Integrated Math I

443 core topics + 126 prerequisite topics taught as needed · approximately 174 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

Expressions & Equations · 7 topics

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

Linear Functions · 10 topics

The Coordinate Plane [M]	Locating points with (x, y) pairs.
Slope of a Line [M]	Rise over run between two points.
Slope-Intercept Form [M]	y = mx + b describes a whole line.
Finding a Line from Points [H]	Reconstructing y = mx + b from data.
Systems of Equations (Substitution) [H]	Two equations, two unknowns.
Elimination: A First Look [H]	Add or subtract equations so one variable cancels.
Systems: Word Problems [H]	Translating two facts into two equations.
Absolute Value Equations [H]	Distance equations have two answers.
Arithmetic Sequences [H]	Add the same amount each step.
Geometric Sequences [H]	Multiply by the same ratio each step.

Geometry · 10 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.

Algebra I: Inequalities & Compound Statements · 13 topics

Two-Step Inequalities with the Flip [E]	Undo the constant, then divide - and flip the sign if the divisor is negative.
Multi-Step Inequalities [M]	Distribute and collect variables just like an equation - then mind the sign.
Which Way Does the Sign Point? [E]	Spot when the inequality flips - and when it doesn't.
Compound AND Inequalities [M]	A sandwiched inequality: do the same steps to all three parts.
Compound OR Inequalities [M]	OR keeps everything either piece allows - only the gap between them fails.
Absolute Value: Less Than [M]	|x − m| < k traps x within k of m - an AND sandwich.
Absolute Value: Greater Than [M]	|x − m| > k pushes x farther than k from m - an OR split.
Multi-Step Absolute Value Inequalities [H]	Isolate the absolute value first; only then split into a sandwich.
At Least / At Most Word Problems [M]	Translate a budget or goal into an inequality, then round the smart way.
Hitting an Average Target [M]	An average of at least T means the total must reach n times T.
Translating Words into Inequalities [E]	At most means ≤, at least means ≥, more than means > - order matters too.
Checking Points Against a System [M]	Substitute the point into each inequality and judge them separately.
All Real Numbers or No Solution? [M]	When the x-terms cancel, only a true-or-false number fact remains.

Algebra I: Descriptive Statistics · 13 topics

Computing the Mean [E]	The mean is the total shared out equally - and totals work backwards too.
Median & Mode [E]	Sort first: the median is the middle, the mode is the most frequent.
Outliers and the Mean [M]	One extreme value drags the mean toward it; the median barely moves.
Choosing a Measure of Center [E]	Skewed or outlier-heavy data wants the median; symmetric data, the mean.
Mean Absolute Deviation [M]	MAD is the average distance of the data from its own mean.
Variance of a Data Set [M]	Square each deviation from the mean, then average the squares.
The Five-Number Summary [M]	Min, Q1, median, Q3, max - five landmarks that sketch a whole data set.
Range, IQR & the Outlier Fence [M]	The IQR measures the spread of the middle half - and builds the outlier fence.
Reading Box Plots [M]	Each piece of a box plot - whisker, half-box, half-box, whisker - holds about 25% of the data.
Two-Way Relative Frequency [M]	Percent of what? The denominator - a row, a column, or everyone - changes the answer.
Interpreting a Fitted Line [M]	Slope is the predicted change per unit of x; the intercept is the prediction at x = 0.
Residuals [M]	A residual is actual minus predicted - the line's miss at one point.
Correlation vs. Causation [E]	Association alone never proves cause - look for lurking variables or a randomized experiment.

Algebra I: Literal Equations & Modeling · 10 topics

Solving a Literal Equation [M]	Isolate a variable using letters instead of numbers.
Rearranging Formulas [M]	Solve a familiar formula for the piece you need.
Consecutive-Integer Problems [M]	Name them x, x+1, x+2 and solve.
Coin & Mixture Problems [M]	One equation for count, one for value.
Percent-Mixture Setups [M]	The amount of an ingredient is percent times total.
Rate-Time-Distance [M]	Distance equals rate times time.
Two Movers [M]	Combine the speeds when they move apart or together.
Geometry Setup Problems [M]	Translate a geometry description into one equation.
Unit Analysis [E]	Multiply by a conversion factor to change units.
Piecewise Cost Models [M]	Different rules apply in different ranges.

Algebra I: Polynomials & Factoring in Depth · 12 topics

Polynomial Sums: Hunting a Coefficient [E]	Add or subtract polynomials and report one requested coefficient.
Monomial Times a Polynomial [E]	Distribute a single term across a trinomial and read off a coefficient.
Binomial Products: Any Term You Like [M]	Expand (ax+b)(cx+d) and pick out the leading, middle, or constant term.
Binomial Times a Trinomial [M]	Expand a binomial against a trinomial and locate a single term.
Squaring a Binomial [M]	Apply (ax±b)² = a²x² ± 2abx + b² and read a coefficient.
The Sum-Times-Difference Product [M]	Recognize (ax+b)(ax−b) = a²x² − b² - the middle term vanishes.
GCF Factoring: the Leftover Trinomial [M]	Pull out the greatest common factor and inspect the quotient's coefficients.
Factoring x² + bx + c to a Factor [M]	Split a monic trinomial and hand back one exact binomial factor.
Factoring by Grouping [H]	Group a four-term polynomial and extract the shared binomial factor.
Factoring a Difference of Squares [M]	Reverse a²x² − b² into the sum and difference of its square roots.
Perfect-Square Trinomials [M]	Recognize a²x² ± 2abx + b² and write it as a single squared binomial.
Factoring ax² + bx + c with a > 1 [H]	Factor a trinomial with a leading coefficient above 1 and return a factor.

Algebra I: Quadratic Functions & Graphs · 13 topics

Evaluating a Quadratic Function [E]	Substitute a number for x and simplify to get the function's output.
The Vertex: x = −b/(2a) [M]	The turning point's x-coordinate comes straight from a and b.
The Vertex y-value [M]	Plug the vertex's x back into the function to get its y.
The Axis of Symmetry [E]	The vertical mirror line runs midway between the two x-intercepts.
The y-Intercept of a Quadratic [E]	A parabola meets the y-axis at its constant term c.
x-Intercepts by Factoring [M]	The graph crosses the x-axis where each factor is zero.
Maximum or Minimum Value [M]	The vertex's y-value is the largest or smallest output the function reaches.
Does It Open Up or Down? [E]	The sign of the leading coefficient decides maximum versus minimum.
Transformations of y = x² [M]	Shifting y = x² moves the vertex from the origin by the shift amounts.
Reading a Shift from Vertex Form [M]	Vertex form spells out exactly how the parent parabola was moved.
Projectile Motion: Time of Peak [M]	A launched object peaks at the vertex time t = v/32.
Projectile Motion: Height at a Time [M]	Evaluate the height model at a given instant.
Comparing Two Parabolas [M]	Read vertex form to compare position, width, and extreme value.

Algebra I: Functions, Domain & Radicals · 12 topics

Evaluating f(x) [E]	f(x) names a rule; f(c) means substitute c for every x and simplify.
Solving f(x) = k [E]	Set the rule equal to the output value and solve the linear equation for x.
Interpreting f(a) = b in Context [E]	f(a) = b says the input a yields the output b - mind which is which.
Reading Function Tables [E]	A table pairs inputs with outputs; a constant step reveals the missing value.
Rate of Change from a Table [M]	Average rate of change is the change in output over the change in input.
Domain & Range of Ordered Pairs [M]	The domain is the set of x-values; the range is the set of y-values.
Domain from a Restriction [M]	Division by zero and negative radicands carve values out of the domain.
Is It a Function? [E]	A relation is a function only if each input maps to exactly one output.
Simplifying Square Roots [M]	Pull the largest perfect-square factor out front: √(k²·m) = k√m.
Adding Like Radicals [M]	Like radicals combine like terms: a√m + b√m = (a + b)√m.
Multiplying Square Roots [M]	Combine under one root - √a · √b = √(ab) - then simplify what results.
Rational Exponents [M]	A fractional exponent is a root: x^(p/q) is the q-th root of x, to the p.

