Printable activity sheets: comparing 'k' values in direct variation

📚 Topic Summary

Direct variation describes a relationship between two variables where one is a constant multiple of the other. This relationship can be expressed as $y = kx$, where $y$ and $x$ are the variables, and $k$ is the constant of variation. The value of $k$ determines the steepness of the line when graphed; a larger $k$ means a steeper line, indicating a greater change in $y$ for a given change in $x$.

Comparing different $k$ values allows us to understand how the relationship between $x$ and $y$ changes. For instance, if $k_1 > k_2$, then for the same value of $x$, $y_1$ (from $y_1 = k_1x$) will be greater than $y_2$ (from $y_2 = k_2x$). This shows that $y$ changes more rapidly with respect to $x$ when $k$ is larger.

🧠 Part A: Vocabulary

• 🧮 Term: Direct Variation
• 📈 Term: Constant of Variation
• 📊 Term: Variable
• 📏 Term: Slope
• 🤝 Term: Relationship

1. Definition: A quantity that may change within the context of a mathematical problem.
2. Definition: The ratio between two variables remains constant.
3. Definition: The constant value ($k$) in a direct variation equation ($y = kx$).
4. Definition: The steepness of a line on a graph, representing the rate of change.
5. Definition: The way in which two or more concepts, objects, or people are connected, or the state of being related.

Match each term with its correct definition.

✍️ Part B: Fill in the Blanks

In a direct variation, the equation is represented as $y = kx$, where $k$ is the _________ of variation. If $k$ increases, the value of $y$ _________ for a given $x$. The graph of a direct variation is a _________ line passing through the _________. Comparing different $k$ values helps us understand how the _________ between $x$ and $y$ changes.

🤔 Part C: Critical Thinking

Explain, using real-world examples, how understanding the constant of variation ($k$) can be useful in everyday situations.