ACARA v9 CONTENT DESCRIPTION “experiment with functions and relations using digital tools, making and testing conjectures and generalising emerging patterns”
Builds on: Exponential Relations (AC9M10A03). Having met linear, quadratic and exponential relations, this unit steps back to the way mathematicians actually work with them: experimenting with a function using digital tools, noticing patterns, forming conjectures, testing them, and generalising what holds into a rule.
Learning a function by playing with it
Some of the best mathematical understanding comes not from being told a rule but from experimenting until you discover it. Digital tools, graphing software, spreadsheets, and interactive sliders, make this kind of experimenting easy: you change a number in a function and watch, instantly, how its graph responds. Take the line y equals m x plus one and slowly increase m. The line tilts steeper and steeper, yet the point where it meets the y-axis never budges from one. From that single experiment you learn what the slope m controls and what it leaves alone, a lesson far stickier than a sentence in a textbook. Experimenting turns a function from a static formula into something you can poke, prod, and come to know.
Change one number, watch the line
Varying a single coefficient and observing the graph is how you experiment with a function.
Experimenting often starts with changing one number and watching what happens. Here the slope changes while the intercept stays put at 1: a bigger slope tilts the line steeper, and a negative slope makes it fall instead of rise. The gold point where it crosses the y-axis never moves, because only the slope is changing.
From observation to conjecture
Experimenting naturally throws up patterns, and a pattern invites a conjecture, which is a careful guess about what is going on. Look at the square numbers one, four, nine, and notice the gaps between them are three and then five. A reasonable conjecture is that the gaps between consecutive squares are the odd numbers, three, five, seven, and so on. Forming a conjecture is a creative act: it is you proposing a rule that the cases so far seem to obey. It is not yet a fact, and that is the point. A conjecture is the hypothesis that the rest of the work, testing and eventually proving, is there to confirm or overturn. Learning to make good conjectures, neither reckless nor timid, is a real mathematical skill.
Observe, conjecture, test
A conjecture is a careful guess from a pattern; testing it on new cases checks whether it holds.
Experimenting begins with noticing a pattern. The square numbers 1, 4, 9 have gaps of 3 and 5 between them. What comes next?
Testing tells you whether to trust it
A conjecture earns trust by surviving tests, so the next move is always to try it on new cases. For the gaps between squares, sixteen minus nine is seven and twenty-five minus sixteen is nine, both odd, so the conjecture passes. Digital tools shine here, because they let you test dozens of cases in seconds rather than grinding through them by hand. But a word of caution rides alongside: passing many tests builds confidence, yet it is not the same as proof. A pattern can hold for case after case and still fail later. Testing can support a conjecture strongly, and it can refute one instantly, but turning a well-tested conjecture into a certainty needs the deductive proof met earlier in the year.
Experimenting with a parabola
Changing the number inside the bracket shifts the parabola sideways, a pattern you can conjecture and generalise.
Experimenting with a parabola, replacing x by x minus a number slides the whole curve sideways: the vertex, marked in gold, moves to that number on the x-axis. From a few cases you might conjecture the rule, that y equals the bracket x minus k squared has its lowest point at x equals k, a generalisation worth testing further and then explaining.
Generalising into a rule
When a conjecture has held up, the goal is to generalise it, to capture every case in a single rule. Add the whole numbers and you get one, three, six, ten, the triangular numbers; experimenting and conjecturing leads to the rule that the sum of the first n whole numbers is n times n plus one, all divided by two. Check it at n equals four: four times five over two is ten, exactly right. A generalisation is powerful because it replaces an endless list of separate cases with one compact statement that handles them all, and it lets you leap straight to the hundredth or thousandth case without listing the ones before. Moving from specific cases to a general rule, and writing that rule with algebra, is the destination this kind of experimenting aims for.
Generalising a pattern into a rule
Generalising turns a pattern seen in several cases into one rule that covers them all.
Adding the whole numbers gives 1, 3, 6, 10, the triangular numbers. Reveal more rows, then look for a single rule that produces them all.
Why testing can never be skipped
It is tempting, once a pattern has held several times, to treat it as settled, but mathematics insists on caution, and a famous kind of example shows why. The expression n squared minus n plus eleven produces a prime number for n equals one, two, three, and indeed for the first ten values, which looks like overwhelming evidence that it always gives a prime. Yet at n equals eleven the value is one hundred and twenty-one, which is eleven times eleven, not prime at all. A single counterexample like this rejects the conjecture completely, no matter how many cases worked before. The lesson is twofold: always keep testing, especially larger and stranger cases, and remember that experimenting and conjecturing, powerful as they are, are the beginning of mathematical certainty rather than the end of it.
One counterexample is enough
A conjecture holding for many cases can still be false; a single counterexample rejects it.
The expression n squared minus n plus 11 gives primes for the first several values of n, tempting us to conjecture it always does. Test a larger value to see what happens.
Quick self-check
1. In the line y = mx + 1, you increase m while keeping the 1. What happens to the graph?
2. A conjecture in mathematics is:
3. Replacing x with (x − 2) in y = x² does what to the parabola?
4. The sums 1, 3, 6, 10, ... are generalised by the rule:
5. A conjecture has held for the first ten cases you tried. The eleventh case fails. The conjecture is: