YFM01: course teaching notes

Version: Issue 3, April 2019; first teaching 2018; unit assessment from 2019

These are original course-owned teaching notes. Objective-level exceptions are listed in the review, and lessons are tier labelled.

FP1 · Complex arithmetic and quadratic roots

What if the root is not real?

• The equation x²+1=0 has no real roots. Extending the number system lets us represent and calculate with both roots.
• This lesson studies complex conjugate 共轭复数: The number obtained by reversing the sign of the imaginary part.

Choose the mathematical structure

• Write z=a+bi with i²=-1. Add components, multiply brackets and use a conjugate to make a division denominator real. Non-real roots of a real-coefficient quadratic occur in conjugate pairs.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• The square root of a sum is not the sum of the square roots. An inverse tangent alone may return the wrong quadrant. Do not confuse modulus with squared modulus.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• FP1 uses complex arithmetic, modulus, Argand representation and quadratic roots. De Moivre powers and roots of unity are reserved for FP2.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP1 · Parabolas and rectangular hyperbolas

How can a rotating parameter trace an ellipse?

• An ellipse can be traced by a rotating parameter even when it is awkward to describe as a single y=f(x).
• This lesson studies parabola · ⁨parabole⁩ 抛物线: The locus of points equally distant from a fixed focus and a fixed directrix.

Choose the mathematical structure

• For y²=4ax, the focus is (a,0), directrix x=-a and vertex (0,0). For xy=c², the coordinate axes are asymptotes. Read the parameter from the coefficient rather than assuming it equals 4a.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• The coefficient of x is 4a, not a. Include both signs when solving y². An asymptote is approached; it is not a finite intercept of xy=c².
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• FP1 coordinate-geometry support for parabolas and rectangular hyperbolas. Ellipse area and ellipse parametrisation are excluded.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP1 · Matrix operations, inverses and plane transformations

Where do the basis vectors go?

• A map stretches one direction and shears another. A matrix records the transformation of each basis vector.
• This lesson studies determinant 行列式: A scalar that determines whether a square matrix is invertible and, in two dimensions, its signed area factor.

Choose the mathematical structure

• A 2×2 matrix [[a,b],[c,d]] has determinant ad-bc and inverse [[d,-b],[-c,a]]/(ad-bc) if the determinant is nonzero. Transform column vectors by left multiplication. The rightmost matrix acts first in a product.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Matrix multiplication is usually not commutative. A determinant of zero means the transformation loses a dimension and has no inverse. Eigenvalues and diagonalisation are advanced scope, not ordinary GCSE vector content.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• FP1 uses 2×2 matrix operations, inverses and plane transformations. Eigenvalues and 3×3 diagonalisation belong to FP3, not this unit.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP1 · Bisection and Newton root finding

How reliable is an approximation?

• A nonlinear equation has no convenient exact solution. An approximation still needs evidence that it is reliable.
• This lesson studies iteration 迭代: A repeated update in which each new approximation is calculated from the previous one.

Choose the mathematical structure

• For a continuous function with a sign change, bisect a root bracket and keep the half with opposite endpoint signs. Newton updates x to x-f(x)/f prime(x), provided the derivative is nonzero.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• A sign change must occur on an interval where the function is continuous. Newton can fail if its derivative is zero, or if its iterates leave the useful domain.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• FP1 numerical methods concern roots: bisection, interpolation and Newton methods. Numerical integration is not included in this lesson.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP1 · Finite sums and induction

Can a polynomial replace an exponential nearby?

• Near a known point, a complicated function can be replaced by a polynomial approximation. The approximation needs a centre and a valid scale.
• This lesson studies sigma notation 求和符号: Notation that adds indexed terms over a stated finite range.

Choose the mathematical structure

• For k=1 to n, sum k=n(n+1)/2, sum k²=n(n+1)(2n+1)/6 and sum k³=[n(n+1)/2]². Differences between consecutive terms can telescope. In induction, verify the starting case and show the next case follows from the assumed case.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• The sum of squares is not the square of the sum. State the index range and check the first case; an inductive step by itself is not a complete induction proof.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• FP1 finite series and induction support only. Maclaurin and Taylor approximation belong to later further-pure units and are excluded.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP1 · Deduction, contradiction and induction

When does a pattern become a proof?

• Checking several integers can suggest a pattern. What turns the pattern into a proof for every integer?
• This lesson studies counterexample 反例: A single valid case that disproves a universal claim.

