Sensim Math · Depth 한국어

2-2 · Finding Patterns

Find the rule of a growing block stack

4.OA.C.54.OA.A.3 · take · grade 4

Archetype: Generalize a Growing Pattern into a Rule · step in a 12-type progression

▶ Practice — 8 problems

Building blocks are stacked following a rule. How many blocks in all are needed to build the stack so that it is 77 layers tall?

The block shapes grow from left to right, one figure after another.

Each figure is an L shape: one side rises one block taller while the bottom row also stretches one block longer. Each time a layer is added, the number of blocks increases by the same amount.

1 3 5 7
Show solution

Understand

Blocks are stacked into L-shaped figures that grow by a fixed rule: 1, 3, 5, 7 blocks for the first four layers. Find the total number of blocks needed to build the stack up to 7 layers tall.

Givens
  • Layer 1 uses 1 block.
  • Layer 2 uses 3 blocks.
  • Layer 3 uses 5 blocks.
  • Layer 4 uses 7 blocks.
  • Each new layer adds the same amount more than the one before.
Unknowns
  • The total number of blocks in all 7 layers combined.
Constraints
  • The number of blocks in each layer follows the pattern 1, 3, 5, 7, ... (the odd numbers).
  • We need every layer from 1 through 7 added together.

Plan

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

The layer counts 1, 3, 5, 7 increase by 2 each time, so I extend that pattern to 7 layers, then add them up (building from the small known sums).

Execute

#5 Look for a Pattern 4.OA.C.5
Each layer has 2 more blocks than the one before, so after 1, 3, 5, 7 the next layers are 7+2=9, 9+2=11, 11+2=13. The seven layers use 1, 3, 5, 7, 9, 11, 13 blocks.
1, 3, 5, 7, 9, 11, 131,\ 3,\ 5,\ 7,\ 9,\ 11,\ 13
Once you see each step adds 2, you can keep counting up by 2 to reach any layer.
#9 Solve an Easier Related Problem 2.OA.B.2
Add the seven layer counts: 1+3=4, +5=9, +7=16, +9=25, +11=36, +13=49.
1+3+5+7+9+11+13=491+3+5+7+9+11+13 = 49
Summing a short list of small numbers step by step keeps the running total easy to track.
Answer: 49 blocks

Review

The sum of the first 7 odd numbers is 7 x 7 = 49, a known shortcut that confirms the layer-by-layer addition. 49 is a sensible total for a 7-layer growing stack.

Pair the outer terms: (1+13)+(3+11)+(5+9)+7 = 14+14+14+7 = 49, the same total found by pairing instead of running addition.

Standards · min grade 4

  • 4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the layer pattern 1, 3, 5, 7 to all seven layers.
  • 2.OA.B.2 Fluently add and subtract within 20 using mental strategies — Adding the seven layer counts together to get the total.
💡 Each layer grows by 2 blocks, and adding up the odd numbers 1+3+5+7+9+11+13 gives a neat 49!