← 3-1 · Write a growing pattern as an expression · Generalize a Growing Pattern into a Rule

Write a growing pattern as an expression · 9 practice problems

3.OA.D.93.OA.D.83.OA.A.3

Generated variants — 9

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

Variant 1 answer: 22 thumbtacks

As shown in the figure, sheets of paper are being pinned to a bulletin board with thumbtacks. How many thumbtacks are needed in all to pin up $10$ sheets?

(Figure) Rectangular sheets of paper are pinned side by side in a single horizontal row, where each pair of neighboring sheets overlaps along its shared edge and uses the same thumbtacks there. A thumbtack is placed at each of the four corners of every sheet, but the thumbtacks along an overlapping edge are shared by the two sheets. Pinning up $1$ sheet needs $4$ thumbtacks, and each additional sheet needs $2$ more thumbtacks.

Show solution

Understand

Sheets of paper are pinned in a single row on a bulletin board. Pinning 1 sheet needs 4 thumbtacks (one at each corner), and each additional sheet shares an edge with its neighbor so it needs only 2 more thumbtacks. We must find how many thumbtacks are needed for 10 sheets.

Givens

1 sheet needs 4 thumbtacks
Each additional sheet needs 2 more thumbtacks (corners on the shared edge are reused)
There are 10 sheets in a row

Unknowns

The total number of thumbtacks for 10 sheets

Constraints

Sheets are in one horizontal row
Tacks on a shared edge serve both neighboring sheets

Plan

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

The figure shows the row growing by one sheet at a time, each adding exactly 2 new tacks. Building small cases reveals the rule 'start with 4, then add 2 for each extra sheet,' which becomes a multiplication-plus-addition expression for any count.

Execute

#9 Solve an Easier Related Problem 3.OA.D.9

Use small cases from the figure: 1 sheet needs 4 tacks; 2 sheets need 4 + 2 = 6; 3 sheets need 4 + 2 + 2 = 8. Each new sheet after the first adds 2 tacks.

$4,\ 6,\ 8,\ \ldots$ (add 2 each time)

Looking at one, two, then three sheets makes the steady jump of 2 easy to spot.

#5 Look for a Pattern 3.OA.D.8

After the first sheet there are 10 - 1 = 9 extra sheets, each adding 2 tacks, on top of the starting 4.

4 + (10 - 1) \times 2

The first sheet sets the 4, and every later sheet contributes the same 2 tacks.

#5 Look for a Pattern 3.OA.A.3

Compute the extra tacks then add the starting amount.

4 + 9 \times 2 = 4 + 18 = 22

9 added sheets bring 18 tacks, plus the original 4 makes 22.

Answer: 22 thumbtacks

Review

Check the rule on a small case: 3 sheets give 4 + 2 times 2 = 8, which matches counting tacks in the figure. For 10 sheets, 22 tacks is between 4 (one sheet) and 10 times 4 = 40 (if none were shared), which is sensible.

Build up by repeated addition: start at 4 for the first sheet, then add 2 for each of the remaining 9 sheets one at a time; this reaches 22 and agrees with the multiplication shortcut.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Finding the steady increase of 2 tacks per added sheet
3.OA.D.8 Solve two-step word problems using four operations within 100 — Writing the rule as 4 + (10 - 1) times 2 and evaluating it
3.OA.A.3 Solve multiplication and division word problems within 100 — Computing 9 times 2 and adding to reach the total

💡 Find how much the count grows each step, start from the first one, then multiply and add: 4 + 9 times 2 = 22!

Variant 2 answer: 46 thumbtacks

As shown in the figure, sheets of paper are being pinned to a bulletin board with thumbtacks. How many thumbtacks are needed in all to pin up $15$ sheets?

Show solution

Understand

Sheets of paper are pinned in a single row on a bulletin board. Pinning 1 sheet needs 4 thumbtacks (one at each corner), and each additional sheet shares an edge with its neighbor so it needs only 3 more thumbtacks. We must find how many thumbtacks are needed for 15 sheets.

