← Find the rule of a growing block stack · Generalize a Growing Pattern into a Rule

Find the rule of a growing block stack · 8 practice problems

4.OA.C.52.OA.B.2

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Generated variants — 8

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy

Building blocks are stacked following a rule. How many blocks in all are needed to build the stack so that it is $6$ layers tall?

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

Layer 1: $1$ block
Layer 2: $4$ blocks
Layer 3: $7$ blocks

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.

Your answer

Show solution

1 · Understandwhat's really being asked

Blocks are stacked into L-shaped figures that grow by a fixed rule: 1, 4, 7 blocks for the first 3 layers. Find the total number of blocks needed to build the stack up to 6 layers tall.

Givens

Layer 1 uses 1 block.
Layer 2 uses 4 blocks.
Layer 3 uses 7 blocks.
Each new layer adds the same amount more than the one before.

Unknowns

The total number of blocks in all 6 layers combined.

Constraints

The number of blocks in each layer follows the pattern 1, 4, 7, ... (adding 3 each step).
We need every layer from 1 through 6 added together.

2 · Planchoose the strategy

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

The layer counts 1, 4, 7 increase by 3 each time, so I extend that pattern to 6 layers, then add them up (building from the small known sums).

3 · Execute2 carry out the plan

1Extend the layer pattern to 6 layers

#5 Look for a Pattern 4.OA.C.5

Each layer has 3 more blocks than the one before, so the 6 layers use 1, 4, 7, 10, 13, 16 blocks.

1,\ 4,\ 7,\ 10,\ 13,\ 16

Once you see each step adds 3, you can keep counting up by 3 to reach any layer.

2Add the blocks layer by layer

#9 Solve an Easier Related Problem 2.OA.B.2

Add the 6 layer counts: 1+4=5+7=12+10=22+13=35+16=51.

1+4+7+10+13+16 = 51

Summing a short list of small numbers step by step keeps the running total easy to track.

Answer: 51 blocks

4 · Reviewdoes it hold up?

51 is a sensible total for a 6-layer growing stack, and the layer-by-layer addition confirms it.

Another way: Pair the outer terms: (1+16)+(4+13)+(7+10) = 51, the same total found by pairing instead of running addition.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the layer pattern 1, 4, 7 to all 6 layers.
2.OA.B.2 Fluently add and subtract within 20 using mental strategies — Adding the layer counts together to get the total.

💡Takeaway. Each layer grows by 3 blocks, and adding up the layer counts gives a neat 51!

Variant 2 easy

Building blocks are stacked following a rule. How many blocks in all are needed to build the stack so that it is $8$ layers tall?

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

Layer 1: $4$ blocks
Layer 2: $5$ blocks
Layer 3: $6$ blocks

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.

Your answer

Show solution

1 · Understandwhat's really being asked

Blocks are stacked into L-shaped figures that grow by a fixed rule: 4, 5, 6 blocks for the first 3 layers. Find the total number of blocks needed to build the stack up to 8 layers tall.

Givens

Layer 1 uses 4 blocks.
Layer 2 uses 5 blocks.
Layer 3 uses 6 blocks.
Each new layer adds the same amount more than the one before.

Unknowns

The total number of blocks in all 8 layers combined.

Constraints

The number of blocks in each layer follows the pattern 4, 5, 6, ... (adding 1 each step).
We need every layer from 1 through 8 added together.

2 · Planchoose the strategy

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

The layer counts 4, 5, 6 increase by 1 each time, so I extend that pattern to 8 layers, then add them up (building from the small known sums).

3 · Execute2 carry out the plan

1Extend the layer pattern to 8 layers

#5 Look for a Pattern 4.OA.C.5

Each layer has 1 more block than the one before, so the 8 layers use 4, 5, 6, 7, 8, 9, 10, 11 blocks.

4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10,\ 11

Once you see each step adds 1, you can keep counting up by 1 to reach any layer.

2Add the blocks layer by layer

#9 Solve an Easier Related Problem 2.OA.B.2

Add the 8 layer counts: 4+5=9+6=15+7=22+8=30+9=39+10=49+11=60.

4+5+6+7+8+9+10+11 = 60

Summing a short list of small numbers step by step keeps the running total easy to track.

Answer: 60 blocks

4 · Reviewdoes it hold up?

60 is a sensible total for a 8-layer growing stack, and the layer-by-layer addition confirms it.

