reduced to the same radicand.
Multiplying Radical Binomials [H]	FOIL as usual, then replace every √m·√m with m and collect like parts.
Rationalizing a Monomial Denominator [H]	Multiply top and bottom by whatever completes the root in the denominator.
Rationalizing with a Conjugate [H]	Multiply top and bottom by the conjugate to make the denominator rational.
Radical Equations with Two Radicals [H]	Two radicals equal to each other square away in one step; a sum needs two.
Checking Candidates of a Radical Equation [H]	Squaring can add candidates, so test each one in the original equation.
Cube Root Equations [H]	Cubing both sides is reversible, so a cube root equation has no extraneous roots.
Solving a Radical Formula [H]	Isolate the radical in a science formula, then square to free the variable.
Radical Equations in Quadratic Form [H]	Substitute u for the radical: x - k√x + m = 0 is a quadratic in √x.
Verifying an Inverse by Composition [H]	A pair are inverses exactly when both compositions return the input x.
The Inverse of a Radical Function [H]	Swap x and y, then square to undo the square root.
The Inverse of a Cubic Function [H]	Undo a cube with a cube root, working from the outside operation inward.
Restricting a Domain to Invert [H]	A parabola inverts only after its domain is cut to one side of the vertex.
Where a Function Meets Its Inverse [H]	For an increasing function, f and its inverse meet where f(x) = x.
The Inverse of an Exponential Function [H]	The inverse of b^x is log_b x: it reports the exponent that produced a value.
The Domain of a Logarithmic Function [H]	A logarithm accepts only positive arguments, and its graph rises from that boundary.
Evaluating a Transformed Logarithmic Function [H]	Undo the shift and stretch around the logarithm, one operation at a time.
Logs of Composite Numbers from Known Values [H]	Factor the argument into known pieces, then apply the product, quotient and power rules.
Log Properties: Which Rules Are Real [H]	Logs convert products to sums, but a log of a SUM cannot be broken apart.
Expanding a Logarithm: Reading the Coefficients [H]	Every factor becomes a term whose coefficient is its exponent, negative below the bar.
Condensing Logarithms of Expressions [H]	Coefficients become exponents; sums become products and differences quotients.
Natural Log Properties [H]	The log laws apply to ln, with the extra simplification ln e = 1.
Writing a Log with the Change of Base Formula [H]	log_b N equals log N over log b in any single convenient base.
Chaining Logarithms [H]	log_a b times log_b c collapses to log_a c: the middle base cancels.
Bounding a Logarithm Between Integers [H]	Trap the argument between two powers of the base to bracket the logarithm.
Exponential Equations with a Quadratic Exponent [H]	Equal powers of one base force equal exponents, even when an exponent is quadratic.
Exponential Equations in Quadratic Form [H]	Since b^(2x) = (b^x)², the substitution u = b^x turns the equation into a quadratic.
Factoring Out a Common Exponential [H]	b^(x+k) is b^x times b^k, so a common power factors out of a sum of exponentials.
Exponential Equations with Unlike Bases [H]	Take a logarithm of both sides so the power law brings the exponent down.
Logarithmic Equations with a Log on Each Side [H]	One-to-one logs: if log_b A = log_b B then A = B, provided both stay positive.
Domain Checks in Logarithmic Equations [H]	Condensing can produce candidates that make a logarithm undefined, and sometimes none survive.
Equations Quadratic in a Logarithm [H]	Substitute u for the logarithm, solve the quadratic, then convert each u back.
Compound Interest with Several Periods a Year [H]	A = P(1 + r/n)^(nt): divide the rate by n and multiply the years by n.
Continuous Compounding [H]	A = Pe^(rt): the growth factor over t years is e^(rt), so rt fixes the multiple.
Population Growth Models [H]	A constant percent increase means multiplying by the same factor 1 + r every year.
Finding a Growth Factor from Two Data Points [H]	The annual factor is the k-th root of the ratio of the two measurements.
Writing a Half-Life Model [H]	Half-life h gives A = A0(1/2)^(t/h): the exponent counts how many half-lives have passed.
Newton's Law of Cooling: Temperature [H]	Only the difference from the surroundings decays; the surrounding temperature is added back.
Newton's Law of Cooling: Time [H]	Count how many times the temperature difference must halve, then multiply by the halving time.

