2nd Grade Rectangle Puzzle Worksheet – Draw Rectangles on a 3x3 Dot Grid

Problem Analysis:

The task is to determine how many rectangles can be formed using the dots on the grid. The grid consists of 9 dots arranged in a 3x3 pattern. A rectangle is defined by selecting two horizontal lines and two vertical lines from the grid, and since a square is also a type of rectangle, we need to count all possible rectangles, including squares.

Step-by-Step Solution:

#### 1. Identify the Structure of the Grid:
- The grid has 3 rows and 3 columns of dots.
- To form a rectangle, we need to select:
- 2 out of the 4 horizontal lines (since there are 3 rows, there are 4 horizontal lines).
- 2 out of the 4 vertical lines (since there are 3 columns, there are 4 vertical lines).

#### 2. Count the Number of Ways to Choose Horizontal Lines:
- There are 4 horizontal lines in total.
- We need to choose 2 out of these 4 lines.
- The number of ways to choose 2 lines from 4 is given by the combination formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \):
\[
\binom{4}{2} = \frac{4!}{2! \cdot 2!} = \frac{4 \times 3}{2 \times 1} = 6
\]

#### 3. Count the Number of Ways to Choose Vertical Lines:
- There are 4 vertical lines in total.
- We need to choose 2 out of these 4 lines.
- Using the same combination formula:
\[
\binom{4}{2} = \frac{4!}{2! \cdot 2!} = \frac{4 \times 3}{2 \times 1} = 6
\]

#### 4. Calculate the Total Number of Rectangles:
- Each pair of horizontal lines can be combined with each pair of vertical lines to form a rectangle.
- Therefore, the total number of rectangles is the product of the number of ways to choose horizontal lines and the number of ways to choose vertical lines:
\[
6 \times 6 = 36
\]

#### 5. Verify the Count:
- We can verify this by considering all possible sizes of rectangles:
- 1x1 squares: There are 4 such squares.
- 1x2 rectangles: There are 6 such rectangles (2 in each row, 3 rows).
- 1x3 rectangles: There are 4 such rectangles (1 in each row, 4 rows).
- 2x1 rectangles: There are 6 such rectangles (2 in each column, 3 columns).
- 2x2 squares: There are 4 such squares.
- 2x3 rectangles: There are 4 such rectangles (1 in each column, 4 columns).
- 3x1 rectangles: There are 4 such rectangles (1 in each row, 4 rows).
- 3x2 rectangles: There are 4 such rectangles (1 in each column, 4 columns).
- 3x3 square: There is 1 such square.
- Summing all these gives:
\[
4 + 6 + 4 + 6 + 4 + 4 + 4 + 4 + 1 = 36
\]

Final Answer:

\[
\boxed{36}
\]