← Find shape and number rule together · Generalize a Growing Pattern into a Rule

Find shape and number rule together · 5 practice problems

4.OA.C.53.OA.D.9

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Generated variants — 5

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

Variant 1 easy

Counters are arranged following a pattern. Draw the shape that belongs in the fifth position, and write the number of counters in each $\square$ .

The counters grow by adding the same number of counters in the up, right, and $\nearrow$ (diagonal) directions each time. The first four shapes are arranged as follows:

1st: $1$ counter
2nd: $4$ counters
3rd: $7$ counters
4th: $10$ counters

Each shape is formed from the one before it by adding counters in the up, right, and diagonal ( $\nearrow$ ) directions.

Your answer

Show solution

1 · Understandwhat's really being asked

Round counters are laid out in a growing right-angled (L-shaped) cluster. The 1st shape has 1 counter, and each new shape is made from the previous one by adding 3 counters: one up, one to the right, and one diagonally. The counts so far are 1, 4, 7, 10. We must draw the 5th shape and give the number of counters that fills the blank $\square$ .

Givens

1st shape: 1 counters
2nd shape: 4 counters
3rd shape: 7 counters
4th shape: 10 counters
Each shape adds 3 counters to the previous one (up, right, and diagonal)

Unknowns

The 5th shape and how many counters it contains

Constraints

Exactly 3 counters are added at each step
Counts form 1, 4, 7, 10, ... going up by 3

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #1 Draw a Diagram#10 Create a Physical Representation

The figure shows the shape growing by the same 3 counters each step, so the counts 1, 4, 7, 10 climb by 3. Continuing that steady add gives the 5th count, and drawing or building the 5th shape confirms it.

3 · Execute3 carry out the plan

1See the number rule

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

Look at the counts under each shape: 1, 4, 7, 10. Each one is 3 more than the one before, because every new shape adds 3 counters (up, right, diagonal).

1, 4, 7, 10 \text{ (add 3 each time)}

The same 3 counters appear at every step, so the count grows by a steady 3.

2Find the 5th count

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

Add 3 to the 4th shape's 10 counters to get the 5th shape's count.

10 + 3 = 13

One more step means one more group of 3 counters added on.

3Draw and build the 5th shape

#1 Draw a Diagram 3.OA.D.9

Start from the 4th shape (10 counters) and place one new counter up, one to the right, and one on the diagonal corner, extending the L into a larger right-angled cluster. Counting all the counters in the drawn shape gives 13.

1 + 3 \times 4 = 13

Drawing it shows the growth literally: the starting 1 plus 4 rounds of 3 makes 13.

Answer: 13 counters

4 · Reviewdoes it hold up?

Counts go 1, 4, 7, 10, 13, each 3 apart, so 13 is correct and bigger than the 4th shape's 10, which makes sense for a growing shape. It also equals 1 + 4 x 3 = 13, the start plus 4 steps of 3.

Another way: Use the position rule (tool 5): the nth shape has 1 + (n - 1) x 3 counters, so the 5th has 1 + 4 x 3 = 13.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Continuing the shape-and-count pattern 1, 4, 7, 10 to the 5th shape
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Explaining the steady add-3 growth as 1 + 4 x 3

💡Takeaway. Each new shape just adds 3 more counters, so after 10 comes 13: count up by 3 and you have the 5th shape!

Variant 2 medium

Counters are arranged following a pattern. Draw the shape that belongs in the fifth position, and write the number of counters in each $\square$ .

The counters grow by adding the same number of counters in the up, right, and $\nearrow$ (diagonal) directions each time. The first four shapes are arranged as follows:

1st: $2$ counters
2nd: $5$ counters
3rd: $8$ counters
4th: $11$ counters

Each shape is formed from the one before it by adding counters in the up, right, and diagonal ( $\nearrow$ ) directions.

