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From the image you provided, we see a series of math pattern problems. Let's break down the patterns and how to create expressions for each:

1. **Problem 10 (Circles):**
   - The pattern shows an increasing number of circles in each step.
   - From left to right:
     - Step 1: 3 circles
     - Step 2: 6 circles
     - Step 3: 9 circles
   - The pattern increases by 3 circles at each step, suggesting an arithmetic progression. The general expression for the number of circles at step \(n\) is:
     \[
     3n
     \]
     This means that in the 7th step, there would be \( 3 \times 7 = 21 \) circles.

2. **Problem 11 (Squares):**
   - The pattern shows squares arranged in a progressively increasing grid.
   - Step 1: 1 square
   - Step 2: 4 squares (2x2)
   - Step 3: 9 squares (3x3)
   - The number of squares follows a quadratic sequence where the number of squares in step \(n\) is \(n^2\). Therefore, the expression is:
     \[
     n^2
     \]
     For the 7th figure, the number of squares would be \(7^2 = 49\).

3. **Problem 12 (Rectangular Tiles):**
   - The pattern shows a grid of tiles increasing by one row and one column each time.
   - Step 1: \(2 \times 2\) grid (4 tiles)
   - Step 2: \(3 \times 3\) grid (9 tiles)
   - Step 3: \(4 \times 4\) grid (16 tiles)
   - Again, this is a quadratic pattern. The number of tiles at step \(n\) is \(n^2\). Thus, the expression is:
     \[
     n^2
     \]
     In the 7th step, the number of tiles is \(7^2 = 49\).

4. **Problem 13 (Linear Tiles):**
   - The pattern shows increasing rows of horizontal tiles.
   - Step 1: 2 tiles
   - Step 2: 3 tiles
   - Step 3: 4 tiles
   - The pattern increases by 1 tile each step. The general expression is:
     \[
     n + 1
     \]
     So for the 7th step, the number of tiles is \(7 + 1 = 8\).

5. **Problem 14 (Circle Clusters):**
   - The pattern shows clusters of circles.
   - Step 1: 9 circles
   - Step 2: 16 circles
   - Step 3: 25 circles
   - This is a quadratic pattern where the number of circles in step \(n\) is \(n^2\). Thus, the expression is:
     \[
     n^2
     \]
     In the 7th step, there would be \(7^2 = 49\) circles.

Would you like any further details or clarifications on these patterns?

Here are 5 related questions for further practice:
1. How would the pattern change if the growth rate doubled?
2. What if you were given geometric sequences instead of arithmetic?
3. How would you modify these expressions for a 10th figure instead of a 7th?
4. Can you create a real-world scenario that fits one of these patterns?
5. How would the expressions change if the increase was non-linear (e.g., exponential growth)?

**Tip:** Always look for common differences or ratios to identify the type of sequence—whether arithmetic, geometric, or quadratic.

Describe each pattern using words. Draw the next likely figure in each pattern. Make a table with a process column to represent each pattern. Write an expression for the number of tiles or circles in the 7th figure.

Learn how to identify arithmetic and quadratic growth patterns in circles and squares. This problem covers expressions such as n^2 and 3n for different types of patterns, suitable for students in Grades 6-8.

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