Algebra II - Series, Conic Sections & Probability Models (Deep) · 47 topics

Arithmetic Totals in Context [H]	Seating, stacking and production totals are arithmetic series in words.
Summing a Block of an Arithmetic Series [H]	The sum of terms m through n is the count of terms times the average of the two ends.
Sigma Notation with a Shifted Lower Limit [H]	A lower limit other than 1 changes the term count, not the term rule.
Re-Indexing a Sum [H]	Shifting the index shifts the limits one way and the term rule the other way.
Summing a Block of a Geometric Series [H]	Add terms m through n of a geometric sequence as S(n) minus S(m-1).
How Many Terms Reach a Geometric Total [H]	Run S = a(rⁿ - 1)/(r - 1) backwards to recover the number of terms.
Recovering the Ratio from a Geometric Sum [H]	Divide the sum by the first term, then solve the resulting polynomial in r.
Does the Infinite Series Converge? [H]	An infinite geometric series has a finite sum exactly when the ratio satisfies |r| < 1.
Infinite Geometric Series in Sigma Notation [H]	Read the first term straight off the lower limit, then apply a/(1 - r).
Total Path of a Bouncing Ball [H]	Rebound heights form an infinite geometric series; the path counts each rise twice.
The Long-Run Level of a Recursive Model [H]	When a fraction r is retained and a fixed amount c is added, the level settles at c/(1 - r).
A Sequence Rebuilt from Two of Its Terms [H]	Two terms fix the step: divide the change by the gap in index positions.
A Circle Recovered from General Form [H]	Complete the square in x and in y to turn Ax² + Ay² + Dx + Ey + F = 0 into standard form.
Focus and Directrix of a Parabola [H]	In (x - h)² = 4p(y - k) the focus sits p units from the vertex and the directrix p units the other way.
Building a Parabola from Focus and Directrix [H]	The vertex is halfway between focus and directrix, and p is the distance to either one.
Horizontal Parabolas [H]	When y is squared the parabola opens left or right and the roles of x and y swap.
Parabolic Dishes and Arches [H]	Put the vertex at the origin, substitute the rim point, and solve for the focal distance.
Center, Vertices and Axes of an Ellipse [H]	The larger denominator names the major axis; its square root is the semi-axis length.
Foci of an Ellipse [H]	For an ellipse the focal distance satisfies c² = a² - b², with a the semi-major axis.
Writing the Equation of an Ellipse [H]	Vertices give a, foci give c, and b² = a² - c² fills in the other denominator.
Elliptical Arches and Whispering Galleries [H]	A semi-elliptical arch of width 2a and height b satisfies x²/a² + y²/b² = 1 for y at least 0.
Vertices and Transverse Axis of a Hyperbola [H]	The variable with the POSITIVE term names the transverse axis, whatever the denominators are.
Foci of a Hyperbola [H]	For a hyperbola the focal distance satisfies c² = a² + b², so the foci lie beyond the vertices.
Asymptotes of a Hyperbola [H]	The asymptotes pass through the center with slopes given by the ratio of the semi-axes.
Writing the Equation of a Hyperbola [H]	Vertices give a, foci or asymptote slope give b, and the positive term follows the transverse axis.
Identifying a Conic from Its Equation [H]	Compare the two squared terms: their count, their signs and whether their coefficients match.
Where a Line Meets an Ellipse [H]	Substitute the line into the ellipse and solve the quadratic that results.
Where a Line Meets a Hyperbola [H]	Substituting a line into a hyperbola gives a quadratic, unless the line matches an asymptote slope.
Two Conics at Once [H]	Treat x² and y² as the two unknowns, eliminate one, then take square roots.
How Many Intersections? [H]	Substitute, collect a quadratic, and let the discriminant count the real intersections.
Arrangements with Repeated Letters [H]	Divide n! by a factorial for each repeated letter, since swapping identical letters changes nothing.
Circular Arrangements [H]	Seat one person to fix the rotation, then arrange the remaining n - 1 people in a line.
Committees with Restrictions [H]	Fix the forced members first, multiply independent choices, and use the complement for at least one.
Counting by Complement [H]	Count everything, then subtract the arrangements that fail the condition.
Pascal's Triangle and Binomial Coefficients [H]	Entry k of row n of Pascal's triangle is exactly the combination C(n, k).
A Specified Term of a Binomial Expansion [H]	The rth term of (a + b)^n is C(n, r - 1)·a^(n-r+1)·b^(r-1).
The Coefficient of a Chosen Power [H]	Choose the index k that makes the powers of x combine to the exponent you want.
Conditional Probability from a Two-Way Table [H]	Conditioning on an event replaces the grand total by that event's own total.
Testing Independence in a Two-Way Table [H]	Two events are independent exactly when conditioning on one leaves the other's probability unchanged.
The Conditional Probability Formula [H]	P(A | B) = P(A and B)/P(B), and the same identity rearranges to P(A and B) = P(A | B)·P(B).
The General Addition Rule [H]	P(A or B) = P(A) + P(B) - P(A and B), because the overlap is otherwise counted twice.
Reversing a Conditional with a Tree [H]	P(B | A) is the branch that produced A divided by every branch that produces A.
The Binomial Probability Model [H]	P(exactly k successes in n independent trials) = C(n, k)p^k(1 - p)^(n-k).
Areas from a Standard Normal Table [H]	Convert the boundary to a z-score, read the area to its left, then adjust for the region asked about.
From a Percentile Back to a Raw Score [H]	Look up the area inside the table to recover z, then undo the standardization with x = mu + z*sigma.
Recovering the Mean or the Standard Deviation [H]	Solve z = (x - mu)/sigma for whichever of the three quantities is unknown.
Equivalent Scores Across Two Distributions [H]	Two scores represent the same standing exactly when they share a z-score.