Another way: Pair the outer terms: (4+11)+(5+10)+(6+9)+(7+8) = 60, the same total found by pairing instead of running addition.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the layer pattern 4, 5, 6 to all 8 layers.
2.OA.B.2 Fluently add and subtract within 20 using mental strategies — Adding the layer counts together to get the total.

💡Takeaway. Each layer grows by 1 block, and adding up the layer counts gives a neat 60!

Variant 3 medium

Building blocks are stacked following a rule. How many blocks in all are needed to build the stack so that it is $6$ layers tall?

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

Layer 1: $1$ block
Layer 2: $2$ blocks
Layer 3: $3$ blocks
Layer 4: $4$ blocks

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.

Your answer

Show solution

1 · Understandwhat's really being asked

Blocks are stacked into L-shaped figures that grow by a fixed rule: 1, 2, 3, 4 blocks for the first 4 layers. Find the total number of blocks needed to build the stack up to 6 layers tall.

Givens

Layer 1 uses 1 block.
Layer 2 uses 2 blocks.
Layer 3 uses 3 blocks.
Layer 4 uses 4 blocks.
Each new layer adds the same amount more than the one before.

Unknowns

The total number of blocks in all 6 layers combined.

Constraints

The number of blocks in each layer follows the pattern 1, 2, 3, 4, ... (adding 1 each step).
We need every layer from 1 through 6 added together.

2 · Planchoose the strategy

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

The layer counts 1, 2, 3, 4 increase by 1 each time, so I extend that pattern to 6 layers, then add them up (building from the small known sums).

3 · Execute2 carry out the plan

1Extend the layer pattern to 6 layers

#5 Look for a Pattern 4.OA.C.5

Each layer has 1 more block than the one before, so the 6 layers use 1, 2, 3, 4, 5, 6 blocks.

1,\ 2,\ 3,\ 4,\ 5,\ 6

Once you see each step adds 1, you can keep counting up by 1 to reach any layer.

2Add the blocks layer by layer

#9 Solve an Easier Related Problem 2.OA.B.2

Add the 6 layer counts: 1+2=3+3=6+4=10+5=15+6=21.

1+2+3+4+5+6 = 21

Summing a short list of small numbers step by step keeps the running total easy to track.

Answer: 21 blocks

4 · Reviewdoes it hold up?

21 is a sensible total for a 6-layer growing stack, and the layer-by-layer addition confirms it.

Another way: Pair the outer terms: (1+6)+(2+5)+(3+4) = 21, the same total found by pairing instead of running addition.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the layer pattern 1, 2, 3, 4 to all 6 layers.
2.OA.B.2 Fluently add and subtract within 20 using mental strategies — Adding the layer counts together to get the total.

💡Takeaway. Each layer grows by 1 block, and adding up the layer counts gives a neat 21!

Variant 4 medium

Building blocks are stacked following a rule. How many blocks in all are needed to build the stack so that it is $7$ layers tall?

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

Layer 1: $1$ block
Layer 2: $3$ blocks
Layer 3: $5$ blocks
Layer 4: $7$ blocks

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.

Your answer

Show solution

1 · Understandwhat's really being asked

Blocks are stacked into L-shaped figures that grow by a fixed rule: 1, 3, 5, 7 blocks for the first 4 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, ... (adding 2 each step).
We need every layer from 1 through 7 added together.

2 · Planchoose the strategy

#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).

3 · Execute2 carry out the plan

1Extend the layer pattern to 7 layers

#5 Look for a Pattern 4.OA.C.5

Each layer has 2 more blocks than the one before, so the 7 layers use 1, 3, 5, 7, 9, 11, 13 blocks.

1,\ 3,\ 5,\ 7,\ 9,\ 11,\ 13

Once you see each step adds 2, you can keep counting up by 2 to reach any layer.

2Add the blocks layer by layer

#9 Solve an Easier Related Problem 2.OA.B.2

Add the 7 layer counts: 1+3=4+5=9+7=16+9=25+11=36+13=49.

1+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

4 · Reviewdoes it hold up?

49 is a sensible total for a 7-layer growing stack, and the layer-by-layer addition confirms it.

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

Standardsmin 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 7 layers.
2.OA.B.2 Fluently add and subtract within 20 using mental strategies — Adding the layer counts together to get the total.

💡Takeaway. Each layer grows by 2 blocks, and adding up the layer counts gives a neat 49!

Variant 5 medium

Building blocks are stacked following a rule. How many blocks in all are needed to build the stack so that it is $5$ layers tall?

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

Layer 1: $2$ blocks
Layer 2: $4$ blocks
Layer 3: $6$ blocks
Layer 4: $8$ blocks

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.

