How many rectangles can you draw on a 3 x 3 geoboard?  Explain

A geoboard is a grid of points, and for a \(3 \times 3\) geoboard, we have 4 points along each side, giving us a \(4 \times 4\) grid of points. Each rectangle on the geoboard can be formed by choosing two distinct horizontal lines and two distinct vertical lines.

To find the number of rectangles:

1. **Count the number of ways to choose 2 distinct horizontal lines:**
   - There are 4 horizontal lines in a \(3 \times 3\) geoboard.
   - The number of ways to choose 2 distinct horizontal lines is given by the combination formula \(\binom{n}{r}\), where \(n\) is the total number of lines and \(r\) is the number of lines chosen.
   \[
   \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6
   \]

2. **Count the number of ways to choose 2 distinct vertical lines:**
   - Similarly, there are 4 vertical lines.
   - The number of ways to choose 2 distinct vertical lines is also:
   \[
   \binom{4}{2} = 6
   \]

3. **Calculate the total number of rectangles:**
   - The total number of rectangles is the product of the number of ways to choose horizontal and vertical lines:
   \[
   6 \times 6 = 36
   \]

Therefore, there are **36 rectangles** that can be drawn on a \(3 \times 3\) geoboard.

Would you like more details or have any questions?

### Related Questions:
1. How many squares can be formed on a \(3 \times 3\) geoboard?
2. How many rectangles can you draw on a \(4 \times 4\) geoboard?
3. What is the general formula for counting rectangles on an \(n \times n\) geoboard?
4. How many rectangles can you draw on a \(2 \times 3\) geoboard?
5. Can you draw a square that is not aligned to the grid lines on a \(3 \times 3\) geoboard?

**Tip:** When counting geometric shapes in grids, it's helpful to use combinations to select lines and then multiply the results for the two dimensions.

How many rectangles can you draw on a 3 x 3 geoboard? Explain

Learn how to count the number of rectangles that can be drawn on a 3x3 geoboard with a detailed explanation. This problem involves using combinatorics and geometric counting, suitable for students in Grades 6-8.

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