Practice · Split a pattern into its fixed and repeating parts · Sensim Math

Variant 1 easy answer: 71 matchsticks

Regular octagons are being built out of matchsticks as shown below. How many matchsticks are needed to make $10$ regular octagons?

The matchstick octagons are joined side by side in a single row. The first octagon uses $8$ matchsticks, and each additional octagon shares one side with the neighboring one, so $7$ more matchsticks are added each time.

Show solution

1 · Understandwhat's really being asked

Regular octagons are made from matchsticks in a row, each sharing one side with the last. We need the number of sticks for 10 of them.

Givens

One octagon on its own takes 8 matchsticks.
Each extra octagon shares a side, so it adds 7.
We want 10 octagons.

Unknowns

The number of matchsticks for 10 octagons.

Constraints

A shared side is one matchstick, not two.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Build the first few and watch what stays the same. The opening octagon is a one-off cost and everything after it repeats, so the count splits neatly into a fixed part and a multiplication.

3 · Execute3 carry out the plan

1Build small cases and spot the rule

#9 Solve an Easier Related Problem 4.OA.C.5

Count the matchsticks for the first few. One octagon: 8. Two: the second shares a side, so it adds 7, giving 15. Three: add 7 again.

8,\; 15,\; 22,\; 29,\; \ldots\; (+7 \text{ each step})

The same 7 sticks every time after the first.

2Separate the fixed part from the changing part

#5 Look for a Pattern 5.OA.A.2

The 8 matchsticks of the first octagon never change -- that is the fixed part. The changing part is the 7 matchsticks added for each octagon after the first.

\text{total} = 8 + 7 \times (\text{number of octagons} - 1)

One rule that works for any number of them.

3Plug in 10 octagons

#5 Look for a Pattern 4.NBT.B.5

There are 10 - 1 = 9 extra octagons, each adding 7 matchsticks. That is 7 x 9 = 63 added matchsticks. Add the fixed 8 from the first octagon.

8 + 7 \times 9 = 8 + 63 = 71

71 matchsticks in all.

Answer: 71 matchsticks

4 · Reviewdoes it hold up?

Count it the other way: 10 separate octagons would take 10 x 8 = 80 sticks, but the 9 shared sides are each counted twice there, so take 9 off: 80 - 9 = 71.

Another way: Notice the rule only needs the first octagon treated specially. Writing it as 7 x 10 + 1 works too, and gives 71 -- the same 71.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Multiplying the repeated cost by the number of extra shapes.
4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.

💡Takeaway. Find the bit of a pattern that never changes, and the rest is just the same step over and over.

Variant 2 easy answer: 73 matchsticks

Regular heptagons are being built out of matchsticks as shown below. How many matchsticks are needed to make $12$ regular heptagons?

The matchstick heptagons are joined side by side in a single row. The first heptagon uses $7$ matchsticks, and each additional heptagon shares one side with the neighboring one, so $6$ more matchsticks are added each time.

Show solution

1 · Understandwhat's really being asked

Regular heptagons are made from matchsticks in a row, each sharing one side with the last. We need the number of sticks for 12 of them.

Givens

One heptagon on its own takes 7 matchsticks.
Each extra heptagon shares a side, so it adds 6.
We want 12 heptagons.

Unknowns

The number of matchsticks for 12 heptagons.

Constraints

A shared side is one matchstick, not two.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Build the first few and watch what stays the same. The opening heptagon is a one-off cost and everything after it repeats, so the count splits neatly into a fixed part and a multiplication.

3 · Execute3 carry out the plan

1Build small cases and spot the rule

#9 Solve an Easier Related Problem 4.OA.C.5

Count the matchsticks for the first few. One heptagon: 7. Two: the second shares a side, so it adds 6, giving 13. Three: add 6 again.

7,\; 13,\; 19,\; 25,\; \ldots\; (+6 \text{ each step})

The same 6 sticks every time after the first.

2Separate the fixed part from the changing part

#5 Look for a Pattern 5.OA.A.2

The 7 matchsticks of the first heptagon never change -- that is the fixed part. The changing part is the 6 matchsticks added for each heptagon after the first.

\text{total} = 7 + 6 \times (\text{number of heptagons} - 1)

One rule that works for any number of them.

