Printable math worksheet for identifying and continuing growing patterns using geometric shapes.

Let's solve each of the growing pattern problems step by step. The goal is to identify the pattern and draw the next figure in each sequence.

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🔹 Problem 1: Green Triangles

Sequence:
- Step 1: One small green triangle.
- Step 2: A larger triangle made of 4 small triangles (a big triangle with a white triangle in the center).
- Step 3: An even larger triangle made of 9 small triangles (3 rows: 1, 3, 5? Wait — let's count carefully.)

Wait — actually, let’s look at it:

- Step 1: 1 small triangle
- Step 2: A triangle composed of 4 small triangles arranged in a larger triangle shape (like a triangle made of 2 rows: 1 on top, 3 below? No — wait)

Actually, looking closely:
- Step 1: 1 triangle
- Step 2: A triangle made of 4 small triangles (a larger triangle divided into 4 smaller ones)
- Step 3: A triangle made of 9 small triangles (a large triangle divided into 9 smaller ones)

So this looks like a triangular number pattern, but more likely it's squares of numbers arranged in triangular shapes?

Wait — better idea: It's a fractal-like growing triangle.

But actually, notice:

- Step 1: 1 triangle
- Step 2: 4 triangles (arranged as a bigger triangle with a white triangle in the center — forming a larger triangle)
- Step 3: 9 triangles

This suggests that each stage has a square number of small triangles:
1 = 1²
4 = 2²
9 = 3²

And the size of the large triangle increases by one row.

So the pattern is: Each step adds a layer, forming a larger triangle made of smaller ones.

So the next figure should have 16 small triangles arranged in a 4×4 triangle shape — i.e., a large triangle divided into 16 small equilateral triangles.

✔ Next figure: A large green triangle divided into 16 small green triangles (4 layers: 1, 3, 5, 7? No — wait, for a triangular arrangement, the total number of small upward-pointing triangles isn't just n²).

Wait — perhaps it's not about counting all small triangles, but how they are arranged.

Looking again:

- Step 1: 1 small green triangle
- Step 2: A triangle made of 4 small triangles: 3 green and 1 white in the center? Or maybe all green?

Actually, the second image shows a larger green triangle made of 4 small green triangles (but arranged in a triangle shape), and the third is a larger triangle made of 9 small green triangles.

Ah! So it's a triangle of triangles:

- Step 1: 1 triangle → size 1
- Step 2: 4 triangles → arranged in a triangle of side length 2 (with 3 on bottom, 1 on top? But 1+2=3 — not 4)

Wait — no. Let’s think differently.

It’s more likely that each figure is a large triangle subdivided into smaller triangles, and the number of small triangles is increasing as squares:

- Step 1: 1 triangle
- Step 2: 4 small triangles forming a larger triangle
- Step 3: 9 small triangles forming an even larger triangle

So this is a triangular grid where the side length increases by 1 each time, and the total number of small upward-pointing triangles is n².

So next step: side length 4, so 16 small triangles.

✔ Answer for #1: Draw a large green triangle made of 16 small green triangles arranged in a triangle shape (like a larger triangle subdivided into 4 rows: 1, 3, 5, 7? No — that would be 16? 1+3+5+7=16 — yes!)

Wait — but 1+3+5+7 = 16, which is correct.

But the shapes are equilateral triangles — so the number of small upward-pointing triangles in a large triangle of side length n is n(n+1)/2? No — that's for dots.

Wait — here’s the key: These appear to be Sierpinski triangle style patterns.

But actually, looking at the images:

- Step 1: One green triangle
- Step 2: A larger triangle made of 4 small triangles (3 green, 1 white?) — no, all seem green.

Wait — in the second figure, there is a white triangle in the center, so it's like a larger triangle made of 4 small triangles, with one removed (white). But no — the white triangle is inside.

Wait — actually, Step 2 shows a large triangle divided into 4 small triangles, all green, arranged in a larger triangle shape.

But then Step 3: a large triangle made of 9 small green triangles, arranged in a 3x3 triangular grid.

So:

- Step 1: 1 triangle (side 1)
- Step 2: 4 triangles (side 2) → 2×2
- Step 3: 9 triangles (side 3) → 3×3

So it's a growing triangle made of smaller triangles, where the side length increases by 1, and the total number of small triangles is n².

So next: side length 4, total 16 small green triangles arranged in a triangular shape.

✔ Draw a large green triangle made of 16 small green triangles (4×4 grid).

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🔹 Problem 2: Pink Squares

Sequence:
- Step 1: 2 pink squares (stacked vertically)
- Step 2: 4 pink squares (2×2 grid)
- Step 3: 6 pink squares (2×3 grid)

Wait — let’s count:

- Step 1: 2 squares (one on top of other)
- Step 2: 4 squares (2 columns, 2 rows)
- Step 3: 6 squares (2 columns, 3 rows)

So pattern: Number of rows increases by 1 each time, and columns stay at 2.

- Step 1: 2 × 1 = 2 squares
- Step 2: 2 × 2 = 4 squares
- Step 3: 2 × 3 = 6 squares

So next: 2 × 4 = 8 squares

✔ Next figure: Two columns of 4 pink squares each → a 2×4 rectangle of pink squares.