Algebra I: Sequences, Data & Probability · 13 topics

Arithmetic Sequences: Any Term [M]	Reach a distant term, find a term's position, or recover the common difference.
Modeling with Arithmetic Sequences [M]	Fixed steps up or down are arithmetic - translate the story into a₁ and d.
Geometric Sequences: Any Term [M]	Multiply, don't add: the nth term uses a whole-number ratio raised to n − 1.
Modeling with Geometric Sequences [M]	Repeated multiplying - doubling, tripling - is geometric growth.
Recursive vs. Explicit Rules [M]	A recursive rule leans on the previous term; an explicit rule jumps straight to term n.
Building Terms from a Recursive Rule [M]	March one term at a time - even when no explicit shortcut exists.
Joint Relative Frequency [M]	A joint relative frequency is one inner cell divided by the grand total.
Marginal Relative Frequency [M]	A marginal relative frequency uses a whole row or column total over the grand total.
Theoretical Probability of Simple Events [M]	Favorable outcomes over equally likely total outcomes, reduced to lowest terms.
Probability of Compound Events [M]	Multiply for independent 'and'; add for non-overlapping 'or'.
Experimental Probability from Data [M]	Count what actually happened over the number of trials - then predict.
Using a Data Display [E]	Read the graph, then combine the values the question actually asks about.
Correlation vs. Causation [E]	Association is not proof of cause - hunt for a lurking variable or a random assignment.

Algebra I - Deep II · 42 topics

Point-Slope Form [H]	One point plus the slope pins down a line: y - y1 = m(x - x1).
Standard Form to Slope-Intercept [H]	Solve Ax + By = C for y to expose the slope and the y-intercept.
Slope-Intercept to Standard Form [H]	Move the x-term next to y and clear any fractions to get Ax + By = C.
Intercepts from Standard Form [H]	Set y = 0 for the x-intercept and x = 0 for the y-intercept.
Parallel & Perpendicular Slopes [H]	Parallel lines share a slope; perpendicular slopes are negative reciprocals.
A Line Through a Point, Parallel or Perpendicular [H]	Borrow the slope (or its negative reciprocal), then solve for b.
A Linear Rule from a Table [H]	The slope is change in y over change in x; then back up to find b.
Linear Cost Models & Break-Even [H]	Cost = fixed fee + rate times amount; set two plans equal to compare.
Slope from a Graph [H]	Read rise over run between two labeled points, sign included.
Elimination with Multiplication [H]	Scale one or both equations so a variable's coefficients cancel.
One, None, or Infinitely Many? [H]	Compare slopes and intercepts: the three cases of a linear system.
Forcing a Special System [H]	Match the coefficient ratios to make a system dependent or inconsistent.
Wind & Current Problems [H]	With and against speeds are s + c and s - c: a built-in system.
Substitution with y Already Solved [H]	Drop the expression for y straight into the other equation.
An Exponential from Two Points [H]	Divide function values to isolate a power of b, then peel back to a.
When Exponential Overtakes Linear [H]	Doubling eventually beats any steady rate: find the exact crossover.
Repeated Fractional Decay [H]	Losing 1/3 each step means multiplying by 2/3 each step.
Linear or Exponential? Reading a Table [H]	Equal differences say linear; equal ratios say exponential.
Quadratics with Rational Roots [H]	Factor or use the formula; a leading coefficient makes fraction roots.
Converting to Vertex Form [H]	Complete the square: half the x-coefficient, square it, rebalance.
Writing a Quadratic from Its Zeros [H]	Zeros p and q give factors (x - p)(x - q); a y-intercept fixes the stretch.
Projectile Motion: Landing Time [H]	Set h = 0 and factor; keep only the positive time.
Area Problems that Become Quadratics [H]	Let w be the width, write length in terms of w, and set w times length equal to the area.
Factoring Completely [H]	Pull the GCF first, then factor what remains all the way down.
Partner Points Across the Axis [H]	Equal heights sit symmetrically: the partner of x is 2h - x.
Radicals as Rational Exponents [H]	The root index goes in the denominator; the power goes in the numerator.
Evaluating Rational Exponents [H]	Root first, then power; a negative exponent means reciprocal.
Rationalizing a Denominator [H]	Multiply top and bottom by the radical to clear it from below.
Radical Equations & Extraneous Roots [H]	Squaring can invent solutions: always check each root in the original.
An Arithmetic Rule from Two Terms [H]	The difference of two terms spread over their index gap gives d.
A Geometric Rule from Two Terms [H]	The ratio of terms two apart is r squared; take the positive root.
Arithmetic, Geometric, or Neither? [H]	Test differences first, then ratios; both must hold at EVERY step.
Two-Step Absolute Value Equations [H]	Isolate the absolute value first, then split into two cases.
Absolute Value as Tolerance [H]	|x - c| <= t means: within t of the target c, both directions.
The Vertex of an Absolute Value Graph [H]	y = a|x - h| + k is a V with corner (h, k); the sign of a points it.
Nested Function Evaluation [H]	Work inside out: evaluate the inner function, feed it to the outer.
A Domain that Fits the Context [H]	The story limits the inputs: counts are whole, measures are real, both are bounded.
Range over a Finite Domain [H]	Evaluate at every allowed input; the range is those outputs.
Domain & Range from a Graph [H]	Domain is the shadow on the x-axis; range is the shadow on the y-axis.
Predicting with a Line of Fit [H]	Find the fit line's slope from two points, then ride it to the target.
Comparing Correlation Strength [M]	Strength lives in |r|; the sign only gives the direction.
The x-Intercept of a Line [H]	Set y = 0 and solve; from two points, find the slope first.

