At a glance
The course in numbers
How to read it. One row is one 45-minute lesson; its pack holds the plan, worksheet and quiz. The week column counts 3 lessons a week until you change the control above; the sheet you download uses the same numbers. Each topic ends at an assessment point, its question bank, which you can assign in the app for a marked score. The learning intention is the first idea of each lesson in the students' own pages; the traps are the examiner traps those pages pre-empt.
Topic 01
Pure Mathematics 1
Weeks 1 to 91.1Quadratics
Weeks 1 to 2| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 1 | 1.1 | Completing the square | In the lab, press Make the square and watch \(x^{2} + bx\) rearranged into a square: there is always a small corner missing, of area \(\left(\frac{b}{2}\right)^{2}\). |
| 1 | 1.1 | The discriminant | The quadratic formula \(x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) only gives real answers when the number under the root is not negative. |
| 1 | 1.1 | Quadratic equations and inequalities | In the lab, pick a quadratic and solve it by factorising, then by the formula, and check that both routes land on the same roots. |
| 2 | 1.1 | Simultaneous equations: one linear, one quadratic | Elimination by adding or subtracting equations does not work when one equation has squared terms. |
| 2 | 1.1 | Equations quadratic in a function of x | Look for two terms where one power is exactly double the other, plus a constant. |
| 2 | Assessment point | Topic question bank, 40 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- The sign inside the bracket matches the sign of \(b\): \(x^{2} - 6x\) gives \((x - 3)^{2}\), not \((x + 3)^{2}\).
- Writing \(2x^{2} - 12x + 7 = 2(x - 3)^{2} - 9 + 7\) is the classic slip: the 9 sits INSIDE the bracket that is multiplied by 2, so it must become 18.
- The vertex of \(a(x + p)^{2} + q\) is at \(x = -p\), not \(x = p\): \((x + 1)^{2}\) is zero when \(x = -1\).
1.2Functions
Weeks 2 to 3| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 2 | 1.2 | Functions: domain, range and composition | In the lab, choose a function and follow the mapping diagram as each input in the domain is sent to its output: the set of outputs is the range. |
| 3 | 1.2 | Inverse functions | In the lab, pick a function and watch its inverse appear as the reflection in \(y = x\): every point \((a, b)\) on \(y = f(x)\) becomes \((b, a)\). |
| 3 | 1.2 | Transformations of graphs | A translation slides the whole graph without changing its shape. |
| 3 | Assessment point | Topic question bank, 24 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- Do not just substitute the two ends of the domain.
- The least \(k\) is the \(x\)-coordinate of the vertex, not its \(y\)-coordinate.
- \(fg(x)\) is NOT \(f(x) \times g(x)\).
1.3Coordinate geometry
Weeks 3 to 4| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 3 | 1.3 | Equations of straight lines | In the lab, drag the two points and watch the gradient and the equation of the line update as you go. |
| 4 | 1.3 | The equation of a circle | A point \((x, y)\) is on the circle when its distance from the centre \((a, b)\) is \(r\). |
| 4 | 1.3 | Intersections of lines and curves | In the lab, draw a curve and a line and read off where they cross: those \(x\)-values are exactly the solutions of the equation formed by setting the two expressions equal. |
| 4 | Assessment point | Topic question bank, 24 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- The gradient of \(3x + 4y = 24\) is not 3 (nor \(\frac{4}{3}\)).
- The perpendicular gradient is the NEGATIVE reciprocal.
- In the length formula, subtract the coordinates before squaring: from \((-1, 3)\) to \((5, -5)\) the changes are 6 and \(-8\), giving \(\sqrt{36 + 64} = 10\).
1.4Circular measure
Weeks 4 to 5| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 4 | 1.4 | Radians | Imagine bending a piece of string the length of the radius around the edge of a circle. |
| 5 | 1.4 | Arc length and sector area | A sector with angle \(\theta\) is the fraction \(\dfrac{\theta}{2\pi}\) of the whole circle, so its arc is \(\dfrac{\theta}{2\pi} \times 2\pi r = r\theta\). |
| 5 | Assessment point | Topic question bank, 16 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- Your calculator must be in the right mode. \(\sin 1.2\) in degree mode gives \(0.0209\) instead of \(0.932\).
- Do not multiply \(\dfrac{7\pi}{12}\) by 180 and stop: that gives \(105\pi\), not \(105^\circ\).
- Never mix units in an angle sum.
1.5Trigonometry
Weeks 5 to 6| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 5 | 1.5 | Graphs of trigonometric functions | In the lab, drag the point round the unit circle and watch its height trace out the sine graph and its across-distance trace out the cosine graph. |
| 5 | 1.5 | Exact values and the two identities | In the lab, step the angle through \(30^\circ\), \(45^\circ\) and \(60^\circ\) and compare the exact values it shows with the sides of the triangles in Fig. 1. |
| 6 | 1.5 | Solving trigonometric equations | In the lab, move the horizontal line \(y = k\) and watch where it cuts the curve. |
| 6 | Assessment point | Topic question bank, 24 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- The tangent graph repeats every \(\pi\), not every \(2\pi\), and it has no maximum or minimum.
- \(y = \sin(x + k)\) moves the graph LEFT by \(k\), not right; and in \(y = \sin 2x\) the period halves, it does not double.
- Watch the ends of the interval.
1.6Series
Weeks 6 to 7| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 6 | 1.6 | The binomial expansion | Multiply out \((a + b)^2\), \((a + b)^3\), \((a + b)^4\) and the coefficients build Pascal's triangle (Fig. 1). |
| 6 | 1.6 | Arithmetic progressions | In the lab, set the first term and the common difference with the sliders and watch each term grow by the same step. |
| 7 | 1.6 | Geometric progressions and sum to infinity | In the lab, choose a first term and a multiplier and watch the terms. |
| 7 | Assessment point | Topic question bank, 24 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- \(\binom{5}{2}\) is not \(\dfrac{5!}{2!} = 60\).
- Bracket the whole second term before squaring: \((-3x)^2 = 9x^2\), not \(-3x^2\) and not \(-9x^2\).
- The \(x^3\) term is the FOURTH term (\(r = 3\)), because counting starts at \(r = 0\).
1.7Differentiation
Weeks 7 to 8| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 7 | 1.7 | The gradient of a curve | In the lab, fix a point on the curve and drag the second point of the chord towards it. |
| 7 | 1.7 | Differentiating powers and the chain rule | The rule \(\dfrac{d}{dx}x^{n} = nx^{n-1}\) works for every rational \(n\): negative, fractional, the lot. |
| 8 | 1.7 | Tangents, normals and rates of change | Three steps every time: find the point (both coordinates), find the gradient by substituting into \(\dfrac{dy}{dx}\), then use \(y - y_{1} = m(x - x_{1})\). |
| 8 | 1.7 | Stationary points and their nature | In the lab, slide the point along the curve and watch the tangent. |
| 8 | Assessment point | Topic question bank, 32 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- Do not put \(h = 0\) into \(\dfrac{12h + 6h^{2} + h^{3}}{h}\) straight away: that gives \(\dfrac{0}{0}\), which means nothing.
- \(\dfrac{d^{2}y}{dx^{2}}\) is NOT \(\left(\dfrac{dy}{dx}\right)^{2}\).
- After solving \(\dfrac{dy}{dx} = 0\) for \(x\), substitute into the equation of the CURVE to get \(y\).
1.8Integration
Weeks 8 to 9| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 8 | 1.8 | Integration as reverse differentiation | Differentiating takes the power down and reduces it by one. |
| 9 | 1.8 | Definite integrals and area | Integrate as usual, write the result in square brackets with the limits, then substitute the top limit and subtract the value at the bottom limit. |
| 9 | 1.8 | Volumes of revolution | Rotate the region under \(y = f(x)\) through \(360^{\circ}\) about the \(x\)-axis. |
| 9 | Assessment point | Topic question bank, 24 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- Dividing by a fraction is multiplying by its reciprocal: \(\dfrac{x^{\frac{3}{2}}}{\frac{3}{2}} = \tfrac{2}{3}x^{\frac{3}{2}}\), not \(\tfrac{3}{2}x^{\frac{3}{2}}\).
- Divide by \(a\); do not multiply. \(\displaystyle\int (2x + 1)^{3}\,dx\) is \(\dfrac{(2x + 1)^{4}}{8}\), not \(\dfrac{2(2x + 1)^{4}}{4}\).
- Find \(c\) straight after integrating, using the equation of the CURVE (\(y\)), not the gradient.