Choose the mathematical structure

• A deductive proof starts from stated definitions or assumptions. A counterexample refutes an all-values claim. Induction requires a base case and an implication from n=k to n=k+1; checking successive cases alone is not induction.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Do not assume the conclusion while proving it. In contradiction, identify the impossible consequence and reject the initial contrary assumption. Induction is not compulsory in AQA 7357; it belongs in its own appropriate further-mathematics scope.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• AQA proof includes deduction, exhaustion and contradiction. IAL P2 introduces exhaustion and counterexample, while P4 introduces contradiction. Keep the method matched to the named unit.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP2 · Quadratics and inequalities

Which widths make enough space?

• A rectangular enclosure has area x(10-x). What widths give at least 21 square metres?
• This lesson studies discriminant 判别式: The quantity b²-4ac that determines the real roots of ax²+bx+c=0.

Choose the mathematical structure

• Factor where possible; otherwise complete the square or use the quadratic formula. A quadratic inequality needs the sign on intervals, not only the roots. The discriminant identifies repeated or missing real roots.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Multiplying an inequality by a negative number reverses its direction. A sketch must show which side of each root satisfies the inequality. Geometry may restrict x further.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Use the vertex to interpret an optimum. Check an endpoint and a point between the roots; the algebra and graph should tell the same story.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP2 · Series, finite differences and Taylor expansions

Can a polynomial replace an exponential nearby?

• Near a known point, a complicated function can be replaced by a polynomial approximation. The approximation needs a centre and a valid scale.
• This lesson studies Taylor polynomial 泰勒多项式: A polynomial formed from a function's derivatives at a chosen centre.

Choose the mathematical structure

• About x=a, f(x)=f(a)+f prime(a)(x-a)+f double prime(a)(x-a)²/2!+... when the expansion is valid. At a=0 it is a Maclaurin expansion. Finite sums may also simplify by cancellation or standard sum formulae.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• The factorial belongs in the denominator of every Taylor coefficient. The expansion centre is not always zero. An approximation close to its centre can be poor far away.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• For a telescoping series, write several terms and identify both surviving ends. For a Taylor approximation, state its order and compare with a known value or a remainder estimate where required.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP2 · Complex numbers and roots

What if the root is not real?

• The equation x²+1=0 has no real roots. Extending the number system lets us represent and calculate with both roots.
• This lesson studies complex conjugate 共轭复数: The number obtained by reversing the sign of the imaginary part.

Choose the mathematical structure

• Write z=a+bi with i²=-1. The conjugate is a-bi and z times its conjugate=a²+b². The modulus is √(a²+b²); the argument needs the correct quadrant. Conjugate roots occur for polynomials with real coefficients.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• The square root of a sum is not the sum of the square roots. An inverse tangent alone may return the wrong quadrant. Do not confuse modulus with squared modulus.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Draw the real axis horizontally and imaginary axis vertically. For products, moduli multiply and arguments add; powers extend this geometric pattern through de Moivre's theorem.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP2 · Differential equations and numerical solutions

What does the starting value decide?

• A growing population's rate is proportional to its current size. The rate equation describes a whole family until an initial population is supplied.
• This lesson studies initial condition 初始条件: A specified value that selects a particular solution from a family.

Choose the mathematical structure

• For dy/dx=ky, separate variables: dy/y=k dx, giving y=Ae^(kx). Use an initial condition to find A. Euler's method takes y next=y+h f(x,y), with a chosen step h.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• An initial condition determines the integration constant; it is not a replacement for integrating. Dividing by y can omit an equilibrium solution y=0. Euler accuracy depends on step size and the equation.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• For second-order linear equations, combine the complementary function with an appropriate particular integral, then apply the required initial conditions. This is Further Pure scope; check the named unit.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP2 · Second-order linear differential equations

Why do we need two initial conditions?

• A displacement model involves both velocity and acceleration. Solving only a first-order rate equation cannot capture both initial conditions.
• This lesson studies complementary function 互补函数: The general solution of the associated homogeneous linear differential equation.

