Algebra help that checks each step you write

alai is a notes app for iPad and the web with Ally, an adaptive AI tutor that teaches alongside you. In Homework help, you write your solution line by line, the way you would on paper, and each time you pause, Ally checks the newest line against the ones above it.

Where algebra goes wrong on paper

An algebra solution is a column of lines, and each line rewrites the one above it. One slip carries into every line below, so a long solution can end in a wrong answer even when most of the steps are right.

Researchers who went through Algebra I students' written work sorted the errors into six groups: variables, negative signs, equality and inequality, operations, fractions, and mathematical properties (Booth and colleagues, 2014). Paul Dawkins wrote a list of the errors that came up often in his Calculus I class at Lamar University, and improper distribution and dropped parentheses are both on it (Common Math Errors).

Solving equations

Each of these slips happens while you rewrite one line into the next:

• Distributing to only the first term. 2(x + 3) = 14 becomes 2x + 3 = 14. The 2 multiplies both terms, so the line should read 2x + 6 = 14.
• Losing a negative sign. −2(x − 4) becomes −2x − 8. The −2 times the −4 is +8, so it should be −2x + 8.
• Doing different things to each side. From 2x + 6 = 14, subtracting 6 from both sides gives 2x = 8. Writing 2x = 14 + 6 takes the 6 away on the left but adds it on the right.
• Dividing part of a side. Dividing 2x + 6 = 14 by 2 gives x + 3 = 7. Writing x + 6 = 7 divides only the first term on the left.

Factoring

• Mixing up the signs. x² + 5x − 14 factors as (x + 7)(x − 2). If you multiply (x − 7)(x + 2) back out, you get x² − 5x − 14.
• Stopping too early. 3x³ − 2x² − 5x = x(3x² − 2x − 5) is correct, but 3x² − 2x − 5 still factors, so the full answer is x(3x − 5)(x + 1).
• Squaring each term. (x + 3)² is x² + 6x + 9. Writing x² + 9 leaves out the middle term.
• Dividing by x. Solving x² = 3x by dividing both sides by x gives x = 3 and loses x = 0. Writing it as x² − 3x = 0 and factoring to x(x − 3) = 0 keeps both solutions.

Systems of equations

A system adds a second variable and more places to lose a sign. Take 2x + y = 7 and x − y = 2. Solving the first equation for y gives y = 7 − 2x, and putting that into the second gives x − (7 − 2x) = 2. The minus sign in front of the brackets has to reach both terms inside, so the next line is x − 7 + 2x = 2, which gives 3x = 9 and x = 3. Writing x − 7 − 2x = 2 instead leads to x = −9. Another slip is stopping at x = 3 without going back to find y, which is 1.

How Ally helps as you work

In Homework help, when you lift your pencil for about two seconds, or tap Check my work, Ally reads the line you just wrote along with everything above it. If every step so far is right, Ally stays quiet and you keep going.

When a line doesn't follow, Ally highlights the term that went wrong and asks about it in one short sentence. The hint is meant to stop one step short: it points at the slip and leaves the corrected line, and the answer, to you. If you're still stuck after it, the next hint is smaller and more concrete.

Say you're solving 2(x + 3) = 14 and your next line is 2x + 3 = 14. Ally highlights the 3 and asks a question like “Nice job getting 2x; does the 2 reach the 3 too?” You look back at the brackets, see that the 2 multiplies the 3 as well, and write 2x + 6 = 14 yourself.

Ally also checks that an answer is finished. If you factor 3x³ − 2x² − 5x as x(3x² − 2x − 5) and stop, nothing you wrote is wrong, but the answer isn't done yet. Ally says so with a question like “Can 3x² − 2x − 5 be factored further?” An answer of x = 6/4 gets the same kind of nudge, because the fraction still reduces.

A few more things that help in algebra:

• Your own method counts. Before calling a line wrong, Ally checks whether it still holds at the solution. Skipping a step or solving in a different order is fine.
• Crossed-out work is ignored. Strike through a line or scribble over it, and Ally leaves it alone.
• Ask why out loud. Hold the talk button and ask. If you ask which two numbers you need for x² + 5x − 14, Ally reminds you that they multiply to −14 and add to 5, and leaves finding them to you.
• Memory of past slips. If you've distributed to only the first term before, Ally can bring up what helped last time.

Common algebra mistakes, one page each

• Distributing to every term: why 2(x + 3) is 2x + 6
• Distributing a negative sign: why −(x − 5) is −x + 5
• Squaring a binomial: why (a + b)² is a² + 2ab + b²
• Canceling terms in a fraction: when you can cancel and when you can't
• Adding fractions: why 1/3 + 1/4 is 7/12
• Flipping the inequality sign: multiplying or dividing by a negative
• Dividing by a variable: how it can lose a solution
• Exponent rules: why x² · x³ is x⁵

Comparing alai with other apps

To see how alai compares with other apps and who each one suits, read alai vs Photomath, alai vs Apple Math Notes, alai vs ChatGPT study mode and alai vs Goodnotes Math Assist. For checking any handwritten math yourself, see how to check handwritten math step by step.