Your answer

Show solution

1 · Understandwhat's really being asked

Blocks are stacked into L-shaped figures that grow by a fixed rule: 2, 4, 6, 8 blocks for the first 4 layers. Find the total number of blocks needed to build the stack up to 5 layers tall.

Givens

Layer 1 uses 2 blocks.
Layer 2 uses 4 blocks.
Layer 3 uses 6 blocks.
Layer 4 uses 8 blocks.
Each new layer adds the same amount more than the one before.

Unknowns

The total number of blocks in all 5 layers combined.

Constraints

The number of blocks in each layer follows the pattern 2, 4, 6, 8, ... (adding 2 each step).
We need every layer from 1 through 5 added together.

2 · Planchoose the strategy

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

The layer counts 2, 4, 6, 8 increase by 2 each time, so I extend that pattern to 5 layers, then add them up (building from the small known sums).

3 · Execute2 carry out the plan

1Extend the layer pattern to 5 layers

#5 Look for a Pattern 4.OA.C.5

Each layer has 2 more blocks than the one before, so the 5 layers use 2, 4, 6, 8, 10 blocks.

2,\ 4,\ 6,\ 8,\ 10

Once you see each step adds 2, you can keep counting up by 2 to reach any layer.

2Add the blocks layer by layer

#9 Solve an Easier Related Problem 2.OA.B.2

Add the 5 layer counts: 2+4=6+6=12+8=20+10=30.

2+4+6+8+10 = 30

Summing a short list of small numbers step by step keeps the running total easy to track.

Answer: 30 blocks

4 · Reviewdoes it hold up?

30 is a sensible total for a 5-layer growing stack, and the layer-by-layer addition confirms it.

Another way: Pair the outer terms: (2+10)+(4+8)+6 = 30, the same total found by pairing instead of running addition.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the layer pattern 2, 4, 6, 8 to all 5 layers.
2.OA.B.2 Fluently add and subtract within 20 using mental strategies — Adding the layer counts together to get the total.

💡Takeaway. Each layer grows by 2 blocks, and adding up the layer counts gives a neat 30!

Variant 6 medium

Building blocks are stacked following a rule. How many blocks in all are needed to build the stack so that it is $8$ layers tall?

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

Layer 1: $3$ blocks
Layer 2: $5$ blocks
Layer 3: $7$ blocks
Layer 4: $9$ blocks

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.

Your answer

Show solution

1 · Understandwhat's really being asked

Blocks are stacked into L-shaped figures that grow by a fixed rule: 3, 5, 7, 9 blocks for the first 4 layers. Find the total number of blocks needed to build the stack up to 8 layers tall.

Givens

Layer 1 uses 3 blocks.
Layer 2 uses 5 blocks.
Layer 3 uses 7 blocks.
Layer 4 uses 9 blocks.
Each new layer adds the same amount more than the one before.

Unknowns

The total number of blocks in all 8 layers combined.

Constraints

The number of blocks in each layer follows the pattern 3, 5, 7, 9, ... (adding 2 each step).
We need every layer from 1 through 8 added together.

2 · Planchoose the strategy

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

The layer counts 3, 5, 7, 9 increase by 2 each time, so I extend that pattern to 8 layers, then add them up (building from the small known sums).

3 · Execute2 carry out the plan

1Extend the layer pattern to 8 layers

#5 Look for a Pattern 4.OA.C.5

Each layer has 2 more blocks than the one before, so the 8 layers use 3, 5, 7, 9, 11, 13, 15, 17 blocks.

3,\ 5,\ 7,\ 9,\ 11,\ 13,\ 15,\ 17

Once you see each step adds 2, you can keep counting up by 2 to reach any layer.

2Add the blocks layer by layer

#9 Solve an Easier Related Problem 2.OA.B.2

Add the 8 layer counts: 3+5=8+7=15+9=24+11=35+13=48+15=63+17=80.

3+5+7+9+11+13+15+17 = 80

Summing a short list of small numbers step by step keeps the running total easy to track.

Answer: 80 blocks

4 · Reviewdoes it hold up?

80 is a sensible total for a 8-layer growing stack, and the layer-by-layer addition confirms it.

Another way: Pair the outer terms: (3+17)+(5+15)+(7+13)+(9+11) = 80, the same total found by pairing instead of running addition.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the layer pattern 3, 5, 7, 9 to all 8 layers.
2.OA.B.2 Fluently add and subtract within 20 using mental strategies — Adding the layer counts together to get the total.