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🔹 Problem 3: Blue Rectangles and Circles

Sequence:
- Step 1: One blue rectangle + one blue circle
- Step 2: One blue rectangle + two blue circles
- Step 3: One blue rectangle + three blue circles

Pattern: The number of circles increases by 1 each time, while the rectangle stays the same.

So next: One blue rectangle + four blue circles

✔ Next figure: One blue rectangle followed by four blue circles in a line.

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🔹 Problem 4: Orange Diamonds (Rhombuses)

Sequence:
- Step 1: One orange diamond
- Step 2: Four orange diamonds arranged in an "X" shape
- Step 3: Nine orange diamonds arranged in a larger "X"

Wait — let’s analyze:

- Step 1: 1 diamond
- Step 2: 4 diamonds forming a cross (like a plus sign, but rotated — X shape)
- Step 3: 9 diamonds forming a larger X

But actually, it's a cross pattern:

- Step 1: 1 diamond (center)
- Step 2: 4 diamonds forming a small X (diagonals)
- Step 3: 9 diamonds forming a larger X

Wait — actually, look at the shapes:

- Step 1: 1 diamond
- Step 2: 4 diamonds arranged in a small X (two diagonals, each with 2 diamonds)
- Step 3: 9 diamonds forming a larger X — likely a diagonal cross with 3 diamonds per arm

Wait — actually, it's more likely:

- Step 1: 1 diamond
- Step 2: 4 diamonds forming a plus sign or X — but arranged like a square rotated?
- Step 3: 9 diamonds forming a larger X

But upon closer inspection:

The second figure has 4 diamonds arranged in a diamond-shaped cross (like a plus sign rotated 45°), forming a small X.

Third figure has 9 diamonds forming a larger X.

Wait — let’s count:

- Step 1: 1 diamond
- Step 2: 4 diamonds (arranged in a 2×2 diamond shape? No — it's a cross)

Actually, it’s likely a growing cross:

- Step 1: 1 diamond (center)
- Step 2: 4 diamonds around it — forming a cross with arms of length 2 (including center)? No.

Wait — better idea:

Look at the number of diamonds:

- Step 1: 1
- Step 2: 4
- Step 3: 9

So: 1 = 1², 4 = 2², 9 = 3²

So next: 16 diamonds

And the shape is a larger X, meaning the arms grow longer.

So it's a square cross or diamond-shaped cross.

In fact, these look like concentric diamond layers.

- Step 1: 1 diamond
- Step 2: 4 diamonds forming a square around it (total 5? But only 4 shown)

Wait — no, in Step 2, there are 4 diamonds forming a cross.

But Step 3 has 9 diamonds — possibly arranged in a 3×3 diamond grid.

Alternatively, it might be each step adds a layer to form a larger diamond.

But the pattern is:

- Step 1: 1 diamond
- Step 2: 4 diamonds in a cross (like a plus sign)
- Step 3: 9 diamonds in a larger cross

Wait — 1, 4, 9 → squares

So next: 16 diamonds

And the shape is a larger X (cross) with arms of length 4.

So it’s a cross with 4 diamonds per arm, including the center.

But center is shared.

So total: 4 arms × 4 diamonds = 16, minus overlaps?

No — better: It's a diamond-shaped cross where each step increases the size.

Alternatively, it's a growing diamond made of smaller diamonds.

But since the numbers are 1, 4, 9, 16 — perfect squares — and the shapes are increasingly larger X's, the next figure should be a 16-diamond X, arranged in a 4×4 diamond grid.

So the pattern is: Each figure is a diamond-shaped grid of size n×n, where n increases.

So:

- Step 1: 1×1 = 1 diamond
- Step 2: 2×2 = 4 diamonds
- Step 3: 3×3 = 9 diamonds
- Step 4: 4×4 = 16 diamonds

And they are arranged in a diamond (rotated square) shape.

✔ Next figure: A 4×4 diamond-shaped grid of orange diamonds (16 total).

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✔ Final Answers:

#### 1. Green Triangles:
- Next: A large green triangle made of 16 small green triangles (arranged in a 4×4 triangular grid).

#### 2. Pink Squares:
- Next: A 2×4 rectangle of pink squares (2 columns, 4 rows).

#### 3. Blue Rectangle and Circles:
- Next: One blue rectangle followed by four blue circles.

#### 4. Orange Diamonds:
- Next: A 4×4 diamond-shaped grid of orange diamonds (16 total), forming a larger X.

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📝 Summary:

| Problem | Pattern | Next Figure |
|--------|--------|-------------|
| 1 | Triangular growth: 1, 4, 9 → next is 16 small green triangles | Large green triangle with 16 small triangles |
| 2 | 2×1, 2×2, 2×3 → next is 2×4 | 2 columns of 4 pink squares |
| 3 | 1 rectangle + 1, 2, 3 circles → next is 4 circles | 1 rectangle + 4 circles |
| 4 | 1, 4, 9 → next is 16 orange diamonds | 4×4 diamond grid |

You can now draw these on the worksheet accordingly.