Your answer

Show solution

1 · Understandwhat's really being asked

Round counters are laid out in a growing right-angled (L-shaped) cluster. The 1st shape has 2 counters, and each new shape is made from the previous one by adding 3 counters: one up, one to the right, and one diagonally. The counts so far are 2, 5, 8, 11. We must draw the 5th shape and give the number of counters that fills the blank $\square$ .

Givens

1st shape: 2 counters
2nd shape: 5 counters
3rd shape: 8 counters
4th shape: 11 counters
Each shape adds 3 counters to the previous one (up, right, and diagonal)

Unknowns

The 5th shape and how many counters it contains

Constraints

Exactly 3 counters are added at each step
Counts form 2, 5, 8, 11, ... going up by 3

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #1 Draw a Diagram#10 Create a Physical Representation

The figure shows the shape growing by the same 3 counters each step, so the counts 2, 5, 8, 11 climb by 3. Continuing that steady add gives the 5th count, and drawing or building the 5th shape confirms it.

3 · Execute3 carry out the plan

1See the number rule

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

Look at the counts under each shape: 2, 5, 8, 11. Each one is 3 more than the one before, because every new shape adds 3 counters (up, right, diagonal).

2, 5, 8, 11 \text{ (add 3 each time)}

The same 3 counters appear at every step, so the count grows by a steady 3.

2Find the 5th count

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

Add 3 to the 4th shape's 11 counters to get the 5th shape's count.

11 + 3 = 14

One more step means one more group of 3 counters added on.

3Draw and build the 5th shape

#1 Draw a Diagram 3.OA.D.9

Start from the 4th shape (11 counters) and place one new counter up, one to the right, and one on the diagonal corner, extending the L into a larger right-angled cluster. Counting all the counters in the drawn shape gives 14.

2 + 3 \times 4 = 14

Drawing it shows the growth literally: the starting 2 plus 4 rounds of 3 makes 14.

Answer: 14 counters

4 · Reviewdoes it hold up?

Counts go 2, 5, 8, 11, 14, each 3 apart, so 14 is correct and bigger than the 4th shape's 11, which makes sense for a growing shape. It also equals 2 + 4 x 3 = 14, the start plus 4 steps of 3.

Another way: Use the position rule (tool 5): the nth shape has 2 + (n - 1) x 3 counters, so the 5th has 2 + 4 x 3 = 14.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Continuing the shape-and-count pattern 2, 5, 8, 11 to the 5th shape
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Explaining the steady add-3 growth as 2 + 4 x 3

💡Takeaway. Each new shape just adds 3 more counters, so after 11 comes 14: count up by 3 and you have the 5th shape!

Variant 3 medium

Counters are arranged following a pattern. Draw the shape that belongs in the fifth position, and write the number of counters in each $\square$ .

The counters grow by adding the same number of counters in the up, right, and $\nearrow$ (diagonal) directions each time. The first four shapes are arranged as follows:

1st: $3$ counters
2nd: $6$ counters
3rd: $9$ counters
4th: $12$ counters

Each shape is formed from the one before it by adding counters in the up, right, and diagonal ( $\nearrow$ ) directions.

Your answer

Show solution

1 · Understandwhat's really being asked

Round counters are laid out in a growing right-angled (L-shaped) cluster. The 1st shape has 3 counters, and each new shape is made from the previous one by adding 3 counters: one up, one to the right, and one diagonally. The counts so far are 3, 6, 9, 12. We must draw the 5th shape and give the number of counters that fills the blank $\square$ .

Givens

1st shape: 3 counters
2nd shape: 6 counters
3rd shape: 9 counters
4th shape: 12 counters
Each shape adds 3 counters to the previous one (up, right, and diagonal)

Unknowns

The 5th shape and how many counters it contains

Constraints

Exactly 3 counters are added at each step
Counts form 3, 6, 9, 12, ... going up by 3

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #1 Draw a Diagram#10 Create a Physical Representation

The figure shows the shape growing by the same 3 counters each step, so the counts 3, 6, 9, 12 climb by 3. Continuing that steady add gives the 5th count, and drawing or building the 5th shape confirms it.