3Plug in 12 heptagons

#5 Look for a Pattern 4.NBT.B.5

There are 12 - 1 = 11 extra heptagons, each adding 6 matchsticks. That is 6 x 11 = 66 added matchsticks. Add the fixed 7 from the first heptagon.

7 + 6 \times 11 = 7 + 66 = 73

73 matchsticks in all.

Answer: 73 matchsticks

4 · Reviewdoes it hold up?

Count it the other way: 12 separate heptagons would take 12 x 7 = 84 sticks, but the 11 shared sides are each counted twice there, so take 11 off: 84 - 11 = 73.

Another way: Notice the rule only needs the first heptagon treated specially. Writing it as 6 x 12 + 1 works too, and gives 73 -- the same 73.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Multiplying the repeated cost by the number of extra shapes.
4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.

💡Takeaway. Find the bit of a pattern that never changes, and the rest is just the same step over and over.

Variant 3 easy answer: 76 matchsticks

Regular hexagons are being built out of matchsticks as shown below. How many matchsticks are needed to make $15$ regular hexagons?

The matchstick hexagons are joined side by side in a single row. The first hexagon uses $6$ matchsticks, and each additional hexagon shares one side with the neighboring one, so $5$ more matchsticks are added each time.

Show solution

1 · Understandwhat's really being asked

Regular hexagons are made from matchsticks in a row, each sharing one side with the last. We need the number of sticks for 15 of them.

Givens

One hexagon on its own takes 6 matchsticks.
Each extra hexagon shares a side, so it adds 5.
We want 15 hexagons.

Unknowns

The number of matchsticks for 15 hexagons.

Constraints

A shared side is one matchstick, not two.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Build the first few and watch what stays the same. The opening hexagon is a one-off cost and everything after it repeats, so the count splits neatly into a fixed part and a multiplication.

3 · Execute3 carry out the plan

1Build small cases and spot the rule

#9 Solve an Easier Related Problem 4.OA.C.5

Count the matchsticks for the first few. One hexagon: 6. Two: the second shares a side, so it adds 5, giving 11. Three: add 5 again.

6,\; 11,\; 16,\; 21,\; \ldots\; (+5 \text{ each step})

The same 5 sticks every time after the first.

2Separate the fixed part from the changing part

#5 Look for a Pattern 5.OA.A.2

The 6 matchsticks of the first hexagon never change -- that is the fixed part. The changing part is the 5 matchsticks added for each hexagon after the first.

\text{total} = 6 + 5 \times (\text{number of hexagons} - 1)

One rule that works for any number of them.

3Plug in 15 hexagons

#5 Look for a Pattern 4.NBT.B.5

There are 15 - 1 = 14 extra hexagons, each adding 5 matchsticks. That is 5 x 14 = 70 added matchsticks. Add the fixed 6 from the first hexagon.

6 + 5 \times 14 = 6 + 70 = 76

76 matchsticks in all.

Answer: 76 matchsticks

4 · Reviewdoes it hold up?

Count it the other way: 15 separate hexagons would take 15 x 6 = 90 sticks, but the 14 shared sides are each counted twice there, so take 14 off: 90 - 14 = 76.

Another way: Notice the rule only needs the first hexagon treated specially. Writing it as 5 x 15 + 1 works too, and gives 76 -- the same 76.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Multiplying the repeated cost by the number of extra shapes.
4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.

💡Takeaway. Find the bit of a pattern that never changes, and the rest is just the same step over and over.

Variant 4 easy answer: 55 matchsticks

Regular squares are being built out of matchsticks as shown below. How many matchsticks are needed to make $18$ regular squares?

The matchstick squares are joined side by side in a single row. The first square uses $4$ matchsticks, and each additional square shares one side with the neighboring one, so $3$ more matchsticks are added each time.

Show solution

1 · Understandwhat's really being asked

Regular squares are made from matchsticks in a row, each sharing one side with the last. We need the number of sticks for 18 of them.

Givens

One square on its own takes 4 matchsticks.
Each extra square shares a side, so it adds 3.
We want 18 squares.

Unknowns

The number of matchsticks for 18 squares.

Constraints

A shared side is one matchstick, not two.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Build the first few and watch what stays the same. The opening square is a one-off cost and everything after it repeats, so the count splits neatly into a fixed part and a multiplication.

3 · Execute3 carry out the plan

1Build small cases and spot the rule

#9 Solve an Easier Related Problem 4.OA.C.5

Count the matchsticks for the first few. One square: 4. Two: the second shares a side, so it adds 3, giving 7. Three: add 3 again.

4,\; 7,\; 10,\; 13,\; \ldots\; (+3 \text{ each step})

The same 3 sticks every time after the first.

2Separate the fixed part from the changing part

#5 Look for a Pattern 5.OA.A.2

The 4 matchsticks of the first square never change -- that is the fixed part. The changing part is the 3 matchsticks added for each square after the first.

\text{total} = 4 + 3 \times (\text{number of squares} - 1)

One rule that works for any number of them.

3Plug in 18 squares

#5 Look for a Pattern 4.NBT.B.5

There are 18 - 1 = 17 extra squares, each adding 3 matchsticks. That is 3 x 17 = 51 added matchsticks. Add the fixed 4 from the first square.

4 + 3 \times 17 = 4 + 51 = 55

55 matchsticks in all.

Answer: 55 matchsticks

4 · Reviewdoes it hold up?

Count it the other way: 18 separate squares would take 18 x 4 = 72 sticks, but the 17 shared sides are each counted twice there, so take 17 off: 72 - 17 = 55.

Another way: Notice the rule only needs the first square treated specially. Writing it as 3 x 18 + 1 works too, and gives 55 -- the same 55.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Multiplying the repeated cost by the number of extra shapes.
4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.

💡Takeaway. Find the bit of a pattern that never changes, and the rest is just the same step over and over.

Variant 5 medium answer: 81 matchsticks

Regular pentagons are being built out of matchsticks as shown below. How many matchsticks are needed to make $20$ regular pentagons?

The matchstick pentagons are joined side by side in a single row. The first pentagon uses $5$ matchsticks, and each additional pentagon shares one side with the neighboring one, so $4$ more matchsticks are added each time.

Show solution

1 · Understandwhat's really being asked

Regular pentagons are made from matchsticks in a row, each sharing one side with the last. We need the number of sticks for 20 of them.

Givens

One pentagon on its own takes 5 matchsticks.
Each extra pentagon shares a side, so it adds 4.
We want 20 pentagons.

Unknowns

The number of matchsticks for 20 pentagons.

Constraints

A shared side is one matchstick, not two.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Build the first few and watch what stays the same. The opening pentagon is a one-off cost and everything after it repeats, so the count splits neatly into a fixed part and a multiplication.

3 · Execute3 carry out the plan

1Build small cases and spot the rule

#9 Solve an Easier Related Problem 4.OA.C.5

Count the matchsticks for the first few. One pentagon: 5. Two: the second shares a side, so it adds 4, giving 9. Three: add 4 again.

5,\; 9,\; 13,\; 17,\; \ldots\; (+4 \text{ each step})

The same 4 sticks every time after the first.

2Separate the fixed part from the changing part

#5 Look for a Pattern 5.OA.A.2

The 5 matchsticks of the first pentagon never change -- that is the fixed part. The changing part is the 4 matchsticks added for each pentagon after the first.

\text{total} = 5 + 4 \times (\text{number of pentagons} - 1)

One rule that works for any number of them.

3Plug in 20 pentagons

#5 Look for a Pattern 4.NBT.B.5

There are 20 - 1 = 19 extra pentagons, each adding 4 matchsticks. That is 4 x 19 = 76 added matchsticks. Add the fixed 5 from the first pentagon.

5 + 4 \times 19 = 5 + 76 = 81

81 matchsticks in all.

Answer: 81 matchsticks

4 · Reviewdoes it hold up?

Count it the other way: 20 separate pentagons would take 20 x 5 = 100 sticks, but the 19 shared sides are each counted twice there, so take 19 off: 100 - 19 = 81.

Another way: Notice the rule only needs the first pentagon treated specially. Writing it as 4 x 20 + 1 works too, and gives 81 -- the same 81.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Multiplying the repeated cost by the number of extra shapes.
4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.

💡Takeaway. Find the bit of a pattern that never changes, and the rest is just the same step over and over.

Variant 6 medium answer: 111 matchsticks

Regular hexagons are being built out of matchsticks as shown below. How many matchsticks are needed to make $22$ regular hexagons?

The matchstick hexagons are joined side by side in a single row. The first hexagon uses $6$ matchsticks, and each additional hexagon shares one side with the neighboring one, so $5$ more matchsticks are added each time.

