ACARA v9 CONTENT DESCRIPTION “select and construct appropriate representations, including tables, graphs, models and mathematical relationships, to organise and process data and information”
Builds on choosing tables and graphs to show a pattern. Here the pattern is not just a shape but a rule you can write down. When measurements rise together in step, the right representation turns a table into a straight line, and the straight line into a mathematical relationship you can calculate with.
From a table to a rule
A class hangs different loads on a spring and measures how far it stretches. The readings start as a table of pairs: load in newtons, extension in centimetres. A table keeps the exact numbers, but it hides the shape of the relationship. Switching to a graph reveals it: the points line up. Choosing that representation is how you begin to process the data into a rule.
Spring extension against load
Extension of a spring, in centimetres, for each load hung on it, in newtons. Start with the table, then switch to the line graph and watch the points fall into a straight line.
In the table you have to compare numbers row by row. As a line graph the relationship is unmistakable: the points sit on a straight line that climbs through the origin. Every extra newton adds the same 4 cm of stretch, so extension is directly proportional to load.
Reading the slope as a number
A straight line through the origin says more than just rises together: it says the extension is a fixed multiple of the load. From the readings, every newton adds 4 cm, so the slope is 4 cm per newton. That constant is the model. Written as extension = 4 times load, or extension = k times load, it lets you calculate the stretch for any load, even one you never hung on the spring. A mathematical relationship is the most compact representation of all.
A stiffer spring, same kind of rule
A second, stiffer spring tested the same way. Switch between the views and read off how much extension each newton now adds.
The line is straight and through the origin again, so this spring is also directly proportional. But it is less steep: each newton adds only 2 cm, so its rule is extension = 2 times load. The slope, captured as a single number, tells the two springs apart at a glance.
A graph makes the odd point stand out
Once you expect a straight line, a point that misses it is easy to see. A different group tested their spring and most points sit neatly on the line, but one load gave an extension far above the trend. In the table that value looked plausible. On the graph it jumps out as a point off the line, the first sign of a measuring slip or a spring stretched past its limit.
Read the graph: find the load that breaks the line
Extension measured for six loads. Five points fall on the same straight line; one does not. Click the load whose extension does not fit.
Click the point that does not fit the pattern of the others.
Does the model say what you think it says?
A mathematical relationship is powerful, but it can be misread. The rule extension = 4 times load holds while the spring behaves proportionally, and only for this spring. Sort each statement by whether the straight-line data and its rule genuinely support it, or whether it claims more than the model can show.
Test conclusions against the spring model
Use the first spring, where extension = 4 cm for every newton of load. Decide which conclusions the straight-line data actually supports.
Claim: The proportional data and its rule support the statement being judged.
Doubling the load doubles the extension.
A 3.5 N load would stretch the spring about 14 cm.
With no load hung on it, the spring shows no extension.
The spring will keep stretching 4 cm per newton no matter how heavy the load.
A second, stiffer spring must follow the very same rule.
Decide whether each statement is evidence for the claim, or not.
Why this matters
A table records data, a graph reveals its shape, and a mathematical relationship turns that shape into a rule you can calculate and predict with. Choosing among them, and moving between them, is how raw measurements become a working model. Engineers sizing a spring, physicists fitting a line and analysts forecasting a trend all rely on picking the representation that processes the data into knowledge.
Quick self-check
1. A spring stretches 4 cm for a 2 N load and 8 cm for a 4 N load. When you double the load, the extension...
2. On the load-and-extension graph the points form a straight line that passes through the origin. This tells you the two quantities are...
3. You measured 4 cm of extension for every 1 N of load. The mathematical relationship that processes this data into a rule is...
4. To predict the extension for a load you never tested, the most useful representation is the...
5. One plotted point sits well above the straight line that all the others follow. The best first response is to...