Givens

1 sheet needs 4 thumbtacks
Each additional sheet needs 3 more thumbtacks (corners on the shared edge are reused)
There are 15 sheets in a row

Unknowns

The total number of thumbtacks for 15 sheets

Constraints

Sheets are in one horizontal row
Tacks on a shared edge serve both neighboring sheets

Plan

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

The figure shows the row growing by one sheet at a time, each adding exactly 3 new tacks. Building small cases reveals the rule 'start with 4, then add 3 for each extra sheet,' which becomes a multiplication-plus-addition expression for any count.

Execute

#9 Solve an Easier Related Problem 3.OA.D.9

Use small cases from the figure: 1 sheet needs 4 tacks; 2 sheets need 4 + 3 = 7; 3 sheets need 4 + 3 + 3 = 10. Each new sheet after the first adds 3 tacks.

$4,\ 7,\ 10,\ \ldots$ (add 3 each time)

Looking at one, two, then three sheets makes the steady jump of 3 easy to spot.

#5 Look for a Pattern 3.OA.D.8

After the first sheet there are 15 - 1 = 14 extra sheets, each adding 3 tacks, on top of the starting 4.

4 + (15 - 1) \times 3

The first sheet sets the 4, and every later sheet contributes the same 3 tacks.

#5 Look for a Pattern 3.OA.A.3

Compute the extra tacks then add the starting amount.

4 + 14 \times 3 = 4 + 42 = 46

14 added sheets bring 42 tacks, plus the original 4 makes 46.

Answer: 46 thumbtacks

Review

Check the rule on a small case: 3 sheets give 4 + 2 times 3 = 10, which matches counting tacks in the figure. For 15 sheets, 46 tacks is between 4 (one sheet) and 15 times 4 = 60 (if none were shared), which is sensible.

Build up by repeated addition: start at 4 for the first sheet, then add 3 for each of the remaining 14 sheets one at a time; this reaches 46 and agrees with the multiplication shortcut.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Finding the steady increase of 3 tacks per added sheet
3.OA.D.8 Solve two-step word problems using four operations within 100 — Writing the rule as 4 + (15 - 1) times 3 and evaluating it
3.OA.A.3 Solve multiplication and division word problems within 100 — Computing 14 times 3 and adding to reach the total

💡 Find how much the count grows each step, start from the first one, then multiply and add: 4 + 14 times 3 = 46!

Variant 3 answer: 19 thumbtacks

As shown in the figure, sheets of paper are being pinned to a bulletin board with thumbtacks. How many thumbtacks are needed in all to pin up $8$ sheets?

(Figure) Rectangular sheets of paper are pinned side by side in a single horizontal row, where each pair of neighboring sheets overlaps along its shared edge and uses the same thumbtacks there. A thumbtack is placed at each of the four corners of every sheet, but the thumbtacks along an overlapping edge are shared by the two sheets. Pinning up $1$ sheet needs $5$ thumbtacks, and each additional sheet needs $2$ more thumbtacks.

Show solution

Understand

Sheets of paper are pinned in a single row on a bulletin board. Pinning 1 sheet needs 5 thumbtacks (one at each corner), and each additional sheet shares an edge with its neighbor so it needs only 2 more thumbtacks. We must find how many thumbtacks are needed for 8 sheets.

Givens

1 sheet needs 5 thumbtacks
Each additional sheet needs 2 more thumbtacks (corners on the shared edge are reused)
There are 8 sheets in a row

Unknowns

The total number of thumbtacks for 8 sheets

Constraints

Sheets are in one horizontal row
Tacks on a shared edge serve both neighboring sheets

Plan

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

The figure shows the row growing by one sheet at a time, each adding exactly 2 new tacks. Building small cases reveals the rule 'start with 5, then add 2 for each extra sheet,' which becomes a multiplication-plus-addition expression for any count.

Execute

#9 Solve an Easier Related Problem 3.OA.D.9

Use small cases from the figure: 1 sheet needs 5 tacks; 2 sheets need 5 + 2 = 7; 3 sheets need 5 + 2 + 2 = 9. Each new sheet after the first adds 2 tacks.

