https://file.aiquickdraw.com/mathbot/file/1729494201834hzbww3x7.jpg

Let's analyze the patterns in the image, and I'll help describe them along with determining the next likely figure for each.

### For Problems 8-11 (Pattern Description and Next Figure)

**8.**
- **Description**: The pattern shows squares starting from one in the top left, then a new square is added diagonally down and to the right in each subsequent figure.
- **Next figure**: The next figure would add another square diagonally down and to the right, forming a 5-square diagonal.

**9.**
- **Description**: The pattern starts with 2 squares in a row, and then in each step, 2 more squares are added to form an extended row.
- **Next figure**: The next figure should have 8 squares in a row (since the current one has 6).

**10.**
- **Description**: This pattern consists of circles, increasing from 3 in the first row to 4 in the second row, forming a triangular shape.
- **Next figure**: The next shape would have a row of 5 circles, forming a triangular arrangement of 5 rows.

**11.**
- **Description**: The pattern starts with a single square, then the number of squares increases by 4 each time, arranged in a 2x2 grid.
- **Next figure**: The next figure should have a 4x4 square grid of 16 squares.

### For Problems 12-13 (Table and Expression)

**12.**
- **Description**: This pattern consists of squares forming a rectangular grid. In each step, the grid reduces by one row and one column.
- **Expression**: Let \( n \) be the step number. The number of tiles in the \( n \)-th figure is given by \( (5 - n + 1)(6 - n + 1) \).

**13.**
- **Description**: This pattern shows a single column of squares, with one less square in each subsequent figure.
- **Expression**: The number of squares in the \( n \)-th figure is simply \( 6 - n \).

Would you like more details or further clarifications? Here are some additional related questions to expand on:

1. What is the explicit process to calculate the number of squares in figure 8?
2. How can triangular numbers help in solving problem 10?
3. Can you generalize the formula for other grid-like patterns similar to 12 and 13?
4. What are some alternative ways to describe the pattern in figure 11?
5. How would you determine the \( n \)-th figure's area if the pattern was three-dimensional?

**Tip**: For pattern recognition, identifying how elements grow or reduce geometrically (or numerically) is key to predicting future shapes.

Describe each pattern and find the next likely figure for each. Also, create a table and expression for the number of tiles or circles in the nth figure.

Explore how to recognize patterns in figures and predict the next one, with expressions for the number of tiles or circles in the nth figure. Suitable for students in Grades 6-8.

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