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 I - Linear Equations, Inequalities & Systems (Deep) · 44 topics

Clearing an Equation of Fractions [H]	Multiplying every term by the least common denominator turns a fractional equation into an integer one.
Fractional Coefficients on Both Sides [H]	Collect fractional x-terms on one side by subtracting the smaller coefficient, or clear all denominators first.
Equations with Decimal Coefficients [H]	Multiplying every term by a power of ten clears decimals the way an LCD clears fractions.
Distributing a Fraction over a Binomial [H]	A fraction in front of parentheses multiplies BOTH terms inside before any collecting happens.
Equations with Nested Grouping [H]	Simplify the innermost grouping first, then distribute the outer factor across every term it produced.
Proportions with Binomial Parts [H]	Cross-multiplying a proportion multiplies each numerator by the other denominator, parentheses included.
One Solution, None, or Every Number? [H]	Compare the simplified x-coefficients: unequal gives one solution, equal gives none or all depending on the constants.
The Coefficient That Removes Every Solution [H]	An equation is contradictory exactly when the x-terms match but the constants do not.
The Constant That Makes an Identity [H]	An identity needs the simplified sides to agree in the x-term and in the constant term.
A Literal Equation with x on Both Sides [H]	Gather every term containing the target variable on one side, factor it out, and divide by the resulting factor.
Rearranging a Formula That Has a Fraction [H]	Clear the fraction by multiplying both sides by its denominator before the target variable is isolated.
Solving for a Variable in a Denominator [H]	Multiply both sides by the denominator first; if the target then appears twice, factor it out.
Rearrange First, Then Substitute [H]	Isolating the wanted variable symbolically before substituting keeps the arithmetic to a single evaluation.
Finding the First Incorrect Step [H]	A rearrangement step is valid only when the same operation is applied to every term on both sides.
A Three-Part Inequality That Reverses [H]	Dividing all three parts of a sandwich inequality by a negative number reverses both inequality signs.
When Every Part Contains the Variable [H]	A sandwich with x in all three parts must be split into two separate inequalities and the two solution sets intersected.
OR Statements That Need a Sign Flip [H]	Each piece of an OR statement is solved on its own, and a negative coefficient reverses only that piece's sign.
Sandwich Inequalities with Fractional Bounds [H]	Multiplying all three parts by a positive denominator clears a fraction without disturbing either inequality sign.
Compound Statements with No or Every Solution [H]	An AND of disjoint pieces has no solution, while an OR of overlapping pieces admits every real number.
Isolating an Absolute Value with a Coefficient on x [H]	Undo everything outside the bars first, then split into two cases and divide by the coefficient of x.
Counting the Solutions of an Absolute Value Equation [H]	Once the bars are alone, a positive right side gives two solutions, zero gives one, and a negative right side gives none.
Absolute Value on Both Sides [H]	Two absolute values are equal when the insides are equal or exact opposites, giving two equations to solve.
Absolute Value Equations with a Variable Outside the Bars [H]	A candidate is genuine only if it makes the expression outside the bars greater than or equal to zero.
Absolute Value Inequalities with a Coefficient on x [H]	A less-than absolute value inequality becomes the sandwich from negative t to t, which is then solved for x in all three parts.
Absolute Value Inequalities That Are Always or Never True [H]	Comparing an absolute value to a negative number settles the inequality without any casework.
Finding a System's Solution in a Table [H]	The solution of a system is the input at which both equations return the same output.
Where Two Graphed Lines Cross [H]	Graphing solves a system by locating the single point that lies on both lines.
Substitution After Isolating a Variable [H]	Isolate the variable whose coefficient is 1 or -1, then substitute that expression into the other equation.
Systems Whose Solution Is Not an Integer [H]	Elimination produces the exact fractional coordinates when no coefficient divides evenly.
Elimination After Clearing Fractions and Decimals [H]	Scaling each equation by its own least common denominator makes the coefficients integers before elimination begins.
Choosing an Efficient Method [H]	The coefficients decide the method: an isolated variable favors substitution, matching or opposite coefficients favor elimination.
Answering a System with One Combination [H]	Adding or subtracting two equations can deliver the requested combination of x and y without solving for either one.
Recovering a Coefficient from a Known Solution [H]	A known solution turns an unknown coefficient into a one-variable equation by substitution.
What Elimination Leaves Behind [H]	When elimination erases both variables, a true remainder means infinitely many solutions and a false one means none.
A Second Point on a Dependent System [H]	A dependent system's solutions are the points of one shared line, so either coordinate determines the other.
Mixing Two Solutions to a Target Strength [H]	A mixture gives two equations: total volume and total amount of the dissolved substance.
A Trip Split Between Two Speeds [H]	Total time and total distance give two equations in the two unknown times.
Splitting Money Between Two Rates [H]	The principals add to the total invested while rate times principal adds to the total interest.
Two Prices Recovered from Two Sales [H]	Two purchases of the same two items give two equations whose unknowns are the prices, not the counts.
The Break-Even Count [H]	Break-even is the number of units at which the revenue equation and the cost equation give the same total.
A Corner of the Feasible Region [H]	Each corner of a feasible region is the intersection of two boundary lines, found by solving them as a system of equations.
The Best Corner of a Feasible Region [H]	A linear objective on a bounded feasible region attains its extreme value at one of the corners, so every corner must be tested.
Counting Whole-Number Points in a Region [H]	Counting lattice points in a feasible region means counting, for each allowed x, the whole-number values of y that fit.
Boundary Style and the Shaded Side [H]	A strict inequality draws a dashed boundary, and the shaded side is decided after the inequality is solved for y.