3 · Execute3 carry out the plan

1See the number rule

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

Look at the counts under each shape: 3, 6, 9, 12. Each one is 3 more than the one before, because every new shape adds 3 counters (up, right, diagonal).

3, 6, 9, 12 \text{ (add 3 each time)}

The same 3 counters appear at every step, so the count grows by a steady 3.

2Find the 5th count

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

Add 3 to the 4th shape's 12 counters to get the 5th shape's count.

12 + 3 = 15

One more step means one more group of 3 counters added on.

3Draw and build the 5th shape

#1 Draw a Diagram 3.OA.D.9

Start from the 4th shape (12 counters) and place one new counter up, one to the right, and one on the diagonal corner, extending the L into a larger right-angled cluster. Counting all the counters in the drawn shape gives 15.

3 + 3 \times 4 = 15

Drawing it shows the growth literally: the starting 3 plus 4 rounds of 3 makes 15.

Answer: 15 counters

4 · Reviewdoes it hold up?

Counts go 3, 6, 9, 12, 15, each 3 apart, so 15 is correct and bigger than the 4th shape's 12, which makes sense for a growing shape. It also equals 3 + 4 x 3 = 15, the start plus 4 steps of 3.

Another way: Use the position rule (tool 5): the nth shape has 3 + (n - 1) x 3 counters, so the 5th has 3 + 4 x 3 = 15.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Continuing the shape-and-count pattern 3, 6, 9, 12 to the 5th shape
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Explaining the steady add-3 growth as 3 + 4 x 3

💡Takeaway. Each new shape just adds 3 more counters, so after 12 comes 15: count up by 3 and you have the 5th shape!

Variant 4 hard

Counters are arranged following a pattern. Draw the shape that belongs in the fifth position, and write the number of counters in each $\square$ .

The counters grow by adding the same number of counters in the up, right, and $\nearrow$ (diagonal) directions each time. The first four shapes are arranged as follows:

1st: $4$ counters
2nd: $7$ counters
3rd: $10$ counters
4th: $13$ counters

Each shape is formed from the one before it by adding counters in the up, right, and diagonal ( $\nearrow$ ) directions.

Your answer

Show solution

1 · Understandwhat's really being asked

Round counters are laid out in a growing right-angled (L-shaped) cluster. The 1st shape has 4 counters, and each new shape is made from the previous one by adding 3 counters: one up, one to the right, and one diagonally. The counts so far are 4, 7, 10, 13. We must draw the 5th shape and give the number of counters that fills the blank $\square$ .

Givens

1st shape: 4 counters
2nd shape: 7 counters
3rd shape: 10 counters
4th shape: 13 counters
Each shape adds 3 counters to the previous one (up, right, and diagonal)

Unknowns

The 5th shape and how many counters it contains

Constraints

Exactly 3 counters are added at each step
Counts form 4, 7, 10, 13, ... going up by 3

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #1 Draw a Diagram#10 Create a Physical Representation

The figure shows the shape growing by the same 3 counters each step, so the counts 4, 7, 10, 13 climb by 3. Continuing that steady add gives the 5th count, and drawing or building the 5th shape confirms it.

3 · Execute3 carry out the plan

1See the number rule

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

Look at the counts under each shape: 4, 7, 10, 13. Each one is 3 more than the one before, because every new shape adds 3 counters (up, right, diagonal).

4, 7, 10, 13 \text{ (add 3 each time)}

The same 3 counters appear at every step, so the count grows by a steady 3.

2Find the 5th count

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

Add 3 to the 4th shape's 13 counters to get the 5th shape's count.

13 + 3 = 16

One more step means one more group of 3 counters added on.

3Draw and build the 5th shape

#1 Draw a Diagram 3.OA.D.9

Start from the 4th shape (13 counters) and place one new counter up, one to the right, and one on the diagonal corner, extending the L into a larger right-angled cluster. Counting all the counters in the drawn shape gives 16.