$5,\ 7,\ 9,\ \ldots$ (add 2 each time)

Looking at one, two, then three sheets makes the steady jump of 2 easy to spot.

#5 Look for a Pattern 3.OA.D.8

After the first sheet there are 8 - 1 = 7 extra sheets, each adding 2 tacks, on top of the starting 5.

5 + (8 - 1) \times 2

The first sheet sets the 5, and every later sheet contributes the same 2 tacks.

#5 Look for a Pattern 3.OA.A.3

Compute the extra tacks then add the starting amount.

5 + 7 \times 2 = 5 + 14 = 19

7 added sheets bring 14 tacks, plus the original 5 makes 19.

Answer: 19 thumbtacks

Review

Check the rule on a small case: 3 sheets give 5 + 2 times 2 = 9, which matches counting tacks in the figure. For 8 sheets, 19 tacks is between 5 (one sheet) and 8 times 5 = 40 (if none were shared), which is sensible.

Build up by repeated addition: start at 5 for the first sheet, then add 2 for each of the remaining 7 sheets one at a time; this reaches 19 and agrees with the multiplication shortcut.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Finding the steady increase of 2 tacks per added sheet
3.OA.D.8 Solve two-step word problems using four operations within 100 — Writing the rule as 5 + (8 - 1) times 2 and evaluating it
3.OA.A.3 Solve multiplication and division word problems within 100 — Computing 7 times 2 and adding to reach the total

💡 Find how much the count grows each step, start from the first one, then multiply and add: 5 + 7 times 2 = 19!

Variant 4 answer: 52 thumbtacks

As shown in the figure, sheets of paper are being pinned to a bulletin board with thumbtacks. How many thumbtacks are needed in all to pin up $25$ sheets?

Show solution

Understand

Givens

1 sheet needs 4 thumbtacks
Each additional sheet needs 2 more thumbtacks (corners on the shared edge are reused)
There are 25 sheets in a row

Unknowns

The total number of thumbtacks for 25 sheets

Constraints

Sheets are in one horizontal row
Tacks on a shared edge serve both neighboring sheets

Plan

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

Execute

#9 Solve an Easier Related Problem 3.OA.D.9

Use small cases from the figure: 1 sheet needs 4 tacks; 2 sheets need 4 + 2 = 6; 3 sheets need 4 + 2 + 2 = 8. Each new sheet after the first adds 2 tacks.

$4,\ 6,\ 8,\ \ldots$ (add 2 each time)

Looking at one, two, then three sheets makes the steady jump of 2 easy to spot.

#5 Look for a Pattern 3.OA.D.8

After the first sheet there are 25 - 1 = 24 extra sheets, each adding 2 tacks, on top of the starting 4.

4 + (25 - 1) \times 2

The first sheet sets the 4, and every later sheet contributes the same 2 tacks.

#5 Look for a Pattern 3.OA.A.3

Compute the extra tacks then add the starting amount.

4 + 24 \times 2 = 4 + 48 = 52

24 added sheets bring 48 tacks, plus the original 4 makes 52.

Answer: 52 thumbtacks

Review

Check the rule on a small case: 3 sheets give 4 + 2 times 2 = 8, which matches counting tacks in the figure. For 25 sheets, 52 tacks is between 4 (one sheet) and 25 times 4 = 100 (if none were shared), which is sensible.

Build up by repeated addition: start at 4 for the first sheet, then add 2 for each of the remaining 24 sheets one at a time; this reaches 52 and agrees with the multiplication shortcut.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Finding the steady increase of 2 tacks per added sheet
3.OA.D.8 Solve two-step word problems using four operations within 100 — Writing the rule as 4 + (25 - 1) times 2 and evaluating it
3.OA.A.3 Solve multiplication and division word problems within 100 — Computing 24 times 2 and adding to reach the total

💡 Find how much the count grows each step, start from the first one, then multiply and add: 4 + 24 times 2 = 52!

Variant 5 answer: 40 thumbtacks

As shown in the figure, sheets of paper are being pinned to a bulletin board with thumbtacks. How many thumbtacks are needed in all to pin up $9$ sheets?