Topic 02
Pure Mathematics 2
Weeks 9 to 173.1Algebra
Weeks 9 to 10| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 9 | 3.1 | The modulus function | Draw the straight line \(y = ax + b\) first, then reflect every part that lies below the \(x\)-axis in the \(x\)-axis. |
| 10 | 3.1 | Polynomial division, factor and remainder theorems | Polynomial division works like numerical long division. |
| 10 | 3.1 | Partial fractions | Everything depends on writing down the correct form before you calculate anything. |
| 10 | 3.1 | The binomial series for any rational n | For a positive integer \(n\) the expansion stops after \(n + 1\) terms. |
| 10 | Assessment point | Topic question bank, 32 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- The vertex of \(y = |2x - 3|\) is at \(x = 1.5\), not \(x = 3\) or \(x = -1.5\).
- In Case 2, \(-(2x - 1)\) is \(-2x + 1\), not \(-2x - 1\).
- Never square when one side might be negative: \(2x + 5 < |x + 1|\) squared gives a wrong answer, because squaring a negative left side flips which side is bigger.
3.2Logarithmic and exponential functions
Weeks 11 to 12| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 11 | 3.2 | Logarithms and the laws of logarithms | \(2^5 = 32\) and \(\log_2 32 = 5\) say the same thing: the power of 2 that gives 32 is 5. |
| 11 | 3.2 | The exponential function and natural logarithms | In the lab, set a positive percentage rate and watch the curve climb faster and faster; then make the rate negative and watch it decay towards zero. |
| 11 | 3.2 | Solving exponential equations and inequalities | Taking logarithms turns an index equation into an ordinary one, because \(\ln a^{k} = k\ln a\). |
| 12 | 3.2 | Reducing relationships to linear form | Take ln of both sides of the model and expand with the laws of logarithms. |
| 12 | Assessment point | Topic question bank, 32 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- \(\log_2 \dfrac{1}{8}\) is \(-3\), not \(\dfrac{1}{3}\) or \(-\dfrac{1}{3}\).
- There is no law for \(\log(x + y)\): \(\log_a(x + y) \ne \log_a x + \log_a y\).
- Do not 'cancel the logs' term by term: \(\log_2(x + 3) + \log_2(x - 1) = 5\) does NOT give \((x + 3) + (x - 1) = 5\) or \(= 32\).
3.3Trigonometry
Weeks 12 to 13| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 12 | 3.3 | Secant, cosecant and cotangent | Each new function is one over an old one. |
| 12 | 3.3 | Compound angle formulae | Fig. 1 shows where \(\sin(A + B)\) comes from: the height of P is built from two pieces. |
| 13 | 3.3 | Double angle formulae | Put \(B = A\) in each compound angle formula. |
| 13 | 3.3 | Expressing a sin x + b cos x as R sin(x + a) | Fig. 1 shows that \(3\sin x + 4\cos x\) is just a sine wave of amplitude 5, shifted left. |
| 13 | Assessment point | Topic question bank, 32 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- \(\sec x\) is NOT \(\cos^{-1} x\).
- The identity is \(\sec^2\theta = 1 + \tan^2\theta\), not \(\sec^2\theta = 1 - \tan^2\theta\) or \(\tan^2\theta = 1 + \sec^2\theta\).
- Never work on both sides at once, and never cross-multiply an identity you are asked to prove: that assumes the result.
3.4Differentiation
Weeks 13 to 14| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 13 | 3.4 | Derivatives of exponentials, logarithms and trigonometric functions | \(e^{x}\) is the one function that is its own derivative. |
| 14 | 3.4 | The product and quotient rules | Name the two factors \(u\) and \(v\), differentiate each (chain rule if needed), then combine: first times derivative of second, plus second times derivative of first. |
| 14 | 3.4 | Parametric and implicit differentiation | When \(x\) and \(y\) both depend on a parameter \(t\), differentiate each with respect to \(t\) and divide. |
| 14 | Assessment point | Topic question bank, 24 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- \(\dfrac{d}{dx}\ln(4x) = \dfrac{4}{4x} = \dfrac{1}{x}\), NOT \(\dfrac{4}{x}\) and not \(\dfrac{1}{4x}\).
- The minus sign belongs to cosine: \(\dfrac{d}{dx}\cos x = -\sin x\), but \(\dfrac{d}{dx}\sin x = +\cos x\).
- \(\tan^{-1} x\) is the inverse function, not \(\dfrac{1}{\tan x} = \cot x\).
