Write a growing pattern as an expression

3.OA.D.93.OA.D.8 · take · grade 3

Archetype: Generalize a Growing Pattern into a Rule · step in a 12-type progression

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?

(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.

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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 25 sheets.

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

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 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 4.

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

Twenty-four 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.

Count columns of tacks (tool 1): a row of 25 sheets has 26 vertical edges, each with 2 tacks, giving 26 times 2 = 52 thumbtacks.

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!