AMC 10 · 2015 · #3
Grade 4 arithmetic
Pick an answer.
AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Counting the whole 5-step figure from scratch is error-prone. Instead, track how the toothpick total grows by one step at a time. The jump from one staircase to the next follows a steady pattern, so I can extend that pattern from 3 steps up to 5 steps and add only the new pieces.
Count smaller staircases
Count small staircases: 1-step=4, 2-step=10, 3-step (given)=18 toothpicks.
Starting small makes the growth visible before the numbers get big.
4.OA.C.5Draw A DiagramFind the growth pattern
Each jump grows by 2: 1→2 step adds 6, 2→3 step adds 8 toothpicks.
Each new step is wider than the last, so each jump adds two more toothpicks than the jump before.
Each new step is wider than the last, so each jump adds two more toothpicks than the jump before.
▸ Why?
The jumps themselves grow by the same fixed amount each time.
▸ Why?
So the running total can be gathered by pairing the first jump with the last.
Extend to steps 4 and 5
Continue the pattern: 3→4 step adds 10, 4→5 step adds 12 toothpicks.
Keep stepping the jumps up by 2 to reach the steps you care about.
4.OA.C.5Look For A PatternAdd the two jumps
Add both jumps: 10 + 12 = 22 toothpicks — choice (D).
Two separate growth steps just stack, so their added toothpicks add together.
4.NBT.B.4Identify SubproblemsWhen a shape grows step by step, watch how much each step adds, then add up only the new pieces.
- Count smaller staircases
- Find the growth pattern
- Extend to steps 4 and 5
- Add the two jumps