Show solution

1 · Understandwhat's really being asked

Regular hexagons are made from matchsticks in a row, each sharing one side with the last. We need the number of sticks for 22 of them.

Givens

One hexagon on its own takes 6 matchsticks.
Each extra hexagon shares a side, so it adds 5.
We want 22 hexagons.

Unknowns

The number of matchsticks for 22 hexagons.

Constraints

A shared side is one matchstick, not two.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Build the first few and watch what stays the same. The opening hexagon is a one-off cost and everything after it repeats, so the count splits neatly into a fixed part and a multiplication.

3 · Execute3 carry out the plan

1Build small cases and spot the rule

#9 Solve an Easier Related Problem 4.OA.C.5

Count the matchsticks for the first few. One hexagon: 6. Two: the second shares a side, so it adds 5, giving 11. Three: add 5 again.

6,\; 11,\; 16,\; 21,\; \ldots\; (+5 \text{ each step})

The same 5 sticks every time after the first.

2Separate the fixed part from the changing part

#5 Look for a Pattern 5.OA.A.2

The 6 matchsticks of the first hexagon never change -- that is the fixed part. The changing part is the 5 matchsticks added for each hexagon after the first.

\text{total} = 6 + 5 \times (\text{number of hexagons} - 1)

One rule that works for any number of them.

3Plug in 22 hexagons

#5 Look for a Pattern 4.NBT.B.5

There are 22 - 1 = 21 extra hexagons, each adding 5 matchsticks. That is 5 x 21 = 105 added matchsticks. Add the fixed 6 from the first hexagon.

6 + 5 \times 21 = 6 + 105 = 111

111 matchsticks in all.

Answer: 111 matchsticks

4 · Reviewdoes it hold up?

Count it the other way: 22 separate hexagons would take 22 x 6 = 132 sticks, but the 21 shared sides are each counted twice there, so take 21 off: 132 - 21 = 111.

Another way: Notice the rule only needs the first hexagon treated specially. Writing it as 5 x 22 + 1 works too, and gives 111 -- the same 111.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Multiplying the repeated cost by the number of extra shapes.
4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.

💡Takeaway. Find the bit of a pattern that never changes, and the rest is just the same step over and over.

Variant 7 medium answer: 51 matchsticks

Regular triangles are being built out of matchsticks as shown below. How many matchsticks are needed to make $25$ regular triangles?

The matchstick triangles are joined side by side in a single row. The first triangle uses $3$ matchsticks, and each additional triangle shares one side with the neighboring one, so $2$ more matchsticks are added each time.

Show solution

1 · Understandwhat's really being asked

Regular triangles are made from matchsticks in a row, each sharing one side with the last. We need the number of sticks for 25 of them.

Givens

One triangle on its own takes 3 matchsticks.
Each extra triangle shares a side, so it adds 2.
We want 25 triangles.

Unknowns

The number of matchsticks for 25 triangles.

Constraints

A shared side is one matchstick, not two.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Build the first few and watch what stays the same. The opening triangle is a one-off cost and everything after it repeats, so the count splits neatly into a fixed part and a multiplication.

3 · Execute3 carry out the plan

1Build small cases and spot the rule

#9 Solve an Easier Related Problem 4.OA.C.5

Count the matchsticks for the first few. One triangle: 3. Two: the second shares a side, so it adds 2, giving 5. Three: add 2 again.

3,\; 5,\; 7,\; 9,\; \ldots\; (+2 \text{ each step})

The same 2 sticks every time after the first.

2Separate the fixed part from the changing part

#5 Look for a Pattern 5.OA.A.2

The 3 matchsticks of the first triangle never change -- that is the fixed part. The changing part is the 2 matchsticks added for each triangle after the first.

\text{total} = 3 + 2 \times (\text{number of triangles} - 1)

One rule that works for any number of them.

3Plug in 25 triangles

#5 Look for a Pattern 4.NBT.B.5

There are 25 - 1 = 24 extra triangles, each adding 2 matchsticks. That is 2 x 24 = 48 added matchsticks. Add the fixed 3 from the first triangle.

3 + 2 \times 24 = 3 + 48 = 51

51 matchsticks in all.

Answer: 51 matchsticks

4 · Reviewdoes it hold up?

