SAT Mathematics — Complete Study Guide

Read this first. It is written for you.

If maths has never quite clicked, this guide is built for exactly that. Nothing here assumes you already understand it. Every rule is explained in plain English before the formula appears, every symbol is spelled out in words, and every method comes with a worked example where every step is shown — nothing is left for you to "just know".

You do not have to be naturally good at maths to do well on this test. The SAT Math section examines a small and fixed list of ideas. It is not trying to surprise you with new mathematics; it is checking whether you can use familiar mathematics accurately, under time pressure, on a problem worded in an unfamiliar way. That is a skill, and skills are built by practice.

What the Test Actually Is

The SAT has two sections: Reading and Writing, and Math. This book covers Math only.

Everything in this section is measured, not remembered. It comes from the eight official College Board practice tests (numbers 4 to 11) and their scoring guides. Where this book says "in every module we checked", it means all sixteen Math modules across those eight papers.

The shape of the Math section

The Math section comes in two modules, done one after the other. On the official paper practice tests, each module has 27 questions, so the section has 54 in total.

The digital test you sit is adaptive: how you do on the first module decides which second module you get. That is why the practice papers are slightly longer than the live test — they have to cover the same ground without adapting. The mathematics is identical either way, and so is everything in this book.

A calculator is allowed on every question

This is the single most important thing to know, and it is printed on the directions page of every Math module:

Use of a calculator is permitted for all questions.

There is a Desmos graphing calculator built into the test app, and you may bring your own approved one. You will never be asked to do arithmetic a calculator cannot handle.

That built-in graphing calculator is worth learning properly, and most students never do. Section 5 is about it.

That has a consequence worth sitting with. If the calculator can do the arithmetic, then the arithmetic is not what is being examined. The marks are in setting the problem up correctly — choosing the right equation, reading the question accurately, knowing what the answer is supposed to mean. A calculator will give you a beautifully precise answer to the wrong equation.

This is why almost every wrong option in this book is a correct calculation of something the question did not ask. Those are the mistakes that survive having a calculator.

The two kinds of question

Student-produced responses are often called grid-ins or fill-ins. There are no options at all. Chapter 8 is devoted to them, because they need a different technique — but there is one thing to learn right now, in Section 4 below, because it is worth marks and takes ten minutes to learn.

Where the grid-ins sit — and they do not move

In all sixteen official modules examined, the student-produced questions were at exactly these positions:

$$\textbf{6, 7, 13, 14, 20, 21, 27}$$

Every module. No exceptions. Seven of the 27 questions, which is where the 26% comes from. Two things follow:

• When you reach question 6, expect no options. It is not a printing error.
• The last question of each module, number 27, is always a grid-in.

There is no penalty for a wrong answer

Nothing is subtracted for being wrong. A blank and a wrong answer score exactly the same: zero. So answer every question, including ones you have not read. On a multiple-choice question a blind guess is worth a 1-in-4 chance; a blank is worth nothing at all.

This applies to grid-ins too. If you have any number at all, enter it.

Section 1 — What the Test Examines

Every SAT Math question belongs to one of four areas. The share each one takes is set by the test's own blueprint.

Notice what that means: 70% of the test is algebra of one kind or another. Geometry is the smallest area, not the largest — which is the opposite of what most people expect, and the opposite of how most people spend their study time.

If you have limited time, that table tells you where to spend it.

What is not on the test

It is worth knowing what you can stop worrying about. The SAT Math section does not examine formal proof, matrices, vectors, logarithms beyond the most basic manipulation, calculus of any kind, or three-dimensional trigonometry.

Section 2 — The Reference Sheet You Are Given

Every Math module begins with a reference sheet. You do not have to memorise anything on it. It is on the screen throughout.

Here is what it gives you.

Areas and circumference

$$A = \pi r^2 \qquad C = 2\pi r \qquad A = \ell w \qquad A = \tfrac{1}{2}bh$$

Pythagoras

$$c^2 = a^2 + b^2$$

Volumes

$$V = \ell w h \qquad V = \pi r^2 h \qquad V = \tfrac{4}{3}\pi r^3 \qquad V = \tfrac{1}{3}\pi r^2 h \qquad V = \tfrac{1}{3}\ell w h$$

Special right triangles

The 30–60–90 triangle, with sides in the ratio $x : x\sqrt{3} : 2x$, and the 45–45–90 triangle, with sides in the ratio $s : s : s\sqrt{2}$.

