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.