Consider the following procedure: Pick a number. Multiply the number by 9, add 15 to the product, divide the sum by 3, and subtract 5. Complete the above procedure for several different numbers. Use inductive reasoning to make a conjecture about the relationship between the size of the resulting number and the size of the original number.

Let's go through the given procedure step by step with a few different numbers to see if we can establish a pattern.

### Procedure:
1. Pick a number: \( x \)
2. Multiply the number by 9: \( 9x \)
3. Add 15 to the product: \( 9x + 15 \)
4. Divide the sum by 3: \( \frac{9x + 15}{3} = 3x + 5 \)
5. Subtract 5: \( 3x + 5 - 5 = 3x \)

### Conclusion:
After performing the procedure for any number \( x \), the final result is always \( 3x \).

### Testing the Procedure:
- If \( x = 1 \):  
  \( 9 \times 1 = 9 \),  
  \( 9 + 15 = 24 \),  
  \( \frac{24}{3} = 8 \),  
  \( 8 - 5 = 3 \),  
  Result: \( 3 \times 1 = 3 \).

- If \( x = 2 \):  
  \( 9 \times 2 = 18 \),  
  \( 18 + 15 = 33 \),  
  \( \frac{33}{3} = 11 \),  
  \( 11 - 5 = 6 \),  
  Result: \( 3 \times 2 = 6 \).

- If \( x = 3 \):  
  \( 9 \times 3 = 27 \),  
  \( 27 + 15 = 42 \),  
  \( \frac{42}{3} = 14 \),  
  \( 14 - 5 = 9 \),  
  Result: \( 3 \times 3 = 9 \).

### Conjecture:
The final result is always three times the original number. In other words, the output is directly proportional to the input, with a proportionality factor of 3.

### Summary:
For any number \( x \), the final result will be \( 3x \). This indicates that the size of the resulting number is three times the size of the original number.

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Do you want any more details or have any questions? 

Here are 5 related questions to consider:
1. How would the procedure change if we doubled the original number at the start?
2. Can you modify the procedure to result in \( 4x \) instead of \( 3x \)?
3. What happens if we add a constant number at a different stage in the process?
4. Can you write a generalized equation for any modification to this procedure?
5. How would the result change if we replaced multiplication by 9 with multiplication by 7?

**Tip**: When working with procedures, always break down each step algebraically to spot patterns or make conjectures.

Explore how multiplying a number by 9, adding 15, dividing by 3, and subtracting 5 results in a linear relationship. This problem demonstrates key algebraic concepts and inductive reasoning, concluding that the final result is always three times the original number. Suitable for students in Grades 6-8.

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