Three facts stated in words

• A circle has $360$ degrees of arc.
• A circle has $2\pi$ radians of arc.
• The angles of a triangle add to $180$ degrees.

What it does not give you

This is the more useful list, because these are the things you must actually know. None of the following appears on the reference sheet:

• The quadratic formula
• Slope: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$
• The equation of a line in any form
• The equation of a circle
• SOHCAHTOA, or any trigonometric identity
• The distance or midpoint formulas
• Anything at all about mean, median, standard deviation or probability
• The exponent rules

The last one catches people out. Learn the list above; ignore the reference sheet until you need a volume.

Section 3 — The Language of the Test

Most people who struggle with SAT maths are not bad at maths. They lose marks on words. Here is the vocabulary the test uses, in plain English.

Everyday maths words

Phrases that decide the answer

These are worth more marks than any formula, because misreading one makes every subsequent step wrong.

Words that tell you which formula

That last pair costs people marks constantly. "Must be true" means you have to rule out every counterexample; "could be true" means one example is enough.

Section 4 — How to Enter a Grid-In Answer

There are no options, so the way you write the answer is the answer. These rules come straight from the test's instructions page, and every one of them throws marks away if broken.

The rules

• Your answer may be up to 5 characters if positive, 6 if negative (the minus sign counts as one).
• If a fraction is too long to fit, write the decimal instead.
• If a decimal is too long to fit, truncate it or round it at the fourth digit. Either is accepted.
• A mixed number like $3\frac{1}{2}$ must be entered as $7/2$ or $3.5$. Writing 3 1/2 scores zero — it gets read as thirty-one halves.
• Do not enter a percent sign, a comma, or a dollar sign. The answer one thousand two hundred and fifty is 1250, never 1,250.

Several forms are usually accepted

This is the part almost nobody is taught. If the answer is $\frac{1}{4}$, then 1/4, 0.25 and .25 are all accepted. You do not have to guess which form the test wants — any correct form scores.

Here are real accepted answers, taken from official answer keys, and what each one is teaching:

That last row deserves attention. Some questions have two valid answers — a quadratic with two roots, for instance. You only enter one. Do not waste time deciding which is "the" answer; enter whichever you found first.

The safest habit

When your answer is a fraction that fits in five characters, enter the fraction. It cannot be wrong through rounding. Only convert to a decimal when the fraction is too long, and then give four digits.

Section 5 — The Built-In Graphing Calculator

The digital SAT gives you a Desmos graphing calculator inside the testing app, on every Math question. It is not a plain calculator with a graph button — it is a proper graphing tool, and knowing how to use it is a skill in its own right.

This section is worth more marks per minute of study than anything else in this guide. A calculator that can graph turns a large class of algebra questions into questions you can see.

The one idea behind almost every use

Anything you can write as an equation, Desmos can graph. Anything you can graph, you can read the answer off.

That single move covers most of what follows.

The four things to practise

1. Solving any equation. Put the left-hand side into one line as $y = \ldots$ and the right-hand side into another. Where the two graphs cross, the equation is true — so the $x$-coordinate of the crossing point is your solution.

If the equation is already "something $= 0$", just graph the something and read off where it crosses the $x$-axis. Desmos labels those points for you: click one and it shows the coordinates.

2. Systems of two equations. Type both in. The intersection is the solution. This is often faster than elimination even when you know elimination perfectly, because you were going to reach for the calculator for the arithmetic anyway.

3. Questions with an unknown constant. This is the least-known technique and the most useful. When a question says something like "where $a$ is a positive constant", type the equation in with the letter still in it. Desmos will offer to add a slider for that letter. Always accept.

Now try values on the slider. If the question is well posed, the answer will be the same for every value the question allows — and that is the answer. You have replaced the algebra with an experiment.

4. Checking a candidate answer. On a multiple-choice question you can type an option in and see whether it fits. On a grid-in, you can substitute your own answer back and confirm the two sides agree.

When not to use it

The temptation, once you can graph, is to graph everything. Do not.

• Simple arithmetic and one-step algebra are faster by hand. Typing $3x + 5 = 20$ into a graphing tool costs more time than solving it.
• Interpretation questions have nothing to graph. "What does the 25 represent?" is answered by thinking about units, not by drawing anything.
• Setting up is still your job. A graph of the wrong equation is a precise answer to a question nobody asked. On a word problem, write the equation down first; only then decide whether to graph it.

