How to Tell Which SAT Math Skill to Practice Next

Common Algebra Slips on the SAT

The most basic skill SAT math depends on is algebra you can do quickly and correctly: rearranging equations, rewriting expressions in different forms, and isolating a variable. It includes “number sense,” a feel for how numbers behave, such as knowing that squaring a positive number less than 1 makes it smaller. When the algebra comes quickly and without doubt, your attention stays on what the question is asking.

In Tom’s experience, many lost points come from the same algebra steps: adding fractions, the two signs that come from taking a square root, solutions that don’t check, the difference of two squares, and a fraction divided by a fraction.

Answer each question below before you open its explanation.

If one of these questions caught you, practice that step on its own. Do a short set of problems on just that step, a few days in a row, until you can do it without stopping to think. The difference of two squares deserves extra practice, because in Tom’s experience it shows up on the SAT often, usually inside a larger question.

The Equations the SAT Tests

Next comes knowing what the test asks: the kinds of questions the SAT uses and the equations that go with each kind. The core equations are linear, quadratic, exponential, and polynomial. The College Board divides the Math section into four areas, shown below with the share of the 44 questions each one gets. Algebra and Advanced Math together make up about 70 percent of the section, so practice on the equations in those two areas covers the most questions.

The reference sheet in Bluebook lists some formulas, such as the Pythagorean theorem and the special right triangles, and leaves others out. It leaves out the equation of a circle, $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius, so you need to know that equation yourself. When a circle’s equation arrives multiplied out, you get it back into that form by “completing the square,” the step Owen was missing in the walkthrough.

When to Use Desmos

Desmos is the graphing calculator built into Bluebook. Using it well means knowing its controls, knowing which questions it solves faster than algebra, and recognizing those questions at a glance. In Tom’s experience, a student who has never trained with Desmos gains points faster by learning it than by practicing any other skill on this page.

Desmos computes and draws graphs, and many questions instead test whether you understand a situation, such as what a slope means in a word problem, which no calculator can decide for you. You learn to tell these two kinds of questions apart by solving practice questions both with Desmos and by hand, and noting which method was faster and more reliable for each kind.

You Can Adapt a Method Once You See Why It Works

Many students who know the basic steps still don’t know why those steps work. Knowing why a method works lets you use it when a question looks different from the ones you practiced, because you can check whether the reason still applies. Keep a simple example of each idea in mind, one you understand completely, and compare new questions against it.

Linear and exponential change are a good place to start, because the SAT tests both and each has a simple example you can keep in mind. “Linear” change starts from an initial amount and adds the same amount at every step, like a savings jar that gets 10 dollars every week. “Exponential” growth starts from an initial amount and increases by the same percent at every step, so it starts slowly and then shoots up, which is why compound interest follows it. Exponential decay decreases by the same percent at every step, so it drops quickly at first while some amount always remains, which is why radioactive decay follows it. Quadratics model yet another shape, such as the height of a ball that rises and then falls.

Knowing why usually comes slowly. Sometimes you have to work a kind of problem a hundred times before it clicks, and it clicks only if you think about what you’re doing along the way: why this step works, and why another step would fail. An AI helps here too: ask it why a method works, and keep asking until the explanation is one you could give in your own words.

Function Inputs and Outputs

Knowing why also prevents mistakes that look careless. Many students forget which part of a function goes on which axis. In $f(x)$, the $x$ inside the parentheses is the input, graphed on the x-axis, and the whole $f(x)$ is the output, graphed on the y-axis, so read it as $y$.

Owen’s second miss in the walkthrough came from mixing up the input and the output. A question that says $f(x) = 14$ has given you an output, so the equation to solve is $3x + 2 = 14$.

Constants and Variables

A “constant” is a single number, even when the question doesn’t tell you its value. A “variable” can take a whole range of values. The digital SAT often builds questions around constants, such as a parabola “where $b$ is a positive constant,” and a student who treats every letter as an unknown can feel lost in a question that is simpler than it looks. When a question says a letter is a constant, it is giving you information: that letter is one fixed number, and the rest of the question often pins it down.

Desmos has its own rules about letters. It treats $x$ and $y$ as the graph’s axes, reserves a few other letters such as $e$, and offers a “slider” for any other letter, which lets you set that letter to one value. Desmos applies those rules whatever the question calls a letter. If a question uses $t$ as its variable, rename it $x$ before you graph. If a question says $x$ is a constant, which the SAT occasionally does, then $x$ is just some number in that question.

You build this kind of understanding by doing problems, thinking about what you’re doing, explaining why each step works, and getting feedback from a tutor or an AI while you explain, so your thinking gets corrected as you go.

Graphs, Tables, and Equations Each Make a Different Fact Easy to See

A relationship between two quantities, such as a phone bill and the data used, can be written as an equation, a table, a graph, or a sentence in words. Strong math students move between these forms to find the one that makes a question easy. Each form makes a different fact easy to see: the equation shows the rule, the table shows exact values, the graph shows the shape, and the words show what the numbers mean.

Switching between forms gets easier with practice. Take a relationship from a problem you solved and write it in the forms the problem didn’t use.

A Personal Reason to Learn Math

The last skill on this page is motivation: having your own reason to learn the math. Find some way to make the math important to you, or at least relevant, whether that is a college plan, a career, or plain curiosity.

Memorization has a place: it helps you recall patterns quickly, which saves time on the test. It often stands in for understanding, though, and a memorized step fails as soon as a question changes its shape. When a topic feels overwhelming, ask what is simple about it, and break it down until you see the idea underneath. Many SAT topics reuse the same simple ideas: linear and exponential change, for example, come down to adding the same amount at each step or multiplying by the same factor.

Your Missed Questions Point to the Skill to Practice First

Most students are strong in some of these skills and weak in others. Look at your recent misses the way the walkthrough looked at Owen’s, and name the skill behind each one. Practice first the weak skill behind most of your misses.

After a few weeks of practice, sort your new misses the same way. Once the first skill is solid, most of your remaining misses will point to a different skill, and that skill is the one to practice next.

Sources

1. 01Psychological Science in the Public InterestImproving Students' Learning With Effective Learning TechniquesA 2013 review by John Dunlosky and colleagues that rates explaining your own steps and asking why a fact is true as moderately useful study techniques.