Using a 25-peg Geoboard, how many different size rectangles (that are not squares) can you make? Show them and give their areas.




To solve this, we first need to establish some key principles regarding the 25-peg geoboard.

### Understanding the 25-Peg Geoboard:
- A **25-peg** geoboard consists of a \( 5 \times 5 \) grid (5 rows and 5 columns of pegs).
- The pegs form the vertices of the grid squares.
- Each rectangle on this grid is formed by selecting two distinct horizontal lines and two distinct vertical lines.

### General Approach:
1. **Horizontal and Vertical Lines:** Since we have 5 rows and 5 columns of pegs, there are 6 possible distinct horizontal lines (row dividers) and 6 possible distinct vertical lines (column dividers) formed between pegs.
2. **Rectangle Criteria:** A rectangle requires two distinct horizontal and two distinct vertical lines. Therefore, the number of ways to choose two horizontal lines from 6 is \( \binom{6}{2} = 15 \), and the number of ways to choose two vertical lines from 6 is \( \binom{6}{2} = 15 \).
3. **Total Rectangles:** The total number of rectangles (including squares) is the product of these, \( 15 \times 15 = 225 \).

### Exclude Squares:
Next, we exclude squares to answer the question about rectangles that **are not squares**:
1. **Squares Count:** A square requires equal horizontal and vertical spacing between its sides. For instance, a \( 1 \times 1 \) square has a unit distance between selected lines, a \( 2 \times 2 \) square has a distance of two units, and so on.
   - \( 1 \times 1 \) squares: There are \( 4 \times 4 = 16 \) of these.
   - \( 2 \times 2 \) squares: There are \( 3 \times 3 = 9 \) of these.
   - \( 3 \times 3 \) squares: There are \( 2 \times 2 = 4 \) of these.
   - \( 4 \times 4 \) squares: There is \( 1 \times 1 = 1 \) of these.
   
   Therefore, the total number of squares is \( 16 + 9 + 4 + 1 = 30 \).

### Number of Non-Square Rectangles:
The number of rectangles that are **not squares** is the total number of rectangles minus the number of squares:
\[
225 - 30 = 195
\]

### Rectangle Areas:
Now, let’s calculate the possible areas of these rectangles. The area of a rectangle is given by multiplying its height by its width (measured in units of the grid spacing).

1. **Possible Heights:** The heights can be 1, 2, 3, or 4 units.
2. **Possible Widths:** The widths can also be 1, 2, 3, or 4 units.

Since we are excluding squares, we only count the cases where height and width are not equal:

- **Height 1, Width 2:** Area = \( 1 \times 2 = 2 \)
- **Height 1, Width 3:** Area = \( 1 \times 3 = 3 \)
- **Height 1, Width 4:** Area = \( 1 \times 4 = 4 \)
- **Height 2, Width 3:** Area = \( 2 \times 3 = 6 \)
- **Height 2, Width 4:** Area = \( 2 \times 4 = 8 \)
- **Height 3, Width 4:** Area = \( 3 \times 4 = 12 \)

The areas of all non-square rectangles on the geoboard are: 2, 3, 4, 6, 8, and 12.

### Conclusion:
- **Total number of rectangles (not squares):** 195
- **Areas of different-sized rectangles:** 2, 3, 4, 6, 8, 12

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. How many squares can be formed on a \( 6 \times 6 \) geoboard?
2. What is the total number of shapes (rectangles + squares) on a \( 4 \times 4 \) geoboard?
3. How would the number of non-square rectangles change on a \( 7 \times 7 \) geoboard?
4. What would be the number of rectangles if diagonals were allowed as sides?
5. How do the areas change if the geoboard is not square but rectangular, say \( 5 \times 6 \)?

**Tip:** Using combinations is a powerful way to count possibilities when forming shapes from grids.

Using a 25-peg Geoboard, how many different size rectangles (that are not squares) can you make? Show them and give their areas.

Discover how to determine the number of non-square rectangles on a 25-peg geoboard and calculate their areas. This problem involves combinatorics and geometry, using selection techniques to count shapes and formulas to determine their areas. Suitable for students in Grades 6-8.

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