Finding your footing with new equations takes patience and clear, encouraging feedback. A Year 10 maths tutor in Sydney can spot exactly where the working goes wrong and show them a clearer way through it.
The biggest step up in Year 10 maths is non-linear relationships and graphing. For students on the Path towards Advanced or Extension maths, that now includes graphing circles, graphing other kinds of functions, and using a graph to solve an equation.
Unpacking the trickiest maths topics in the Year 10 syllabus
NESA’s Stage 5 Path content asks students to graph circles of the form x² + y² = r² and to find the equation of a circle with any centre and radius. It also asks them to use graphing technology to solve a pair of equations where one isn’t a straight line.
Recognising the shape from the equation
| Equation | Shape | What to read from it |
|---|---|---|
| y = mx + c | Straight line | Gradient m, y-intercept c |
| y = ax² + bx + c | Parabola | Opens up if a is positive, down if negative |
| y = 2ˣ | Exponential curve | Grows faster and faster; never touches the x-axis |
| x² + y² = r² | Circle | Centre at the origin, radius r |
| (x − a)² + (y − b)² = r² | Circle | Centre at (a, b), radius r |
Using a graph to solve: a worked example
Where do the line y = x + 1 and the circle x² + y² = 25 meet?
Graph both, and they cross at two points: (3, 4) and (−4, −3). Those two points are the solutions, because they’re the only points that make both equations true at once. Check one: 3² + 4² = 9 + 16 = 25, and 4 = 3 + 1.
That idea, that an intersection is a solution, is what Year 10 students find hardest to put into words. Once it clicks, a lot of harder questions become much less mysterious.
Trigonometry also steps up, with new kinds of problems involving triangles that aren’t right-angled. Like the new graphs, it asks students to work with relationships they may not have seen before.
How we make non-linear graphs make sense in an online session
Every student starts with a free 30-minute intro session, where the tutor learns how they’re feeling about school, the marks they’ve been getting and want to get, and the rhythm of their week.
Sessions run on Google Meet with a shared digital whiteboard, and Desmos comes in whenever a graph helps. For non-linear relationships, that means:
- Sketch first. Your child predicts the shape from the equation before graphing it.
- Graph it in Desmos and compare it with the prediction.
- Change one number and watch what moves: the centre, the radius, the steepness.
- Find where graphs meet and check those points in both equations.
Marks and confidence sit on the tracker page at the top of the shared Google Doc.
How to step out a graphing question clearly under exam conditions
Before handing in any graphing question, check:
- Axes are labelled, with a sensible, even scale.
- Key points are marked: intercepts, turning points, the centre of a circle.
- The shape matches the type of equation.
- Any solutions from intersections have been checked in both equations.
- The answer states what the solution means in the question’s context.