Calculating "if 1 costs $1, then 3 costs $3" when shopping is actually relying on direct proportion. On the flip side, "double the speed for the same distance and the time it takes is cut in half" is inverse proportion. We use both ideas unconsciously in everyday life, but understanding the precise mathematical distinction between them lets you apply the concepts to a far wider range of problems.
Direct proportion is a relationship where one value increases and the other increases at the same rate. It's written as y=kx: double x, and y doubles; triple x, and y triples. Graph it and you get a straight line through the origin. Inverse proportion, on the other hand, is a relationship where one value increases while the other decreases at the same rate, written as y=k/x, with a graph shaped like a smooth curve.
The easiest way to tell the two apart is to check "is it constant when you multiply the two numbers, or when you divide them?" In a direct proportion, y divided by x is always constant; in an inverse proportion, the two values multiplied together are always constant. Whenever you're unsure which relationship a table of data shows, doing this calculation gives you the answer right away.
As real-life examples, when a car's fuel efficiency is constant, the amount of fuel and the distance traveled are directly proportional, and when a fixed amount of work needs doing, the number of workers and the time it takes are inversely proportional — the more workers there are, the less each person has to do, so the total time shrinks. Explaining it through examples like these makes the abstract formulas feel much more concrete.
When it's hard to tell whether a table of data is direct or inverse proportion, learning a few classic situations ahead of time makes judging much faster. For instance, "when each person does the same amount of work per day, the number of workers versus the number of days to finish the whole job" is a go-to setup for inverse proportion — double the workers, and the days needed are cut in half. On the other hand, "how far a car that gets 15km per liter of gas can travel, based on how much gas is in the tank" is a classic example of direct proportion. Having a few of these recurring situations organized ahead of time makes it much easier to recognize a similar pattern when a new problem comes along. Playing a game with your child to spot direct and inverse proportion situations in everyday life is also a great way to make the concept stick.
Comparing the shapes of the two graphs with your own eyes also helps a lot in cementing the concept. On our activity page, you can switch between direct and inverse proportion with tabs and immediately watch how the graph (a line versus a curve) changes with the constant k. Put both graphs side by side and you'll come away certain of exactly why one is a straight line and the other a curve. Comparing how completely different the graphs look for direct versus inverse proportion, even with the same value of k, makes it much more intuitive why the two relationships behave in exactly opposite ways. It's also worthwhile to assign homework where you find and write down one example each of direct and inverse proportion from the world around you. Realizing that far more situations than you'd expect are explained by one of these two relationships makes math feel like it's still at work well outside the classroom.