Common mistakes and how to catch them
Each page in this library covers one common slip. It shows the wrong line next to the right one, explains the idea behind the slip, and gives you one question to ask yourself so you can catch it in your own work. Every page ends with three practice problems and their answers.
The library comes from alai, an iPad app that reads your handwriting as you work. In Homework help, when you pause after writing a step, the tutor, Ally, reads it. When a step is wrong, Ally underlines the part that slipped and asks a question about it, so you find the fix yourself. The questions on these pages are written the same way: each one points at one part of the work and leaves the answer to you.
The pages are grouped by subject:
- Algebra: distributing, minus signs, squaring a binomial, canceling, adding fractions, inequalities, dividing by a variable and exponent rules
- Calculus: the chain rule, the product rule, the constant of integration and 0/0 in limits
- Physics: unit conversions, the sign of gravity and vector components
- Chemistry: balancing equations without changing subscripts, mole ratios and significant figures
Studying a wrong step next to the right one is something researchers have tested in classrooms. AlgebraByExample, a set of Algebra 1 assignments from the SERP Institute, pairs worked examples, some correct and some with a mistake in them, with a problem to try on your own. In a year-long study, students who usually struggle in algebra scored an average of 10 percentage points higher on conceptual knowledge than peers in control classrooms taught by the same teacher.
- Distributing to every termThe number outside the parentheses multiplies every term inside, so 2(x + 3) = 2x + 6, and writing 2x + 3 means the 2 skipped the 3.
- Distributing a negative signA minus sign in front of parentheses multiplies every term inside by −1, so −(x − 4) = −x + 4 and 5 − 2(x − 3) = 11 − 2x.
- Squaring a binomial(a + b)² means (a + b)(a + b), which multiplies out to a² + 2ab + b², so writing a² + b² leaves out the middle term, 2ab.
- Canceling terms in a fractionYou can cancel only a factor of the whole top and the whole bottom, so the 6 in (x + 6)/6 can't be canceled, because it's added to x.
- Adding fractionsTo add fractions, rewrite them over a common denominator and then add the numerators: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
- Flipping the inequality signFlip the inequality sign whenever you multiply or divide both sides by a negative number, so −3x < 12 becomes x > −4.
- Dividing by a variableDividing both sides of x² = 5x by x throws away the solution x = 0, so move everything to one side and factor: x(x − 5) = 0.
- Exponent rulesMultiplying powers of the same base adds the exponents, x² · x³ = x⁵, while raising a power to a power multiplies them, (x²)³ = x⁶.
- Forgetting the chain ruleTo differentiate (3x + 1)⁵, multiply by the derivative of the inside, so the derivative is 5(3x + 1)⁴ · 3 = 15(3x + 1)⁴.
- The product ruleThe derivative of f(x)g(x) is f′(x)g(x) + f(x)g′(x), so the derivative of x² sin x is 2x sin x + x² cos x.
- The constant of integrationAn indefinite integral stands for every antiderivative, and those differ by a constant, so ∫ 2x dx = x² + C.
- 0/0 in limitsGetting 0/0 when you plug into a limit means you should simplify first: (x² − 9)/(x − 3) becomes x + 3, so the limit as x → 3 is 6.
- Unit conversionsA common unit conversion slip is an upside-down factor, so write units on every factor and check that the old unit cancels before you multiply.
- The sign of gravityWith up as positive, anything in free fall has a = −9.80 m/s², and the slip is writing +9.80 or putting −9.80 into a formula that already subtracts g.
- Vector componentsUse cosine for the vector component next to the angle and sine for the one across from it, wherever the angle is measured from.
- Changing subscripts to balanceChanging a subscript turns one substance into another, like H₂O into H₂O₂, so balance a chemical equation by changing only the coefficients.
- Mole ratiosThe coefficients in a balanced equation compare moles, so convert grams to moles before you use the mole ratio, then convert back to grams.
- Significant figuresWhen you multiply or divide, keep the fewest significant figures, and when you add or subtract, keep the fewest decimal places.