PSAT Math

287 core topics + 63 prerequisite topics taught as needed · approximately 110 hours of instruction including spaced review · premium unlock $19.99

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

SAT & ACT Test Prep · 8 topics

Linear Equations on the Test [M]	Fee-plus-rate problems: read off slope and intercept.
Ratios, Rates & Percents on the Test [M]	Scale a ratio; apply a percent; divide for a unit rate.
Systems on the Test [M]	Two totals, two unknowns - count and value.
Quadratics & Functions on the Test [M]	Factor fast; evaluate compositions inside-out.
Exponential Models on the Test [M]	Repeated multiplication: y = a·bᵗ.
Geometry on the Test [M]	Pythagorean triples and area-perimeter conversions.
Data Analysis & Tables [M]	Weighted averages and table percentages.
Statistics on the Test [M]	Means shift predictably; medians come from expanded frequency tables.

SAT/ACT Math: Advanced Practice A · 50 topics

Linear Equations [H]	Authentic Digital-SAT / ACT exam-style practice.
Linear Word Problems [H]	Authentic Digital-SAT / ACT exam-style practice.
Linear Inequalities [H]	Authentic Digital-SAT / ACT exam-style practice.
Systems of Equations [H]	Authentic Digital-SAT / ACT exam-style practice.
Systems: Word Problems [H]	Authentic Digital-SAT / ACT exam-style practice.
Slope of a Line [H]	Authentic Digital-SAT / ACT exam-style practice.
Equation of a Line [H]	Authentic Digital-SAT / ACT exam-style practice.
Parallel & Perpendicular Slopes [H]	Authentic Digital-SAT / ACT exam-style practice.
Interpreting Linear Models [H]	Authentic Digital-SAT / ACT exam-style practice.
Absolute-Value Equations [H]	Authentic Digital-SAT / ACT exam-style practice.
Percent [H]	Authentic Digital-SAT / ACT exam-style practice.
Percent Increase & Decrease [H]	Authentic Digital-SAT / ACT exam-style practice.
Ratios & Proportions [H]	Authentic Digital-SAT / ACT exam-style practice.
Rates & Unit Rates [H]	Authentic Digital-SAT / ACT exam-style practice.
Basic Probability [H]	Authentic Digital-SAT / ACT exam-style practice.
Two-Way Tables [H]	Authentic Digital-SAT / ACT exam-style practice.
Mean of a Data Set [H]	Authentic Digital-SAT / ACT exam-style practice.
Median of a Data Set [H]	Authentic Digital-SAT / ACT exam-style practice.
Scatterplots & Line of Best Fit [H]	Authentic Digital-SAT / ACT exam-style practice.
Exponential Growth [H]	Authentic Digital-SAT / ACT exam-style practice.
Solving Quadratics [H]	Authentic Digital-SAT / ACT exam-style practice.
Parabola Vertex & Minimum [H]	Authentic Digital-SAT / ACT exam-style practice.
Function Notation [H]	Authentic Digital-SAT / ACT exam-style practice.
Composition of Functions [H]	Authentic Digital-SAT / ACT exam-style practice.
Expanding Polynomials [H]	Authentic Digital-SAT / ACT exam-style practice.
Exponent Rules [H]	Authentic Digital-SAT / ACT exam-style practice.
Rational Expressions [H]	Authentic Digital-SAT / ACT exam-style practice.
The Discriminant [H]	Authentic Digital-SAT / ACT exam-style practice.
Exponential Functions [H]	Authentic Digital-SAT / ACT exam-style practice.
Angle Relationships [H]	Authentic Digital-SAT / ACT exam-style practice.
Area of a Triangle [H]	Authentic Digital-SAT / ACT exam-style practice.
The Pythagorean Theorem [H]	Authentic Digital-SAT / ACT exam-style practice.
Similar Triangles [H]	Authentic Digital-SAT / ACT exam-style practice.
Circle Area & Circumference [H]	Authentic Digital-SAT / ACT exam-style practice.
Arc Length [H]	Authentic Digital-SAT / ACT exam-style practice.
Right-Triangle Trig Ratios [H]	Authentic Digital-SAT / ACT exam-style practice.
Solving with Trig Ratios [H]	Authentic Digital-SAT / ACT exam-style practice.
Volume of Solids [H]	Authentic Digital-SAT / ACT exam-style practice.
Distance in the Plane [H]	Authentic Digital-SAT / ACT exam-style practice.
Adding Complex Numbers [H]	Authentic Digital-SAT / ACT exam-style practice.
Multiplying Complex Numbers [H]	Authentic Digital-SAT / ACT exam-style practice.
Powers of i [H]	Authentic Digital-SAT / ACT exam-style practice.
Modulus of a Complex Number [H]	Authentic Digital-SAT / ACT exam-style practice.
Literal Equations [H]	Authentic Digital-SAT / ACT exam-style practice.
Average Rate [H]	Authentic Digital-SAT / ACT exam-style practice.
Compound Probability [H]	Authentic Digital-SAT / ACT exam-style practice.
Range of a Data Set [H]	Authentic Digital-SAT / ACT exam-style practice.
Quadratic Word Problems [H]	Authentic Digital-SAT / ACT exam-style practice.
Arithmetic & Geometric Sequences [H]	Authentic Digital-SAT / ACT exam-style practice.
Midpoint of a Segment [H]	Authentic Digital-SAT / ACT exam-style practice.

