AMC 10 · 2021 · #3
Grade 7 algebraPick an answer.
Neither city's temperature is given, so Tool #4 (Introduce a Variable) names them M and L and turns "warmer by N" into M-L=N. The two individual temperatures then cancel, so Tool #16 (Change Focus) says to track only the gap between the cities — the gap is the one quantity the problem actually controls. Tool #13 (Convert to Algebra) turns the phrase "differ by 2 degrees" into an absolute-value equation, which splits into two cases; Tool #2 (Make a Systematic List) works through both cases so neither is missed, and Tool #3 (Eliminate Possibilities) checks that both survive before the values are multiplied.
Name the noon temperatures
Name the noon temperatures.
Naming both temperatures makes "warmer by N" into a subtraction you can carry through the rest of the problem.
6.EE.B.6Introduce A VariableWrite the 4:00 temperatures
Write the four o'clock temperatures.
A temperature change just adds a signed number to the old reading.
7.NS.A.1Convert To AlgebraTrack the gap, not the temperatures
Track the gap, not the temperatures.
The gap between two cities changes by the sum of the two moves toward each other, so only the gap matters.
The gap between two cities changes by the sum of the two moves toward each other, so only the gap matters.
▸ Why?
When two quantities both change, the gap between them changes at the difference of their moves.
▸ Why?
Whatever both share cancels out of the difference, so the starting temperatures never have to be found.
Turn "differ by 2" into an equation
Differing by two is an absolute value.
"Differ by" measures how far apart two numbers are, and a distance ignores which one is larger.
6.NS.C.7Convert To AlgebraList both cases
List both cases.
An absolute-value equation always opens into exactly two ordinary equations.
6.EE.B.7Make A Systematic ListCheck both, then multiply
Multiplying gives 60.
Neither case contradicts anything the problem said, so both values count toward the product.
7.NS.A.3Eliminate Possibilities"Differ by 2" only fixes how far apart the numbers are, not which one is bigger, so an absolute value opens two cases and both belong in the answer.
- Name the noon temperatures
- Write the 4:00 temperatures
- Track the gap, not the temperatures
- Turn "differ by 2" into an equation
- List both cases
- Check both, then multiply
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