💡Takeaway. Each layer grows by 2 blocks, and adding up the layer counts gives a neat 80!

Variant 7 hard

Building blocks are stacked following a rule. How many blocks in all are needed to build the stack so that it is $9$ layers tall?

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

Layer 1: $1$ block
Layer 2: $3$ blocks
Layer 3: $5$ blocks
Layer 4: $7$ blocks

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.

Your answer

Show solution

1 · Understandwhat's really being asked

Blocks are stacked into L-shaped figures that grow by a fixed rule: 1, 3, 5, 7 blocks for the first 4 layers. Find the total number of blocks needed to build the stack up to 9 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 9 layers combined.

Constraints

The number of blocks in each layer follows the pattern 1, 3, 5, 7, ... (adding 2 each step).
We need every layer from 1 through 9 added together.

2 · Planchoose the strategy

#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 9 layers, then add them up (building from the small known sums).

3 · Execute2 carry out the plan

1Extend the layer pattern to 9 layers

#5 Look for a Pattern 4.OA.C.5

Each layer has 2 more blocks than the one before, so the 9 layers use 1, 3, 5, 7, 9, 11, 13, 15, 17 blocks.

1,\ 3,\ 5,\ 7,\ 9,\ 11,\ 13,\ 15,\ 17

Once you see each step adds 2, you can keep counting up by 2 to reach any layer.

2Add the blocks layer by layer

#9 Solve an Easier Related Problem 2.OA.B.2

Add the 9 layer counts: 1+3=4+5=9+7=16+9=25+11=36+13=49+15=64+17=81.

1+3+5+7+9+11+13+15+17 = 81

Summing a short list of small numbers step by step keeps the running total easy to track.

Answer: 81 blocks

4 · Reviewdoes it hold up?

81 is a sensible total for a 9-layer growing stack, and the layer-by-layer addition confirms it.

Another way: Pair the outer terms: (1+17)+(3+15)+(5+13)+(7+11)+9 = 81, the same total found by pairing instead of running addition.

Standardsmin 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 9 layers.
2.OA.B.2 Fluently add and subtract within 20 using mental strategies — Adding the layer counts together to get the total.

💡Takeaway. Each layer grows by 2 blocks, and adding up the layer counts gives a neat 81!

Variant 8 hard

Building blocks are stacked following a rule. How many blocks in all are needed to build the stack so that it is $7$ layers tall?

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

Layer 1: $2$ blocks
Layer 2: $5$ blocks
Layer 3: $8$ blocks
Layer 4: $11$ blocks

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.

Your answer

Show solution

1 · Understandwhat's really being asked

Blocks are stacked into L-shaped figures that grow by a fixed rule: 2, 5, 8, 11 blocks for the first 4 layers. Find the total number of blocks needed to build the stack up to 7 layers tall.

Givens

Layer 1 uses 2 blocks.
Layer 2 uses 5 blocks.
Layer 3 uses 8 blocks.
Layer 4 uses 11 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 2, 5, 8, 11, ... (adding 3 each step).
We need every layer from 1 through 7 added together.

2 · Planchoose the strategy

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

The layer counts 2, 5, 8, 11 increase by 3 each time, so I extend that pattern to 7 layers, then add them up (building from the small known sums).

3 · Execute2 carry out the plan

1Extend the layer pattern to 7 layers

#5 Look for a Pattern 4.OA.C.5

Each layer has 3 more blocks than the one before, so the 7 layers use 2, 5, 8, 11, 14, 17, 20 blocks.

2,\ 5,\ 8,\ 11,\ 14,\ 17,\ 20

Once you see each step adds 3, you can keep counting up by 3 to reach any layer.

2Add the blocks layer by layer

#9 Solve an Easier Related Problem 2.OA.B.2

Add the 7 layer counts: 2+5=7+8=15+11=26+14=40+17=57+20=77.

2+5+8+11+14+17+20 = 77

Summing a short list of small numbers step by step keeps the running total easy to track.

Answer: 77 blocks

4 · Reviewdoes it hold up?

77 is a sensible total for a 7-layer growing stack, and the layer-by-layer addition confirms it.

Another way: Pair the outer terms: (2+20)+(5+17)+(8+14)+11 = 77, the same total found by pairing instead of running addition.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the layer pattern 2, 5, 8, 11 to all 7 layers.
2.OA.B.2 Fluently add and subtract within 20 using mental strategies — Adding the layer counts together to get the total.

💡Takeaway. Each layer grows by 3 blocks, and adding up the layer counts gives a neat 77!

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