Formulas tell you what to use. This guide tells you when, why, and what the question is actually asking. Each concept explained with the thinking process — not just the steps.
~35% of the math section. Linear equations, systems, inequalities, and how to interpret them in context. Most of these questions are doable in under a minute if you recognise the type.
Algebra
Linear Equations in Context High Freq
Word problems that describe a real situation using y = mx + b. You need to identify what m and b represent.
What the SAT actually asks: "What does the number 3.5 represent in the equation C = 3.5h + 20?"
The thinking process:
The slope (m) is the rate — how much y changes per unit of x. In context: cost per hour, miles per gallon, students per classroom.
The y-intercept (b) is the starting value — what y equals when x = 0. In context: initial fee, starting amount, base cost.
SAT Example
A plumber charges a $50 call-out fee plus $80 per hour. Total cost: C = 80h + 50. The 80 represents the hourly rate. The 50 represents the fixed fee before any hours are worked. If the SAT asks "what does 50 represent?" — it's the cost when h = 0, i.e. the initial fee.
Common wrong answer
Students pick answers that describe the slope when the question asks about the intercept, and vice versa. Read exactly what value the question is asking about.
Algebra
Systems of Equations High Freq
Two equations, two unknowns. The SAT also asks about the number of solutions a system has.
Solving systems — two methods:
Substitution: Isolate one variable in one equation, substitute into the other. Elimination: Add or subtract the equations to cancel one variable. Multiply first if needed.
Number of solutions:
One solution: Lines intersect at one point. Different slopes.
No solution: Lines are parallel. Same slope, different y-intercepts.
Infinite solutions: Lines are identical. Same slope, same y-intercept (equations are multiples of each other).
No solution test
y = 2x + 3 and y = 2x + 7 — same slope (2), different intercepts. Parallel lines → no solution.
Algebra
Linear Inequalities
Same as equations — but the direction of the inequality flips when you multiply or divide by a negative.
The one rule that matters: When you multiply or divide both sides by a negative number, flip the inequality sign.
Example
-2x < 8 → divide both sides by -2, flip: x > -4
On a number line: open circle = strict inequality (< or >). Closed circle = or-equal-to (≤ or ≥).
System of inequalities: The solution is the region that satisfies both. On Desmos, shade both and find the overlap. On paper, test a point.
Algebra
Interpreting Graphs of Linear Functions
What does the x-intercept, y-intercept, or slope mean in the real-world context?
The SAT loves asking what each part of a graph represents in context.
x-intercept: the value of x when y = 0. In context: when the quantity reaches zero (e.g. when a tank is empty, when debt is fully paid).
y-intercept: the value of y when x = 0. In context: the starting amount, initial value.
A graph shows litres of water in a tank over time. The x-intercept is when the tank is empty. The y-intercept is how much water was in the tank at the start. The slope (negative) is how fast water is draining.
Advanced Math
~35% of the math section. Quadratics, polynomials, exponential functions, and function notation. These are the questions most students lose the most marks on.
Advanced Math
Quadratic Models High Freq
Recognising when a situation is quadratic and pulling the right information from each form.
Three forms — each reveals different information:
Standard formax² + bx + c: c is the y-intercept. Use the quadratic formula for roots.
Factored forma(x−p)(x−q): p and q are the x-intercepts (roots). Easy to read off.
Vertex forma(x−h)² + k: (h, k) is the vertex. k is the minimum or maximum value.
SAT Thinking Process
If the question asks for the maximum height of a projectile — use vertex form and read off k. If it asks when the object hits the ground — use factored form and read off the positive root. Convert between forms if needed, or use Desmos to graph and read the point directly.
Advanced Math
Exponential Growth & Decay High Freq
Recognising exponential patterns and reading the growth rate, initial value, and decay factor.
The form:y = a(b)^x
a = initial value (when x = 0)
b = growth or decay factor
b > 1 = growth. 0 < b < 1 = decay
Percentage form:y = a(1 + r)^t for growth, y = a(1 − r)^t for decay. r is the rate as a decimal.
SAT Example
A population model: P = 500(1.08)^t. Initial population = 500. Growth rate = 8% per year. After t = 3 years: P = 500(1.08)³ ≈ 630.
Key question type
"What does the number 1.08 represent?" → The population increases by 8% each year. Not: the population is 108. Not: the population grows by 1.08 people.
Advanced Math
Function Notation & Composition
f(x), f(a), f(g(x)) — what the notation means and how to evaluate it.
f(a) means: substitute a wherever x appears in the function.
Example
If f(x) = 2x² − 3, then f(4) = 2(16) − 3 = 29.
Composition f(g(x)): Work from the inside out. Find g(x) first, then plug that result into f.
Example
f(x) = x + 1, g(x) = 3x. Find f(g(2)): first g(2) = 6, then f(6) = 7.
Reading from a graph: If the question gives a graph and asks for f(3), find x = 3 on the graph and read the y-value.
Advanced Math
Polynomial Division & Remainders
The Remainder Theorem lets you find remainders without long division.
Remainder Theorem: When polynomial p(x) is divided by (x − a), the remainder equals p(a).
Example
What is the remainder when x³ − 4x + 2 is divided by (x − 3)?
Sub in x = 3: 27 − 12 + 2 = 17. Remainder = 17.
Factor Theorem: If p(a) = 0, then (x − a) is a factor of p(x). This means a is an x-intercept of the graph.
Advanced Math
Rational Expressions & Equations
Fractions with polynomials — simplify, solve, and watch for excluded values.
Simplifying: Factor numerator and denominator, then cancel common factors.
Example
(x²−9)/(x−3) = (x+3)(x−3)/(x−3) = x+3 (where x ≠ 3)
Solving: Multiply both sides by the LCD to clear denominators. Then solve the resulting polynomial equation.
