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SAT Math Lesson Library
The math the Digital SAT tests, grouped by unit — worked problems across algebra, advanced math, data analysis, geometry and trigonometry, plus how the two adaptive modules actually work. Math only; we do not cover the Reading and Writing section.
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How the two adaptive modules work and what a score means
Two modules of 22 questions in 35 minutes each, how Module 1 chooses Module 2, why every question in a module is worth the same, and exactly what to do in the last five minutes.
Read the lesson »The built-in Desmos calculator
The Digital SAT keeps a Desmos graphing calculator on screen for all 70 minutes. Where it saves a minute, where it costs one, and how to keep the right coordinate.
Read the lesson »Student-produced responses: fractions, decimals and negatives
A quarter of the SAT maths section is typed, not clicked. Five characters, no mixed numbers, no units or commas, fill the box on a long decimal, and keep the minus sign.
Read the lesson »Back-solving and plugging in
Test the answer choices when the algebra is ugly, start in the middle because SAT numeric choices ascend, pick a friendly number when the choices are expressions — and know when one line of algebra or Desmos is faster.
Read the lesson »Pacing 22 questions in 35 minutes
Ninety-five seconds a question is the average, not the plan. Bank time on the early easy items, flag and return through the review screen, and never leave a blank.
Read the lesson »Linear equations in one variable
Clear the fractions, distribute, collect the variable on one side, and rearrange a formula for one letter — the four moves behind every SAT linear equation item.
Read the lesson »Equations with no solution or infinitely many
When the variable cancels, the equation is telling you something. Compare the two sides as multiples of x, and find the constant that makes an SAT equation have no solution or infinitely many.
Read the lesson »Linear functions: what slope and intercept mean in context
Read f(x) = 12x + 250 as a starting amount plus a rate, say what the slope and the intercept mean in the situation's own words, and build the function from the sentence.
Read the lesson »Writing a line: slope-intercept, standard and point-slope forms
Slope from two points, point-slope when you have a point and a slope, and standard form Ax + By = C whose intercepts are one division away — plus how to tell whether two differently-written equations are the same line.
Read the lesson »Parallel and perpendicular lines
Parallel lines share a slope and perpendicular slopes multiply to −1. Dig the slope out of standard form, build a line through a given point, and handle the SAT parameter version.
Read the lesson »Linear models from tables and word problems
Spot a constant rate of change in a table, run the table back to x = 0 for the starting value, build a model from a sentence, and predict beyond the last row.
Read the lesson »Systems by substitution and elimination
Substitute when a letter is already alone, eliminate when the columns line up, and learn the SAT shortcut that answers x + y without solving the system at all.
Read the lesson »Systems with no or infinitely many solutions
No solution means parallel lines: same slope, different intercept. Infinitely many means one line written twice. Test both straight from standard form and solve for the missing constant.
Read the lesson »Systems from graphs
The crossing point is the solution. Read it x first then y, check it in both equations, recognise parallel and coincident lines on sight, and build the system back out of the picture.
Read the lesson »Word problems that become systems
Name the variables in a sentence, write one equation for the count and one for the value, or set two rates equal. Then check the answer against the words, not against your own algebra.
Read the lesson »Linear inequalities in one variable
Solve a linear inequality like an equation, flip the sign when you divide by a negative, read a compound inequality, and round the right way for the greatest or least integer.
Read the lesson »Inequalities in two variables and their graphs
Draw the boundary solid or dashed, shade the side a test point proves, read a shaded region back into an inequality, and check whether an ordered pair is a solution.
Read the lesson »Systems of inequalities
The solution to a system of inequalities is where the shaded regions overlap: test a point in every inequality, build a region from a budget and a capacity, and find its corner.
Read the lesson »Interpreting constants and coefficients
What the constant, the coefficient and the whole term mean in a linear model — and how to tell the one right interpretation from three choices that are true but answer a different question.