Algebra I - Functions, Sequences & Exponential Models (Deep) · 46 topics

Evaluating a Function at an Expression [H]	Substituting an expression for x works exactly as substituting a number: replace every x, then simplify.
The Rule for a Composition [H]	To build f(g(x)), substitute the whole rule for g in place of every x in f.
Order Inside a Composition [H]	Composition is not commutative: f(g(a)) and g(f(a)) are usually different numbers.
Solving an Equation Built from a Composition [H]	Undo a composition from the outside in: strip the outer function first, then the inner one.
Composition from Two Tables [H]	Read the inner output from its column, then look that number up as the next input.
Net Change Written in Function Notation [H]	The expression f(b) - f(a) is the total change in output between the two inputs.
Chaining Two Models in Context [H]	When one model's output is the next model's input, compose them and keep the units in order.
Domain and Range as Intervals from a Graph [H]	The domain is the span of x-values covered; the range is the span of y-values reached.
Counting the Range from a Table [H]	The range is the SET of outputs, so a repeated output is listed only once.
Output Extremes over a Restricted Domain [H]	On a restricted domain the extreme outputs come from testing the whole allowed input set.
Where a Graph Lies Above or Below the Axis [H]	A graph is above the x-axis exactly where its output is positive, and below it where the output is negative.
The Range of an Absolute Value Function [H]	An absolute value graph turns at its vertex, so its range starts or stops at the vertex y-value.
Average Rate of Change from a Table [H]	Average rate of change is the change in output divided by the change in input over the same interval.
Comparing Average Rates over Several Intervals [H]	Compare intervals by computing every rate, since a large change over a long interval can still be slow.
Interpreting an Average Rate of Change [H]	An average rate of change carries output units per input unit and says how fast the output changed on average.
Recovering a Value from an Average Rate [H]	Change in output equals the average rate times the change in input, so a known rate recovers a missing value.
Evaluating a Piecewise Function [H]	Choose the piece whose condition the input satisfies, then evaluate only that piece.
Which Piece Owns the Boundary [H]	At a boundary input, only the piece whose inequality includes that number applies.
Solving a Piecewise Equation [H]	Solve each piece separately and keep only the roots that lie inside that piece's own condition.
Three-Tier Rate Models [H]	In a tiered rate each block of usage is charged at its own price, so the total is a sum of completed tiers plus the partial one.
Step Functions in Context [H]	A step charge rounds a partial period up to a whole one, so the cost jumps at each boundary.
Reading a Value off a Piecewise Graph [H]	On a piecewise linear graph, find the segment containing the input and follow that segment's constant slope.
Describing a Transformation of y = |x| [H]	In y = a|x - h| + k the value h shifts the graph horizontally, k shifts it vertically, and a stretches or reflects it.
Writing an Absolute Value Rule from a Description [H]	Translate each described move into its parameter: horizontal shifts change h, vertical shifts change k, stretches and reflections change a.
Evaluating a Transformed Absolute Value Rule [H]	Evaluate the inside of the bars first, take the absolute value, then apply the outside multiplier and constant.
The Image of a Point Under a Transformation [H]	Under y = a|x - h| + k a point (p, q) of the parent graph moves to (p + h, aq + k).
x-Intercepts of an Absolute Value Graph [H]	Setting a|x - h| + k = 0 gives |x - h| = -k/a, which has two, one, or no solutions depending on that value's sign.
Shifting a Line Horizontally and Vertically [H]	Replacing x by x - h slides a line right h units, which changes its intercept but never its slope.
Reflecting a Line Across an Axis [H]	The rule -f(x) reflects a graph across the x-axis, while f(-x) reflects it across the y-axis.
From an Explicit Rule to a Recursive Rule [H]	An explicit rule names its first term and its step, which are exactly the two parts of the recursive rule.
A Distant Term from a Recursive Rule [H]	Convert a recursive rule to its explicit form to jump straight to a distant term.
Which Term Has a Given Value [H]	Setting the nth-term rule equal to a value and solving for n identifies the term's position.
Geometric Sequences with a Fractional Ratio [H]	A common ratio between 0 and 1 shrinks the terms, and the nth term is the first term times the ratio to the (n - 1) power.
Sequences with an Alternating Sign [H]	A negative common ratio flips the sign at every step, so odd and even positions carry opposite signs.
Inserting the Missing Terms Between Two Terms [H]	Count the gaps between the known terms: an arithmetic sequence splits the difference evenly, a geometric one splits the ratio into equal factors.
Writing a Recursive Rule from a List [H]	Test consecutive terms for a common difference or a common ratio, then state the first term with the operation that produces the next.
Writing an Exponential Model from a Story [H]	An exponential model is y = a b^t, where a is the starting amount and b is the factor applied once per time period.
Interpreting the Initial Value and the Growth Factor [H]	In y = a b^t the coefficient a is the amount at time zero and b is the fraction of the previous amount kept each period.
Recovering the Initial Value of a Model [H]	Since a later value equals a times b^t, dividing that value by b^t recovers the initial amount a.
Percent Change over Several Periods [H]	Percent changes compound rather than add, so multiply the factors and convert the product back to a percent.
Models Whose Period Is Not One Time Unit [H]	When a factor applies once every k time units, the number of factors is the elapsed time divided by k.
Comparing Several Models at One Instant [H]	At a specified time the winner is decided by evaluating every model, since a larger base can still trail a larger starting amount.
The First Whole Period Past a Threshold [H]	Tabulate the model period by period and stop at the first whole time value that crosses the threshold.
Average Rates: Linear against Exponential [H]	A linear function has the same average rate of change on every interval, while an exponential function's rate grows as the interval moves right.
The Gap Between Linear and Exponential Growth [H]	Evaluate both models at the same time and subtract to measure how far apart linear and exponential growth have moved.
Choosing Between a Linear and an Exponential Model [H]	Repeated addition of a fixed amount is linear; repeated multiplication by a fixed factor is exponential.