SAT/ACT Math: Advanced Practice B · 50 topics

Linear Equations with Fractions [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Systems: Combined Values [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Compound Inequalities [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Linear Modeling [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
x- and y-Intercepts [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
One, None, or Infinite Solutions [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Systems with a Parameter [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Absolute-Value Inequalities [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Reading Linear Graphs [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Simple Interest [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Reverse Percent [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Dividing in a Ratio [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Unit Conversion [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Density & Rates [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Conditional Probability [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
How Changes Affect Statistics [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Comparing Spread [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Interpreting Scatterplots [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Weighted Averages [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Exponential Decay [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Vertex Form of a Parabola [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Sum & Product of Roots [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
The Remainder Theorem [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Simplifying Rational Expressions [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Radical Equations [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Exponential Equations [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Inverse Functions [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Function Transformations [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Nonlinear Systems [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Doubling & Compound Growth [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Parallel Lines & Transversals [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Special Right Triangles [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Equation of a Circle [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Inscribed & Central Angles [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Area of Similar Figures [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Volume of Spheres & Cones [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Surface Area [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Midpoint of a Segment [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Trigonometric Identities [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Radians & Degrees [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Dividing Complex Numbers [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Modulus of a Complex Number [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Complex Roots of Quadratics [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Sums of Powers of i [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Successive Percent Changes [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
At-Least-One Probability [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Arithmetic Series [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Geometric Series [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Solving Right Triangles with Trig [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).
Combined Work Rates [H]	Authentic Digital-SAT / ACT exam-style practice (advanced).