(Figure) Rectangular sheets of paper are pinned side by side in a single horizontal row, where each pair of neighboring sheets overlaps along its shared edge and uses the same thumbtacks there. A thumbtack is placed at each of the four corners of every sheet, but the thumbtacks along an overlapping edge are shared by the two sheets. Pinning up $1$ sheet needs $8$ thumbtacks, and each additional sheet needs $4$ more thumbtacks.

Show solution

Understand

Sheets of paper are pinned in a single row on a bulletin board. Pinning 1 sheet needs 8 thumbtacks (one at each corner), and each additional sheet shares an edge with its neighbor so it needs only 4 more thumbtacks. We must find how many thumbtacks are needed for 9 sheets.

Givens

1 sheet needs 8 thumbtacks
Each additional sheet needs 4 more thumbtacks (corners on the shared edge are reused)
There are 9 sheets in a row

Unknowns

The total number of thumbtacks for 9 sheets

Constraints

Sheets are in one horizontal row
Tacks on a shared edge serve both neighboring sheets

Plan

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

The figure shows the row growing by one sheet at a time, each adding exactly 4 new tacks. Building small cases reveals the rule 'start with 8, then add 4 for each extra sheet,' which becomes a multiplication-plus-addition expression for any count.

Execute

#9 Solve an Easier Related Problem 3.OA.D.9

Use small cases from the figure: 1 sheet needs 8 tacks; 2 sheets need 8 + 4 = 12; 3 sheets need 8 + 4 + 4 = 16. Each new sheet after the first adds 4 tacks.

$8,\ 12,\ 16,\ \ldots$ (add 4 each time)

Looking at one, two, then three sheets makes the steady jump of 4 easy to spot.

#5 Look for a Pattern 3.OA.D.8

After the first sheet there are 9 - 1 = 8 extra sheets, each adding 4 tacks, on top of the starting 8.

8 + (9 - 1) \times 4

The first sheet sets the 8, and every later sheet contributes the same 4 tacks.

#5 Look for a Pattern 3.OA.A.3

Compute the extra tacks then add the starting amount.

8 + 8 \times 4 = 8 + 32 = 40

8 added sheets bring 32 tacks, plus the original 8 makes 40.

Answer: 40 thumbtacks

Review

Check the rule on a small case: 3 sheets give 8 + 2 times 4 = 16, which matches counting tacks in the figure. For 9 sheets, 40 tacks is between 8 (one sheet) and 9 times 8 = 72 (if none were shared), which is sensible.

Build up by repeated addition: start at 8 for the first sheet, then add 4 for each of the remaining 8 sheets one at a time; this reaches 40 and agrees with the multiplication shortcut.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Finding the steady increase of 4 tacks per added sheet
3.OA.D.8 Solve two-step word problems using four operations within 100 — Writing the rule as 8 + (9 - 1) times 4 and evaluating it
3.OA.A.3 Solve multiplication and division word problems within 100 — Computing 8 times 4 and adding to reach the total

💡 Find how much the count grows each step, start from the first one, then multiply and add: 8 + 8 times 4 = 40!

Variant 6 answer: 31 thumbtacks

As shown in the figure, sheets of paper are being pinned to a bulletin board with thumbtacks. How many thumbtacks are needed in all to pin up $30$ sheets?

(Figure) Rectangular sheets of paper are pinned side by side in a single horizontal row, where each pair of neighboring sheets overlaps along its shared edge and uses the same thumbtacks there. A thumbtack is placed at each of the four corners of every sheet, but the thumbtacks along an overlapping edge are shared by the two sheets. Pinning up $1$ sheet needs $2$ thumbtacks, and each additional sheet needs $1$ more thumbtacks.

Show solution

Understand

Sheets of paper are pinned in a single row on a bulletin board. Pinning 1 sheet needs 2 thumbtacks (one at each corner), and each additional sheet shares an edge with its neighbor so it needs only 1 more thumbtacks. We must find how many thumbtacks are needed for 30 sheets.