3.5Integration
Weeks 14 to 16| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 14 | 3.5 | Integrating exponentials, 1/(ax + b) and trigonometric functions | Each result below is a derivative read backwards. |
| 15 | 3.5 | Integration using trigonometric identities | Rearrange \(\cos 2x = 1 - 2\sin^2 x = 2\cos^2 x - 1\) to express each square in terms of \(\cos 2x\). |
| 15 | 3.5 | Integration using partial fractions and f'(x)/f(x) | Differentiate \(\ln|f(x)|\) with the chain rule and you get \(\dfrac{f'(x)}{f(x)}\). |
| 15 | 3.5 | Integration by substitution | Treat \(\dfrac{du}{dx}\) like a fraction to swap \(dx\) for \(du\). |
| 16 | 3.5 | Integration by parts | Integrate the product rule \(\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx} + v\dfrac{du}{dx}\) and rearrange to get the formula. |
| 16 | 3.5 | The trapezium rule | Split \([a, b]\) into \(n\) equal intervals (strips). |
| 16 | Assessment point | Topic question bank, 48 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- \(\displaystyle\int \dfrac{1}{2x+1}\,dx\) is NOT \(\ln|2x+1|\).
- \(\displaystyle\int \sin x\,dx = -\cos x\), with a MINUS.
- Do not forget the \(\dfrac{1}{a}\) in front of \(\tan^{-1}\), and do not leave \(\tan^{-1}1\) in degrees: \(\tan^{-1}1 = \dfrac{\pi}{4}\), never 45, inside a calculus answer.
3.6Numerical solution of equations
Weeks 16 to 17| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 16 | 3.6 | Locating roots by a change of sign | In the lab, choose a curve and move the line across it. |
| 17 | 3.6 | Iterative methods | An iterative formula \(x_{n+1} = F(x_n)\) builds a sequence \(x_1, x_2, x_3, \ldots\): each term is \(F\) of the one before. |
| 17 | Assessment point | Topic question bank, 16 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- In a sketch-and-count question the marks are for two correct graph shapes in the right places AND a stated conclusion ("one point of intersection, so one root").
- Compare with zero, not with each other: you must rearrange to \(f(x) = 0\) first.
- No sign change does NOT mean no root. \(f(x) = (x-1)^2\) has a root at \(x = 1\) but is positive on both sides; and two roots between \(a\) and \(b\) cancel out the sign change.
Topic 03
Pure Mathematics 3
Weeks 17 to 203.7Vectors
Weeks 17 to 18| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 17 | 3.7 | Vectors in two and three dimensions | In the lab, build a vector from its components and watch its length update: the length is Pythagoras on the components. |
| 17 | 3.7 | The vector equation of a line | Read \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\) as a set of directions: from the origin go to \(A\), then walk \(t\) lots of \(\mathbf{b}\). |
| 18 | 3.7 | Parallel, intersecting and skew lines | Picture the edges of a box (Fig. 1). |
| 18 | 3.7 | The scalar product | The scalar (dot) product has two forms. |
| 18 | Assessment point | Topic question bank, 32 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- \(\overrightarrow{AB}\) is \(\mathbf{b} - \mathbf{a}\), not \(\mathbf{a} - \mathbf{b}\).
- Parallel vectors need ONE common factor for every component. \((2, 4, 6)\) and \((1, 2, 4)\) look alike, but the factors are \(2, 2, 1.5\), so they are not parallel.
- The midpoint is half the SUM of the position vectors, \(\tfrac{1}{2}(\mathbf{a} + \mathbf{b})\).
3.8Differential equations
Weeks 18 to 19| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 18 | 3.8 | Forming differential equations | Every modelling statement has the same three parts: WHICH quantity changes (the top of the derivative), with respect to WHAT (usually time \(t\)), and what the rate depends on. |
| 19 | 3.8 | Solving by separating the variables | Treat \(\dfrac{dy}{dx}\) as if it were a fraction: divide by the \(y\)-part and 'multiply' by \(dx\), so that each side has only one variable. |
| 19 | Assessment point | Topic question bank, 16 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- 'Rate of change' with no other words means with respect to TIME.
- If you write \(\dfrac{dh}{dt} = -k\sqrt{h}\), then \(k\) must come out POSITIVE.
