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Let's go over the content of the worksheet.

### Task 1: Understanding the Constant of Proportionality
It seems you are asked to fill in a concept map for the **constant of proportionality**, with four sections:
1. **What is it?**: This refers to a value that represents the relationship between two proportional quantities. It's the ratio between the two variables. For instance, if $ y = kx $, then $ k $ is the constant of proportionality.
   
2. **What I Know About It**: You can mention that it's a fixed ratio in proportional relationships. For example, in the equation $ y = 2x $, the constant is 2. The relationship stays constant no matter the values of $ x $ and $ y $.
   
3. **Examples**: One example could be the equation of a proportional relationship, like $ y = 5x $, where 5 is the constant. Another real-life example could be the price of apples: if 1 apple costs $2, then 5 apples cost $10, where the constant of proportionality is 2.

4. **Non-Examples**: A non-example could be a nonlinear relationship, such as $ y = x^2 + 3 $. Here, there is no constant ratio between $ y $ and $ x $.

### Task 2: Explaining Hiroaki's Mistake
Hiroaki says that the constant of proportionality must be a whole number and cannot be a fraction or decimal. However, this is incorrect because the constant can be any real number, including fractions or decimals. For example, in the proportional relationship $ y = \frac{1}{2}x $, the constant of proportionality is $ \frac{1}{2} $, which is a fraction. The important property is that the ratio between $ y $ and $ x $ remains constant, not the form of the constant itself.

Would you like any further clarification on this? Here are five related questions that might help deepen your understanding:

1. How can you identify the constant of proportionality in a table of values?
2. Can the constant of proportionality be negative? If so, what would that mean in a real-world scenario?
3. What happens to the graph of a proportional relationship when the constant is a fraction?
4. How does the constant of proportionality relate to the slope of a line in a graph of a proportional relationship?
5. Why is it important for the constant of proportionality to remain the same in a proportional relationship?

**Tip**: To identify the constant of proportionality from a table, divide the value of one variable by the corresponding value of the other (e.g., $ y/x $)—the result should be the same for all data points.

Prepare for Proportional Relationship Problems: Think about what you know about proportional relationships and constants of proportionality. Fill in each box using words, numbers, and pictures. Hiroaki says that a constant of proportionality must be a whole number and cannot be a fraction or decimal. Explain why Hiroaki is incorrect.

Explore proportional relationships and the constant of proportionality. Learn how to fill in a concept map on the topic and correct common misconceptions, such as why the constant can be a fraction or decimal.

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