Givens

1 sheet needs 2 thumbtacks
Each additional sheet needs 1 more thumbtacks (corners on the shared edge are reused)
There are 30 sheets in a row

Unknowns

The total number of thumbtacks for 30 sheets

Constraints

Sheets are in one horizontal row
Tacks on a shared edge serve both neighboring sheets

Plan

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

The figure shows the row growing by one sheet at a time, each adding exactly 1 new tacks. Building small cases reveals the rule 'start with 2, then add 1 for each extra sheet,' which becomes a multiplication-plus-addition expression for any count.

Execute

#9 Solve an Easier Related Problem 3.OA.D.9

Use small cases from the figure: 1 sheet needs 2 tacks; 2 sheets need 2 + 1 = 3; 3 sheets need 2 + 1 + 1 = 4. Each new sheet after the first adds 1 tacks.

$2,\ 3,\ 4,\ \ldots$ (add 1 each time)

Looking at one, two, then three sheets makes the steady jump of 1 easy to spot.

#5 Look for a Pattern 3.OA.D.8

After the first sheet there are 30 - 1 = 29 extra sheets, each adding 1 tacks, on top of the starting 2.

2 + (30 - 1) \times 1

The first sheet sets the 2, and every later sheet contributes the same 1 tacks.

#5 Look for a Pattern 3.OA.A.3

Compute the extra tacks then add the starting amount.

2 + 29 \times 1 = 2 + 29 = 31

29 added sheets bring 29 tacks, plus the original 2 makes 31.

Answer: 31 thumbtacks

Review

Check the rule on a small case: 3 sheets give 2 + 2 times 1 = 4, which matches counting tacks in the figure. For 30 sheets, 31 tacks is between 2 (one sheet) and 30 times 2 = 60 (if none were shared), which is sensible.

Build up by repeated addition: start at 2 for the first sheet, then add 1 for each of the remaining 29 sheets one at a time; this reaches 31 and agrees with the multiplication shortcut.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Finding the steady increase of 1 tacks per added sheet
3.OA.D.8 Solve two-step word problems using four operations within 100 — Writing the rule as 2 + (30 - 1) times 1 and evaluating it
3.OA.A.3 Solve multiplication and division word problems within 100 — Computing 29 times 1 and adding to reach the total

💡 Find how much the count grows each step, start from the first one, then multiply and add: 2 + 29 times 1 = 31!

Variant 7 answer: 14 thumbtacks

As shown in the figure, sheets of paper are being pinned to a bulletin board with thumbtacks. How many thumbtacks are needed in all to pin up $12$ sheets?

(Figure) Rectangular sheets of paper are pinned side by side in a single horizontal row, where each pair of neighboring sheets overlaps along its shared edge and uses the same thumbtacks there. A thumbtack is placed at each of the four corners of every sheet, but the thumbtacks along an overlapping edge are shared by the two sheets. Pinning up $1$ sheet needs $3$ thumbtacks, and each additional sheet needs $1$ more thumbtacks.

Show solution

Understand

Sheets of paper are pinned in a single row on a bulletin board. Pinning 1 sheet needs 3 thumbtacks (one at each corner), and each additional sheet shares an edge with its neighbor so it needs only 1 more thumbtacks. We must find how many thumbtacks are needed for 12 sheets.

Givens

1 sheet needs 3 thumbtacks
Each additional sheet needs 1 more thumbtacks (corners on the shared edge are reused)
There are 12 sheets in a row

Unknowns

The total number of thumbtacks for 12 sheets

Constraints

Sheets are in one horizontal row
Tacks on a shared edge serve both neighboring sheets

Plan

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

The figure shows the row growing by one sheet at a time, each adding exactly 1 new tacks. Building small cases reveals the rule 'start with 3, then add 1 for each extra sheet,' which becomes a multiplication-plus-addition expression for any count.

Execute

#9 Solve an Easier Related Problem 3.OA.D.9

Use small cases from the figure: 1 sheet needs 3 tacks; 2 sheets need 3 + 1 = 4; 3 sheets need 3 + 1 + 1 = 5. Each new sheet after the first adds 1 tacks.