4 + 3 \times 4 = 16

Drawing it shows the growth literally: the starting 4 plus 4 rounds of 3 makes 16.

Answer: 16 counters

4 · Reviewdoes it hold up?

Counts go 4, 7, 10, 13, 16, each 3 apart, so 16 is correct and bigger than the 4th shape's 13, which makes sense for a growing shape. It also equals 4 + 4 x 3 = 16, the start plus 4 steps of 3.

Another way: Use the position rule (tool 5): the nth shape has 4 + (n - 1) x 3 counters, so the 5th has 4 + 4 x 3 = 16.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Continuing the shape-and-count pattern 4, 7, 10, 13 to the 5th shape
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Explaining the steady add-3 growth as 4 + 4 x 3

💡Takeaway. Each new shape just adds 3 more counters, so after 13 comes 16: count up by 3 and you have the 5th shape!

Variant 5 hard

Counters are arranged following a pattern. Draw the shape that belongs in the fifth position, and write the number of counters in each $\square$ .

The counters grow by adding the same number of counters in the up, right, and $\nearrow$ (diagonal) directions each time. The first four shapes are arranged as follows:

1st: $5$ counters
2nd: $8$ counters
3rd: $11$ counters
4th: $14$ counters

Each shape is formed from the one before it by adding counters in the up, right, and diagonal ( $\nearrow$ ) directions.

Your answer

Show solution

1 · Understandwhat's really being asked

Round counters are laid out in a growing right-angled (L-shaped) cluster. The 1st shape has 5 counters, and each new shape is made from the previous one by adding 3 counters: one up, one to the right, and one diagonally. The counts so far are 5, 8, 11, 14. We must draw the 5th shape and give the number of counters that fills the blank $\square$ .

Givens

1st shape: 5 counters
2nd shape: 8 counters
3rd shape: 11 counters
4th shape: 14 counters
Each shape adds 3 counters to the previous one (up, right, and diagonal)

Unknowns

The 5th shape and how many counters it contains

Constraints

Exactly 3 counters are added at each step
Counts form 5, 8, 11, 14, ... going up by 3

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #1 Draw a Diagram#10 Create a Physical Representation

The figure shows the shape growing by the same 3 counters each step, so the counts 5, 8, 11, 14 climb by 3. Continuing that steady add gives the 5th count, and drawing or building the 5th shape confirms it.

3 · Execute3 carry out the plan

1See the number rule

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

Look at the counts under each shape: 5, 8, 11, 14. Each one is 3 more than the one before, because every new shape adds 3 counters (up, right, diagonal).

5, 8, 11, 14 \text{ (add 3 each time)}

The same 3 counters appear at every step, so the count grows by a steady 3.

2Find the 5th count

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

Add 3 to the 4th shape's 14 counters to get the 5th shape's count.

14 + 3 = 17

One more step means one more group of 3 counters added on.

3Draw and build the 5th shape

#1 Draw a Diagram 3.OA.D.9

Start from the 4th shape (14 counters) and place one new counter up, one to the right, and one on the diagonal corner, extending the L into a larger right-angled cluster. Counting all the counters in the drawn shape gives 17.

5 + 3 \times 4 = 17

Drawing it shows the growth literally: the starting 5 plus 4 rounds of 3 makes 17.

Answer: 17 counters

4 · Reviewdoes it hold up?

Counts go 5, 8, 11, 14, 17, each 3 apart, so 17 is correct and bigger than the 4th shape's 14, which makes sense for a growing shape. It also equals 5 + 4 x 3 = 17, the start plus 4 steps of 3.

Another way: Use the position rule (tool 5): the nth shape has 5 + (n - 1) x 3 counters, so the 5th has 5 + 4 x 3 = 17.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Continuing the shape-and-count pattern 5, 8, 11, 14 to the 5th shape
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Explaining the steady add-3 growth as 5 + 4 x 3

💡Takeaway. Each new shape just adds 3 more counters, so after 14 comes 17: count up by 3 and you have the 5th shape!

Where next?

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