- 'Growing fastest' means the RATE \(\dfrac{dN}{dt}\) is a maximum.
3.9Complex numbers
Weeks 19 to 20| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 19 | 3.9 | Complex numbers: arithmetic in Cartesian form | No real number squares to \(-1\), so we invent one: \(i = \sqrt{-1}\). |
| 19 | 3.9 | The Argand diagram, modulus-argument and exponential form | On an Argand diagram the real part goes across and the imaginary part goes up. |
| 20 | 3.9 | Complex roots and square roots | Take the conjugate of a whole polynomial equation with real coefficients: the coefficients do not change, but every \(z\) becomes \(z^{*}\). |
| 20 | 3.9 | Loci in the Argand diagram | Rewrite every modulus in the form \(|z - a|\), so you can read off the centre \(a\). |
| 20 | Assessment point | Topic question bank, 32 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- \(\sqrt{-36}\) is \(6i\), not \(-6\) or \(6\).
- \(zz^{*} = x^{2} + y^{2}\), always real and never negative.
- You cannot divide parts separately: \(\dfrac{3 + 4i}{1 - 2i}\) is NOT \(3 - 2i\).
Topic 04
Mechanics
Weeks 20 to 244.1Forces and equilibrium
Weeks 20 to 21| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 20 | 4.1 | Forces, components and resultants | Before any calculation, draw a force diagram: the particle as a dot, every force as an arrow starting from it, each labelled. |
| 21 | 4.1 | Equilibrium of a particle | In the lab, set the hanging weight to 40 N and the string angles to \(30^\circ\) and \(50^\circ\). |
| 21 | 4.1 | Friction and limiting equilibrium | When two rough surfaces touch, the contact force has a component perpendicular to the surface (the normal reaction \(R\)) and a component along it (friction \(F\)). |
| 21 | Assessment point | Topic question bank, 24 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- Weight and normal reaction are NOT a third-law pair: they act on the same body and are different kinds of force.
- Do not learn "horizontal is cos".
- Keep the signs.
4.2Kinematics of motion in a straight line
Weeks 21 to 22| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 21 | 4.2 | Displacement, velocity and acceleration graphs | In the lab, give the particle a positive initial velocity and a negative acceleration, and run it for long enough that it turns round. |
| 22 | 4.2 | Constant acceleration: the suvat equations | In the lab, release the ball from a chosen height and step through the strobe images. |
| 22 | 4.2 | Variable acceleration: using calculus | Given displacement as a function of time, differentiate once for velocity and again for acceleration. "At rest" (or "instantaneously at rest") means \(v = 0\); "maximum velocity" usually means \(a = 0\). |
| 22 | Assessment point | Topic question bank, 24 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- Average speed is total DISTANCE divided by time, not the average of the start and end speeds (that only works for constant acceleration).
- On a displacement-time graph the HEIGHT is position, not speed.
- Area below the time axis is negative DISPLACEMENT but positive DISTANCE.
4.3Momentum
Week 22| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 22 | 4.3 | Momentum and the conservation of momentum | In the lab, give A and B velocities in opposite directions and watch the total momentum readout before and after the collision. |
| 22 | Assessment point | Topic question bank, 8 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- Subtracting the speeds gives \(0.2(15 - 12) = 0.6\) N s, which is wrong.
- Every unknown velocity after the collision should be written as a signed variable, and the sign of the answer tells you the direction.
- Do not do both collisions in one equation unless all the particles really do end up moving together.
4.4Newton's laws of motion
Week 23| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 23 | 4.4 | Newton's laws of motion | In the lab, set the friction coefficient to zero and double the applied force: the acceleration doubles. |
| 23 | 4.4 | Motion on an inclined plane | On a smooth plane the only force along the slope is the weight component \(mg\sin\alpha\), so \(a = g\sin\alpha\) down the plane whichever way the particle is moving. |
| 23 | 4.4 | Connected particles and pulleys | For a car towing a trailer in a straight line, you can write \(F = ma\) for the car, for the trailer, or for the whole system. |
| 23 | Assessment point | Topic question bank, 24 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- \(F = ma\) uses the RESULTANT force.
- Keep signs consistent.
- "Moving upwards" does not mean "accelerating upwards".
4.5Energy, work and power
Week 24| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 24 | 4.5 | Work done by a force | In the lab, keep the pulling force fixed and raise the angle of the rope from \(0^\circ\) towards \(90^\circ\). |
| 24 | 4.5 | Kinetic and potential energy, the work-energy principle | In the lab, release the ball on the slope with friction set to zero and watch the bar chart: potential energy drains into kinetic energy and the total bar stays the same height. |
| 24 | 4.5 | Power and driving force | In the lab, set the road slope to zero, start the car and watch the speed-time graph level off. |
| 24 | Assessment point | Topic question bank, 24 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- Do not use the full force when it is at an angle: \(60 \times 8 = 480\) J is the work only if the rope is horizontal.
- Work done by the weight on a slope uses the VERTICAL height, \(mg \times d\sin\alpha\), not \(mg \times d\).
- With the rope at an angle, the normal reaction is NOT \(mg\).
Topic 05
Probability & Statistics 1
Weeks 25 to 285.1Representation of data
Week 25| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 25 | 5.1 | Stem-and-leaf, box plots, histograms and cumulative frequency | Open the histogram tab of the lab and widen one class. |
| 25 | 5.1 | Measures of central tendency and variation | In the frequency-table tab of the lab, change one frequency and watch which averages move. |
| 25 | 5.1 | Variance and standard deviation, including coded data | Each value's deviation is \(x - \bar{x}\). |
| 25 | Assessment point | Topic question bank, 24 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- Plotting frequency (not frequency density) on the vertical axis when the class widths differ.
- Forgetting to reverse the left-hand leaves.
- Plotting cumulative frequencies at the class midpoints.
5.2Permutations and combinations
Week 26| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 26 | 5.2 | Permutations: arrangements in a line and with restrictions | Fill the places one at a time: \(n\) choices for the first place, \(n - 1\) for the next, and so on, giving \(n! = n(n-1)\cdots 2 \times 1\). |
| 26 | 5.2 | Combinations: selections | Choosing Ali, Ben and Cy is the same selection as choosing Cy, Ali and Ben. |
| 26 | Assessment point | Topic question bank, 16 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- Dividing by \(3! + 2!\) instead of \(3! \times 2!\).
- Forgetting the internal arrangements of the block.
- Using 'total minus together' for three or more people who must all be apart.
5.3Probability
Weeks 26 to 27| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 26 | 5.3 | Probability: sample spaces, exclusive events and tree diagrams | In the lab's sample-space tab, roll two dice many times and compare the frequency of each total with the grid of 36 outcomes. |
| 27 | 5.3 | Conditional probability and independence | In the lab's conditional tab, click on a condition and watch the sample space shrink to just those outcomes. |
| 27 | Assessment point | Topic question bank, 16 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- Saying P(total 8) \(= \dfrac{1}{11}\) because there are 11 possible totals.
- Adding probabilities of events that are NOT exclusive.
- Keeping the same denominators on the second branches.
5.4Discrete random variables
Weeks 27 to 28| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 27 | 5.4 | Discrete probability distributions, E(X) and Var(X) | A probability distribution lists every value a discrete random variable \(X\) can take, with its probability. |
| 27 | 5.4 | The binomial distribution | Check the four conditions every time, and say them in the context of the question when asked to justify the model. |
| 28 | 5.4 | The geometric distribution | The conditions are the same as for the binomial (independent trials, two outcomes, constant \(p\)) except that there is NO fixed number of trials: you stop at the first success. |
| 28 | Assessment point | Topic question bank, 24 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- Forgetting that \(\text{P}(X \ge 3)\) includes 3. 'At least 3' and 'more than 2' are the same event for whole-number values; 'more than 3' is not.
- Missing a route in \(\text{P}(X = 1)\).
- Stopping at \(\text{E}(X^2)\), or writing \(\text{Var}(X) = \text{E}(X^2) - \text{E}(X)\).
5.5The normal distribution
Week 28| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 28 | 5.5 | The normal distribution | A normal distribution is fixed by its mean \(\mu\) and variance \(\sigma^2\), written \(\text{N}(\mu, \sigma^2)\). |
| 28 | 5.5 | The normal approximation to the binomial | A binomial distribution is roughly bell-shaped when \(n\) is large and \(p\) is not too close to 0 or 1. |
| 28 | Assessment point | Topic question bank, 16 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- Dividing by the variance. \(\text{N}(50, 16)\) means \(\sigma^2 = 16\), so \(\sigma = 4\); \(\dfrac{56 - 50}{16}\) is wrong.
- Writing \(\Phi(0.75) - \Phi(1)\) for \(\text{P}(46 < X < 53)\).
- Using \(z = 0.1\) or \(z = 0.8\) in the equation.
Topic 06
Probability & Statistics 2
Weeks 29 to 316.1The Poisson distribution
Week 29| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 29 | 6.1 | The Poisson distribution | The Poisson distribution counts events in a fixed interval of time, length, area or volume. |
| 29 | 6.1 | Approximations: binomial to Poisson, Poisson to normal | When \(n\) is large and \(p\) is small, successes are rare events scattered among many trials, which is exactly the Poisson picture. |
| 29 | Assessment point | Topic question bank, 16 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- Do not just recite "independent and random".
- "More than 2" means \(X \ge 3\), so it is \(1 - \text{P}(X \le 2)\), not \(1 - \text{P}(X \le 3)\).
- In a two-stage question, do not add the rates to \(\lambda = 6\) for the week and find \(\text{P}(X = 0)\): that is the probability of no faults all week, a different event.
6.2Linear combinations of random variables
Week 29| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 29 | 6.2 | Linear combinations of random variables | Multiplying by \(a\) stretches the spread by \(|a|\), so the variance is multiplied by \(a^{2}\). |
| 29 | Assessment point | Topic question bank, 8 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- \(\text{Var}(3X - 5)\) is not \(3 \times 4 - 5\), and not \(3 \times 4\).
- The commonest Paper 6 error: \(\text{Var}(A - B) = 9 - 16\).
- Before adding Poisson parameters, put them on the same interval.
6.3Continuous random variables
Week 30| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 30 | 6.3 | Probability density functions, mean and variance | For a continuous variable, \(f(x)\) is not a probability: it is a height, and probabilities are areas under it. |
| 30 | Assessment point | Topic question bank, 8 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- \(f(2) = \tfrac{3}{8}\) is not \(\text{P}(X = 2)\).
- \(\text{Var}(X) = \int x^{2} f(x)\,dx\) is NOT the variance until you subtract \(\mu^{2}\).
- The median is not half-way along the domain.
6.4Sampling and estimation
Week 30| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 30 | 6.4 | Populations, samples and the distribution of the sample mean | The population is every item of interest; a sample is the part you actually observe. |
| 30 | 6.4 | Unbiased estimates and confidence intervals | An estimate is unbiased if, although it varies from sample to sample, it is right on average. |
| 30 | Assessment point | Topic question bank, 16 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- "It is too small" or "it is not random" alone rarely scores.
- \(\text{Var}(\bar{X}) = \dfrac{\sigma^{2}}{n}\), so the standard deviation is \(\dfrac{\sigma}{\sqrt{n}}\), not \(\dfrac{\sigma}{n}\).
- Use the CLT only when the population is NOT normal and \(n\) is large, and say so.
6.5Hypothesis tests
Week 31| Week | Ref | Lesson | Learning intention |
|---|---|---|---|
| 31 | 6.5 | Hypothesis tests with binomial and Poisson distributions | Every test follows the same script. |
| 31 | 6.5 | Hypothesis tests for a population mean | Use this when the population is normal (or the sample is large) and \(\sigma\) is given. |
| 31 | 6.5 | Type I and Type II errors | Every decision could be wrong. |
| 31 | Assessment point | Topic question bank, 24 multiple-choice questions: print or review, or assign in the app for a marked score. |
Examiner traps to pre-empt:
- Compare the TAIL probability \(\text{P}(X \ge 9)\) with 5%, never the single probability \(\text{P}(X = 9)\).
- Write hypotheses about the PARAMETER (\(p\) or \(\lambda\)), not about the observed count, and scale \(\lambda\) to the period of the observation: 2 months at 3.2 per month means \(H_0: \lambda = 6.4\).
- The continuity correction points AWAY from the tail: \(X \ge 95\) becomes \(Y > 94.5\), and \(X \le 8\) becomes \(Y < 8.5\).
Also for this course
Beyond the scheme
Lesson packs93 packs: plan, worksheet, quiz
Question bankBuild a test from every course
PlannerBlock out the term
Classes and gradebookAssign the banks, track everyone
Student syllabus pageA-Level Maths 9709
Original work by the TheLucidSTEM team. Week numbers assume 3 lessons a week unless you change the control; the download carries the numbers shown.