$3,\ 4,\ 5,\ \ldots$ (add 1 each time)

Looking at one, two, then three sheets makes the steady jump of 1 easy to spot.

#5 Look for a Pattern 3.OA.D.8

After the first sheet there are 12 - 1 = 11 extra sheets, each adding 1 tacks, on top of the starting 3.

3 + (12 - 1) \times 1

The first sheet sets the 3, and every later sheet contributes the same 1 tacks.

#5 Look for a Pattern 3.OA.A.3

Compute the extra tacks then add the starting amount.

3 + 11 \times 1 = 3 + 11 = 14

11 added sheets bring 11 tacks, plus the original 3 makes 14.

Answer: 14 thumbtacks

Review

Check the rule on a small case: 3 sheets give 3 + 2 times 1 = 5, which matches counting tacks in the figure. For 12 sheets, 14 tacks is between 3 (one sheet) and 12 times 3 = 36 (if none were shared), which is sensible.

Build up by repeated addition: start at 3 for the first sheet, then add 1 for each of the remaining 11 sheets one at a time; this reaches 14 and agrees with the multiplication shortcut.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Finding the steady increase of 1 tacks per added sheet
3.OA.D.8 Solve two-step word problems using four operations within 100 — Writing the rule as 3 + (12 - 1) times 1 and evaluating it
3.OA.A.3 Solve multiplication and division word problems within 100 — Computing 11 times 1 and adding to reach the total

💡 Find how much the count grows each step, start from the first one, then multiply and add: 3 + 11 times 1 = 14!

Variant 8 answer: 63 thumbtacks

As shown in the figure, sheets of paper are being pinned to a bulletin board with thumbtacks. How many thumbtacks are needed in all to pin up $20$ sheets?

(Figure) Rectangular sheets of paper are pinned side by side in a single horizontal row, where each pair of neighboring sheets overlaps along its shared edge and uses the same thumbtacks there. A thumbtack is placed at each of the four corners of every sheet, but the thumbtacks along an overlapping edge are shared by the two sheets. Pinning up $1$ sheet needs $6$ thumbtacks, and each additional sheet needs $3$ more thumbtacks.

Show solution

Understand

Sheets of paper are pinned in a single row on a bulletin board. Pinning 1 sheet needs 6 thumbtacks (one at each corner), and each additional sheet shares an edge with its neighbor so it needs only 3 more thumbtacks. We must find how many thumbtacks are needed for 20 sheets.

Givens

1 sheet needs 6 thumbtacks
Each additional sheet needs 3 more thumbtacks (corners on the shared edge are reused)
There are 20 sheets in a row

Unknowns

The total number of thumbtacks for 20 sheets

Constraints

Sheets are in one horizontal row
Tacks on a shared edge serve both neighboring sheets

Plan

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

The figure shows the row growing by one sheet at a time, each adding exactly 3 new tacks. Building small cases reveals the rule 'start with 6, then add 3 for each extra sheet,' which becomes a multiplication-plus-addition expression for any count.

Execute

#9 Solve an Easier Related Problem 3.OA.D.9

Use small cases from the figure: 1 sheet needs 6 tacks; 2 sheets need 6 + 3 = 9; 3 sheets need 6 + 3 + 3 = 12. Each new sheet after the first adds 3 tacks.

$6,\ 9,\ 12,\ \ldots$ (add 3 each time)

Looking at one, two, then three sheets makes the steady jump of 3 easy to spot.

#5 Look for a Pattern 3.OA.D.8

After the first sheet there are 20 - 1 = 19 extra sheets, each adding 3 tacks, on top of the starting 6.

6 + (20 - 1) \times 3

The first sheet sets the 6, and every later sheet contributes the same 3 tacks.

#5 Look for a Pattern 3.OA.A.3

Compute the extra tacks then add the starting amount.

6 + 19 \times 3 = 6 + 57 = 63

19 added sheets bring 57 tacks, plus the original 6 makes 63.