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 · 5 topics

Decimal Addition & Subtraction	Line up the decimal points.
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.

Ratios, Data & Geometry (Middle School) · 8 topics

Integer Operations	Fluent four-operation arithmetic with negative numbers.
Absolute Value & Distance	Absolute value is distance from zero.
Unit Rates	Per-one comparisons: dollars per item, miles per hour.
Ratio Tables & Equivalent Ratios	Scaling both parts of a ratio keeps it equivalent.
Solving Proportions	Cross-multiply to find the missing value.
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.

Grade 8: Functions, Exponents & Geometry · 2 topics

Transformations	Translations slide, reflections flip, rotations turn.
Dilations & Similarity	Dilations scale from a center; similar figures share shape.

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.

Algebra II: Statistics & Probability · 12 topics

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

Algebra II: Rational Expressions & Equations · 11 topics

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

Algebra II: Sequences & Series · 11 topics

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

Algebra II: Logarithms in Depth · 9 topics

Exponential ↔ Logarithmic Form	Every logarithm is an exponent: b^k = x and log_b(x) = k say the same thing.
Evaluating Logarithms Exactly	Read log_b(x) as a question: to what power must b be raised to give x?
Change of Base	log_b(x) = log_c(x)/log_c(b) - rewrite over any convenient common base.
Expanding with the Log Laws	Products become sums, quotients differences, powers coefficients.
Condensing into a Single Logarithm	Run the log laws backward: sums into products, coefficients into powers.
The Power Law in Detail	log_b(x^k) = k·log_b(x) - even when the exponent is a root.
Solving Exponential Equations with Logs	Take a logarithm of both sides - or match a common base - to free the exponent.
Solving Logarithmic Equations	Rewrite in exponential form, or combine logs first, then solve for x.
Application: Doubling Time	Repeated doubling is a logarithm base 2: n doublings multiply by 2^n.

Algebra II: Polynomial & Rational Functions · 12 topics

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

Algebra II: Introduction to Trigonometry · 3 topics

Sine as a Ratio	sin of an angle is the opposite side over the hypotenuse.
Cosine as a Ratio	cos of an angle is the adjacent side over the hypotenuse.
The Pythagorean Identity	sin^2 + cos^2 = 1 turns one ratio into the other.

Algebra II: Complex Numbers & Matrices · 4 topics

Adding Complex Numbers	Combine real parts with real parts and imaginary parts with imaginary parts.
Multiplying Complex Numbers	FOIL the two binomials, then replace i² with −1 and collect parts.
Complex Conjugates	The conjugate of a + bi keeps the real part and flips the sign of i.
The Product (a + bi)(a − bi)	A number times its conjugate is always the real value a² + b².

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