Algebra I

What people get wrong

1. Forgetting to reverse the inequality symbol when multiplying or dividing both sides by a negative number. Instead, flip the sign every time you multiply or divide by a negative, and check with a test value if unsure.
2. Distributing a negative sign incorrectly, for example treating $-3(x+2)$ as $-3x+2$ instead of $-3x-6$. Instead, multiply every term inside the parentheses by the full coefficient, sign included.
3. When rearranging a formula, dividing only one term on a side instead of the entire side by the divisor, writing $w = P - 2l/2$ instead of $w = (P-2l)/2$. Instead, treat the whole side being divided as one quantity in parentheses.
4. Accepting any algebraic solution to a constraint without checking it makes sense in context, such as a negative number of tickets or a fractional number of people. Instead, always test the solution against the real-world limits of the variable.

What people get wrong

1. Forgetting to flip the inequality symbol when multiplying or dividing both sides by a negative number while solving a system of inequalities — always check the sign of the coefficient before dividing.
2. Treating f(-2) as -f(2), i.e. evaluating the function then negating it, instead of substituting -2 directly for every x in the rule — always replace the variable itself with the given input.
3. Reading a graph's domain or range and including an endpoint that has an open circle — an open circle means that value is not included, so the inequality must be strict ( ), not ≤ or ≥.
4. Applying the geometric sequence formula (multiplying by r) to a sequence that is actually arithmetic, or vice versa — check whether consecutive terms have a constant difference or a constant ratio before choosing a formula.

What people get wrong

1. Using the mean and standard deviation to describe a skewed data set or one with an outlier. Instead, check the shape first (a box plot or dot plot showing asymmetry or an extreme value) and switch to median and IQR when the data are not roughly symmetric.
2. Including the overall median in both halves when finding Q1 and Q3 for an odd-sized data set. Instead, split the data into a strict lower half and upper half, excluding the middle value itself, before taking each half's median.
3. Computing a residual as predicted minus observed. Instead, always use residual = observed y − predicted y (ŷ), so a positive residual means the actual point lies above the line.
4. Reading a two-way table by dividing by the grand total when the question asks for a conditional relative frequency. Instead, identify whether the question restricts attention to one row or one column, and divide only by that row's or column's total.

What people get wrong

1. When subtracting one polynomial from another, students distribute the negative sign to only the first term of the second polynomial. Instead, rewrite the subtraction as adding the opposite of every term before combining like terms.
2. When completing the square, students add $(b/2)^2$ to only one side of the equation, which breaks the equality. Instead, add the same value to both sides, or move the constant to the other side first and add there too.
3. When factoring a trinomial with a leading coefficient other than 1 (like $2x^2+7x+3$), students apply the simple $(x+p)(x+q)$ pattern that only works when $a=1$. Instead, use the ac-method or check for a common factor first, and always verify by multiplying back.
4. When applying the quadratic formula, students mishandle the negative sign on $b$, especially when squaring it, writing $b^2$ instead of correctly computing $(-b)^2$ or $b^2-4ac$ with a negative $b$. Instead, substitute all values in parentheses first, then simplify.

What people get wrong

1. Dropping the negative sign in x=-b/(2a) — always write the formula with the negative explicitly before substituting values, since a common error is computing b/(2a) instead.
2. Getting the sign of h backwards in vertex form — in f(x)=a(x-h)^2+k, a term like (x+3)^2 means h=-3 (shift left 3), not h=3; rewrite (x+3) as (x-(-3)) before reading off h.
3. Assuming a square-root function has domain 'all real numbers' — always set the radicand ≥0 and solve for x before stating the domain.
4. Misreading which piece of a piecewise function applies at a boundary point — check whether the inequality on each piece is strict or non-strict, since the point can only belong to one piece.