Excluded values: Any value that makes a denominator equal zero is excluded from the domain. Always check your solutions — plug back in to make sure they don't create a zero denominator.
Advanced Math
Equivalent Forms of Expressions
Manipulating an expression to match a given form — completing the square, factoring, expanding.
The SAT often gives an expression and asks which form is equivalent, or what constant must be added to complete the square.
Completing the square: x² + bx → add (b/2)² to complete. The expression becomes (x + b/2)² − (b/2)².
Tip: When the question asks "which is equivalent to..." — expand every answer choice and see which matches. Or use Desmos to graph both and confirm they're the same.
Problem Solving & Data Analysis
~15% of the math section. Ratios, proportions, percentages, statistics, and interpreting data. These are often the fastest questions on the test once you recognise the type.
Data Analysis
Ratios, Rates & Proportions High Freq
Setting up and solving proportions, unit rates, and multi-step rate problems.
The proportion setup:a/b = c/d → cross multiply → ad = bc.
Unit rate: Divide to find "per one" (miles per hour, cost per item). Then multiply by the quantity needed.
SAT Example
If 3 pens cost $4.50, how much do 7 pens cost? Unit rate = $1.50/pen. 7 × $1.50 = $10.50. Or set up proportion: 3/4.50 = 7/x → x = 10.50.
Multi-step: Convert units first when units don't match, then set up the proportion.
Data Analysis
Reading & Interpreting Tables/Graphs
Two-way tables, scatterplots, and bar charts — extracting and comparing data accurately.
Two-way tables: Be precise about which row and column the question is asking about. "Of those who..." means you only look at that subset — don't use the total.
Common error
"What fraction of students who prefer cats also prefer morning classes?" — divide cat-preferrers who prefer morning by total cat-preferrers. Not by the grand total.
Scatterplots: Line of best fit shows the trend. A point above the line = actual value is higher than predicted. Below = lower than predicted. The SAT sometimes asks which point is best (or worst) predicted by the model.
Data Analysis
Statistics: Mean, Median, Spread
What each measure tells you — and when the SAT uses one to mislead you about the other.
Mean vs Median:
Mean is pulled toward outliers. A very high or low value shifts the mean significantly.
Median is resistant to outliers — it only depends on the middle value(s).
If a dataset has outliers, the median is a better measure of "typical."
Standard deviation measures spread. Larger SD = data is more spread out. Two datasets can have the same mean but very different SDs.
SAT Trap
A question shows a dataset where one value is changed. If the changed value is an outlier, the mean changes significantly but the median may stay the same. Know which one you're being asked about.
Data Analysis
Margin of Error & Sample Validity
What makes a survey or study's conclusions valid — and when they can't be generalised.
Random sampling: For conclusions to apply to a broader population, the sample must be randomly selected from that population.
The SAT tests this directly: If students at one school are surveyed, conclusions can only be drawn about students at that school — not all students nationally.
Margin of error: An estimate of ±x means the true value is likely within that range. Larger sample → smaller margin of error.
Key distinction
A survey can show correlation — it cannot prove causation. "Ice cream sales correlate with drowning rates" — both rise in summer. The SAT will test whether you can distinguish "associated with" from "causes."
Geometry & Trigonometry
~15% of the math section. These are covered fully in the Formula Blueprint. Here's the thinking layer — what the question is really asking.
Geometry
Setting Up Geometry Problems
Most geometry errors are setup errors — choosing the wrong formula or the wrong measurement.
The process for every geometry question:
Draw and label the figure if one isn't given.
Mark what you know directly on the figure.
Identify which formula connects what you know to what you need.
Check units — if one measurement is in cm and another in m, convert first.
Most common error
Using diameter when the formula needs radius (or vice versa). Before plugging into any circle formula, confirm whether you have r or d.
Geometry
Trigonometry in Context
When to use sin, cos, and tan — and the co-function identity the SAT loves.
Choose the ratio based on what you have and what you need:
Have opposite, need hypotenuse (or vice versa) → use sin
"In a right triangle, sin(x°) = cos(y°). What is x + y?" → Since sin(x) = cos(90−x), we need y = 90−x → x + y = 90.
Geometry
Similar Triangles & Scale
Corresponding sides are proportional. Scale factor applies to lengths, squares to area, cubes to volume.
Two triangles are similar if: their angles are equal (AA), or their sides are proportional.
Setting up the proportion: Match corresponding sides (short to short, long to long).
Example
Triangle A has sides 3, 4, 5. Similar triangle B has shortest side 9. Scale factor = 3. Remaining sides = 12 and 15.
Scale factor and area: If linear scale factor is k, area scales by k². If the sides double (k=2), the area quadruples.
The SAT Math Mindset
Read the question last
Read the question before the answer choices. Decide what you'd expect the answer to look like — then find the match. Don't let the choices distract your thinking.
Plug in numbers
When a question uses variables in the answer choices, plug in a specific number (avoid 0 and 1). Evaluate each choice with that number and eliminate the ones that don't work.
Backsolve from answer choices
For "what is x?" questions, try the answer choices directly. Start with the middle value — if it's too big, try a smaller one. Faster than solving algebraically for many questions.
Use Desmos strategically
If solving by hand would take more than 20 seconds — graph it. Systems of equations, parabola vertices, circle equations. Don't use Desmos for simple one-step algebra.
Label everything you know
Before solving, write out all given values. Label the diagram. This prevents "I had everything I needed but forgot to use it" — the most common type of careless error.
Check the question again
After solving, re-read the final question. Did they ask for x or 2x+1? The diameter or the radius? The year or the number of years? This saves marks every time.