Mistake Master
Solve, then answer the ask, not the algebra
The most expensive SAT mistake costs nothing mathematically: perfect work, wrong report. The variable gets gridded when an expression was asked, the width answers a perimeter question, the other person's share stands in for the one named. Every distractor for it is pre-written. This lesson builds the two-second reread that keeps solved work worth its points.
§1
What this topic is about
▸
The most expensive mistake on the SAT costs nothing mathematically: the work is right and the reported number is not the one the question named. This unit-opening topic trains the two halves of the habit: never stopping at a solved variable or intermediate value, and never answering for the wrong quantity, the other person, the other variable, the total.
§2
The equation is a step, not the ask
▸
Most SAT algebra questions deliberately ask for an expression built FROM the variable ($2x$, $x + y$, the perimeter) rather than the variable itself. The distractor for the honest solver is always $x$.
- Solve, then substitute into the expression the question named.
- Intermediate values (a width, a total, one evaluation of a function) are candidates, never answers.
- The given constants of the problem reappear as distractors; a familiar-looking number is not a safe one.
Worked example. If $3x + 5 = 26$, what is the value of $2x$?
$3x = 21$, so $x = 7$.
$2x = 14$. Gridding $7$ (the variable) or $21$ (the $3x$ from the equation) hands in a step instead of the answer.
§3
Which quantity was named
▸
Problems with two people, two variables, or three consecutive integers make you solve for one quantity on the way to another. The setup usually delivers the UNNAMED one first.
- Underline the named quantity before solving: Jon's share, $y$, the largest.
- In a ratio split, the equation's natural output is one part; check whose part.
- Systems hand you $x$ first when the question wants $y$ almost every time.
Worked example. $x + y = 18$ and $x - y = 4$. What is $y$?
Adding gives $2x = 22$, so $x = 11$.
$y = 18 - 11 = 7$. The $11$ arrives first and looks final; the question named $y$.
§4
Multi-step asks
▸
Geometry and statistics wrap the ask one layer deeper: the width is needed for the perimeter, the total for the missing score. The intermediate value is genuinely required, which is exactly what makes it feel like the destination.
- Name the chain before computing: area gives width, width gives perimeter.
- After each step, ask: is this the named quantity or fuel for the next step?
- The last computation should output the question's own words.
Worked example. A rectangle has area $48$ and length $12$. What is its perimeter?
The width is $48 \div 12 = 4$.
$P = 2(12 + 4) = 32$. Stopping at $4$, or reporting the area $48$ back, are the two coded traps.
§5
The reread
▸
The whole topic compresses into one habit: after the last computation, reread the question's final sentence and check the number in your hand against the words.
Solve, then substitute into what was named; underline the asked quantity before starting; treat every intermediate value as fuel; and spend two seconds rereading the ask before gridding.
§6
Two patterns that cost real points
▸
Two patterns cover this topic. They are the same ones the diagnostic routes on.
The solved variable or a middle step gets reported.
$x = 7$ gets gridded when $2x$ was asked; a width answers a perimeter question; a total answers a question about one score.
Fix. The equation is a step. After the algebra, substitute into the expression the question actually named, and keep going until the output matches the ask's own words.
The wrong quantity gets reported.
The other person's share, the other variable, the total, or the smallest of the set stands in for the quantity the question named.
Fix. Underline the named quantity first. When the setup delivers a different quantity on the way, label it as someone else's answer and finish the trip.
Ten quick checks across the two patterns: expressions built from the variable, named quantities in splits and systems, multi-step geometry and averages, and comparison asks. Pick or type your answer, then check. Progress is saved.