Proportional Relationships

TL;DR

Proportional relationships describe how two quantities change together at a constant rate. You can spot them because their graph is a straight line through the origin (0,0). The constant ratio between these quantities is called the constant of proportionality.

1. The Mental Model

Imagine you're buying candy by weight. If you buy twice as much candy, you pay twice as much. This direct, consistent scaling is what a proportional relationship is all about.

2. The Core Material

A proportional relationship exists between two quantities when one quantity is a constant multiple of the other. This means if you double one quantity, the other quantity also doubles. If you halve one, the other halves.

You can express a proportional relationship in a few ways:

• Equation: y = kx
  • y and x are your two quantities.
  • k is the constant of proportionality. It's the number you multiply x by to get y.
• Ratio: y/x = k
  • This shows that the ratio between y and x is always constant.
• Graph: A straight line that passes through the origin (0,0).

2.1 Finding the Constant of Proportionality (k)

Photo by Noelle Otto on Pexels

To find k, you just divide y by x. It's that simple!

Example: If you earn $12 for 2 hours of work, what's your constant of proportionality (hourly wage)?
k = y/x = $12 / 2 hours = $6 per hour
So, k = 6. Your equation would be y = 6x.

2.2 Identifying Proportional Relationships from Tables

Photo by Skylar Kang on Pexels

Look at the ratio y/x for each pair of values. If y/x is the same for every pair, it's a proportional relationship.

Example:

Since y/x is always 7, this is a proportional relationship with k = 7.

2.3 Understanding the Graph

Photo by Sergey Meshkov on Pexels

A graph of a proportional relationship will always be:

• A straight line.
• It will always pass through the origin (0,0). This means when x is 0, y must also be 0 (if you work 0 hours, you earn $0).

graph TD
    A["Start with a Relationship"] --> B{Is it a straight line?};
    B -- "No" --> C["Not Proportional"];
    B -- "Yes" --> D{Does it pass through (0,0)?};
    D -- "No" --> C;
    D -- "Yes" --> E["It's Proportional!"];
    E --> F["Find 'k' (constant of proportionality) by y/x or slope"];

3. Worked Example

Let's say you're buying apples by the pound. The store charges $2.50 per pound.

1. Identify quantities: x = pounds of apples, y = total cost.
2. Determine the constant of proportionality: The price per pound is $2.50. This is your k. So, k = 2.5.
3. Write the equation: y = 2.5x
4. Create a table of values:

5. Describe the graph: If you were to plot these points, you'd get a straight line starting at the origin (0 pounds cost $0.00) and going upwards, showing that the cost increases steadily with each pound of apples.

4. Key Takeaways

• A proportional relationship means two quantities change together at a constant rate.
• Its equation is always in the form y = kx, where k is the constant of proportionality.
• The constant k is found by dividing y by x (k = y/x).
• On a graph, a proportional relationship is a straight line that always goes through the origin (0,0).
• If you see a table, calculate y/x for each row; if it's the same number every time, it's proportional.
• The constant of proportionality tells you how much y changes for every one unit change in x.

Common Mistakes to Avoid:
- Don't confuse a straight line that doesn't go through the origin with a proportional relationship; it's a linear relationship, but not proportional.
- Don't forget that k must be consistent across all data points, not just one.
- Mistaking y/x for x/y when calculating k; k is always y/x.
- Assuming any relationship where one variable increases as the other increases is proportional; it must be at a constant rate and start at zero.

5. Now Try It

You're tracking how much water drains from a leaky faucet. After 5 minutes, 150 ml of water has dripped out. After 12 minutes, 360 ml has dripped.

1. Is this a proportional relationship? How do you know?
2. What is the constant of proportionality?
3. Write an equation for this relationship.
4. How much water would have dripped out after 8 minutes?

What success looks like: You can confidently determine if the relationship is proportional, find k, write the equation, and use it to predict future outcomes.