Split a pattern into its fixed and repeating parts
4.NBT.B.54.OA.C.55.OA.A.2
Generated variants — 12
Regular octagons are being built out of matchsticks as shown below. How many matchsticks are needed to make regular octagons?
The matchstick octagons are joined side by side in a single row. The first octagon uses matchsticks, and each additional octagon shares one side with the neighboring one, so 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
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
2Separate the fixed part from the changing part
3Plug in 10 octagons
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.
Standardsmin grade 5
4.NBT.B.5Multiply 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.5Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.5.OA.A.2Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.
Regular heptagons are being built out of matchsticks as shown below. How many matchsticks are needed to make regular heptagons?
The matchstick heptagons are joined side by side in a single row. The first heptagon uses matchsticks, and each additional heptagon shares one side with the neighboring one, so 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
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
2Separate the fixed part from the changing part
3Plug in 12 heptagons
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.
Standardsmin grade 5
4.NBT.B.5Multiply 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.5Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.5.OA.A.2Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.
Regular hexagons are being built out of matchsticks as shown below. How many matchsticks are needed to make regular hexagons?
The matchstick hexagons are joined side by side in a single row. The first hexagon uses matchsticks, and each additional hexagon shares one side with the neighboring one, so 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
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
2Separate the fixed part from the changing part
3Plug in 15 hexagons
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.
Standardsmin grade 5
4.NBT.B.5Multiply 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.5Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.5.OA.A.2Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.
Regular squares are being built out of matchsticks as shown below. How many matchsticks are needed to make regular squares?
The matchstick squares are joined side by side in a single row. The first square uses matchsticks, and each additional square shares one side with the neighboring one, so 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
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
2Separate the fixed part from the changing part
3Plug in 18 squares
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.
Standardsmin grade 5
4.NBT.B.5Multiply 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.5Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.5.OA.A.2Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.
Regular pentagons are being built out of matchsticks as shown below. How many matchsticks are needed to make regular pentagons?
The matchstick pentagons are joined side by side in a single row. The first pentagon uses matchsticks, and each additional pentagon shares one side with the neighboring one, so 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
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
2Separate the fixed part from the changing part
3Plug in 20 pentagons
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.
Standardsmin grade 5
4.NBT.B.5Multiply 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.5Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.5.OA.A.2Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.
Regular hexagons are being built out of matchsticks as shown below. How many matchsticks are needed to make regular hexagons?
The matchstick hexagons are joined side by side in a single row. The first hexagon uses matchsticks, and each additional hexagon shares one side with the neighboring one, so 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
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
2Separate the fixed part from the changing part
3Plug in 22 hexagons
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.
Standardsmin grade 5
4.NBT.B.5Multiply 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.5Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.5.OA.A.2Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.
Regular triangles are being built out of matchsticks as shown below. How many matchsticks are needed to make regular triangles?
The matchstick triangles are joined side by side in a single row. The first triangle uses matchsticks, and each additional triangle shares one side with the neighboring one, so 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
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
2Separate the fixed part from the changing part
3Plug in 25 triangles
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.
Standardsmin grade 5
4.NBT.B.5Multiply 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.5Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.5.OA.A.2Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.
Regular squares are being built out of matchsticks as shown below. How many matchsticks are needed to make regular squares?
The matchstick squares are joined side by side in a single row. The first square uses matchsticks, and each additional square shares one side with the neighboring one, so 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
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
2Separate the fixed part from the changing part
3Plug in 30 squares
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.
Standardsmin grade 5
4.NBT.B.5Multiply 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.5Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.5.OA.A.2Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.
Regular pentagons are being built out of matchsticks as shown below. How many matchsticks are needed to make regular pentagons?
The matchstick pentagons are joined side by side in a single row. The first pentagon uses matchsticks, and each additional pentagon shares one side with the neighboring one, so 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
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
2Separate the fixed part from the changing part
3Plug in 33 pentagons
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.
Standardsmin grade 5
4.NBT.B.5Multiply 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.5Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.5.OA.A.2Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.
Regular pentagons are being built out of matchsticks as shown below. How many matchsticks are needed to make regular pentagons?
The matchstick pentagons are joined side by side in a single row. The first pentagon uses matchsticks, and each additional pentagon shares one side with the neighboring one, so 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
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
2Separate the fixed part from the changing part
3Plug in 40 pentagons
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.
Standardsmin grade 5
4.NBT.B.5Multiply 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.5Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.5.OA.A.2Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.
Regular squares are being built out of matchsticks as shown below. How many matchsticks are needed to make regular squares?
The matchstick squares are joined side by side in a single row. The first square uses matchsticks, and each additional square shares one side with the neighboring one, so 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
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
2Separate the fixed part from the changing part
3Plug in 45 squares
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.
Standardsmin grade 5
4.NBT.B.5Multiply 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.5Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.5.OA.A.2Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.
Regular triangles are being built out of matchsticks as shown below. How many matchsticks are needed to make regular triangles?
The matchstick triangles are joined side by side in a single row. The first triangle uses matchsticks, and each additional triangle shares one side with the neighboring one, so 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
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
2Separate the fixed part from the changing part
3Plug in 50 triangles
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.
Standardsmin grade 5
4.NBT.B.5Multiply 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.5Generate a number or shape pattern that follows a given rule — Generating the first few counts to find the step.5.OA.A.2Write simple expressions that record calculations with numbers — Writing the count as a fixed part plus a repeating part.
Where next?
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