AMC 10 · 2015 · #3
Easy mode Grade 4Ann built a staircase shape out of toothpicks. It has 3 steps and uses 18 toothpicks, just like the picture shows. She wants to make it taller, with 5 steps. How many more toothpicks does she need to add?
Pick an answer.
AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A 3-step toothpick staircase is built from 18 toothpicks. Find how many more toothpicks must be added to turn it into a 5-step staircase.
Givens: A 3-step staircase uses 18 toothpicks.; The staircase grows one step at a time, each step made of toothpick edges.
Unknowns: The number of extra toothpicks needed to go from a 3-step to a 5-step staircase.
Understand
Restated: A 3-step toothpick staircase is built from 18 toothpicks. Find how many more toothpicks must be added to turn it into a 5-step staircase.
Givens: A 3-step staircase uses 18 toothpicks.; The staircase grows one step at a time, each step made of toothpick edges.
Plan
Primary tool: #5 Look for a Pattern
Secondary: #1 Draw a Diagram, #7 Identify Subproblems
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.
Execute — Answer: D
4.OA.C.5 Step 1 Count smaller staircases
- Build up from the smallest staircases and count toothpicks.
- A 1-step staircase is a single square = 4 toothpicks.
- A 2-step staircase = 10 toothpicks.
- The given 3-step staircase = 18 toothpicks.
💡 Starting small makes the growth visible before the numbers get big.
3.OA.D.9 Step 2 Find the growth pattern
- Look at how much each staircase grows over the one before it.
- Going 1-step to 2-step adds 10 - 4 = 6 toothpicks.
- Going 2-step to 3-step adds 18 - 10 = 8 toothpicks.
- The amount added goes up by 2 each time.
💡 Each new step is wider than the last, so each jump adds two more toothpicks than the jump before.
4.OA.C.5 Step 3 Extend to steps 4 and 5
- Continue the pattern of jumps.
- Going 3-step to 4-step adds 8 + 2 = 10 toothpicks.
- Going 4-step to 5-step adds 10 + 2 = 12 toothpicks.
💡 Keep stepping the jumps up by 2 to reach the steps you care about.
4.NBT.B.4 Step 4 Add the two jumps
- Getting from the 3-step staircase to the 5-step staircase means making both jumps, so add the two amounts: 10 + 12 = 22 toothpicks.
- That is the number of toothpicks Ann must add, so the answer is (D).
💡 Two separate growth steps just stack, so their added toothpicks add together.
4.OA.C.5 Build up from the smallest staircases and count toothpicks. A 1-step staircase i 3.OA.D.9 Look at how much each staircase grows over the one before it. Going 1-step to 2- 4.OA.C.5 Continue the pattern of jumps. Going 3-step to 4-step adds 8 + 2 = 10 toothpicks 4.NBT.B.4 Getting from the 3-step staircase to the 5-step staircase means making both jump Review
Reasonableness: Check against a direct count: an n-step staircase uses (n+1)(n+2) - 2 toothpicks, giving 18 for n=3 and (6)(7) - 2 = 40 for n=5. Then 40 - 18 = 22, matching the answer. 22 also sits sensibly between the choices 20 and 24.
Alternative: Use the formula (n+1)(n+2) - 2 directly: compute the 5-step total as 40, the 3-step total as 18, and subtract to get 22 without listing every jump.
CCSS standards used (min grade 4)
4.OA.C.5Generate a number or shape pattern following a given rule (Building the sequence of staircase toothpick totals and extending the shape pattern to 5 steps.)3.OA.D.9Identify arithmetic patterns and explain using properties of operations (Spotting that the toothpicks added each jump increase by a steady 2.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Combining the two jump amounts (10 + 12) and checking against the total difference 40 - 18.)
⭐ When a shape grows step by step, watch how much each step adds, then add up only the new pieces.
⭐ When a shape grows step by step, watch how much each step adds, then add up only the new pieces.
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