SAT Maths — Sat Math Overview Complete Chapter Guide

Chapter Overview & Weightage

SAT Math covers four domains, and knowing the weightage of each is the first step to smart preparation. The exam has 44 questions total — 27 multiple choice and 17 student-produced response (grid-in).

SAT Math Section Breakdown (2024 Digital SAT format):

Module 1: 22 questions, 35 minutes
Module 2: 22 questions, 35 minutes (adaptive — harder or easier based on Module 1 performance)
Total: 44 questions, 70 minutes
Score range: 200–800

Domain	Questions (approx.)	Weightage
Algebra	13–15	~35%
Advanced Math	13–15	~35%
Problem Solving & Data Analysis	5–7	~15%
Geometry & Trigonometry	5–7	~15%

The 2023–24 shift to Digital SAT changed the format significantly. Paper SAT had a no-calculator and calculator section; Digital SAT allows a built-in Desmos calculator for all questions. This changes strategy — use Desmos aggressively for graphs and checking answers.

Key Concepts You Must Know

Ranked by frequency across recent SAT administrations:

Algebra (highest priority — ~35% of paper)

Linear equations in one and two variables
Systems of linear equations (substitution, elimination, graphical interpretation)
Linear inequalities and their graphs
Interpreting linear functions in context (slope as rate of change)
Absolute value equations

Advanced Math (~35% of paper)

Quadratic equations — factoring, completing the square, quadratic formula
Parabola properties: vertex form, axis of symmetry, roots
Exponential functions: growth/decay, interpreting a·bˣ in context
Polynomial operations and remainder theorem
Rational expressions and equations
Function notation, composition, and transformations

Problem Solving & Data Analysis (~15%)

Ratios, rates, proportional relationships
Percentages — percent change, percent of a value
Unit conversions
Mean, median, mode from tables and graphs
Scatterplot interpretation — line of best fit, correlation vs. causation
Two-way tables and conditional probability basics

Geometry & Trigonometry (~15%)

Area and perimeter of triangles, circles, rectangles
Pythagorean theorem and special right triangles (30-60-90, 45-45-90)
Similarity and congruence in triangles
Circle theorems: arc length, sector area, central angles
Basic trig ratios: sin, cos, tan in right triangles
Radian measure

Students aiming for 700+ should master Advanced Math thoroughly — it’s where the hard Module 2 questions cluster. Students aiming for 600+ should lock down Algebra first since those questions appear in both modules.

Important Formulas

The SAT provides a reference sheet at the start of each Math module. Know what’s on it so you don’t waste time re-deriving, and know what’s NOT on it so you memorise those separately.

Area of circle: $A = \pi r^2$
Circumference: $C = 2\pi r$
Area of rectangle, triangle
Pythagorean theorem: $a^2 + b^2 = c^2$
Special right triangles (30-60-90 and 45-45-90 ratios)
Volume formulas: rectangular prism, cylinder, sphere, cone, pyramid
The quadratic formula: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

These are given — don’t memorise, just recognise.

Slope formula:

m = \frac{y_2 - y_1}{x_2 - x_1}

Slope-intercept form: $y = mx + b$

Standard form of a line: $Ax + By = C$

Vertex form of a parabola: $y = a(x - h)^2 + k$ , vertex at $(h, k)$

Discriminant: $\Delta = b^2 - 4ac$

$\Delta > 0$ : two real roots
$\Delta = 0$ : one real root (touches x-axis)
$\Delta < 0$ : no real roots

Exponential growth/decay: $y = a(1 + r)^t$ or $y = a(1 - r)^t$

Percent change: $\dfrac{\text{new} - \text{old}}{\text{old}} \times 100$

Distance formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

Midpoint formula: $\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)$

Arc length: $s = r\theta$ (θ in radians)

SOHCAHTOA: $\sin\theta = \dfrac{\text{opp}}{\text{hyp}}$ , $\cos\theta = \dfrac{\text{adj}}{\text{hyp}}$ , $\tan\theta = \dfrac{\text{opp}}{\text{adj}}$

When to use each:

Discriminant → any question about “number of solutions” or “how many x-intercepts”
Vertex form → questions about maximum/minimum value of a quadratic
Slope as rate → any “word problem with a linear model” — slope IS the rate of change
Exponential form → “doubles every n years”, “decays by 3% annually”

Solved Previous Year Questions

PYQ 1 — Linear Systems (SAT March 2024)

Question: A bakery sells muffins for $2 each and cookies for $3 each. On Monday, they sold 50 items total and collected $120. How many muffins were sold?

Solution:

Set up the system. Let $m$ = muffins, $c$ = cookies.

m + c = 50 \quad \text{...(i)}

2m + 3c = 120 \quad \text{...(ii)}

Multiply (i) by 2: $2m + 2c = 100$

Subtract from (ii): $c = 20$

So $m = 50 - 20 = \mathbf{30}$

On Digital SAT, you can also plug answer choices back into both equations to verify. With a built-in calculator available, this check takes 20 seconds and eliminates careless errors.

PYQ 2 — Quadratic / Vertex Form (SAT May 2024)

Question: The function $f(x) = -2(x-3)^2 + 8$ represents the height (in metres) of a ball. What is the maximum height?

Solution:

The function is already in vertex form $f(x) = a(x-h)^2 + k$ .

Vertex is at $(h, k) = (3, 8)$ .

Since $a = -2 < 0$ , the parabola opens downward — so the vertex is a maximum.