SAT & ACT - Advanced Problem Set · 37 topics

No-Solution Equations with a Parameter [H]	A linear equation has no solution when the x-terms match but the constants do not.
Identities: Infinitely Many Solutions [H]	Infinitely many solutions means both sides are the identical expression.
Blend & Mixture Systems [H]	Count equation plus value equation - solve the blend by elimination.
Perimeter Relations as Equations [H]	Translate the length-width relation, then solve the perimeter equation.
Budget Inequalities: Greatest Whole Number [H]	Subtract the fee, divide by the unit cost, and round DOWN.
Writing Absolute-Value Inequalities [H]	Within t of c means |x - c| <= t: center inside the bars, tolerance outside.
Distribute Both Sides, Fraction Answers [H]	Expand both sides fully, collect x, and accept a fractional answer.
Percent of a Percent [H]	Chain the percents: take the second percent of the first group, not of the total.
Matching Two Percents [H]	Set (a/100)m = (b/100)n and solve for the unknown amount.
Ratio Totals, Differences & Equalizing [H]	Find the value of one ratio part from the total, then answer the twist.
Hitting a Target Mean [H]	Work with totals: new total needed minus current total is the required score.
Removing a Value from a Mean [H]	Totals again: subtract the removed value from n x mean, divide by n - 1.
Mean vs Median under a Change [H]	The mean feels every change; the median only moves if the middle moves.
Interpreting a Margin of Error [H]	The margin of error brackets the population MEAN, not every individual.
Sampling, Assignment & Conclusions [H]	Random selection lets you generalize; random assignment lets you claim cause.
Two-Step Rate Conversions [H]	Convert the time unit and the distance unit one factor at a time.
Reading Exponential Models [H]	In a(b)^t, a is the start; b - 1 is the percent change per period.
Compounding a Percent Change [H]	Apply the growth factor once per period - never add the percents.
Linear or Exponential? [H]	Equal AMOUNTS per step are linear; equal FACTORS or percents are exponential.
One Real Solution: Solving for the Parameter [H]	Set the discriminant b^2 - 4ac equal to zero and solve for the constant.
Symmetric Functions of the Roots [H]	Build r^2 + s^2 and 1/r + 1/s from the root sum and product - no solving.
Function Notation Traps: f(2x) and Friends [H]	Find the x that makes the INPUT expression equal the requested value.
Composing with a Quadratic [H]	Evaluate the inner function first, then feed its output to the outer one.
Finding a Constant from a Zero [H]	A zero at x = k means p(k) = 0: substitute k and solve for the constant.
Exponent Rules in Structure Form [H]	Add exponents to multiply, subtract to divide, multiply for a power of a power.
Solving Simple Rational Equations [H]	Isolate the fraction, multiply both sides by the denominator, and solve.
Radical Equations & Extraneous Roots [H]	Square both sides, solve the quadratic, then CHECK - squaring invents roots.
Circle Equations: Completing the Square [H]	Halve each linear coefficient, square it, add to both sides, read off (h, k) and r.
Points on a Circle [H]	Substitute the known coordinate and solve the Pythagorean relationship.
Solving sin = cos Equations [H]	sin equals cos for acute angles exactly when the two angles sum to 90.
One Trig Ratio from Another [H]	Sketch the right triangle the given ratio describes; find the third side.
Angle of Elevation with Tangent [H]	Height = tan(angle) x horizontal distance; add the eye height if given.
Area of a Sector [H]	A sector is the fraction angle/360 of the full circle's area pi r^2.
Determinants with an Unknown Entry [H]	Set ad - bc equal to the given determinant and solve; scaling by k scales det by k^2.
Solving Simple Matrix Equations [H]	Matrix equations work entry by entry: subtract, then divide by the scalar.
Logarithm Quickies [H]	log_b(m) asks: b to what power gives m? Reciprocals give negative logs.
Adding & Subtracting Logarithms [H]	Log of a product adds; log of a quotient subtracts - then evaluate the power.