Choose the mathematical structure

• For y double prime+ay prime+by=f(x), solve the auxiliary quadratic for the homogeneous part. Add a suitable particular integral for the forcing term. Repeated and complex roots require their correct forms.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Two arbitrary constants need two independent conditions. If the trial particular integral duplicates a complementary-function term, multiply the trial by x as required. Do not discard a valid oscillatory solution because the auxiliary roots are complex.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Differentiate the final expression and substitute into the original differential equation. Check both initial conditions separately, and interpret the permitted solution interval.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP2 · Polar curves and area

Why describe a curve by direction?

• A petal-shaped curve is simpler when its radius depends on direction. Cartesian coordinates can hide this structure.
• This lesson studies polar coordinate 极坐标: A position described by distance r and angle θ from a chosen origin and reference ray.

Choose the mathematical structure

• Use x=r cosθ and y=r sinθ. A polar area is one half the integral of r² with respect to θ over a correctly chosen interval. Identify symmetry and repeated tracing before selecting limits.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Negative r places a point in the opposite direction; it is not an ordinary negative distance along the same ray. A parametrisation may trace the same region more than once, so a full parameter interval can overcount area.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Sketch enough points to establish orientation and bounds. For an enclosed region between two curves, determine intersections and which radial square contributes the outer boundary on each interval.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP3 · Hyperbolic functions and inverse relations

How can growth and decay make a symmetric curve?

• A hanging cable has a curved profile related to exponentials. Hyperbolic functions combine growth and decay symmetrically.
• This lesson studies hyperbolic cosine 双曲余弦: The function cosh x=(e^x+e^(-x))/2.

Choose the mathematical structure

• Define sinh x=(e^x-e^(-x))/2 and cosh x=(e^x+e^(-x))/2. Their identity is cosh²x-sinh²x=1. Derivatives are sinh prime=cosh and cosh prime=sinh.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• The hyperbolic identity has a minus sign. cosh is not one-to-one on all real inputs; its usual inverse uses x≥0. Ordinary circular-trigonometric identities cannot be substituted unchanged.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Use exponential definitions to prove identities and solve equations. State domain restrictions for inverse functions before differentiating or integrating them.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP3 · Conics, parametric curves and tangent reasoning

How can a rotating parameter trace an ellipse?

• An ellipse can be traced by a rotating parameter even when it is awkward to describe as a single y=f(x).
• This lesson studies parametric equation 参数方程: A representation in which coordinates are expressed using a common parameter.

Choose the mathematical structure

• For x=a cos t,y=b sin t, eliminating t gives x²/a²+y²/b²=1. For parametric curves, dy/dx=(dy/dt)/(dx/dt), where dx/dt≠0. A zero denominator may signal a vertical tangent.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Eliminating a parameter may lose a domain restriction or direction of travel. A vertical tangent cannot be assigned a finite dy/dx. Distinguish semiaxes from full widths.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• For advanced coordinate geometry, use the defining equation and a consistent parameter. For IAL P4 parametric integration, use the specification's restricted requirements rather than importing all Further Pure conics.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP3 · Chain, product, quotient and implicit differentiation

Does the inside expression change too?

• A cost curve is a power of a changing expression. Differentiating the outer power alone misses the rate of its input.
• This lesson studies chain rule 链式法则: The rule that multiplies the outer derivative by the inner derivative for a composite function.

Choose the mathematical structure

• For y=f(g(x)), y prime=f prime(g(x))g prime(x). For uv, differentiate to u prime v+uv prime. For u/v, use (u prime v-uv prime)/v², where v≠0.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• A derivative of a product is not the product of the derivatives. In implicit differentiation, every differentiated function of y brings a dy/dx factor. A quotient's denominator is squared.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Choose a useful form before differentiating: expanding a short polynomial may be simpler. For related rates, write the relation in symbols, differentiate with respect to time, then substitute measured values.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP3 · Substitution, parts and partial fractions

Can we integrate the two factors separately?

• A simple-looking product such as xe^x cannot be integrated by separately integrating each factor.
• This lesson studies integration by parts 分部积分法: An integration method based on the derivative of a product.

Choose the mathematical structure

• Use substitution when an inner derivative appears as a factor. By parts, integral u v prime =uv-integral u prime v. For a rational function, divide first if needed, then decompose into partial fractions.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Choosing u and v prime well matters: the remaining integral should become simpler. In a definite substitution, either change the limits or return to x before applying the original limits.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Check by differentiating. For volumes of revolution around the x-axis use V=π integral y² dx; do not confuse the square of a function with the integral of the function.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP3 · Spatial vectors, lines and angles

Can nonparallel paths still miss each other?

• Two paths in space can be neither parallel nor intersecting. A two-dimensional sketch may conceal their separation.
• This lesson studies direction vector 方向向量: A nonzero vector giving the direction of a line.

Choose the mathematical structure

• A line has r=a+λb with position vector a and nonzero direction b. To find an intersection, equate all components and solve for both line parameters. For an angle use a·b=|a||b|cosθ where scalar products are in the named syllabus.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Satisfying two component equations does not guarantee the third. Nonparallel lines may be skew. A line's direction vector is not its position vector, and a scalar parameter has no coordinate units.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• State whether a requested angle is acute, directed or an angle between lines. Use the syllabus-specific plane or distance method only when it is actually required for that unit.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

FP3 · Matrix transformations, inverses and eigenvalues

Where do the basis vectors go?

• A map stretches one direction and shears another. A matrix records the transformation of each basis vector.
• This lesson studies determinant 行列式: A scalar that determines whether a square matrix is invertible and, in two dimensions, its signed area factor.

Choose the mathematical structure

• A 2×2 matrix [[a,b],[c,d]] has determinant ad-bc and inverse [[d,-b],[-c,a]]/(ad-bc) if the determinant is nonzero. Transform column vectors by left multiplication. The rightmost matrix acts first in a product.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Matrix multiplication is usually not commutative. A determinant of zero means the transformation loses a dimension and has no inverse. Eigenvalues and diagonalisation are advanced scope, not ordinary GCSE vector content.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Find an eigenvector by solving (A-λI)v=0 with v≠0. Explain the geometrical meaning: this vector keeps its line direction under the transformation.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit FP3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

M1 · Kinematics and Newton's laws

Which force accelerates the trolley?

• A trolley accelerates while friction opposes its motion. The engine's force alone does not equal mass times acceleration.
• This lesson studies resultant force 合力: The vector sum of all external forces acting on the modelled object.

Choose the mathematical structure

• Choose a positive direction and draw a force diagram. Use resultant F=ma. For constant acceleration, v=u+at and s=ut+at²/2. A particle model ignores size; a smooth surface ignores friction.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• The normal reaction need not equal weight on a slope. Connected-particle models need one equation per particle and a consistent acceleration relation. The constant-acceleration formulae cannot be used for arbitrary variable acceleration.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Explain what each idealisation permits and what it leaves out. For a velocity-time graph, gradient represents acceleration and signed area represents displacement.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit M1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

M1 · Vectors and transformation geometry

Why is displacement shorter than the walk?

• Walking 4 m east and 3 m north gives a displacement of 5 m, even though the travelled distance is 7 m.
• This lesson studies resultant 合向量: The vector sum representing the combined displacement or force.

Choose the mathematical structure

• Add corresponding vector components and subtract position vectors to find a displacement. A translation moves every point by the same vector; a scalar multiple changes length and possibly direction.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• The order of subtraction matters: BA=-AB. Proving parallelism needs a scalar-multiple relation; a sketch alone is insufficient. Negative enlargement reverses position about its centre.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Use vector components, scalar multiples and magnitudes for displacements and forces. This unit does not require the scalar product.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit M1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

M1 · Moments, equilibrium and centres of mass

Why does a longer spanner help?

• A long spanner turns a nut more easily. The turning effect depends on perpendicular distance as well as force.
• This lesson studies moment · ⁨donné⁩ 力矩: Force multiplied by the perpendicular distance from its line of action to a pivot.

Choose the mathematical structure

• For equilibrium, both the resultant force and resultant moment are zero. Take moments about a useful pivot to remove unknown forces through that point. A combined centre of mass is a mass-weighted mean position.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Use the perpendicular lever arm, not a diagonal distance to the point of application. A zero resultant force alone does not prevent rotation. A support that would need a negative normal reaction may lose contact.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• For a lamina made from parts, choose a common origin and tabulate each signed mass-area contribution. A removed region has negative mass in that calculation, not a new physical negative material.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit M1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

M1 · Inclined planes, friction and connected particles

Is friction already at its maximum?

• A crate on a slope is at rest. Friction can adjust to balance the force down the slope; it need not already be at its maximum.
• This lesson studies limiting friction 极限摩擦力: The maximum static friction before slipping, equal to μ times the normal reaction in the model.

Choose the mathematical structure

• Resolve parallel and perpendicular to the plane. Weight components are mg sinθ and mg cosθ. Static friction satisfies F≤μR and equals μR only at the limiting case. Write separate equations for connected particles.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Friction opposes motion or the tendency to move, not always the coordinate direction. A taut light inextensible string over a smooth pulley gives equal tension and a common acceleration magnitude; each assumption has a job.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Start with force diagrams and a proposed direction of motion. If the calculated direction conflicts with the friction assumption, revisit the model instead of keeping inconsistent signs.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit M1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

M2 · Kinematics and Newton's laws

Which force accelerates the trolley?

• A trolley accelerates while friction opposes its motion. The engine's force alone does not equal mass times acceleration.
• This lesson studies resultant force 合力: The vector sum of all external forces acting on the modelled object.

Choose the mathematical structure

• Choose a positive direction and draw a force diagram. Use resultant F=ma. For constant acceleration, v=u+at and s=ut+at²/2. A particle model ignores size; a smooth surface ignores friction.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• The normal reaction need not equal weight on a slope. Connected-particle models need one equation per particle and a consistent acceleration relation. The constant-acceleration formulae cannot be used for arbitrary variable acceleration.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Explain what each idealisation permits and what it leaves out. For a velocity-time graph, gradient represents acceleration and signed area represents displacement.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit M2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

M2 · Moments, equilibrium and centres of mass

Why does a longer spanner help?

• A long spanner turns a nut more easily. The turning effect depends on perpendicular distance as well as force.
• This lesson studies moment · ⁨donné⁩ 力矩: Force multiplied by the perpendicular distance from its line of action to a pivot.

Choose the mathematical structure

• For equilibrium, both the resultant force and resultant moment are zero. Take moments about a useful pivot to remove unknown forces through that point. A combined centre of mass is a mass-weighted mean position.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Use the perpendicular lever arm, not a diagonal distance to the point of application. A zero resultant force alone does not prevent rotation. A support that would need a negative normal reaction may lose contact.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• For a lamina made from parts, choose a common origin and tabulate each signed mass-area contribution. A removed region has negative mass in that calculation, not a new physical negative material.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit M2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

M2 · Work, energy, power and collisions

What does a slower rebound tell us?

• A ball rebounds slower after hitting a wall. Momentum and kinetic energy answer different parts of the collision story.
• This lesson studies coefficient of restitution 恢复系数: The ratio of relative separation speed to relative approach speed along the line of impact.

Choose the mathematical structure

• Work is force times displacement in its direction. Kinetic energy is mv²/2 and gravitational potential energy is mgh. Momentum is conserved for an isolated collision; restitution gives the relative speed relation, not conservation of kinetic energy.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Conserve momentum only when the external impulse is negligible. Kinetic energy is generally lost in an inelastic collision. Power is a rate of energy transfer, not an amount of energy.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• State the direction signs before writing collision equations. For a connected motion problem, choose the whole system for energy when internal tensions cancel; account for frictional work explicitly.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit M2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

M2 · Projectile motion and variable acceleration

Why does horizontal motion share vertical time?

• A ball is launched at an angle. Its horizontal and vertical motion share time but follow different equations.
• This lesson studies projectile 抛体: A particle moving under gravity after launch in a model that neglects air resistance.

Choose the mathematical structure

• Resolve initial velocity into horizontal u cosθ and vertical u sinθ. With no air resistance, horizontal acceleration is zero and vertical acceleration is -g. Use the same time in both components.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Zero vertical velocity at the highest point does not mean zero total speed. Launch and landing heights need not be equal. Do not use a range formula that assumes equal heights without checking them.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• For variable acceleration, integrate a(t) to get v(t) and use the initial velocity to find the constant, then integrate for displacement. Confirm units and the physical time interval.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit M2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

M3 · Elasticity, circular motion and variable dynamics

Which length stretches the spring?

• A stretched spring stores energy. A particle moving in a circle needs a resultant force towards the centre, even at constant speed.
• This lesson studies extension 伸长量: The increase in an elastic string's or spring's length from its natural length.

Choose the mathematical structure

• For an elastic string, tension=λx/l when stretched, with natural length l and modulus λ. Elastic energy=λx²/(2l). For circular motion, radial acceleration=v²/r; resolve actual forces in the radial direction.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Centripetal force is the name of the inward resultant, not an extra force to add to the diagram. A slack string has no tension. Distinguish extension from total stretched length.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• For variable acceleration, use a=dv/dt or a=v dv/dx when appropriate. Combine the chosen motion relation with the force or energy model, then check the permitted range of motion.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit M3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

M3 · Differential equations and numerical solutions

What does the starting value decide?

• A growing population's rate is proportional to its current size. The rate equation describes a whole family until an initial population is supplied.
• This lesson studies initial condition 初始条件: A specified value that selects a particular solution from a family.

Choose the mathematical structure

• For dy/dx=ky, separate variables: dy/y=k dx, giving y=Ae^(kx). Use an initial condition to find A. Euler's method takes y next=y+h f(x,y), with a chosen step h.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• An initial condition determines the integration constant; it is not a replacement for integrating. Dividing by y can omit an equilibrium solution y=0. Euler accuracy depends on step size and the equation.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• For second-order linear equations, combine the complementary function with an appropriate particular integral, then apply the required initial conditions. This is Further Pure scope; check the named unit.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit M3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

M3 · Moments, equilibrium and centres of mass

Why does a longer spanner help?

• A long spanner turns a nut more easily. The turning effect depends on perpendicular distance as well as force.
• This lesson studies moment · ⁨donné⁩ 力矩: Force multiplied by the perpendicular distance from its line of action to a pivot.

Choose the mathematical structure

• For equilibrium, both the resultant force and resultant moment are zero. Take moments about a useful pivot to remove unknown forces through that point. A combined centre of mass is a mass-weighted mean position.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Use the perpendicular lever arm, not a diagonal distance to the point of application. A zero resultant force alone does not prevent rotation. A support that would need a negative normal reaction may lose contact.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• For a lamina made from parts, choose a common origin and tabulate each signed mass-area contribution. A removed region has negative mass in that calculation, not a new physical negative material.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit M3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

S1 · Data summaries, histograms and interpretation

Can one average tell the whole story?

• Two groups have the same median but different spread. One summary cannot describe both location and consistency.
• This lesson studies frequency density · ⁨densité de fréquence⁩ 频率密度: Frequency divided by class width, used as histogram height.

Choose the mathematical structure

• Compare an appropriate average and spread in context. A histogram uses area for frequency, so height=frequency/class width. Grouped estimates assume representative values within intervals.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• The tallest histogram bar need not contain the most observations. A grouped mean is an estimate. Correlation does not prove causation, and extrapolation extends beyond the observed range.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Choose a display that fits the data type. Give both a numerical comparison and what it means for the population; do not infer more precision than the sample supports.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit S1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

S1 · Probability, trees and conditional reasoning

What changes after the first draw?

• A bag has 3 red and 2 blue counters. Taking two without replacement changes the chance of the second colour.
• This lesson studies conditional probability 条件概率: The probability of an event after restricting the sample space to a stated condition.

Choose the mathematical structure

• Multiply along a tree branch and add disjoint branches. With replacement, the composition stays fixed. Conditional probability is P(A given B)=P(A∩B)/P(B), for P(B)>0.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Mutually exclusive means no overlap; independent means that knowing one event does not change the other's probability. Two disjoint events with positive probability are not independent.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• A two-way table makes the restricted denominator visible. Before using a product P(A)P(B), justify independence from the context or the supplied information.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit S1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

S1 · Regression, financial models and residuals

Is the prediction supported by the data?

• A line predicts bus travel time from distance. A small calculation error matters less than using a model outside its evidence.
• This lesson studies residual 残差: The observed value minus the value predicted by a fitted model.

Choose the mathematical structure

• A linear model y=a+bx has intercept a and slope b with contextual units. Inspect residuals and the data range. For repeated percentage change use a geometric model; for a loan distinguish principal, rate, repayment and period.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• A good fit does not prove a causal mechanism. A correlation coefficient measures linear association, not the gradient. Calculator output must be translated into a model, checked and interpreted.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• In IB AI, record the data source, domain, assumptions and calculator method. Compare an alternative model and judge predictions against residuals. A financial answer must state payment timing and whether interest is compounded.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit S1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

S1 · Binomial, normal and Poisson models

What makes a count predictable?

• A quality inspector counts defective items. The number is random, but a model can describe its likely range.
• This lesson studies expected value 期望值: The probability-weighted mean of a random variable.

Choose the mathematical structure

• For a discrete random variable, sum the probabilities to 1 and calculate E(X)=sum xP(X=x). For the normal model use z=(x-μ)/σ and calculate an appropriate area.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Not every count is binomial: changing p or dependence can invalidate it. For a continuous variable, the probability of one exact value is zero. Continuity correction matters when approximating a discrete distribution by a normal one.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• S1 includes discrete random variables and the normal distribution. Named binomial and Poisson models are S2 content.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit S1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

S2 · Binomial, normal and Poisson models

What makes a count predictable?

• A quality inspector counts defective items. The number is random, but a model can describe its likely range.
• This lesson studies expected value 期望值: The probability-weighted mean of a random variable.

Choose the mathematical structure

• A binomial model needs fixed n, independent trials, two outcomes and constant p. E(X)=np and Var(X)=np(1-p). For a normal model use z=(x-μ)/σ. A Poisson model describes counts with a constant rate and appropriate independence assumptions.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Not every count is binomial: changing p or dependence can invalidate it. For a continuous variable, the probability of one exact value is zero. Continuity correction matters when approximating a discrete distribution by a normal one.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Write the event as an inequality before using calculator distribution functions. Distinguish P(X<k), P(X≤k) and a tail complement. State assumptions in context.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit S2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

S2 · Sampling and a large data set

Who is missing from the data?

• A weather database has a missing entry and several stations. Treating each row as identical can distort a comparison.
• This lesson studies sampling frame 抽样框: The list or population definition from which a sample is selected.

Choose the mathematical structure

• Identify the population, sampling unit and frame. Distinguish random, systematic, stratified, quota and opportunity sampling. Missing data is not zero; verify units, dates and variable definitions before comparing samples.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• A large biased sample remains biased. Stratification is not the same as selecting whoever is available from each group. AQA large-data-set familiarity requires the actual supplied data and metadata, not invented weather values.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• For AQA, use the official large data set in a supervised spreadsheet task: identify a variable, justify a comparison, inspect missing values, create a display and explain a limitation. Save the decisions with the analysis.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit S2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

S2 · Hypothesis testing and contextual conclusions

Could chance explain the result?

• A factory claims that only 10% of items are defective. A sample contains more defects, but chance alone may explain some difference.
• This lesson studies significance level 显著性水平: The chosen probability threshold for rejecting a null hypothesis.

Choose the mathematical structure

• State H₀ and H₁ in population parameters before inspecting the outcome. Calculate the appropriate tail probability under H₀. Reject H₀ when the evidence meets the specified significance rule; otherwise say there is insufficient evidence.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Failing to reject H₀ is not proof that H₀ is true. Choose the tail from H₁, not from whichever tail gives a small result. Statistical significance does not measure the practical size of an effect.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Finish with a sentence about the population and the original claim. State the model's assumptions and consider whether the sampling procedure supports them.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit S2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

S2 · Continuous distributions and density functions

Is a height the same as a probability?

• A waiting time can take any value in an interval. Its probability comes from area, not the graph's height at one instant.
• This lesson studies probability density 概率密度: A nonnegative function whose integral over an interval gives its probability.

Choose the mathematical structure

• A density f must be nonnegative and integrate to 1 over its support. The cumulative distribution F(x) is the integral up to x. For a continuous variable P(X=x)=0 and E(X) is the integral of xf(x).
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Density can exceed 1 without being invalid; total area must equal 1. A cumulative distribution cannot decrease. Include the zero-density region outside the support when defining a complete model.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• For a uniform distribution, density is reciprocal interval length. Find medians or percentiles using the cumulative probability, and use a support-aware integral for moments.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit S2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

S3 · Confidence intervals, chi-square and regression

How uncertain is the sample mean?

• Two samples give different means. We need a statement about sampling uncertainty, not a promise that one sample equals the population.
• This lesson studies confidence interval 置信区间: An interval calculated by a procedure designed to cover the population parameter at a stated long-run rate.

Choose the mathematical structure

• For a normal mean with known σ, a 95% interval is sample mean±1.96σ/√n. For chi-square, sum (observed-expected)²/expected with the correct degrees of freedom. Choose a test that fits the data and assumptions.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• A 95% confidence procedure does not give a 95% probability that a fixed population mean moves within this one interval. Expected counts and estimated parameters affect the validity and degrees of freedom of chi-square tests.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Report uncertainty in the context. In regression, inspect residuals, avoid unjustified extrapolation, and distinguish a fitted relationship from a causal explanation. S3 and IB AI HL require different verified scope boundaries.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit S3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

S3 · Sampling and a large data set

Who is missing from the data?

• A weather database has a missing entry and several stations. Treating each row as identical can distort a comparison.
• This lesson studies sampling frame 抽样框: The list or population definition from which a sample is selected.

Choose the mathematical structure

• Identify the population, sampling unit and frame. Distinguish random, systematic, stratified, quota and opportunity sampling. Missing data is not zero; verify units, dates and variable definitions before comparing samples.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• A large biased sample remains biased. Stratification is not the same as selecting whoever is available from each group. AQA large-data-set familiarity requires the actual supplied data and metadata, not invented weather values.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• For AQA, use the official large data set in a supervised spreadsheet task: identify a variable, justify a comparison, inspect missing values, create a display and explain a limitation. Save the decisions with the analysis.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit S3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

S3 · Regression, financial models and residuals

Is the prediction supported by the data?

• A line predicts bus travel time from distance. A small calculation error matters less than using a model outside its evidence.
• This lesson studies residual 残差: The observed value minus the value predicted by a fitted model.

Choose the mathematical structure

• A linear model y=a+bx has intercept a and slope b with contextual units. Inspect residuals and the data range. For repeated percentage change use a geometric model; for a loan distinguish principal, rate, repayment and period.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• A good fit does not prove a causal mechanism. A correlation coefficient measures linear association, not the gradient. Calculator output must be translated into a model, checked and interpreted.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• In IB AI, record the data source, domain, assumptions and calculator method. Compare an alternative model and judge predictions against residuals. A financial answer must state payment timing and whether interest is compounded.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit S3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

D1 · Algorithms, networks and linear programming

Cheapest network or shortest route?

• A school must connect buildings with cable. The shortest route between two buildings and the cheapest whole network are different problems.
• This lesson studies minimum spanning tree 最小生成树: A connected cycle-free network joining every vertex with the smallest possible total edge weight.

Choose the mathematical structure

• Kruskal selects edges in increasing weight while avoiding cycles. Dijkstra updates shortest tentative distances from a start. Linear programming optimizes a linear objective over a feasible region; inspect vertices and integer restrictions when required.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• A spanning tree has no cycles and joins all vertices. The largest single edge is not automatically excluded from every optimal solution. A shortest-path algorithm cannot replace a minimum-spanning-tree algorithm.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• For critical paths, calculate earliest and latest event times and identify zero-float activities. State units and interpret the optimum. The existing archive has no D1 pairs, so these original tasks do not establish a reviewed D1 past-paper bank.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit D1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

D1 · Critical paths and scheduling

Must every activity wait for every other?

• A school event has activities that can run in parallel. Adding every duration overestimates the minimum completion time.
• This lesson studies float · ⁨flotter⁩ 浮动时间: The time an activity can be delayed without delaying completion under the network model.

Choose the mathematical structure

• For an activity network, calculate earliest event times forward and latest event times backward. Total float for activity i to j is L_j-E_i-duration. A critical activity has zero total float; more than one critical path may exist.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• Do not add durations of independent parallel activities. A zero-float activity belongs to a critical path, but there may be several such paths. Resource limits can require a schedule longer than the network's theoretical minimum.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Draw precedence relationships before assigning times. Distinguish activity duration from event time. Explain what a delay does to the completion date, and record any resource assumptions.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit D1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

D1 · Sorting, packing and network algorithms

Is a fast packing method always optimal?

• A packing method quickly fills boxes, but a fast valid arrangement need not use the smallest number of boxes.
• This lesson studies algorithm 算法: A finite set of ordered instructions that solves a defined class of problems.

Choose the mathematical structure

• Trace the named algorithm exactly, including its tie rules. In first-fit packing, place each item in the first available bin that can hold it. First-fit decreasing sorts before applying first-fit. A heuristic may be valid without being optimal.
• State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

• Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Test a tempting shortcut

• An example of success does not prove a heuristic is always optimal. Keep intermediate lists in a sorting trace; do not jump from input to a sorted final list. A shortest-path update must retain predecessor information if a route is requested.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• For Dijkstra, choose the smallest unsettled tentative label and update its neighbours. For route-inspection problems, distinguish a closed route from an open one and identify odd vertices before pairing them.
• A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

• edexcel IAL further mathematics; official unit D1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
• Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.