Algebra I - Polynomials, Factoring & Quadratic Functions (Deep) · 46 topics

Degree and Standard Form [H]	The degree is the largest exponent, or in two variables the largest sum of exponents in a single term.
Subtracting a Polynomial in Full [H]	A subtraction sign in front of a polynomial changes the sign of every term inside it, not only the first.
Perimeter and Area as Polynomials [H]	Adding polynomial side lengths gives a perimeter; multiplying two of them gives an area.
Expanding Three Binomials [H]	Multiply two binomials first, then distribute the third across the resulting trinomial.
Squaring a Two-Variable Binomial [H]	(ax + by)squared equals a squared x squared plus 2abxy plus b squared y squared, so the middle term is twice the product of the two parts.
Difference of Squares in Arithmetic [H]	Two numbers equally far from a round number multiply to that round number squared minus the offset squared.
Differences of Squares, Higher Powers [H]	Any expression of the form A squared minus B squared factors as (A - B)(A + B), even when A or B carries a power or a second variable.
Telling Special Products Apart [H]	A sum times a difference loses its middle term, while a squared binomial keeps a middle term equal to twice the product of its parts.
The Missing Middle Term [H]	In a perfect-square trinomial the middle coefficient is twice the product of the square roots of the two end terms.
The GCF of Several Monomials [H]	The GCF of monomials pairs the greatest common divisor of the coefficients with the smallest exponent each variable carries.
Factoring Out a Negative Common Factor [H]	Pulling out a negative common factor changes the sign of every term that remains inside the parentheses.
A Common Binomial Factor [H]	A whole binomial can serve as the common factor, and reversing the order inside a binomial costs one factor of -1.
A Missing Dimension from an Area [H]	Dividing a polynomial area by one side length means factoring that side out of the area.
Factoring Trinomials in Two Variables [H]	A trinomial in x and y factors by the same reasoning as one in x alone, with y attached to the constant part of each binomial.
The ac-Method: Splitting the Middle Term [H]	To factor ax squared plus bx plus c, find two integers whose product is ac and whose sum is b, then use them to split the middle term.
Factorable or Prime? [H]	A trinomial splits into binomials with integer coefficients only when b squared minus 4ac is a perfect square.
Constants That Make a Trinomial Factor [H]	x squared plus bx plus c factors over the integers exactly when b and c come from a pair of integers with sum b and product c.
Factoring a Negative Leading Term [H]	Factor -1 (with any common factor) out of a trinomial that leads with a negative term, then factor the positive trinomial that remains.
Quartics in Quadratic Form [H]	A polynomial in x to the fourth, x squared and a constant factors like a trinomial in x squared, and each resulting factor may split again.
Grouping After Rearranging [H]	Four terms can be reordered before grouping so that each pair shares a common factor.
Grouping with a Negative Second Pair [H]	When the third and fourth terms are negative, factor a negative number out of the second pair so both pairs leave the same binomial.
Solutions from a Factored Equation [H]	A product equals zero exactly when one of its factors equals zero, and a constant factor contributes no solution.
Set It Equal to Zero First [H]	The zero-product property applies only after every term is moved to one side so the other side is zero.