SAT & ACT - Algebra and Advanced Math Under Time · 48 topics

Two Plans: The Break-Even Number [H]	Set the two cost expressions equal; the solution is the break-even number of items.
Equations That Hide a Unit Conversion [H]	Convert so the rate and the total use the same units before solving.
Writing a System of Inequalities [H]	Each sentence of a constraint becomes one inequality with matching units.
Back-Solving from the Answer Choices [H]	When the choices are numbers, testing them can be faster than solving.
Discount and Fee Equations [H]	A percent off multiplies the list price by (100 - d)/100; undo it by dividing.
Counting the Whole Numbers a Constraint Allows [H]	Divide both bounds by the step, round the low end up and the high end down.
Scaling an Entire Equation [H]	If the target expression is a multiple of the given one, multiply the equation.
Systems in Context: Answer the Asked Quantity [H]	Solve the system, then compute the quantity the question actually names.
Systems That Never Agree [H]	Two linear models never agree when their rates match but their fees do not.
Systems That Are the Same Line [H]	Infinitely many solutions means one equation is a constant multiple of the other.
Reading a System's Solution from a Graph [H]	The solution of a system is the point where the two graphs cross.
Systems: Reading the Requested Expression [H]	Solve the system, then substitute into the expression the question names.
Predicted Change from a Slope [H]	The slope is the change in the output for each one-unit change in the input.
Interpreting a Line of Best Fit [H]	The intercept is the predicted output at input 0; the slope is the predicted change per unit.
Building a Linear Model from a Table [H]	Two rows give the slope; extend backward to the input value 0 for the intercept.
When Does the Model Reach a Value [H]	Set the model equal to the target value and solve for the input.
Recognizing a Perfect Square [H]	When the left side is a perfect square trinomial, take square roots instead of factoring.
Solving by Square Roots [H]	Isolate the squared factor, then take both square roots of each side.
Factoring When the Lead Coefficient Is Not 1 [H]	Factor out any common factor first, then split the lead coefficient across two binomials.
Reading the Radicand of the Quadratic Formula [H]	In the quadratic formula the number under the radical is b^2 - 4ac.
Maximum Value of a Quadratic Model [H]	A downward parabola peaks at its vertex, whose input is -b/(2a).
A Parabola's Vertex from Its Intercepts [H]	The vertex sits halfway between the x-intercepts; substitute that input for its value.
How Many Real Solutions [H]	The sign of b^2 - 4ac decides two, one, or no real solutions.
Parameter Ranges from the Discriminant [H]	Translate the wanted number of solutions into an inequality in the discriminant.
Coefficients in a Product of Polynomials [H]	Collect only the products of terms whose degrees add to the degree you need.
Factoring by Structure [H]	A difference of squares or a perfect square trinomial factors from its square roots.
Combining Rational Expressions [H]	Rewrite each fraction over the common denominator, then collect the numerator.
Quotient Plus Remainder Form [H]	Divide, then match coefficients: the quotient times the divisor plus the remainder rebuilds the numerator.
Distinct Real Zeros from Factored Form [H]	Each distinct factor contributes its own real zeros; a repeated factor still counts once.
Radical Equations: Which Roots Survive [H]	Squaring can create roots that fail the original equation, so test every candidate.
Rational Equations and Excluded Values [H]	A value that makes a denominator zero can never be a solution, even if the algebra produces it.
Equations with Rational Exponents [H]	Raise both sides to the reciprocal power to undo a rational exponent.
Half-Life Models [H]	Each half-life multiplies the amount by 1/2, so count the number of half-lives.
Choosing the Model for a Given Rate [H]	A percent change per period becomes a growth factor of 1 plus or minus the rate.
Solving an Exponential Model for the Time [H]	Divide off the initial amount, write both sides with the same base, then equate exponents.
An Exponential Pattern in a Table [H]	In an exponential table consecutive outputs have a constant ratio, not a constant difference.
Converting a Rate to Another Period [H]	The base applies over the period in the exponent's denominator; other periods need that base raised to a power.
Reading a Function from a Table [H]	A table gives inputs and outputs; a composition uses one row's output as the next row's input.
Reading Values from a Graph [H]	f(a) is the height of the graph above a; solving f(x) = c reads the graph the other way.
Solving an Equation with a Composition [H]	Work from the outside in: undo the outer function first, then solve the inner equation.
Piecewise-Defined Functions [H]	Choose the branch whose condition the input satisfies, then use only that rule.
What f(a) = b Says in Context [H]	In f(a) = b the input a and the output b carry the units the problem assigns them.
Writing the Equation of a Transformed Graph [H]	Horizontal shifts go inside the function and act opposite to their sign; vertical changes go outside.
Where a Point Goes Under a Transformation [H]	Track the input that produces the same output: inside changes move x, outside changes move y.
Transforming the Parent Parabola [H]	Write the shifted parabola in vertex form, then expand to read b and c.
A Line and a Parabola [H]	Set the two expressions equal to get one quadratic whose roots are the x-coordinates.
When a Line Meets a Parabola Once [H]	Exactly one solution means the combined quadratic has discriminant zero.
A Line and a Circle [H]	Compare the distance from the center to the line with the radius.

