Math Puzzles for Kids - Shape Puzzles - Free Printable

The image you've uploaded is a "Rectangle Up Puzzle Grids" worksheet designed for 2nd-grade students, from Math-Salamanders.com. The goal of this activity is to use the dot grids to draw different rectangles by connecting the dots.

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🔍 What You See:

There are four identical 3×3 dot grids (3 rows and 3 columns of dots), arranged in a 2×2 layout on the page. Each grid has 9 dots, forming a square lattice.

The instructions say:
> *"Use these grids to help you draw different rectangles."*

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✔ Objective:

To draw rectangles using the dots as vertices (corners). Students should connect the dots to form rectangles of various sizes and orientations.

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🧩 How to Solve This Puzzle:

#### Step 1: Understand the Grid
Each grid has 3 rows and 3 columns of dots:
```
• • •
• • •
• • •
```
You can only use the dots as corners — you cannot add new points.

#### Step 2: Draw Rectangles
A rectangle must have:
- Four corners,
- Opposite sides equal and parallel,
- All angles 90°,
- Corners at the dots.

Let’s explore possible rectangles you can make on one grid.

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📐 Possible Rectangles on a 3×3 Dot Grid

On a 3×3 grid, here are all the possible rectangles you can draw:

#### 1. 1×1 squares
- These are small squares formed by adjacent dots.
- Example: Top-left corner dot → right → down → left.
- There are 4 such squares (one in each corner of the 3×3 grid).

#### 2. 1×2 rectangles
- Width = 2 units, Height = 1 unit.
- Can be horizontal or vertical.
- Horizontal: 2 per row × 3 rows = 6 total.
- Vertical: 2 per column × 3 columns = 6 total.
- But wait: in a 3×3 dot grid, you can only fit 2 dots horizontally or vertically apart → so:
- Horizontal 1×2 rectangles: 2 per row × 3 rows = 6
- Vertical 1×2 rectangles: 2 per column × 3 columns = 6

Wait — actually, let's clarify:

In a 3×3 dot grid, the distance between dots is 1 unit. So:

- You can have rectangles that span:
- 1 unit wide × 1 unit tall → 1×1 square
- 1 unit wide × 2 units tall → 1×2 rectangle (vertical)
- 2 units wide × 1 unit tall → 2×1 rectangle (horizontal)
- 2 units wide × 2 units tall → 2×2 square

So let's count:

#### ✔ Types of Rectangles:

| Rectangle Size | Number of Positions |
|----------------|---------------------|
| 1×1 (square) | 4 (top-left, top-right, bottom-left, bottom-right) |
| 1×2 (vertical) | 3 columns × 2 positions per column = 6 |
| 2×1 (horizontal)| 3 rows × 2 positions per row = 6 |
| 2×2 (square) | Only 1 possible — uses outer 4 corners |

But wait — actually, we can do better:

Let’s label the dots:

```
(1,1) (1,2) (1,3)
(2,1) (2,2) (2,3)
(3,1) (3,2) (3,3)
```

Now list all rectangles:

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🧮 Counting All Possible Rectangles

A rectangle is defined by choosing:
- Two distinct horizontal lines (rows) → choose 2 out of 3 → C(3,2) = 3
- Two distinct vertical lines (columns) → choose 2 out of 3 → C(3,2) = 3

So total number of rectangles = 3 × 3 = 9

These are:
- 3 choices of row pairs: (1,2), (2,3), (1,3)
- 3 choices of column pairs: (1,2), (2,3), (1,3)

Each combination gives a unique rectangle.

So the 9 rectangles are:

| Row Pairs | Column Pairs | Rectangle Type |
|----------|--------------|----------------|
| (1,2) | (1,2) | 1×1 square |
| (1,2) | (2,3) | 1×1 square |
| (1,2) | (1,3) | 1×2 rectangle |
| (2,3) | (1,2) | 1×1 square |
| (2,3) | (2,3) | 1×1 square |
| (2,3) | (1,3) | 1×2 rectangle |
| (1,3) | (1,2) | 2×1 rectangle |
| (1,3) | (2,3) | 2×1 rectangle |
| (1,3) | (1,3) | 2×2 square |

So there are 9 different rectangles possible on one 3×3 dot grid.

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✔ How to Use the Worksheet

Since there are four grids, the student can:
- Draw different rectangles on each grid.
- Try to draw all 9 rectangles across the four grids.
- Or focus on drawing various shapes (e.g., 1×1, 1×2, 2×1, 2×2).
- Practice identifying squares vs rectangles.

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💡 Tips for Students:

- Use a ruler to draw straight lines.
- Start with small rectangles (1×1).
- Then try larger ones (2×1, 2×2).
- Make sure opposite sides are equal and parallel.
- Label the rectangles if needed.

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🎯 Final Answer Summary:

- On each 3×3 dot grid, you can draw 9 different rectangles.
- These include:
- 4 small 1×1 squares
- 2 horizontal 2×1 rectangles
- 2 vertical 1×2 rectangles
- 1 large 2×2 square
- Use the four grids to practice drawing these rectangles.

✔ Task: Draw different rectangles on each grid using the dots as corners.

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Let me know if you'd like a printable version or a visual guide!