Direct and inverse proportion

What is it about?

In a relation (or assignment) each $x$ value goes with exactly one $y$ value. Two special relations come up all the time in everyday life. In the simulation every quantity has its own colour: $x$ blue, $y$ purple, the constant $k$ orange.

Directly proportional – “twice as much, twice the price”: doubling (tripling, …) $x$ doubles (triples, …) $y$. All quotients are equal (constant quotient):

$k$ is called the constant of proportionality. The graph is a ray through the origin; the slope triangles “1 to the right, $k$ up” have the same shape everywhere.

Inversely proportional – “twice as many, half the time”: doubling $x$ halves $y$, tripling $x$ gives a third of $y$. All products are equal (constant product):

The graph is a branch of a hyperbola. All rectangles under its points have area $k$ – they become narrower and taller or wider and flatter, but never bigger or smaller.

The rule of three uses exactly these properties: first work out one unit, then the wanted amount. Examples: apples ($4\,\text{kg}$ cost $€10$) and a pool that $6$ pumps fill in $10\,\text{h}$.

Careful, trap: not every “the more – the more” relation is proportional. A taxi costs a base fare of $€3$ plus $€2$ per km. $5\,\text{km}$ cost $€13$, but $10\,\text{km}$ cost only $€23$ – not $€26$. The quotients $13 : 5 = 2.6$ and $23 : 10 = 2.3$ differ, and the line $y = 2x + 3$ does not pass through the origin.

How to check a table of values: work out the quotients $y : x$ – all equal? Then directly proportional. Otherwise work out the products $x \cdot y$ – all equal? Then inversely proportional. Otherwise neither.

Try it

1. Apples on the scales: drag $P$ along the graph or tap a column of the table. What always stays the same in the row $y : x$? Compare the small orange triangles.
2. Press “Double it”: the mass is doubled. What happens to the price on the scales, in the table (arrows below) and in the graph?
3. Pumps at the pool: press “Double it”. How fast does the pool fill now? Then drag $P$ and watch the rectangle: which number never changes?
4. Six pumps, ten hours: run “Rule of three step by step”. Why is there “$\cdot 6$” on the right when there is “$: 6$” on the left?
5. Taxi trap: press “Double it”. Where is the point “€26?” and where is the real fare? What do the dashed rays from the origin show?
6. Test: sharing pizza: tap your guess first, then “Check the table”. “Next table” takes you on – including two traps.

Tasks

Task 1: Buying apples

$4\,\text{kg}$ of apples cost $€10$. How much do $7\,\text{kg}$ cost? Load task

Task 2: Pumps at the pool

$6$ pumps of equal power fill a pool in $10$ hours. How long do $4$ pumps take, how long do $5$ pumps take? Load task

Task 3: Doubling and tripling

$3$ pumps fill a pool in $4$ hours. How long do $6$ pumps take, how long do $9$ pumps take? Give the second answer in hours and minutes. Load task

Task 4: Lisa and the taxi

A taxi costs a base fare of $€3$ plus $€2$ per km. Lisa says: “$5\,\text{km}$ cost $€13$, so $10\,\text{km}$ cost twice as much, that is $€26$.” Is she right? Load task

Task 5: Check three tables

Which table belongs to a directly proportional relation, which to an inversely proportional one, and which to neither? For the proportional ones, give the value of $k$.

(a) $(1 \mid 3),\ (2 \mid 6),\ (5 \mid 15)$ (b) $(2 \mid 12),\ (3 \mid 8),\ (6 \mid 4)$ (c) $(1 \mid 5),\ (2 \mid 7),\ (3 \mid 9)$

Load task

Task 6: Getting there faster

For a distance of $120\,\text{km}$ you need exactly $3\,\text{h}$ at $40\,\tfrac{\text{km}}{\text{h}}$. Explain why speed and travel time are inversely proportional, and work out the travel time at $50\,\tfrac{\text{km}}{\text{h}}$ and at $90\,\tfrac{\text{km}}{\text{h}}$. Load task

Ideas for the classroom

• Use: proportional and inversely proportional relations with the rule of three (grade 6 or 7, depending on school type and federal state) and their return with linear functions (direct proportion) and hyperbolas (inverse proportion).
• Three representations: picture, table and graph react together. It can pay off to cover one of them and let students predict what “Double it” will do there.
• Tasks from the picture: with “Show the solution” off (show=0) unknown values appear as “?”. In the test, students can tap a guess first and then check it; “Next table” works well for a short round with the whole class.
• Sorting task: students collect everyday examples (fuel price, workers and working time, age and shoe size, phone contract with a base fee …) and sort them into the three cases before checking.
• Model: for the pumps only whole numbers make sense; the branch of the hyperbola connects the meaningful points. The age and height values are approximate averages.
• Common stumbling blocks:
  • “The more – the more” is taken to mean directly proportional (taxi, age and height).
  • In the inverse rule of three the right side is treated the same way as the left – the memory aid “opposite direction” helps here.
  • The graph of an inverse proportion is drawn as a falling straight line; the products show that this does not fit.
  • Only one quotient is worked out. To decide, you need all pairs of the table.