SAT & ACT - Problem Solving and Data Analysis · 48 topics

Scaling a Rate Across Workers and Time [H]	Reduce to one machine for one hour, then multiply by the new number of machines and hours.
Workers Times Time Is Constant [H]	The total work in worker-hours is fixed, so more workers means proportionally fewer hours.
Fuel Economy, Distance and Cost [H]	Convert miles to gallons with the mileage rate, then gallons to dollars with the price.
Unit Price Across Related Units [H]	Rewrite the price as a price per single unit, then multiply by the number of those units.
Converting Squared Units [H]	A length factor of k between units becomes a factor of k squared between the areas.
Map Scale in Both Directions [H]	Multiply by the scale to go from map to ground and divide to go from ground to map.
Scale Drawings: Finding a Third Measurement [H]	Find the meters-per-centimeter scale from the matched pair, then apply it to the other measurement.
Proportional Relationships and the Constant of Proportionality [H]	In a proportional relationship y = kx, so k is y divided by x and stays the same for every pair.
Rates per Unit of Body Mass [H]	Multiply the per-kilogram rate by the mass for the daily amount, then divide by the number of doses or days.
Population Density [H]	Population density is population divided by area, so population is density times area.
Reversing a Discount and a Tax [H]	Multiply by (100 - d)/100 and then by (100 + t)/100 going forward, and divide by those same factors going backwards.
Percentage Points Versus Percent Change [H]	A change in a percent is measured in percentage points; the percent change divides that gap by the starting percent.
The Percent Decrease That Undoes an Increase [H]	Undoing a percent change compares the change to the NEW amount, so the two percents differ.
Percents of a Total Read from a Table [H]	Divide the count by the total that the question names, then multiply by 100.
Percent Change Between Table Entries [H]	Percent change is the difference divided by the EARLIER value, times 100.
Percent Greater and Percent Less [H]	The quantity named after the word 'than' is the base, so switching direction changes the percent.
Unspecified Totals: Substitute a Convenient Value [H]	When no starting amount is given, let the starting amount be 100 and follow the changes.
The Mean With and Without an Outlier [H]	Work with totals: the mean of the remaining values is the total minus the removed value, divided by the new count.
Effect of an Outlier on Mean, Median and Range [H]	The mean and the range react to an extreme value; the median moves only if the middle position moves.
Shifting and Scaling a Data Set [H]	Adding the same amount to every value shifts the center but not the spread; multiplying scales both.
Quartiles and the Interquartile Range [H]	The interquartile range is the median of the upper half minus the median of the lower half.
Median from a Frequency Table [H]	Add the frequencies until the running total reaches the middle position, and read the value in that row.
Skew: Comparing the Mean and the Median [H]	A long tail pulls the mean toward it, so the mean sits on the tail side of the median.
Combining the Means of Two Groups [H]	Add the two group totals and divide by the combined count; the combined mean is pulled toward the larger group.
Shares Read from a Bar Graph [H]	Add the bars the question describes, then divide by the total of all the bars.
Residuals: Observed Minus Predicted [H]	A residual is the observed value minus the value the fitted line predicts at that x.
Interpreting the Slope of a Line of Best Fit [H]	The slope is the predicted change in the y quantity for a one-unit increase in the x quantity.
Interpreting the Intercept of a Fitted Model [H]	The constant term is the value the model predicts when the input variable equals zero.
Using a Fitted Model in Both Directions [H]	Substitute the input to predict an output, and solve the equation to recover an input from an output.
Counting Points About a Line of Best Fit [H]	A point above the fitted line was underestimated by the model; a point below it was overestimated.
Choosing the Type of Model for a Scatterplot [H]	Constant differences between successive values indicate a linear model; a constant ratio indicates an exponential model.
The Limits of Extrapolation [H]	A fitted model is evidence only across the range of the data that produced it.
Comparing Two Series Over Time [H]	Compare the two values year by year, or set the two models equal to find when the faster one passes the other.
Completing a Two-Way Table for a Probability [H]	Recover a missing cell from the row and column totals before computing any probability.
Which Total Goes in the Denominator [H]	A conditional probability divides by the total of the group named in the condition, not by the grand total.
Comparing Proportions Between Groups [H]	Divide each group's count by that group's own total before comparing, since the group sizes differ.
Estimating a Count in the Full Population [H]	Multiply the population size by the proportion observed in the random sample.
Recovering a Count from a Stated Probability [H]	Set the count over the total equal to the stated probability and solve for the unknown count.
Building the Interval from a Margin of Error [H]	The plausible values for the population mean run from the estimate minus the margin of error to the estimate plus it.
Comparing Two Estimates with Margins of Error [H]	Two estimates support a claim of a real difference only when their plausible intervals do not overlap.
What a Confidence Statement Does and Does Not Claim [H]	A confidence interval describes plausible values for a population parameter, not the spread of individuals.
Judging a Sampling Method [H]	Only a random sample of the population in question supports an estimate for that population.
Which Design Supports a Causal Conclusion [H]	Random assignment to groups supports a cause-and-effect claim; random selection from a population supports generalizing it.
Choosing the Setup Whose Units Cancel [H]	Arrange the given rates so every unwanted unit cancels and only the requested unit remains.
Concentration of a Mixture [H]	Track the amount of the dissolved substance and the total volume separately, then divide.
Diluting or Strengthening a Solution [H]	Adding water keeps the amount of substance fixed while the total grows; adding pure substance increases both.
Back-Solving from the Answer Choices [H]	When the choices are numbers and the condition is easy to test, test the choices instead of solving.
Estimating to Choose the Closest Value [H]	When the choices differ by powers of ten, round the numbers and compare orders of magnitude.

