YMA01: 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.

P1 · Indices, surds and standard form

How small is a microscopic length?

• A microscope records a length of 0.000072 metres. A compact representation must keep its size correct.
• This lesson studies index 指数: The power to which a base is raised.

Choose the mathematical structure

• For the same positive base, multiplication adds indices and division subtracts them. A negative index means reciprocal; a fractional index represents a root. Standard form has 1≤a<10.
• 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

• Index laws do not turn a sum into a single power: 2^3+2^4=24, not 2^7. Do not round a surd when an exact answer is requested.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Check powers of ten against the original quantity. Use surds for exact geometry, and round only the final length when the question asks for a decimal.
• 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 mathematics; official unit P1. 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.

P1 · 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 mathematics; official unit P1. 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.

P1 · Coordinate geometry and tangents

How does a path's slope become an equation?

• A path rises 6 metres over a horizontal distance of 3 metres. Its gradient connects a diagram to an equation.
• This lesson studies gradient 斜率: The change in y divided by the corresponding change in x.

Choose the mathematical structure

• A line through (x₁,y₁) with gradient m has y-y₁=m(x-x₁). Parallel lines have equal gradients. Finite perpendicular gradients multiply to -1. A circle has (x-a)²+(y-b)²=r².
• 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 vertical line has no finite gradient; do not force it into y=mx+c. Read the signs of a circle's centre carefully. The radius to a tangent is perpendicular to the tangent.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Before solving a line-circle intersection, predict whether there are zero, one or two intersections. Substitution produces a quadratic whose discriminant checks the prediction.
• 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 mathematics; official unit P1. 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.

P1 · Right triangles and non-right triangles

Which side does the ladder need?

• A ladder reaches a height of 4 m while its foot is 3 m from a wall. Which sides are known, and which angle do we need?
• This lesson studies hypotenuse 斜边: The side opposite the right angle in a right-angled triangle.

Choose the mathematical structure

• In a right triangle a²+b²=c²; sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse and tanθ=opposite/adjacent. For other triangles, use the sine or cosine rule, or area=ab sin C/2.
• 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

• Label sides relative to the chosen angle. Pythagoras needs a right angle. A calculator angle mode error can produce a plausible but wrong result. Keep unrounded values for later steps.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Use a plan or elevation for a three-dimensional problem before applying a triangle rule. Explain why the chosen triangle contains the required length or angle.
• 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 mathematics; official unit P1. 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.

P1 · Radians, identities and trigonometric equations

How far does the rim travel?

• A rotating wheel sweeps an arc. Radians let its angle connect directly to the distance travelled along the rim.
• This lesson studies radian 弧度: The angle subtended by an arc whose length equals the radius.

Choose the mathematical structure

• For θ in radians, arc length s=rθ and sector area A=r²θ/2. Use sin²θ+cos²θ=1 and angle identities to simplify or solve. List every solution in the stated interval.
• 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 calculator's inverse sine gives one principal value. It does not give every solution. Dividing by sin x can discard solutions where sin x=0; check that case first.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Choose an identity that reduces the equation to one trigonometric function. Keep exact familiar angles where possible, and check each solution against the original equation.
• 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 mathematics; official unit P1. 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.

P1 · Derivatives and stationary points

What is the slope at one point?

• A curved road has different slopes at different positions. An average gradient cannot describe every point.
• This lesson studies derivative 导数: The instantaneous rate of change, also the gradient of a tangent.

Choose the mathematical structure

• For y=ax^n, dy/dx=anx^(n-1). A stationary point satisfies dy/dx=0. Check the sign change of the derivative, or the second derivative when it is nonzero, to classify it.
• 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 gives a gradient, not the ordinate. Evaluate y and dy/dx separately at the supplied x-coordinate; the normal gradient is the negative reciprocal of a nonzero tangent gradient.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• P1 covers polynomial derivatives, gradients, tangents and normals. Optimization and stationary-point classification are taught in P2; they are not introduced here.
• 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 mathematics; official unit P1. 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.

P1 · Integrals, area and accumulation

How much change has accumulated?

• A velocity graph tells us motion at an instant. How can we recover the displacement accumulated over time?
• This lesson studies antiderivative 原函数: A function whose derivative equals the given integrand.

Choose the mathematical structure

• Integrate polynomial powers by increasing the index by one and dividing by the new index. Include an arbitrary constant and use a supplied point to determine it.
• 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 indefinite integral needs a constant. Integrating each term changes its power; copying the derivative rule gives the wrong result.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• P1 uses indefinite polynomial integration and a point to determine the constant. Definite integrals, areas and the trapezium rule are P2 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 mathematics; official unit P1. 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.

P2 · 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 mathematics; official unit P2. 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.

P2 · 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 mathematics; official unit P2. 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.

P2 · Coordinate geometry and tangents

How does a path's slope become an equation?

• A path rises 6 metres over a horizontal distance of 3 metres. Its gradient connects a diagram to an equation.
• This lesson studies gradient 斜率: The change in y divided by the corresponding change in x.