Count it the other way: 25 separate triangles would take 25 x 3 = 75 sticks, but the 24 shared sides are each counted twice there, so take 24 off: 75 - 24 = 51.

Another way: Notice the rule only needs the first triangle treated specially. Writing it as 2 x 25 + 1 works too, and gives 51 -- the same 51.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Multiplying the repeated cost by the number of extra shapes.
4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.

💡Takeaway. Find the bit of a pattern that never changes, and the rest is just the same step over and over.

Variant 8 medium answer: 91 matchsticks

Regular squares are being built out of matchsticks as shown below. How many matchsticks are needed to make $30$ regular squares?

The matchstick squares are joined side by side in a single row. The first square uses $4$ matchsticks, and each additional square shares one side with the neighboring one, so $3$ more matchsticks are added each time.

Show solution

1 · Understandwhat's really being asked

Regular squares are made from matchsticks in a row, each sharing one side with the last. We need the number of sticks for 30 of them.

Givens

One square on its own takes 4 matchsticks.
Each extra square shares a side, so it adds 3.
We want 30 squares.

Unknowns

The number of matchsticks for 30 squares.

Constraints

A shared side is one matchstick, not two.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Build the first few and watch what stays the same. The opening square is a one-off cost and everything after it repeats, so the count splits neatly into a fixed part and a multiplication.

3 · Execute3 carry out the plan

1Build small cases and spot the rule

#9 Solve an Easier Related Problem 4.OA.C.5

Count the matchsticks for the first few. One square: 4. Two: the second shares a side, so it adds 3, giving 7. Three: add 3 again.

4,\; 7,\; 10,\; 13,\; \ldots\; (+3 \text{ each step})

The same 3 sticks every time after the first.

2Separate the fixed part from the changing part

#5 Look for a Pattern 5.OA.A.2

The 4 matchsticks of the first square never change -- that is the fixed part. The changing part is the 3 matchsticks added for each square after the first.

\text{total} = 4 + 3 \times (\text{number of squares} - 1)

One rule that works for any number of them.

3Plug in 30 squares

#5 Look for a Pattern 4.NBT.B.5

There are 30 - 1 = 29 extra squares, each adding 3 matchsticks. That is 3 x 29 = 87 added matchsticks. Add the fixed 4 from the first square.

4 + 3 \times 29 = 4 + 87 = 91

91 matchsticks in all.

Answer: 91 matchsticks

4 · Reviewdoes it hold up?

Count it the other way: 30 separate squares would take 30 x 4 = 120 sticks, but the 29 shared sides are each counted twice there, so take 29 off: 120 - 29 = 91.

Another way: Notice the rule only needs the first square treated specially. Writing it as 3 x 30 + 1 works too, and gives 91 -- the same 91.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Multiplying the repeated cost by the number of extra shapes.
4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.

💡Takeaway. Find the bit of a pattern that never changes, and the rest is just the same step over and over.

Variant 9 hard answer: 133 matchsticks

Regular pentagons are being built out of matchsticks as shown below. How many matchsticks are needed to make $33$ regular pentagons?

The matchstick pentagons are joined side by side in a single row. The first pentagon uses $5$ matchsticks, and each additional pentagon shares one side with the neighboring one, so $4$ more matchsticks are added each time.

Show solution

1 · Understandwhat's really being asked

Regular pentagons are made from matchsticks in a row, each sharing one side with the last. We need the number of sticks for 33 of them.

Givens

One pentagon on its own takes 5 matchsticks.
Each extra pentagon shares a side, so it adds 4.
We want 33 pentagons.

Unknowns

The number of matchsticks for 33 pentagons.

Constraints

A shared side is one matchstick, not two.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Build the first few and watch what stays the same. The opening pentagon is a one-off cost and everything after it repeats, so the count splits neatly into a fixed part and a multiplication.

3 · Execute3 carry out the plan

1Build small cases and spot the rule

#9 Solve an Easier Related Problem 4.OA.C.5

Count the matchsticks for the first few. One pentagon: 5. Two: the second shares a side, so it adds 4, giving 9. Three: add 4 again.

5,\; 9,\; 13,\; 17,\; \ldots\; (+4 \text{ each step})

The same 4 sticks every time after the first.

2Separate the fixed part from the changing part

#5 Look for a Pattern 5.OA.A.2

The 5 matchsticks of the first pentagon never change -- that is the fixed part. The changing part is the 4 matchsticks added for each pentagon after the first.