SAT & ACT - Geometry, Trigonometry and Test Strategy · 46 topics

Transversal Angles Labeled with Expressions [H]	Equal angles give an equation; same-side interior angles give a sum of 180.
The Exterior Angle Relationship [H]	An exterior angle of a triangle equals the sum of the two remote interior angles.
Isosceles Triangle Angle Chains [H]	Equal sides face equal angles, and the three angles of a triangle total 180 degrees.
Triangle Angles Given as a Ratio [H]	Ratio parts of the 180 degree angle sum: divide 180 by the total number of parts.
Interior and Exterior Angles of a Regular Polygon [H]	Exterior angles of any polygon total 360 degrees; interior angles total 180(n - 2).
A Segment Parallel to a Side of a Triangle [H]	A line parallel to one side cuts off a smaller triangle similar to the whole.
Indirect Measurement with Similar Triangles [H]	Objects and their shadows at the same time form similar triangles, so height over shadow is constant.
Perimeter and Area Scale Factors [H]	Lengths and perimeters scale by k, while areas scale by k squared.
Which Criterion Proves Congruence [H]	SSS, SAS, ASA and AAS prove congruence; two sides with a non-included angle and three angles do not.
The Triangle Midsegment Relationship [H]	The segment joining two midpoints is parallel to the third side and half its length.
The Pythagorean Theorem in Applied Settings [H]	In a right triangle the two legs and the hypotenuse satisfy leg squared plus leg squared equals hypotenuse squared.
Diagonals of a Rectangular Box [H]	The diagonal of a box satisfies length squared plus width squared plus height squared equals diagonal squared.
45-45-90 Triangles and Square Diagonals [H]	In a 45-45-90 triangle the hypotenuse is the leg times the square root of 2.
30-60-90 Triangles in Applied Settings [H]	In a 30-60-90 triangle the sides are in the ratio 1 to the square root of 3 to 2.
Altitude and Area of an Equilateral Triangle [H]	An altitude splits an equilateral triangle into two 30-60-90 triangles, giving altitude s times root 3 over 2.
Area and Perimeter of Rectilinear Composites [H]	Add or subtract rectangles for the area; an L-shaped region has the same perimeter as its bounding rectangle.
Composites with Semicircular Regions [H]	Add the half-circle area or half-circumference to the straight parts, using the radius as half the given diameter.
Shaded Regions Built from Circles [H]	A shaded region is the containing area minus the removed area; four quarter circles of equal radius make one whole circle.
Solving for a Dimension from a Volume [H]	Substitute the known measures into V = pi r squared h and solve for the missing one.
Surface Area of Boxes and Cubes [H]	A box has surface area 2(lw + lh + wh); a cube of edge s has surface area 6s squared and volume s cubed.
How Scaling a Solid Changes Its Volume [H]	Multiplying every length by k multiplies the volume by k cubed.
How Scaling a Solid Changes Its Surface Area [H]	Multiplying every length by k multiplies every area by k squared, so a volume ratio k cubed gives an area ratio k squared.
Volume with a Unit Conversion [H]	Converting a volume cubes the length conversion factor: 1 cubic yard is 27 cubic feet and 1 cubic foot is 1728 cubic inches.
Two Solids of Equal Volume [H]	A cone is one third of the cylinder with the same base and height, so equal volumes force a height ratio of 3 to 1.
Arc Length Run Backwards [H]	Arc length is the central angle over 360 times the circumference, so any one of the three can be found from the other two.
Perimeter of a Sector or Ring Region [H]	The perimeter of a sector is its arc plus the two straight radii, not the arc alone.
The Circle Through the Ends of a Diameter [H]	The center of a circle is the midpoint of any diameter and the radius is half the diameter's length.
Circles Tangent to an Axis [H]	A circle is tangent to the x-axis when its radius equals the distance from its center to that axis.
Proportional Reasoning with Arcs and Sectors [H]	An arc or sector is the same fraction of the circle as its central angle is of 360 degrees.
Finding a Side from a Given Trig Ratio [H]	A trig ratio identifies which two sides are compared, so one known side scales the whole triangle.
Sine and Cosine of Complementary Angles [H]	For complementary angles the sine of one equals the cosine of the other.
Wires and Ramps at 30, 45 and 60 Degrees [H]	A 45 degree right triangle has equal legs; a 30 or 60 degree right triangle has sides in the ratio 1 to root 3 to 2.
Area of a Right Triangle from a Trig Ratio [H]	Find both legs from the ratio and the hypotenuse, then take half their product.
Arc Length in Radian Measure [H]	In radian measure the arc length equals the radius times the angle.
Sector Area in Radian Measure [H]	A sector of angle theta radians in a circle of radius r has area one half r squared theta.
An Unknown Coordinate from a Distance [H]	Square the distance formula: the horizontal and vertical differences must satisfy the Pythagorean relationship.
Missing Endpoints and Missing Vertices [H]	A midpoint averages the endpoints, so an endpoint is twice the midpoint minus the known endpoint.
Perpendicular Bisectors in the Plane [H]	The perpendicular bisector passes through the midpoint with slope the negative reciprocal of the segment's slope.
Lines Through a Point, Parallel or Perpendicular [H]	Parallel lines share a slope; perpendicular slopes multiply to negative one.
Areas of Figures Given by Coordinates [H]	Use a horizontal or vertical side as the base; the height is the perpendicular distance to the opposite vertex.
Back-Solving from the Answer Choices [H]	Testing the answer choices is often faster than solving, and it automatically rejects extraneous solutions.
Substituting a Convenient Value [H]	To compare algebraic expressions, evaluate each at one convenient number and keep only the match.
Estimating to Eliminate Choices [H]	When the choices differ by a factor of three or more, a rough estimate identifies the answer without exact computation.
Answering Without Solving for Everything [H]	When a question asks for a combination such as x + y, combine the equations instead of solving for each variable.
Answering the Quantity the Question Asks For [H]	Solve for the variable, then compute the expression the question actually requests.
Catching the Unit Change in the Last Line [H]	Check the unit the answer must be in: an area conversion uses the square of the length factor.