A reasonable rule: if you can see the answer within about twenty seconds by hand, do it by hand. If you cannot, or if the algebra looks long or error-prone, graph it.

Practise on the real thing

The testing app is called Bluebook, and its Desmos is a specific version. Practise in the same place you will sit the test, so nothing about the interface is new on the day. Free official practice tests run inside Bluebook.

The short version: learn to find an intersection, find a zero, and add a slider. Those three things cover most of what the calculator is for, and most students never learn the third one at all.

Section 6 — Strategy on the Day

Order

The questions get harder as the module goes on, roughly. They are not in strict difficulty order, but question 25 is usually harder than question 3.

You are not required to work in order. If a question is not moving after about 90 seconds, mark it, put down something, and move on. There are no bonus marks for the hard ones — every question is worth exactly the same.

Timing

Two modules, 27 questions each. That works out at a little over a minute per question, which sounds tight and is actually comfortable if you do not get stuck. The single biggest time loss is refusing to abandon a question.

When you are stuck on a multiple-choice question

Three techniques, in order of usefulness:

1. Work backwards from the options. Put each option into the question and see which one fits. On many algebra questions this is faster than solving. It is unavailable on grid-ins, which is one reason they need separate practice.
2. Pick a number. If a question is entirely in letters — "which expression is equivalent to..." — choose a value like $x = 2$, work out what the question gives, then test each option with the same value. Anything that does not match is out. Avoid 0 and 1: they make too many things look equal.
3. Estimate. Especially on geometry with a figure, a rough answer often eliminates two options immediately. Figures are drawn to scale unless the question says otherwise — the directions page says so explicitly.

Read the last line twice

More marks are lost to answering the wrong question than to any mathematical error. The question asks for $2x$, you find $x$ and stop. It asks for the perimeter, you find the area. It asks how many were left, you find how many were taken.

Before you enter an answer, look at the final sentence of the question again and check you have produced the thing it named.

Section 7 — How Your Score Is Worked Out

The Math section is scored 200 to 800. Your Reading and Writing score is also 200 to 800, and the two add to give a total out of 1600.

The number of questions you get right — your raw score — is converted to the 200–800 scale by a table published with each test. Here is what that conversion looks like, from official practice test 10:

Two things are worth reading off that table.

Every question is worth roughly the same. There is no bonus for hard ones. So the fastest way to raise a score is to stop losing the easy ones — which is mostly about reading carefully, not about learning more mathematics.

Nothing is worth as much as the middle. Going from 27 right to 38 right is about 120 points. Going from 44 to 50 is about the same again, but takes far more work. If your score is in the middle, the gains available to you are large.

The conversion is slightly different for every test, because papers are equated against each other for difficulty. The mock exams in Chapter 9 use these real tables, so the score they report is a genuine estimate rather than a guess.

Section 8 — How to Use This Book

Each chapter has the same shape:

1. Teaching — the idea in plain English, with the formulas you need.
2. Questions — attempt these on paper before reading on.
3. Full solutions — every question re-taught from the beginning, including why each wrong option is wrong. Each wrong option is a real mistake that real students make; learning to recognise them is most of the work.
4. A takeaway for each question — the transferable part, worth more than the answer.

The routine that makes this book work: attempt the question honestly first, even if you only get partway. Then read the full solution even when you got it right — the solutions carry the shortcuts and connections that turn one question into ten marks' worth of understanding. When you get one wrong, find your mistake among the wrong-option notes. Recognising your own mistake pattern is the fastest improvement available to you.

A study plan

If you have eight weeks:

If you have two weeks, do chapters 1 and 2, then chapter 8, then one mock. That is the highest-value slice of the book.

If you remember only seven things from this guide: 1. Answer every question. A blank scores zero; a guess might not. 2. 70% of the test is algebra. Study accordingly. 3. You have a calculator, so the marks are in the set-up, not the sums. 4. Learn the built-in graphing calculator: intersection, zero, slider. 5. Questions 6, 7, 13, 14, 20, 21 and 27 have no options. 6. On a grid-in, 3 1/2 scores zero. Write 7/2 or 3.5. 7. Re-read the last line of the question before you answer it.