Lesson slides and practice tools for Cambridge International 9709. Open them on any device, as many times as you like.
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The full Session 1 lesson. Work through it at your own pace, drag the sliders, and tap any practice question to check your answer.
The full Session 2 lesson. Two routes to every answer, side by side, with a rule for picking one. Tap any practice question to check your working.
The full Session 3 lesson. Simultaneous equations where one of them is a curve, and quadratics hiding inside bigger equations. Slide the line in the built-in lab, and tap any practice question to check your working.
Slide a straight line across a circle or a parabola and watch the algebra keep up, one line at a time. The discriminant tells you whether you get two answers, one, or none, before you work any of them out. New for 1.3: drag a point around a circle and see why the tangent is perpendicular to the radius.
Move the critical values, flip the parabola, switch the inequality. The sketch highlights the answer and the sign table gets there the other way, so you can watch both routes land on the same thing.
When the number in front of x² is not 1, you can factor it out or divide by it. Pick a quadratic and run both routes side by side, one step at a time, and see exactly where they meet.
Drag the vertex and watch the equation change. Predict the roots or the expanded form first, then tick the box to check yourself. Roots are given in exact form.
Twelve moves that should be automatic. Beat the clock on these and you buy yourself time for the hard questions in the exam.
Run a snow cone stand and pick your price. The profit curve is a parabola, and its vertex, its roots and its discriminant each answer a real question about the business.
The full Session 1 lesson. Find the domain and range of a function, and see why the two ends of the domain are not always the two ends of the range. A labeled figure, a Range Finder you can drag, and worked exam questions.
The full Session 2 lesson. Composite functions, both notations Cambridge and Khan Academy use, and the condition that decides when a composite can even be formed. A draggable widget shows the verdict flip in real time.
The full Session 3 lesson. One-one functions, finding an inverse two ways, the square-root sign trap, and why every sketch needs the mirror line y = x. A Mirror Lab lets you drag a point and watch its reflection.
The full Session 4 lesson. Translations, stretches and reflections of a graph, why a change inside the bracket acts backwards, and the exact words Cambridge marks. A Transformation Lab lets you predict where each point lands, then check.
The full Session 5 lesson. When two transformations act on different variables the order does not matter, when they share one it does, how to read y = af(x - p) + q, and a full 5-mark question from June 2024. An Order Lab lets you compare both orders on the same graph.
Set a, b, c and d in y = a f(b(x + c)) + d and watch every marked point travel. It writes the description the way Cambridge marks it, and a second tab does two moves in both orders so you can see when the order changes the answer.
The full Session 1 lesson. Gradient, midpoint and length, three forms of a line and when to use each, parallel and perpendicular gradients, where two lines meet, and the perpendicular bisector, finished on the last 4 marks of a June 2024 exam question. A Bisector Lab lets you try four ways of building the line and test each one.
The full Session 2 lesson. Where the circle equation comes from, how to read its center and radius, getting back from the expanded form, inside, on or outside, and the tangent at a point, finished on the first 3 marks of a June 2023 exam question. A Circle Reader and a Tangent Lab let you test the common mistakes and watch them miss.
The full Session 3 lesson. The curve shapes to know, one move for any line and curve (substitute, rearrange, read the discriminant), finding k for a tangent, and a 5-mark June 2023 exam question worked mark by mark. The Meeting Lab lets you slide a line across a circle, a parabola or a reciprocal curve and watch it cut, touch or miss.
The full Session 4 lesson. Why the center of a circle sits on the perpendicular bisector of every chord, the right angle in a semicircle and how Cambridge uses it backwards, the rest of the June 2023 circle question, and a June 2024 question that pulls all of 1.3 together. Drag points in the Chord Lab and the Semicircle Lab and watch the gradients multiply to −1.
Learn circles on your own, at your own pace. Where the equation comes from, reading the center and radius, the expanded form, inside, on or outside (with a St. Croix hurricane), and the tangent at a point. Predict before you reveal, drag points, watch the marks on a real Cambridge question light up one by one, and check yourself with 12 questions. Your progress is saved in your browser.
Drag two points, or type in the ones from your homework, and get the gradient, midpoint, length and the equation in all three forms, each with the working written out. Switch on the perpendicular bisector, a parallel or perpendicular line, or a second line to see where they meet.
Twelve quick straight-line moves against the clock: perpendicular gradients, midpoints, exact lengths, integer form, intersections. A fresh set every run, with the reason shown after every answer.
Drag a storm track past St. Croix and work out how close it comes, whether it enters the warning circle, and when. Uses every idea in coordinate geometry, one step at a time.
Drop points and color them by a distance test. The ones the same distance from A and B line up into the perpendicular bisector; the ones a fixed distance from a center make a circle. Then the algebra shows why.
More topics arrive as we cover them. Everything here stays up, so you can come back to it before a test.
These are grouped by what you are stuck on. Each one says which method it uses, so pick a method you like and stay with it. Watching four videos that all do it a different way will leave you more confused than when you started. If something in the lesson did not land, hearing someone else say it usually helps.
Nothing matches that. Try a shorter word, or press Everything to start over.
From Session 1. These are the videos that helped when I was relearning this myself. Still here because the discriminant work builds straight on top of them.
A slow, full walkthrough of the method, step by step, with nothing skipped. This is the divide-by-a route, the one we use in class. Watch it if you can follow the steps in the lesson but lose the thread when you try it alone.
▶Where the name comes from: you are literally completing a square. Watch this one if the method feels like a rule you memorized and never really understood.
