ACT Math study guide · Chapter 44
Choose the Best Solving Route
Every question on this section can be solved by one or more of four routes: hand, working the algebra or arithmetic directly with no tool; handheld, using an approved physical calculator; Desmos, using the online graphing tool; or hybrid, combining hand reasoning for the setup with a tool for one specific sub-step. None of these four routes is correct by default. The fastest reliable route depends on what the specific question is actually asking, not on a fixed preference for one tool over another.
| Signal in the question | Route this signal favors |
|---|---|
| Simple arithmetic or a short, visible algebraic step | Hand. A calculator adds keystrokes to something already fast. |
| Repeated arithmetic, trig, or statistics computation on known values | Handheld. Fast and familiar once practiced. |
| Finding an intersection, a root, or a regression line; reading a graph's shape | Desmos. This is exactly what it is built to do. |
| A word problem requiring translation before any calculation begins | Hand for the setup, then any tool for the arithmetic. Hybrid. |
| A conceptual judgment: which theorem applies, what a study design supports, whether a number is rational | Hand. No tool decides this for you. |
▸ Route selection on a familiar problem type
Worked Example A quadratic equation has a negative discriminant. Should this be solved by hand, handheld, Desmos, or hybrid?
The algebra itself, the quadratic formula with a negative number under the radical, is a short, fixed sequence of steps.
Graphing this quadratic on Desmos would show no real x-intercepts, but it would not display the complex solutions themselves.
Answer: hand. This is fast by hand and Desmos cannot finish the job.
▸ Route selection on a system
Worked Example A system of two linear equations needs to be solved. Should this be solved by hand, handheld, Desmos, or hybrid?
Substitution or elimination by hand is fast if the coefficients are simple.
Graphing both lines on Desmos and reading the intersection is equally fast, and reduces the chance of an arithmetic slip.
Answer: either hand or Desmos reliably works here; this is a case where personal speed and comfort, not a fixed rule, should decide.
A harder version of this material asks you to recognize when a route that usually works stops being reliable for one specific problem, such as a system with fractional coefficients that make Desmos's default viewing window unhelpful without adjustment, at which point switching routes costs less time than fighting the original one.
There isn't a Desmos shortcut for deciding which route to use. Recognizing what a question is actually asking, from the opening chapter of this book onward, is what makes this decision fast.