Integrated Math II

146 core topics + 31 prerequisite topics taught as needed · approximately 44 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

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.

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.

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 · 4 topics

Equivalent Fractions	Different fractions can name the same amount.
Simplifying Fractions	Reducing a fraction to lowest terms.
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 · 5 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.

Linear Functions · 3 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.

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

Integer Operations	Fluent four-operation arithmetic with negative numbers.
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.

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.

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