Expressing Patterns Algebraically worksheet with visual patterns and algebraic formula exercises.

Since I can't view or access images directly, I can’t see the specific patterns in your uploaded image. However, based on the title and structure of the worksheet you've described — "Expressing Patterns Algebraically" — I can guide you through how to solve such problems step by step.

The task involves analyzing a sequence of shapes (like squares, circles, triangles) arranged in a pattern, counting the number of each shape in each figure, identifying the pattern, and then writing an algebraic expression that represents the number of shapes in the *n*th figure.

Let’s go through a typical example like the ones shown in your worksheet:

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Example Problem (based on common patterns):

Pattern:
- Figure 1: 1 square, 2 circles
- Figure 2: 2 squares, 4 circles
- Figure 3: 3 squares, 6 circles

We are asked to:
1. Fill in a table with the number of squares (S) and circles (C) for each figure.
2. Write a formula for the pattern.

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Step-by-step Solution:

#### Step 1: Create a Table

| Figure Number (n) | Squares (S) | Circles (C) |
|-------------------|-------------|-------------|
| 1 | 1 | 2 |
| 2 | 2 | 4 |
| 3 | 3 | 6 |

#### Step 2: Look for a Pattern

- Squares: 1, 2, 3 → increases by 1 each time → S = n
- Circles: 2, 4, 6 → increases by 2 each time → C = 2n

So:
- Squares: S = n
- Circles: C = 2n

#### Step 3: General Formula

For the *n*th figure:
- Number of squares = n
- Number of circles = 2n

You would write this as:
> S = n
> C = 2n

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Now Let's Apply This to Your Worksheet

Assuming the patterns in your image follow similar logic, here's how to approach each one:

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#### Problem 1:
If the pattern is:
- Figure 1: 1 square, 2 circles
- Figure 2: 2 squares, 4 circles
- Figure 3: 3 squares, 6 circles

Then:
- S = n
- C = 2n

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#### Problem 2:
Suppose it's:
- Figure 1: 2 triangles, 1 circle
- Figure 2: 4 triangles, 2 circles
- Figure 3: 6 triangles, 3 circles

Then:
- Triangles = 2n
- Circles = n

So:
- T = 2n
- C = n

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#### Problem 3:
Maybe:
- Figure 1: 3 squares, 2 circles
- Figure 2: 5 squares, 4 circles
- Figure 3: 7 squares, 6 circles

Look at differences:
- Squares: +2 each time → starts at 3 → S = 2n + 1
- Circles: +2 each time → starts at 2 → C = 2n

So:
- S = 2n + 1
- C = 2n

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Tips for Solving These Problems:

1. Count carefully: Make sure you count all shapes in each figure.
2. Make a table: List n (figure number), S (squares), C (circles), etc.
3. Find the pattern:
- Is the number increasing linearly? Then it’s likely of the form `an + b`.
- If it doubles, maybe multiply by 2.
- Check if the difference between terms is constant (linear).
4. Test your formula: Plug in n=1, 2, 3 and see if it matches.

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Final Answer Format:

For each problem, you’ll fill in the table and write:

> S = ___
> C = ___

Or whatever shapes are used (e.g., T for triangle).

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If you can describe one of the patterns from the image (e.g., “Figure 1 has one square and two circles, Figure 2 has two squares and four circles…”), I can help you solve it exactly!

Let me know!