Proportional relationships are everywhere. They describe situations where two quantities scale up or down together at a consistent rate. Think about buying apples: if one apple costs 50p, two cost £1, and ten cost £5. The price and the number of apples are proportional. The core of this relationship is the unit rate, which tells us how much of one quantity corresponds to just one unit of another.

Sometimes, calculating a unit rate involves fractions. Imagine you're baking and a recipe calls for $\frac{1}{2}$ a cup of sugar for a batch that makes $\frac{2}{3}$ of a dozen cookies. To find the amount of sugar needed per dozen, you need to find the unit rate. This means we need to divide the amount of sugar by the number of dozens.

To solve this, we use the rule for dividing fractions: multiply by the reciprocal of the denominator. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.

That unit rate has another name: the constant of proportionality . It's the number that consistently relates the two quantities. We often represent it with the letter $k$. For any proportional relationship, you can find one quantity by multiplying the other quantity by $k$. This gives us a simple but powerful equation.

Let's see how this works in a table. Suppose a car travels at a constant speed. The table below shows the distance travelled over time.

To check if this relationship is proportional, we can calculate the ratio $\frac{y}{x}$ for each pair of values. If the ratio is the same every time, that value is our constant of proportionality, $k$.

The ratio is always 80. So, the constant of proportionality is 80. The unit rate is 80 km per hour. The equation for this relationship is $y = 80x$.

When you plot a proportional relationship on a graph, it has two distinct features:

1. It is a straight line.
2. It passes through the origin (the point (0, 0)).

If a graph is a straight line but doesn't go through the origin, it's a linear relationship, but not a proportional one. The point (0, 0) is crucial. It means that if you have zero of one quantity (like time), you have zero of the other (like distance).

On the graph of a proportional relationship, the point (1, r) is special. The value 'r' is the unit rate, or $k$. In our car example, the point (1, 80) shows that in 1 hour, the car travels 80 km. This point gives you the unit rate just by looking at the y-coordinate when the x-coordinate is 1. This 'r' is just another way to refer to the constant of proportionality, $k$, specifically in the context of a graph.

Let's check your understanding of these concepts.

Understanding proportional relationships is a key skill. It helps you analyze data, make predictions, and solve problems involving scaling, from recipes to maps to financial calculations.