Quadratics with Zero as a Solution [H]	When a quadratic has no constant term, factoring out x shows that zero is one of its two solutions.
Cubics Solved by Factoring [H]	A cubic that factors into three linear factors has one solution for each factor, found by the zero-product property.
Sum and Product of the Roots [H]	For ax squared plus bx plus c equal to zero, the solutions add to -b/a and multiply to c/a.
Solving a Squared Binomial [H]	Isolate the squared binomial, take both square roots, and then solve the two resulting linear equations.
Counting Solutions Before Solving [H]	Once a squared binomial is isolated, the sign of the number it equals decides whether there are two, one, or no real solutions.
The Number That Completes the Square [H]	x squared plus bx becomes a perfect square when (b/2) squared is added.
The Rewritten Completing-Square Step [H]	Completing the square rewrites a quadratic equation as (x + h) squared equal to k, with h half the x-coefficient and k whatever the balancing leaves.
Completing the Square with a Stretch [H]	When the leading coefficient is not 1, factor it out of the x-terms before completing the square inside the parentheses.
Exact Roots from the Formula [H]	The quadratic formula returns exact roots, which are simplified by reducing the radical and then dividing out any common factor.
Rational, Irrational, or Non-Real [H]	A positive perfect-square discriminant gives two rational solutions, a positive non-square gives two irrational ones, zero gives one repeated solution, and a negative gives none.
Forcing a Repeated Solution [H]	A quadratic has exactly one real solution when its discriminant is zero, which is one equation in the unknown coefficient.
Choosing a Solution Method [H]	The structure of a quadratic decides the efficient method: no linear term invites square roots, a perfect-square discriminant invites factoring, and anything else needs the formula.
Vertex Form from a Vertex and a Point [H]	The vertex fixes h and k, and one more point on the parabola determines the stretch factor a.
An Axis from Two Equal Outputs [H]	Two inputs with the same output sit symmetrically about the axis, so the axis is their midpoint.
The Vertex of Intercept Form [H]	In f(x) = a(x - p)(x - q) the vertex sits above or below the midpoint of the two x-intercepts.
The Range of a Quadratic Function [H]	A parabola's range starts at its vertex y-value and runs upward when it opens up or downward when it opens down.
Increasing and Decreasing Intervals [H]	A parabola changes direction only at its vertex, so an interval that avoids the vertex is entirely increasing or entirely decreasing.
Average Rate of Change of a Quadratic [H]	The average rate of change over an interval is the slope of the line joining the two endpoints of the graph.
Vertex Form Back to Standard Form [H]	Expanding a(x - h) squared plus k gives ax squared minus 2ahx plus ah squared plus k.
Projectile Motion: Maximum Height [H]	A projectile's greatest height is the y-value of the vertex of its height model, reached at time t = v/32.
Reaching a Given Height [H]	Setting a height model equal to a given height gives a quadratic whose two solutions are the times going up and coming down.
Fencing That Maximizes Area [H]	Writing area as a quadratic in one dimension turns a fencing problem into a vertex problem.
A Uniform Border Around a Rectangle [H]	A border of uniform width adds twice that width to each dimension, so the total area is a quadratic in the width.

