Practice questions with worked solutions for Australian students

A student who scores four out of five and never sees why the fifth was wrong has learned almost nothing from the exercise. Worse, a student who guessed two of the four correct answers now has a score that flatters them and hides the gap. Every question on these pages shows its method after you answer, so the feedback is about the reasoning rather than the tally.

That is also why we do not hide the working behind a sign-up. A parent checking whether their child understands index laws before Thursday's test needs the answer now, not after creating an account and confirming an email address. If the explanation helps and you want a tutor to take it further, the enquiry form is at the bottom of every page.

The questions are deliberately the ones students get wrong for a reason rather than by carelessness. Subtracting instead of adding when Pythagoras gives you the hypotenuse, negating instead of inverting a negative index, retelling the plot instead of analysing it — these are patterns, and naming the pattern is what stops it recurring.

Each topic lists the Australian Curriculum v9.0 content descriptors it maps to, so you can see the link between a practice set and what the school is actually teaching that term. The descriptors are ACARA's own identifiers, which means you can search them against your child's reporting or the school's scope and sequence document.

The year bands on each page are indicative rather than fixed. Schools sequence topics differently, and states differ again at senior level, so a topic marked Years 8 to 9 may appear earlier in an extension class or later in a standard one. Use the descriptor rather than the year band if you want to check exactly where a topic sits.

The most useful starting point is rarely the topic a student is currently struggling with in class. If Year 10 quadratics are going badly, the underlying problem is often index laws or negative numbers from two years earlier, and drilling quadratics will keep producing the same wrong answers for reasons that never surface.

A practical approach is to start one topic below where the trouble is showing. If that set goes smoothly, the gap is genuinely in the current work and you can move up with confidence. If it does not, you have found the actual issue in ten minutes rather than after a term of frustration.

For students who are coping but want to get ahead, the reverse applies: pick the topic the class is about to start. Arriving at a new unit having already seen the method once changes the experience of the first lesson entirely, because the student is consolidating rather than meeting it cold.

There is no timer, no leaderboard and no streak. Those mechanics increase time on page, which is useful to us and not to the student, and they turn a diagnostic exercise into a performance one. A student who is anxious about mathematics does not need a countdown clock attached to it.

There is also no adaptive sequencing that decides what a student should see next. Adaptive systems work well at scale and badly at the point where a student has a specific misunderstanding, because they infer from answer patterns rather than from asking. That inference is what a tutor does better than software.

The most effective way to use these sets is spaced out. Working through one topic on Monday and revisiting it on Thursday produces markedly better retention than doing the same total number of questions in a single sitting, because the effort of recalling something you have partly forgotten is what strengthens the memory.

Reading the working on questions you answered correctly is worth the extra minute. Students often arrive at a right answer by a longer route or by recognising the shape of the question rather than the underlying method, and the difference shows up under exam pressure when the question is phrased unfamiliarly.

If a topic goes badly, the useful next step is usually one level down rather than more of the same. A student who cannot solve simultaneous equations often has an issue with substitution or with negative numbers rather than with the method itself, and repeating the same set will not surface that.