Read the lesson »Exponent rules, including rational exponents
Product, quotient and power rules, zero and negative exponents, and rational exponents — reading a 3/2 exponent as the square root of x cubed and matching one of four equivalent forms.
Read the lesson »Radicals
Simplify radicals by pulling out perfect squares, multiply and divide them, add only like radicals, rationalise a denominator with a conjugate, and confirm every exact answer with a decimal check.
Read the lesson »Polynomial operations and factoring
Add, subtract and multiply polynomials, factor a difference of squares, a trinomial with or without a leading coefficient, and a sum or difference of cubes — then match an expanded expression to one of four forms.
Read the lesson »Rational expressions
Factor before you cancel, cancel only whole brackets, combine over one denominator, and keep the excluded values the SAT hands you.
Read the lesson »Completing the square and vertex form
Halve and square the x-coefficient, handle a leading coefficient that is not 1, and read the vertex straight out of vertex form - or get it faster from x = -b/2a.
Read the lesson »Quadratic functions: three forms and what each shows
Standard form shows the y-intercept, factored form the x-intercepts, vertex form the vertex - how to convert between them and how the SAT asks which form displays what.
Read the lesson »Discriminant and number of solutions
One number, b² − 4ac, tells you whether a quadratic has two real solutions, one or none — and it is how the SAT asks you to solve for a missing constant.
Read the lesson »Solving quadratics
Factoring and the zero-product property, the square-root method, the quadratic formula — and how to pick the right one in a few seconds on the SAT.
Read the lesson »Radical and rational equations
Isolate and square, or clear the denominators — then substitute every candidate back, because both moves can invent solutions the original equation never had.
Read the lesson »Exponential functions, growth and decay
Read a and b out of y = a(b) to the x, turn a percent into a multiplier, handle an exponent like t/2 that counts periods rather than years, and say what each part means in context.
Read the lesson »Linear versus exponential change
Adding the same amount each step is linear; multiplying by the same factor is exponential. Test a table both ways, turn a percent into a multiplier, and read growth or decay off the base.
Read the lesson »Polynomial functions: zeros, factors and end behaviour
A factor (x − r) and a zero at r are the same fact. Read zeros off a factored form, tell a crossing from a touch, and use degree and the leading coefficient to pick the graph.
Read the lesson »Rational functions and asymptotes
Where a rational function is undefined, how the degrees decide the horizontal asymptote, and why a cancelling factor leaves a hole instead of a wall.
Read the lesson »Absolute value functions
Absolute value as distance from zero: solving |ax + b| = c in two cases, the V-shaped graph and its vertex, and when an inequality is a band or two rays.
Read the lesson »Function notation, composition and transformations
Reading f(3) from an equation, a table and a graph, solving f(x) = 7 backwards, composing functions in the right order, and turning a shift sentence into a movement.
Read the lesson »Linear-quadratic systems
Substitute the line into the parabola to get one quadratic, then let the discriminant count the intersections: two, one tangency, or none.
Read the lesson »Ratios and rates
A ratio counts equal parts, not amounts: split a total in a given ratio, set up a proportion with the labels lined up, build a unit rate, and tell direct from inverse.
Read the lesson »Unit conversion and dimensional analysis
Multiply by a fraction equal to 1 so the unwanted unit cancels: chain factors, convert a rate top and bottom, and square or cube the factor for square and cubic units.
Read the lesson »Percentages, percent change and reverse percent
Percent of a number, increases and decreases as multipliers, percent change against the original, why successive discounts do not add, and how to reverse a percent.
Read the lesson »Mean, median and mode from data displays
Read the mean, the median and the mode off a frequency table or a dot plot, handle an even-sized set, and know which measure an outlier moves and which it leaves alone.
Read the lesson »Spread: range, standard deviation and comparing distributions
Find a range, quartiles and an interquartile range, read a box plot, and decide which of two distributions has the greater standard deviation without computing one.