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.

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 · 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.

Quadratics & Polynomials · 13 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.
The Discriminant	b² − 4ac tells you how many real solutions exist.
Vertex of a Parabola	The turning point at x = −b/2a.
Graphs of Quadratics	Intercepts and symmetry of a parabola.
Quadratic Models	Projectile motion and other parabolic models.

Radicals & Exponentials · 8 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.
Operations with Radicals	Adding like radicals and multiplying square roots.
Rational Exponents	Fractional exponents are roots: x^(p/q) is the q-th root of x, raised to the p.
Radical Equations	Isolate the radical, then square both sides.
Exponential Growth & Decay	Quantities that multiply by the same factor each time step: y = a·bᵗ.

Functions & Algebra II · 2 topics

Function Notation & Evaluation	Reading f(x) notation and plugging in inputs.
Transformations of Functions	How f(x − h) + k slides a graph around the plane.

Ratios, Data & Geometry (Middle School) · 9 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.
Reading Data Displays	Pulling answers out of dot plots, tables, and bar graphs.
Probability Basics	Favorable outcomes over total outcomes.
Compound Probability	Independent events multiply.

Grade 7: Proportions, Geometry & Statistics · 1 topics

Two-Step Inequalities

Solve like an equation; flip when you multiply by a negative.

Grade 8: Functions, Exponents & Geometry · 3 topics

Transformations	Translations slide, reflections flip, rotations turn.
Dilations & Similarity	Dilations scale from a center; similar figures share shape.
Scatter Plots & Association	Positive, negative, or no association between two variables.

Geometry: Congruence, Triangles & Circles · 12 topics

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

Algebra I: Exponential Functions · 6 topics

Evaluating Exponential Functions	Plug integer inputs into f(x) = a·bˣ - including 0 and negatives.
Growth Rates & Growth Factors	Growing r% per step means multiplying by 1 + r/100 each step.
Decay Rates & Decay Factors	Losing r% per step means multiplying by 1 − r/100 each step.
Modeling Percent Growth & Decay	Turn a percent rate into a factor, then multiply once per time step.
Doubling Contexts	Doubling every period is y = a·2ᵗ with t counting periods, not hours.
Linear vs. Exponential Growth	A linear head start never survives repeated doubling.

Geometry: Right-Triangle Trigonometry · 12 topics

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

Geometry: Transformations & Symmetry · 11 topics

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

Geometry: Solids, Cross-Sections & Modeling · 9 topics

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

Geometry: Constructions & Loci · 7 topics

Copying a Segment	A compass transfers a length exactly, so a copied segment matches the original.
Copying an Angle	Transferring an angle's arc and chord reproduces its measure exactly.
Bisecting a Segment	Equal arcs from both endpoints locate the midpoint and the perpendicular bisector.
Bisecting an Angle	The angle bisector cuts an angle into two congruent halves.
The Perpendicular Bisector Property	A point is equidistant from two endpoints exactly when it lies on their perpendicular bisector.
Loci: Sets of Equidistant Points	A locus is the full set of points meeting a distance condition - a bisector, a circle, or parallels.
Points of Concurrency	The centroid cuts each median 2:1, and the circumcenter is equidistant from all three vertices.

Geometry: Coordinate Geometry in Depth · 12 topics

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

Geometry: Arc Length, Sectors & Radians · 5 topics

What Fraction of the Circle?	A central angle takes the fraction theta/360 of the whole circle.
Arc Length as a Piece of the Circumference	Arc length is (theta/360) of the circumference 2*pi*r.
Sector Area as a Slice of the Circle	Sector area is (theta/360) of the circle area pi*r^2.
Inscribed & Central Angles on One Arc	The central angle equals its arc; the inscribed angle is half of it.
The Tangent-Chord Angle	A tangent-chord angle is half the arc it cuts off.

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