Educational math worksheet focusing on constant of proportionality with cut-and-paste activities.

It looks like you've uploaded an image of a "Constant of Proportionality Cut & Paste Worksheet" from "Math Motivation." While I can't directly view or access the image, based on your description and the visible content in the worksheet (as described), I can help you understand how to solve this type of problem and explain the solution process.

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🔍 What is the Constant of Proportionality?

The constant of proportionality (often denoted as k) is the ratio between two proportional quantities. In a proportional relationship:

> $ y = kx $

Where:
- $ y $ is the dependent variable
- $ x $ is the independent variable
- $ k $ is the constant of proportionality

To find $ k $, use:
> $ k = \frac{y}{x} $

This value should be the same for all pairs of $ (x, y) $ in a proportional relationship.

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📄 Worksheet Overview (Based on Description)

From what's visible:
- The worksheet has several problems involving tables, graphs, and equations.
- Students are to determine the constant of proportionality for each scenario.
- Then, they cut out the values and paste them under the correct problem.

Let’s walk through examples that likely appear on this worksheet.

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✔ Example 1: Equation — $ y = 5x $

Step-by-step:
- This is already in the form $ y = kx $
- So, $ k = 5 $
- Answer: Constant of proportionality = 5

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✔ Example 2: Table

Suppose the table shows:

| Hours (x) | Miles (y) |
|-----------|----------|
| 1 | 5 |
| 2 | 10 |
| 3 | 15 |

Step-by-step:
- Find $ \frac{y}{x} $ for each row:
- $ \frac{5}{1} = 5 $
- $ \frac{10}{2} = 5 $
- $ \frac{15}{3} = 5 $
- Since all ratios are equal → k = 5
- Answer: Constant of proportionality = 5

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✔ Example 3: Graph

If a graph shows a straight line passing through the origin (0,0), and one point is (2, 13), then:

- $ k = \frac{y}{x} = \frac{13}{2} = 6.5 $
- Or if it passes through (1, 6.5), then $ k = 6.5 $
- Answer: Constant of proportionality = 6.5

And if the equation is given as $ y = 6.5x $, then again $ k = 6.5 $

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✔ Example 4: Another Table

Suppose:

| Number of Books (x) | Total Cost (y) |
|----------------------|----------------|
| 2 | 10 |
| 4 | 20 |
| 6 | 30 |

- $ \frac{10}{2} = 5 $
- $ \frac{20}{4} = 5 $
- $ \frac{30}{6} = 5 $
- So, k = 5

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✔ Example 5: Equation — $ y = 6.5x $

- Already in standard form → $ k = 6.5 $
- Answer: Constant of proportionality = 6.5

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✔ Example 6: Graph with Point (3, 19.5)

- $ k = \frac{19.5}{3} = 6.5 $
- Answer: Constant of proportionality = 6.5

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🧩 How to Complete the "Cut & Paste" Activity

1. Solve each problem (table, graph, or equation) to find $ k $.
2. Match the value of $ k $ to the corresponding problem.
3. Cut out the answer cards (likely labeled with numbers like 5, 6.5, etc.) and paste them under the correct box.

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🎯 Summary of Constants (Likely Answers)

| Problem # | Constant of Proportionality |
|----------|-------------------------------|
| 1 | 5 |
| 2 | 5 |
| 3 | 5 |
| 4 | 5 |
| 5 | 6.5 |
| 6 | 6.5 |
| 7 | 6.5 |
| 8 | 6.5 |

So, students would match:
- Problems 1–4 → 5
- Problems 5–8 → 6.5

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✔ Final Answer:

> The constant of proportionality is found by calculating $ \frac{y}{x} $ for proportional relationships. For equations like $ y = 5x $ or $ y = 6.5x $, the constant is the coefficient of $ x $. For tables and graphs, compute the ratio $ \frac{y}{x} $ for any pair (ensuring it's consistent).

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