How to check your handwritten math step by step

The method in five steps

1. Estimate the answer before you start.
2. Check each line against the line before it.
3. Substitute your answer into the original problem.
4. Check the units.
5. Look through the common slips for the topic.

1. Estimate first

A rough answer before you start gives you something to compare with at the end. Round the numbers and ask what size and sign the answer should have.

Say a car travels 150 km in 2.5 hours. That's 150 km in somewhere between 2 and 3 hours, so the speed should be between 50 and 75 km/h. If your work ends at 375 km/h, you multiplied where you should have divided. The real answer is 150 ÷ 2.5 = 60 km/h.

Estimates help most with word problems, physics and chemistry. For a pure algebra problem, the substitution check in step 3 does more.

2. Check each line against the line before it

Every line of a solution should say the same thing as the line before it, in a new form. You can test that without knowing the answer.

For an expression, put the same small number into both lines. They should give the same value. Take 2(x + 3) and the next line 2x + 3, and try x = 2:

• 2(2 + 3) = 10
• 2(2) + 3 = 7

The values don't match, so that step is wrong. The 2 has to multiply the 3 as well, which gives 2x + 6. Use a number like 2 or 3 for this. Zero and one can hide mistakes, because many different expressions are equal there: x² and x are both 1 when x = 1.

For an equation, a correct step keeps the same solution. If your final answer doesn't work in the original problem, put it into each of your lines from the top. The first line where it starts working comes right after the slip.

The slip is between lines 1 and 2: the 3 reached the x but missed the 4. Fixing it gives 3x + 12 = 2x + 19, so x = 7, and 3(7 + 4) = 33 = 2(7) + 19.

A few kinds of step can change the solutions even when the algebra looks right, so give these a second look:

• Dividing by something with a variable in it can lose a solution. Dividing x² = 3x by x leaves x = 3 and loses x = 0.
• Squaring both sides can add a solution that doesn't work. Squaring √(x + 2) = x gives x = 2 or x = −1, and only x = 2 works in the original.
• Clearing a fraction with a variable in the denominator can give a value that makes the denominator zero, and that value can't be a solution.
• Multiplying or dividing an inequality by a negative number flips the direction of the sign.

3. Substitute your answer into the original problem

Put your answer back into the problem exactly as it was given. One of your own lines won't do, because a slip in an early line carries through to everything after it.

• More than one answer: check each one. This is how you catch the extra solution from squaring, or a value that makes a denominator zero.
• Word problems: check the answer against the words. A length can't be negative, and a number of people has to be a whole number.
• Systems of equations: check every equation, since an answer can fit one and miss another.
• Factoring: multiply the factors back out and compare with the original expression.
• Integrals: differentiate your answer. You should get back the function you integrated.
• Inequalities: test one number inside your answer and one outside it.

4. Check the units

In physics and chemistry, write the units on every line, all the way through. Units follow the same algebra as the numbers, so they catch slips the numbers hide. If a speed comes out in m·s, you multiplied where you should have divided, since speed is in m/s. Put every quantity in the same units before you combine them: 72 km/h is 20 m/s.

5. Look through the common slips

Many mistakes are one of a handful of familiar slips. When a check fails, these are good places to look first. Each one has its own page.

Algebra

• Distributing to every term: 2(x + 3) = 2x + 6. The number outside multiplies every term inside.
• Distributing a negative sign: −(x − 4) = −x + 4. The minus changes the sign of every term inside.
• Squaring a binomial: (x + 3)² = x² + 6x + 9. The middle term is the one that goes missing.
• Cancelling terms in a fraction: you can cancel a factor of the whole top and the whole bottom, so (2x + 6) ÷ 2 = x + 3. In (x + 6) ÷ 6, the 6 can't cancel on its own.
• Adding fractions: ½ + ⅓ = 5/6. Find a common denominator first.
• Flipping the inequality sign: dividing −2x < 6 by −2 gives x > −3.
• Dividing by a variable: dividing x² = 3x by x loses the solution x = 0.
• Exponent rules: x² × x³ = x⁵, and (x²)³ = x⁶.

Calculus

• Forgetting the chain rule: the derivative of sin(3x) is 3cos(3x).
• The product rule: the derivative of x² sin x is 2x sin x + x² cos x.
• The constant of integration: ∫ 2x dx = x² + C.
• Zero over zero in limits: getting 0/0 means you haven't found the limit yet. Factor and simplify, then try again.

Physics

• Unit conversions: 72 km/h = 20 m/s.
• The sign of gravity: choose which way is positive and keep it. With up as positive, the acceleration from gravity is −9.8 m/s².
• Vector components: a force F at an angle θ above the horizontal has a horizontal part F cos θ and a vertical part F sin θ.

Chemistry

• Balancing equations with subscripts: change the coefficients and leave the subscripts alone, as in 2H₂ + O₂ → 2H₂O.
• Mole ratios: take the ratio from the balanced equation. In 2H₂ + O₂ → 2H₂O, 2 mol of H₂ react with 1 mol of O₂.
• Significant figures: a product keeps the significant figures of the least precise measurement, so 2.0 × 3.14 = 6.3.

Check AI answers the same way

If you use an AI tool, check what it gives you with the same steps. In a 2025 PNAS study, a ChatGPT-style tutor built on GPT-4 and asked in 2023 for the answers to 57 high school practice problems was right 51% of the time. The authors note that models have improved a lot since then, and the same checks work on whatever an AI gives you. There's more in homework help that gives hints instead of answers.

How alai checks each step while you work

alai is an iPad notes app with a tutor, Ally, who reads your handwriting as you work. Ally checks your lines each time you pause and leaves the fixing to you.

• When Ally reads: in Homework help, when you lift your pencil for about two seconds, Ally reads the page. Right after you erase, Ally waits a little longer, since you'll probably rewrite.
• What Ally judges: only your handwriting. Crossed-out lines and doodles are ignored, a line you're still writing isn't treated as wrong, and a correct method that differs from the answer key still counts. Before calling a line wrong, Ally is set up to check it the way this page does: a correct step keeps the same solution.
• What Ally says: nothing while your steps are right. At the first wrong line, Ally highlights the exact term or line and gives one short hint, usually a question, without saying the corrected step. If a line is too messy to read, Ally asks you to rewrite it or say it out loud.
• When you want a check now: the Check my work button asks Ally to look right away. If you stop writing for a while, Ally may offer the smallest next step as a question.
• When it's finished: Ally only calls a problem solved once the answer is in the form the question asks for, such as a reduced fraction, a fully factored polynomial or a physics answer with its units.

For the line 3x + 4 under 3(x + 4) = 2x + 19, Ally underlines the +4 and asks, “Did the 3 reach every term inside the brackets?” Ally can still misread handwriting or get a check wrong, so substituting your final answer back in is worth doing every time.

Other apps read handwritten math too. Apple's Math Notes solves expressions you write with an equals sign and marks what it can't read or solve. Goodnotes' Math Assist solves equations you write and can underline lines it reads as incorrect. The comparisons with Apple Math Notes and Goodnotes Math Assist go through how each one works.