Eight square tiles are laid out on a table so that they make a solid pattern.  Each tile must touch at least one other tile along an entire edge. The squares all have sides of length 1. Here is an example: The figures have a perimeter of 14 and 16 respectively. Draw or Describe the arrangement that has the smallest perimeter?

To find the arrangement of eight square tiles that minimizes the perimeter, we need to ensure that the squares form a compact shape. Each tile shares as many edges as possible with adjacent tiles to reduce the overall boundary exposed to the outside. 

A possible optimal configuration is a **2 by 4 rectangle**:

1. Imagine arranging the eight tiles in two rows of four tiles each.
2. Each tile has a side length of 1, so the rectangle has a total width of 4 and a height of 2.

### Perimeter Calculation:
- Width of the rectangle = 4
- Height of the rectangle = 2

The formula for the perimeter of a rectangle is:
\[
P = 2 \times (\text{width} + \text{height})
\]
\[
P = 2 \times (4 + 2) = 2 \times 6 = 12
\]

Thus, the arrangement that forms a 2x4 rectangle will have a **perimeter of 12**, which is the smallest possible perimeter for eight tiles.

Would you like me to illustrate the arrangement or clarify any steps further?

Here are some related questions for you to consider:
1. How does changing the arrangement affect the perimeter of the tiles?
2. Why does a rectangular arrangement minimize the perimeter in this case?
3. Can a different shape other than a rectangle give a smaller perimeter for the same number of tiles?
4. What would the perimeter be if you had 9 tiles arranged in the smallest perimeter configuration?
5. How would the perimeter change if the tiles were arranged in an "L" shape?

**Tip:** When aiming to minimize the perimeter, focus on arranging tiles to share as many sides as possible.

Eight square tiles are laid out on a table so that they make a solid pattern. Each tile must touch at least one other tile along an entire edge. The squares all have sides of length 1. Here is an example: The figures have a perimeter of 14 and 16 respectively. Draw or describe the arrangement that has the smallest perimeter?

Explore how to arrange eight square tiles to minimize their perimeter using basic geometry and optimization techniques. Learn how a 2x4 rectangle offers the smallest perimeter of 12 units. This detailed explanation is suitable for middle school students learning about perimeter and optimization.

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