Maximum height $= \mathbf{8}$ metres, achieved at $x = 3$ .

Students often expand this and try to use $x = -b/2a$ . That works, but costs 3× more time. Recognise vertex form instantly — it’s a 5-second question if you do.

PYQ 3 — Data Analysis / Scatterplot (SAT October 2023)

Question: A scatterplot shows data for 10 cities, with population density (per sq km) on the x-axis and average commute time (minutes) on the y-axis. The line of best fit has equation $y = 0.04x + 18$ . A city has a population density of 2500. What does the model predict for commute time?

Solution:

Substitute $x = 2500$ :

y = 0.04(2500) + 18 = 100 + 18 = \mathbf{118} \text{ minutes}

Now interpret the slope: each additional person per sq km is associated with a $0.04$ -minute increase in commute time.

SAT loves asking you to “interpret the meaning of the slope in context.” Always state it as: “For every 1-unit increase in [x-variable], [y-variable] increases/decreases by [slope].” This phrasing matches what the answer choices use.

Difficulty Distribution

Understanding how difficulty distributes helps you allocate time during the actual test.

Difficulty	Share of Questions	Where They Appear
Easy	~30% (≈13 questions)	First half of each module
Medium	~40% (≈18 questions)	Middle of each module
Hard	~30% (≈13 questions)	Last 5–6 of each module + all of hard Module 2

On the Digital SAT, Module 2 is adaptive. If you do well on Module 1, Module 2 gets harder — but the scoring ceiling is higher. Most students targeting 700+ should aim to score 18+/22 on Module 1 to unlock the hard module, where the top scores live.

The hard questions in Module 2 are not “harder topics” — they’re harder versions of the same topics. A hard quadratic question still tests quadratics; it just has more algebraic complexity or a less obvious setup. Your concept base doesn’t need to change — your problem-solving patience does.

Expert Strategy

Phase 1: Lock Down Algebra First (Weeks 1–2)

Every student, regardless of target score, should start here. Linear equations and systems appear in both modules, at every difficulty level. A shaky foundation here costs points across 30% of the paper.

Practice until you can solve any two-variable system in under 90 seconds without second-guessing.

Phase 2: Master Advanced Math Concepts (Weeks 3–4)

This is where 650→750 gains happen. The SAT tests quadratics, polynomials, and exponential functions with a consistent set of tricks. Work through 50–60 official College Board practice questions in this domain specifically.

The College Board releases free official practice tests on their website. Use those — not third-party books that often have subtly wrong answer logic. The SAT has very specific question patterns that only official material captures accurately.

Phase 3: Desmos Fluency (Ongoing)

The built-in Desmos graphing calculator is your biggest unfair advantage — if you use it well. Practice:

Graphing a parabola to find its roots visually
Checking if a system of equations has 0, 1, or 2 solutions
Evaluating functions at specific values instantly

Students who learn to use Desmos for checking answers (not just initial solving) consistently improve accuracy by 3–5 questions.

Phase 4: Timed Practice Sets

Two weeks before the exam, shift to full timed modules. The time pressure on SAT Math is real — 44 questions in 70 minutes is about 95 seconds per question. Budget 60 seconds for easy questions so you have 2+ minutes for the hard ones at the end.

Mark and move strategy: If a question takes more than 90 seconds and you’re not close to an answer, mark it, move on, and return. On Digital SAT, every question is worth the same. Never spend 4 minutes on one hard question while three easy ones sit unanswered.

Common Traps

Trap 1: Sign Errors in Vertex Form

When $f(x) = (x+3)^2 + 5$ , the vertex is at $(-3, 5)$ — not $(3, 5)$ .

The formula is $f(x) = (x - h)^2 + k$ , so $x + 3 = x - (-3)$ , giving $h = -3$ .

This exact trap appears in roughly 1–2 questions per official test.

Rewrite any vertex form as $f(x) = (x - h)^2 + k$ before reading off the vertex. This mechanical step catches sign errors every time.

Trap 2: Confusing “No Solution” vs. “Infinitely Many Solutions” in Linear Systems

No solution: lines are parallel (same slope, different y-intercepts) → $0 = \text{nonzero}$
Infinitely many solutions: same line (same slope, same y-intercept) → $0 = 0$

SAT asks “for what value of $k$ does the system have no solution?” regularly. Set slopes equal, then verify the intercepts are different.

Trap 3: Percent Increase Then Decrease

A price increases 20% then decreases 20%. Students assume it returns to the original. It doesn’t.

If original = 100, after 20% increase = 120, after 20% decrease = $120 \times 0.8 = 96$ . Net change: -4%.

Never add or subtract percentage changes directly. Always apply them multiplicatively: $100 \times 1.20 \times 0.80 = 96$ .

Trap 4: Reading Scatterplot Questions Too Quickly

SAT data questions often ask for the value “predicted by the model” (use the line of best fit equation) versus the “actual value” (read the data point). These are deliberately set up to give different answers.

Read the question stem carefully: “according to the equation” vs. “according to the data.”

Trap 5: Extraneous Solutions in Rational Equations

When solving equations with variables in denominators, always check your solutions back in the original equation. If a solution makes the denominator zero, it’s extraneous and must be rejected.

SAT includes answer choices that are these extraneous solutions specifically to catch students who skip the check.

Final week strategy: Do one full timed practice test, then spend twice as long reviewing wrong answers as you spent taking the test. Each wrong answer is a pattern the SAT uses — once you understand why you got it wrong (concept gap? misread? careless?), that pattern stops costing you points.