Find shape and number rule together

4.OA.C.5 · take · grade 4

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

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.

Show solution

Understand

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 three counters: one up, one to the right, and one diagonally. The counts so far are 1, 4, 7, 10. We must draw the fifth shape and give the number of counters that fills the blank box.

Givens

1st shape: 1 counter
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 fifth 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

Plan

#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 fifth count, and drawing or building the fifth shape confirms it.

Execute

#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).

$1,\ 4,\ 7,\ 10$ (add 3 each time)

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

#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.

#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 + 3 + 3 + 3 = 13

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

Answer: 13 counters

Review

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 four steps of 3.

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.

Standards · min 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 fifth 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

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