\text{total} = 5 + 4 \times (\text{number of pentagons} - 1)

One rule that works for any number of them.

3Plug in 33 pentagons

#5 Look for a Pattern 4.NBT.B.5

There are 33 - 1 = 32 extra pentagons, each adding 4 matchsticks. That is 4 x 32 = 128 added matchsticks. Add the fixed 5 from the first pentagon.

5 + 4 \times 32 = 5 + 128 = 133

133 matchsticks in all.

Answer: 133 matchsticks

4 · Reviewdoes it hold up?

Count it the other way: 33 separate pentagons would take 33 x 5 = 165 sticks, but the 32 shared sides are each counted twice there, so take 32 off: 165 - 32 = 133.

Another way: Notice the rule only needs the first pentagon treated specially. Writing it as 4 x 33 + 1 works too, and gives 133 -- the same 133.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Multiplying the repeated cost by the number of extra shapes.
4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.

💡Takeaway. Find the bit of a pattern that never changes, and the rest is just the same step over and over.

Variant 10 hard answer: 161 matchsticks

Regular pentagons are being built out of matchsticks as shown below. How many matchsticks are needed to make $40$ regular pentagons?

The matchstick pentagons are joined side by side in a single row. The first pentagon uses $5$ matchsticks, and each additional pentagon shares one side with the neighboring one, so $4$ more matchsticks are added each time.

Show solution

1 · Understandwhat's really being asked

Regular pentagons are made from matchsticks in a row, each sharing one side with the last. We need the number of sticks for 40 of them.

Givens

One pentagon on its own takes 5 matchsticks.
Each extra pentagon shares a side, so it adds 4.
We want 40 pentagons.

Unknowns

The number of matchsticks for 40 pentagons.

Constraints

A shared side is one matchstick, not two.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Build the first few and watch what stays the same. The opening pentagon is a one-off cost and everything after it repeats, so the count splits neatly into a fixed part and a multiplication.

3 · Execute3 carry out the plan

1Build small cases and spot the rule

#9 Solve an Easier Related Problem 4.OA.C.5

Count the matchsticks for the first few. One pentagon: 5. Two: the second shares a side, so it adds 4, giving 9. Three: add 4 again.

5,\; 9,\; 13,\; 17,\; \ldots\; (+4 \text{ each step})

The same 4 sticks every time after the first.

2Separate the fixed part from the changing part

#5 Look for a Pattern 5.OA.A.2

The 5 matchsticks of the first pentagon never change -- that is the fixed part. The changing part is the 4 matchsticks added for each pentagon after the first.

\text{total} = 5 + 4 \times (\text{number of pentagons} - 1)

One rule that works for any number of them.

3Plug in 40 pentagons

#5 Look for a Pattern 4.NBT.B.5

There are 40 - 1 = 39 extra pentagons, each adding 4 matchsticks. That is 4 x 39 = 156 added matchsticks. Add the fixed 5 from the first pentagon.

5 + 4 \times 39 = 5 + 156 = 161

161 matchsticks in all.

Answer: 161 matchsticks

4 · Reviewdoes it hold up?

Count it the other way: 40 separate pentagons would take 40 x 5 = 200 sticks, but the 39 shared sides are each counted twice there, so take 39 off: 200 - 39 = 161.

Another way: Notice the rule only needs the first pentagon treated specially. Writing it as 4 x 40 + 1 works too, and gives 161 -- the same 161.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Multiplying the repeated cost by the number of extra shapes.
4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.

💡Takeaway. Find the bit of a pattern that never changes, and the rest is just the same step over and over.

Variant 11 hard answer: 136 matchsticks

Regular squares are being built out of matchsticks as shown below. How many matchsticks are needed to make $45$ regular squares?

The matchstick squares are joined side by side in a single row. The first square uses $4$ matchsticks, and each additional square shares one side with the neighboring one, so $3$ more matchsticks are added each time.

Show solution

1 · Understandwhat's really being asked

Regular squares are made from matchsticks in a row, each sharing one side with the last. We need the number of sticks for 45 of them.

Givens

One square on its own takes 4 matchsticks.
Each extra square shares a side, so it adds 3.
We want 45 squares.

Unknowns

The number of matchsticks for 45 squares.