▶Worked example after worked example, including the harder cases where the number in front of the x squared is not 1. Watch this one when you want more practice than the worksheet gives you.
Expressions like x² + 7x + 12, with nothing in front of the x². Find two numbers that multiply to the last number and add to the middle one. Start here if factoring itself is the problem, before you touch anything harder.
Three worked ones in under a minute, including the sign rules for when the last number is negative. The fastest way to see the whole idea.
▶Takes it one step further than factoring: factors x² − 10x + 16 and then uses it to solve. Watch this one if you can factor but do not see what factoring is for.
▶The same job, done in a two-by-two grid so you can see it on paper. Worth watching even if factor pairs already work for you, because the box is what scales up to the harder ones below.
▶Fourteen seconds, one example, no talking. Good as a last-second reminder before a test. On-screen text is partly in Hindi.
▶Splitting the middle term, which is the method the harder ones are built on. The narration is in Hindi, but the written work carries it on its own.
✎This one you read instead of watch. It does the thing the videos skip: it shows why the two numbers have to multiply to the last term and add to the middle one. Practice questions at the bottom mark themselves.
Expressions like 2x² + 7x + 3 or 10t² − t − 3. This is the one that trips people up, and it shows up constantly in exam questions. Pick one of the two methods below and get good at it. They both work every time.
The clearest one here, and the only one that names the problem out loud. Multiply the outside two numbers, split the middle, factor in pairs, check by expanding. If you watch one video on this page, watch this one.
▶The same method on 2x² + 11x + 5, tighter and faster. Watch it straight after the one above to see the method twice on different numbers, which is what makes it stick.
▶The box and the AC method combined on 5x² + 17x + 6, narrated the whole way. This is the one to watch if the silent box video went too fast, because it says out loud where each number in the grid comes from.
▶3x² + 7x + 2 done silently in a box, start to finish. Good if you would rather see the shape of it than listen to someone talk. You do not need sound for this one.
▶Eleven seconds. 3x² + 8x + 4 through a cross grid with no explanation at all. Too fast to learn from, but a good reminder once you already know the method.
✎Two full worked examples, 2x² + 7x + 3 and 6x² − 5x − 4, and then the part no short has room for: why grouping works at all. Read this when you can do it but do not trust it.
▶The unhurried version of the shorts above. Every line explained as it is written. Watch this when a 60-second video went past too fast to follow.
▶Two different routes to the same answer, side by side: guessing the brackets and checking, or splitting the middle term properly. Useful for deciding which one suits you.
These get the right answer fast, and they are worth knowing. But read the warning under each one before you use it in a written answer, because Cambridge marks your method, not just your final line.
The catch: the second line he writes is not equal to the expression you started with. He multiplies it by 2, factors the easier version, then divides back at the end. It works, and he does not skip that last step, but if you write that middle line in an exam you cannot justify it and you will lose method marks. Use it to check an answer you already have. Do not use it to write one.
▶The same shortcut on a nastier example, 10x² − 11x + 3. Same catch as above: the middle line is a different expression from the one in the question. Same advice too. Check with it, do not write it down.
▶Handles a genuinely ugly one, 72x² + 11x − 35, and the method itself is fine. But it stops at the roots and never writes out the brackets. That answers a "solve" question. It does not answer a "factor" question.
Full lessons and practice sets. Go here when the shorts made sense while you watched them and then fell apart the moment you tried one yourself. That happens to everybody.
Covers the easy case and the harder case in one sitting, slowly. She says the reasoning out loud, which most videos skip. The best one here if you are starting from not really getting it.
▶Half an hour, example after example, including common factors and the difference of two squares. You are not meant to watch all of it. Skip to the part you need.
▶Worked the way an examiner wants to see it written. Watch this one when you can get the answer but are not sure how much working to show.
✎Print it and mark it yourself. The answers are a separate PDF on the same page, so you will not see them by accident while you work.
✎Real exam questions on the harder case only, so every question is the one you actually need practice at. Answers are built in.
Things like x² − 5x + 4 < 0, where the answer is a whole stretch of the number line. Everything below assumes you can already factor, so if that is shaky, go back up this page first.
x² + 3x − 4 ≤ 0, sketched, with the part of the curve below the axis colored in. The clearest picture of what a "between" answer actually means. It is also the only one of the three that models the ≤ case.
▶The other shape of answer: x > −1 or x < −3, written as two pieces joined by "or". This is the only video here that shows the outside case, and it is the case people get wrong.
▶The other route, with no picture of a curve at all. He puts the critical values on a number line and tests an actual number in each region. Watch this if sketching the parabola does not click for you.
✎Our own. It gives you an inequality, you tap the stretches of the number line you think are the answer, and it tells you which half you got wrong and why. The place to go once the videos make sense and you need reps.
▶The full version of the number-line route, with several examples and the notation written out properly. Closest to how we do it in class.
▶Longer, and it uses real exam questions. Good once you have the method and want to see harder wording.
✎Nine worked examples, then eight questions to try, then three exam questions. If you want one page that takes you from nothing to exam-ready, this is it.
✎Real past-paper questions from 2008 to 2023, each with a video of someone working it. Free, and you do not need an account. This is as close as you get to sitting the real exam.
✎The homework set. Nineteen questions, answers on the same page, mark yourself and bring in anything that did not come out.
A note on the endpoints. None of the three inequality shorts explains why x = 1 is sometimes included in the answer and sometimes not. Here is the rule. If the question uses < or >, the critical values are out, because at those values the expression equals exactly zero and zero is not less than zero. If it uses ≤ or ≥, they are in. Slide 8 of the lesson covers this properly.