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 · 6 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 Functions · 4 topics

The Coordinate Plane	Locating points with (x, y) pairs.
Slope of a Line	Rise over run between two points.
Slope-Intercept Form	y = mx + b describes a whole line.
Systems of Equations (Substitution)	Two equations, two unknowns.

Quadratics & Polynomials · 8 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 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.

Radicals & Exponentials · 6 topics

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

Geometry · 6 topics

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

Functions & Algebra II · 5 topics

Function Notation & Evaluation	Reading f(x) notation and plugging in inputs.
Complex Numbers	The imaginary unit i = √(−1) and numbers of the form a + bi.
Operations with Complex Numbers	Multiplying complex numbers with FOIL and i² = −1.
Logarithms	log_b(x) asks: to what power must b be raised to get x?
Properties of Logarithms	Logs turn products into sums, quotients into differences, powers into multiples.

Trigonometry · 1 topics

Right-Triangle Trigonometry

SOH-CAH-TOA: the three trig ratios of an acute angle in a right triangle.

Probability & Statistics · 1 topics

Mean, Median & Range

Center and spread in one pass.

Algebra II: Complex Numbers & Matrices · 4 topics

Adding 2×2 Matrices	Add two matrices of the same size entry by matching entry.
Scalar Multiplication of a Matrix	Multiply a matrix by a number by scaling every entry.
Multiplying 2×2 Matrices	Each product entry is a row of A dotted with a column of B.
The Determinant of a 2×2 Matrix	For [[a, b], [c, d]] the determinant is ad − bc.

AP Statistics: Collecting Data · 1 topics

Population vs Sample; Parameter vs Statistic

Who we study versus who we measure; the truth versus our estimate.

AP Statistics: Statistical Inference · 4 topics

Sampling Distribution of x̄	Same center as the population, but less spread.
Sampling Distribution of p̂	Centered at p, spread √(p(1 − p)/n).
Standard Error Formulas	Estimate how much a statistic bounces from sample to sample.
Margin of Error	How far the estimate can plausibly be from the parameter.

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