AMC 10 · 2004 · #3
Grade 4 arithmeticPick an answer.
AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The problem hands you the END of the chain (the fifth practice, 48) and asks for the START (the first practice). Tool #11 (Work Backwards) fits exactly: instead of building up, undo the rule one step at a time. Forward the rule is "multiply by 2," so backward it is "divide by 2." Tool #5 (Look for a Pattern) names the structure — a doubling chain — so you know there are exactly four backward steps from the fifth practice to the first.
See the doubling chain
Line up the five practices: each is double the one before, so four doublings run from the first up to the fifth, which is 48.
"Twice as many" is a times-2 jump, and there are four jumps from the first practice to the fifth.
Twice as many is a times-two jump, and four jumps separate the first practice from the fifth.
▸ Why?
Twice as many is one equal group repeated twice, which is exactly what multiplying records.
▸ Why?
Halving undoes doubling exactly, so stepping backwards from a known later value recovers the earlier one.
Undo one doubling
Undo one doubling by halving: 48÷2=24, so the fourth practice was 24 free throws.
Halving reverses doubling, so one step back from 48 is 24.
3.OA.C.7Work BackwardsKeep halving back to the first
Keep halving back: 24÷2=12, then 12÷2=6, then 6÷2=3 at the first practice.
Each half-step peels off one doubling until only the first practice is left.
3.OA.C.7Work BackwardsRead off and check the answer
So the first practice was 3 free throws, choice (A). Forward-check: 3→6→12→24→48 hits 48 at the fifth.
If halving down and doubling back up match the given 48, the starting value is confirmed.
3.OA.C.7Work BackwardsWhen the end of a doubling chain is given and you want the start, walk backward by halving one step at a time.
- See the doubling chain
- Undo one doubling
- Keep halving back to the first
- Read off and check the answer