AMC 10 · 2015 · #3

Easy mode Grade 4

Problem

Ann 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.

(A)

(B)

(C)

(D)

(E)

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.

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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

#1 Draw a Diagram 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.

$$1\text{-step}=4,\quad 2\text{-step}=10,\quad 3\text{-step}=18$$

💡 Starting small makes the growth visible before the numbers get big.

#5 Look for a Pattern 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.

$$+6,\ +8,\ \ldots\ (\text{each jump is } 2 \text{ more than the last})$$

💡 Each new step is wider than the last, so each jump adds two more toothpicks than the jump before.

#5 Look for a Pattern 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.

$$3\to4:\ +10,\qquad 4\to5:\ +12$$

💡 Keep stepping the jumps up by 2 to reach the steps you care about.

#7 Identify Subproblems 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).

$$10 + 12 = 22$$

💡 Two separate growth steps just stack, so their added toothpicks add together.

[1] #1 4.OA.C.5 Build up from the smallest staircases and count toothpicks. A 1-step staircase i

[2] #5 3.OA.D.9 Look at how much each staircase grows over the one before it. Going 1-step to 2-

[3] #5 4.OA.C.5 Continue the pattern of jumps. Going 3-step to 4-step adds 8 + 2 = 10 toothpicks

[4] #7 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.5 Generate 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.9 Identify arithmetic patterns and explain using properties of operations (Spotting that the toothpicks added each jump increase by a steady 2.)
4.NBT.B.4 Fluently 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.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Count smaller staircases

Find the growth pattern

Extend to steps 4 and 5

Add the two jumps

Review

CCSS standards used (min grade 4)

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