Read the lesson »Reading histograms, dot plots and box plots
What each data display shows and what it throws away: counting from a histogram bar and a dot plot, the five-number summary in a box plot, and choosing the display that answers the question.
Read the lesson »Scatterplots and the line of best fit
Read a scatterplot, use the line of best fit to predict, get its slope and intercept off the graph and say what they mean, and measure a residual as actual minus predicted.
Read the lesson »Linear and exponential models of data
Decide from a scatterplot or a table whether a linear or an exponential model fits, build it from the data, read growth or decay off the base, and stop the model at the edge of the data.
Read the lesson »Probability from two-way tables
Read a two-way table with its margins and grand total, turn a cell into a probability, and handle and versus or by subtracting the overlap exactly once.
Read the lesson »Conditional probability
Given that replaces the grand total with one row or one column. Spot the conditioning phrase, pick the right denominator, and keep P(A given B) apart from P(B given A).
Read the lesson »Sampling, margin of error and inference
A random sample estimates a population, and the margin of error turns that estimate into an interval of plausible values. Learn which conclusions the interval supports and what a larger sample does to it.
Read the lesson »Evaluating statistical claims
How a sample was drawn decides how far a result generalises, and random assignment is what licenses a cause-and-effect claim. Spot confounding and reject the three overstated choices.
Read the lesson »Data word problems with the calculator
Percent of a subtotal from a two-way table, weighted averages, a rate applied to a graph reading and two tables combined — extract the numbers, write the expression, let Desmos compute.
Read the lesson »Lines and angles
Complementary and supplementary pairs, vertical angles, linear pairs, and a transversal across two parallel lines — which angles are equal, which sum to 180°, and how to chase one angle to another.
Read the lesson »Triangle angles and the triangle inequality
The 180° angle sum, the exterior-angle shortcut, isosceles base angles, and the triangle inequality that decides which lengths could be the third side.
Read the lesson »Similar and congruent triangles
Similar gives equal angles and proportional sides; congruent gives everything. AA, the correspondence sentence, the parallel-line setup, and why area scales by the square.
Read the lesson »Area and perimeter of composite figures
The area formulas, composite shapes built by adding or subtracting pieces, shaded regions as big shape minus small shape, keeping π exact, and why a composite perimeter leaves out internal edges.
Read the lesson »Volume and surface area
The SAT gives you the volume formulas, so the work is choosing one and rearranging it. Base × height, a third of it when a solid points, counting faces for surface area, and turning a circumference into a radius.
Read the lesson »Right triangles and the Pythagorean theorem
a² + b² = c² with the hypotenuse alone on one side, the triples that save you the arithmetic, finding a leg instead, and the right triangle hiding inside a rectangle, an isosceles triangle or the xy-plane.
Read the lesson »Special right triangles
1 : 1 : √2 and 1 : √3 : 2, which ratio number belongs to which side, how to work backwards from the hypotenuse, and where these triangles hide inside a square and an equilateral triangle.
Read the lesson »Trig ratios and the complementary-angle identity
Sine, cosine and tangent as side ratios, named from the angle you are working with — plus the SAT's signature identity sin(x°) = cos(90° − x°) and where it comes from.
Read the lesson »Radians and the unit circle
A radian measured by the radius, the π/180 factor both ways, the common angles in both units, and a labelled unit circle where every point is (cos θ, sin θ).
Read the lesson »Equation of a circle
Standard form (x − h)² + (y − k)² = r², the sign trap inside the brackets, completing the square to get there, and finding the radius from a point on the circle.
Read the lesson »Arcs, sectors and chords
A central angle cuts an arc; the angle out of 360° is the fraction of the circle. Arc length, sector area, the radian versions, the bisected chord and the tangent-radius right angle.
Read the lesson »Coordinate geometry: distance, midpoint and shapes
The distance formula is the Pythagorean theorem in disguise, the midpoint is an average, the missing endpoint undoes that average, and slopes plus distances name a shape from its vertices.
Read the lesson »Stuck on one of these?
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