AMC 10 · 2010 · #6
Grade 6 algebraPick an answer.
The question asks for the biggest and smallest a quantity can be, which is the signature of Tool #14 (Extreme Principle): solve by finding the boundary cases. Tool #15 (Organize Information in More Ways) sorts every student into a four-way start-to-end table so nobody is double counted. Tool #4 (Introduce a Variable) names the one free choice — how many students flip from "Yes" to "No" — turning the switch count into a single expression whose two ends give the answer.
Sort students into four groups
Four groups describe every student exactly.
Splitting the class into start-then-end groups turns "changed their answer" into an exact, countable set instead of a vague idea.
6.RP.A.3Organize Information In More WaysName the flips and link them
The two flip counts differ by a fixed amount.
The Yes total only shifts by the difference between students walking in and students walking out, so that difference is locked at 20.
The Yes total only shifts by the difference between students walking in and students walking out, so that difference is locked.
▸ Why?
Both counts are changed by their own flows, so their difference records exactly the net movement.
▸ Why?
The class is exactly its four groups put together, so the group sizes account for every student.
Write the switch count as one expression
So the switch rate rides on one unknown.
With one variable controlling both flip groups, the whole answer rides on the single number s.
6.EE.B.6Introduce A VariablePush s to its extremes
That unknown runs from zero to 30.
Squeezing a group down to 0 — either no forced flips, or no leftover stayers — is exactly what pins each boundary case.
6.EE.B.8Extreme PrincipleCompare the extremes
The extremes differ by 60, choice (D).
The gap between the biggest and smallest switch counts is the whole question — it comes out to 60, choice (D).
6.EE.B.8Extreme PrincipleSort everyone into who-switched groups, notice the Yes side only moves by the net in-minus-out, then push the free flip count to its lowest and highest to bracket the answer.
- Sort students into four groups
- Name the flips and link them
- Write the switch count as one expression
- Push s to its extremes
- Compare the extremes