Constraints

A shared side is one matchstick, not two.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Build the first few and watch what stays the same. The opening square is a one-off cost and everything after it repeats, so the count splits neatly into a fixed part and a multiplication.

3 · Execute3 carry out the plan

1Build small cases and spot the rule

#9 Solve an Easier Related Problem 4.OA.C.5

Count the matchsticks for the first few. One square: 4. Two: the second shares a side, so it adds 3, giving 7. Three: add 3 again.

4,\; 7,\; 10,\; 13,\; \ldots\; (+3 \text{ each step})

The same 3 sticks every time after the first.

2Separate the fixed part from the changing part

#5 Look for a Pattern 5.OA.A.2

The 4 matchsticks of the first square never change -- that is the fixed part. The changing part is the 3 matchsticks added for each square after the first.

\text{total} = 4 + 3 \times (\text{number of squares} - 1)

One rule that works for any number of them.

3Plug in 45 squares

#5 Look for a Pattern 4.NBT.B.5

There are 45 - 1 = 44 extra squares, each adding 3 matchsticks. That is 3 x 44 = 132 added matchsticks. Add the fixed 4 from the first square.

4 + 3 \times 44 = 4 + 132 = 136

136 matchsticks in all.

Answer: 136 matchsticks

4 · Reviewdoes it hold up?

Count it the other way: 45 separate squares would take 45 x 4 = 180 sticks, but the 44 shared sides are each counted twice there, so take 44 off: 180 - 44 = 136.

Another way: Notice the rule only needs the first square treated specially. Writing it as 3 x 45 + 1 works too, and gives 136 -- the same 136.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Multiplying the repeated cost by the number of extra shapes.
4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.

💡Takeaway. Find the bit of a pattern that never changes, and the rest is just the same step over and over.

Variant 12 hard answer: 101 matchsticks

Regular triangles are being built out of matchsticks as shown below. How many matchsticks are needed to make $50$ regular triangles?

The matchstick triangles are joined side by side in a single row. The first triangle uses $3$ matchsticks, and each additional triangle shares one side with the neighboring one, so $2$ more matchsticks are added each time.

Show solution

1 · Understandwhat's really being asked

Regular triangles are made from matchsticks in a row, each sharing one side with the last. We need the number of sticks for 50 of them.

Givens

One triangle on its own takes 3 matchsticks.
Each extra triangle shares a side, so it adds 2.
We want 50 triangles.

Unknowns

The number of matchsticks for 50 triangles.

Constraints

A shared side is one matchstick, not two.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Build the first few and watch what stays the same. The opening triangle is a one-off cost and everything after it repeats, so the count splits neatly into a fixed part and a multiplication.

3 · Execute3 carry out the plan

1Build small cases and spot the rule

#9 Solve an Easier Related Problem 4.OA.C.5

Count the matchsticks for the first few. One triangle: 3. Two: the second shares a side, so it adds 2, giving 5. Three: add 2 again.

3,\; 5,\; 7,\; 9,\; \ldots\; (+2 \text{ each step})

The same 2 sticks every time after the first.

2Separate the fixed part from the changing part

#5 Look for a Pattern 5.OA.A.2

The 3 matchsticks of the first triangle never change -- that is the fixed part. The changing part is the 2 matchsticks added for each triangle after the first.

\text{total} = 3 + 2 \times (\text{number of triangles} - 1)

One rule that works for any number of them.

3Plug in 50 triangles

#5 Look for a Pattern 4.NBT.B.5

There are 50 - 1 = 49 extra triangles, each adding 2 matchsticks. That is 2 x 49 = 98 added matchsticks. Add the fixed 3 from the first triangle.

3 + 2 \times 49 = 3 + 98 = 101

101 matchsticks in all.

Answer: 101 matchsticks

4 · Reviewdoes it hold up?

Count it the other way: 50 separate triangles would take 50 x 3 = 150 sticks, but the 49 shared sides are each counted twice there, so take 49 off: 150 - 49 = 101.

Another way: Notice the rule only needs the first triangle treated specially. Writing it as 2 x 50 + 1 works too, and gives 101 -- the same 101.

Standardsmin grade 5

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number — Multiplying the repeated cost by the number of extra shapes.
4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.

💡Takeaway. Find the bit of a pattern that never changes, and the rest is just the same step over and over.

Split a pattern into its fixed and repeating parts · 12 practice problems

Generated variants — 12

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints