Desmos or by hand? A question-by-question table for digital SAT math

Short answer: graph it when the question gives you an equation, a system, a function or a table and asks where, how many, or what constant. Do it by hand when it is arithmetic, a one-step equation, a rewrite of an expression, or an answer written with a root or pi. The calculator is open on every math question on the digital SAT, so the question is never whether you may use it. It is whether it is faster, and the decision should take about five seconds.

Here is the whole decision as one table, then the reasoning behind the rows that students get wrong most often.

The table

Question type	Faster	The move	Trap to watch
System of two linear equations	Desmos	Type both as written, click the crossing	Answering x when it asked for x − y
Line and curve: how many solutions	Desmos	Graph both, zoom out, count crossings	The second crossing is off the starting screen
Zeros, vertex, max or min of a quadratic	Desmos	Type y = f(x), click the gray dots	Reporting the x of the vertex when it asked for the value
Radical or rational equation	Desmos	Each side as its own graph, read the crossings	None: graphs never show extraneous roots
“For what k does it have one solution?” (one equation)	Desmos	Replace k with y, click the turning point	Typing k and fighting a slider
“For what k are the lines parallel / tangent?” (two graphs)	Desmos	Type k, accept the slider, drag	Slider range defaults to −10 to 10
System of inequalities: which point works	Desmos	Graph both, type each choice as a point	Dashed lines do not count as included
Table of values to an equation	Desmos	Table, then y₁ ~ mx₁ + b or y₁ ~ ab^x₁	A letter left over as a slider breaks the fit
Circle in general form	Desmos	Type it exactly as printed	Completing the square by hand instead
Circle through three points	Usually Desmos	Table, then (x₁−h)² + (y₁−k)² ~ r²	A right angle in the points makes it a ten-second hand question
Mean, median, standard deviation of a list	Desmos	L = [ … ], then mean(L)	Frequency tables: do those by hand
One- or two-step linear equation	By hand	Two lines of arithmetic	Opening the calculator out of habit
Percents, discounts, rates	By hand	One multiplication or division	None, if you set it up first
“Infinitely many solutions” for a linear equation	By hand	Match the two sides	A slider cannot show you “the same line” quickly
Equivalent expressions	By hand	Expand or factor	Graphing all four choices takes a minute
Exact answers with √ or π	By hand	Simplify, or compare decimals choice by choice	Desmos gives 1.414 and the choice says √2
Setting up a word problem	By hand first	Write the equation, then graph it if it helps	Desmos cannot read the words
“Find 8x + 3” when x is easy	By hand	Solve for the expression, not x	The calculator answers the wrong question confidently

How do you decide in five seconds?

Look at what the question hands you, not what it asks. If it hands you something you could type, an equation, two equations, a function, a table, then graphing is probably faster. If it hands you a number, a percent or a sentence, it probably is not. Then check the exception rows: an answer choice with a root or pi, or an expression to rewrite, goes back to the pencil.

That rule is the one the fifteen-second rule grows out of. If Desmos is not giving you the answer within fifteen seconds of starting to type, you chose wrong, and switching now is cheaper than finishing.

Why does Desmos win the constant questions?

Because it turns algebra into looking. A question about a constant, for what value of k does this have exactly one solution, is really a question about where a horizontal line touches a curve. Replace k with y and Desmos draws the curve; the turning point is the answer. No discriminant, no arithmetic in the head, nothing to get wrong. The question review on this move works one example both ways with the clock running.

The trap is the slider. Typed with the k left in, Desmos offers a slider, and dragging one looking for an exact value is slow. Sliders are for two-graph conditions: a line tangent to a parabola, two lines parallel.

Why is the radical equation a Desmos question?

Because squaring both sides by hand can create a solution that is not there, and the graph never squares anything. Graph each side, and the crossings are the solutions, with no extraneous root to check for. The first question review shows the same equation solved both ways: seventy seconds by hand, fifteen graphed, and the graphed one cannot fall for the trap.

Why are equivalent expressions a hand question?

Because there is nothing to look at. You can graph the original and all four choices and see which one overlaps, and it works, but it is five lines of typing and a minute of time for a question that is two lines of expanding. The same goes for percents and one-step equations. The calculator is not wrong on these; it is just slower than the pencil.

When does by hand beat Desmos on a question Desmos can do?

When the question has a shortcut it was written around. The circle through three points is the clean example: a tilde regression always works in about twenty seconds, but when the three points form a right angle, the hypotenuse is a diameter and the answer takes ten. The SAT puts those right angles in on purpose. Learn the regression because it always works, then learn to spot the shortcut.

What to practice

The table is not something to memorize. It is something to drill until the five-second decision is automatic. The way I do it with students: take twenty mixed questions from the official Question Bank, and before solving any of them, write “D” or “H” next to each. Then solve, timing each one. The questions where you wrote the wrong letter are the ones to study, and they are almost always the same three or four rows of the table for any given student.

And practice the keystrokes in the same calculator the test uses, because a move you know but cannot type quickly is a move you will not use on test day.