Use estimation to judge whether an answer is reasonable, substitute solutions back to verify, and apply units and benchmarks to catch errors before submitting.

What this topic is asking

A reliable test-taker does not just compute an answer; they judge whether it is reasonable and check it. On the Grade 10 MCAS, estimation and verification catch arithmetic slips, rule out impossible multiple-choice options, and confirm constructed-response answers. This section is about building those habits so errors are caught before they cost points.

Estimating to judge reasonableness

Before trusting an exact answer, form a rough estimate of what it should be. This is especially valuable on the calculator-free session and as a guard against keying errors on the calculator session.

• Square roots: trap between perfect squares. $\sqrt{80}$ lies between $\sqrt{64} = 8$ and $\sqrt{81} = 9$, so it is about 8.9.
• Percentages: 19% of 200 is roughly 20% of 200 = 40, so the exact answer should be near 38.
• Products and quotients: round to friendly numbers. $48 \times 21$ is about $50 \times 20 = 1000$, so an answer of 1008 is plausible but 108 is not.

If your exact answer is far from the estimate, you have likely made an error and should recheck.

Substituting back to verify

The most powerful check for an equation is to substitute the solution back into the original equation and confirm it works.

For a quadratic, substitute each root back; for a system, check the ordered pair in the equation you did not use to solve. Verification is quick and catches sign and arithmetic slips reliably.

Using units and context

Context and units rule out impossible answers immediately:

• A length, time, or count cannot be negative; reject a negative root in such a context.
• A probability is always between 0 and 1; an answer of 1.4 or $-0.2$ is impossible.
• A count of objects must be a whole number; 6.5 buses is not an answer, though it tells you 7 are needed.
• Units must match: an answer in square units when the question asks for a length signals a wrong formula (area instead of perimeter).

Checking that the answer's sign, size, and units fit the situation is a fast final guard.

Eliminating options on multiple choice

On selected-response items, you can often rule out wrong options before computing exactly. If the answer must be positive, eliminate negatives; if it must be a little more than 6, eliminate 14 and 40. This narrows the choices and, on a multiple-select, prevents adding an obviously wrong option that would lose the all-or-nothing point.

Working backward from the options

A particular strength of multiple-choice items is that the answer is in front of you, so when solving forward is slow, you can test the options. For "which value of $x$ satisfies $2x + 5 = 17$", substituting the candidates is sometimes faster than solving: $x = 6$ gives $2(6) + 5 = 17$, a match. This back-substitution is especially useful on the calculator-free session and as a check on an answer you found algebraically. It does not replace knowing how to solve, but it is a reliable safety net.

Checking units and the question asked

A final, often-missed check is whether you answered the actual question. A problem may ask for the perimeter, but it is easy to compute the area instead; or it may ask for the time the ball lands, while you found the maximum height. Re-read the question after solving and confirm your answer matches what was asked, in the right units. This catches a whole class of errors that are not arithmetic at all but come from answering a slightly different question than the one posed.

Try this

Q1. Between which integers does $\sqrt{55}$ lie?

• Cue. $7^2 = 49$ and $8^2 = 64$, so between 7 and 8.

Q2. A probability is computed as $1.3$. What does that tell you?

• Cue. It is impossible; recheck, since probability is between 0 and 1.