Answer: 63 thumbtacks

Review

Check the rule on a small case: 3 sheets give 6 + 2 times 3 = 12, which matches counting tacks in the figure. For 20 sheets, 63 tacks is between 6 (one sheet) and 20 times 6 = 120 (if none were shared), which is sensible.

Build up by repeated addition: start at 6 for the first sheet, then add 3 for each of the remaining 19 sheets one at a time; this reaches 63 and agrees with the multiplication shortcut.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Finding the steady increase of 3 tacks per added sheet
3.OA.D.8 Solve two-step word problems using four operations within 100 — Writing the rule as 6 + (20 - 1) times 3 and evaluating it
3.OA.A.3 Solve multiplication and division word problems within 100 — Computing 19 times 3 and adding to reach the total

💡 Find how much the count grows each step, start from the first one, then multiply and add: 6 + 19 times 3 = 63!

Variant 9 answer: 40 thumbtacks

As shown in the figure, sheets of paper are being pinned to a bulletin board with thumbtacks. How many thumbtacks are needed in all to pin up $18$ sheets?

(Figure) Rectangular sheets of paper are pinned side by side in a single horizontal row, where each pair of neighboring sheets overlaps along its shared edge and uses the same thumbtacks there. A thumbtack is placed at each of the four corners of every sheet, but the thumbtacks along an overlapping edge are shared by the two sheets. Pinning up $1$ sheet needs $6$ thumbtacks, and each additional sheet needs $2$ more thumbtacks.

Show solution

Understand

Sheets of paper are pinned in a single row on a bulletin board. Pinning 1 sheet needs 6 thumbtacks (one at each corner), and each additional sheet shares an edge with its neighbor so it needs only 2 more thumbtacks. We must find how many thumbtacks are needed for 18 sheets.

Givens

1 sheet needs 6 thumbtacks
Each additional sheet needs 2 more thumbtacks (corners on the shared edge are reused)
There are 18 sheets in a row

Unknowns

The total number of thumbtacks for 18 sheets

Constraints

Sheets are in one horizontal row
Tacks on a shared edge serve both neighboring sheets

Plan

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

The figure shows the row growing by one sheet at a time, each adding exactly 2 new tacks. Building small cases reveals the rule 'start with 6, then add 2 for each extra sheet,' which becomes a multiplication-plus-addition expression for any count.

Execute

#9 Solve an Easier Related Problem 3.OA.D.9

Use small cases from the figure: 1 sheet needs 6 tacks; 2 sheets need 6 + 2 = 8; 3 sheets need 6 + 2 + 2 = 10. Each new sheet after the first adds 2 tacks.

$6,\ 8,\ 10,\ \ldots$ (add 2 each time)

Looking at one, two, then three sheets makes the steady jump of 2 easy to spot.

#5 Look for a Pattern 3.OA.D.8

After the first sheet there are 18 - 1 = 17 extra sheets, each adding 2 tacks, on top of the starting 6.

6 + (18 - 1) \times 2

The first sheet sets the 6, and every later sheet contributes the same 2 tacks.

#5 Look for a Pattern 3.OA.A.3

Compute the extra tacks then add the starting amount.

6 + 17 \times 2 = 6 + 34 = 40

17 added sheets bring 34 tacks, plus the original 6 makes 40.

Answer: 40 thumbtacks

Review

Check the rule on a small case: 3 sheets give 6 + 2 times 2 = 10, which matches counting tacks in the figure. For 18 sheets, 40 tacks is between 6 (one sheet) and 18 times 6 = 108 (if none were shared), which is sensible.

Build up by repeated addition: start at 6 for the first sheet, then add 2 for each of the remaining 17 sheets one at a time; this reaches 40 and agrees with the multiplication shortcut.

Standards · min grade 3

3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Finding the steady increase of 2 tacks per added sheet
3.OA.D.8 Solve two-step word problems using four operations within 100 — Writing the rule as 6 + (18 - 1) times 2 and evaluating it
3.OA.A.3 Solve multiplication and division word problems within 100 — Computing 17 times 2 and adding to reach the total

💡 Find how much the count grows each step, start from the first one, then multiply and add: 6 + 17 times 2 = 40!