Choose the mathematical structure

• A line through (x₁,y₁) with gradient m has y-y₁=m(x-x₁). Parallel lines have equal gradients. Finite perpendicular gradients multiply to -1. A circle has (x-a)²+(y-b)²=r².
• 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 vertical line has no finite gradient; do not force it into y=mx+c. Read the signs of a circle's centre carefully. The radius to a tangent is perpendicular to the tangent.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Before solving a line-circle intersection, predict whether there are zero, one or two intersections. Substitution produces a quadratic whose discriminant checks the prediction.
• 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 mathematics; official unit P2. 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.

P2 · Sequences, series and recurrence

Does the change add or multiply?

• A saving plan adds ¥30 more each week; a population model grows by 5% each year. Equal differences and equal ratios need different models.
• This lesson studies common ratio 公比: The constant multiplier between consecutive terms of a geometric sequence.

Choose the mathematical structure

• For an arithmetic progression, u_n=a+(n-1)d and S_n=n(2a+(n-1)d)/2. For a geometric progression, u_n=ar^(n-1) and S_n=a(1-r^n)/(1-r). An infinite geometric sum exists only if |r|<1.
• 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 first term has index 1, so the exponent is n-1. A sequence is a list; a series is a sum. A geometric sequence can alternate in sign and still converge.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Explain whether the context justifies additive or multiplicative change. In finance, distinguish a single deposit from a stream of deposits before choosing a sum formula.
• 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 mathematics; official unit P2. 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.

P2 · Binomial expansion and valid approximations

How does a small change affect a power?

• A small measurement change affects a power of a quantity. An expansion can show the size of first and second effects.
• This lesson studies binomial coefficient 二项式系数: The number of ways to choose a specified number of objects from a set.

Choose the mathematical structure

• For positive integer n, (a+b)^n is a finite binomial expansion. For noninteger n, expand (1+x)^n as 1+nx+n(n-1)x²/2+... with |x|<1. Factor out constants before using this form.
• 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 expansion in 2x requires |2x|<1 for the infinite series, not merely |x|<1. Positive-integer expansions are finite and do not have that convergence restriction.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• State the range of validity alongside an approximation. Retain enough terms to support the requested accuracy, and distinguish a coefficient from the whole term containing x.
• 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 mathematics; official unit P2. 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.

P2 · Exponentials, logarithms and modelling

Why does a decay model stay positive?

• A medicine concentration falls by the same percentage each hour. A constant subtraction would eventually predict a negative amount.
• This lesson studies half-life 半衰期: The time for a decaying quantity to fall to half its initial value.

Choose the mathematical structure

• For y=Ae^(kt), k is a proportional rate. Taking logs gives ln y=ln A+kt. Logarithms require positive arguments, and log(x+y) is not log x+log y.
• 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 fitted exponential is a model, not a guarantee. Specify the time units and range of use. A negative k describes decay; a negative starting amount usually contradicts the context.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Compare actual observations with the model. Systematic departures may indicate changing conditions. In a report, explain what the rate and initial value mean, rather than giving a bare equation.
• 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 mathematics; official unit P2. 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.

P2 · Radians, identities and trigonometric equations

How far does the rim travel?

• A rotating wheel sweeps an arc. Radians let its angle connect directly to the distance travelled along the rim.
• This lesson studies radian 弧度: The angle subtended by an arc whose length equals the radius.

Choose the mathematical structure

• For θ in radians, arc length s=rθ and sector area A=r²θ/2. Use sin²θ+cos²θ=1 and angle identities to simplify or solve. List every solution in the stated interval.
• 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 calculator's inverse sine gives one principal value. It does not give every solution. Dividing by sin x can discard solutions where sin x=0; check that case first.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Choose an identity that reduces the equation to one trigonometric function. Keep exact familiar angles where possible, and check each solution against the original equation.
• 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 mathematics; official unit P2. 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.

P2 · Derivatives and stationary points

What is the slope at one point?

• A curved road has different slopes at different positions. An average gradient cannot describe every point.
• This lesson studies derivative 导数: The instantaneous rate of change, also the gradient of a tangent.

Choose the mathematical structure

• For y=ax^n, dy/dx=anx^(n-1). A stationary point satisfies dy/dx=0. Check the sign change of the derivative, or the second derivative when it is nonzero, to classify it.
• 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 zero derivative does not always mean a maximum or minimum: y=x³ is stationary at 0 but continues increasing. An endpoint can also produce an extreme value on a restricted domain.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• At GCSE/IGCSE use only the polynomial scope allowed by the tier; do not add chain, product or quotient rules there. At advanced level, connect the derivative to rates and optimization with a valid domain.
• 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 mathematics; official unit P2. 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.

P2 · Integrals, area and accumulation

How much change has accumulated?

• A velocity graph tells us motion at an instant. How can we recover the displacement accumulated over time?
• This lesson studies antiderivative 原函数: A function whose derivative equals the given integrand.

Choose the mathematical structure

• For n≠-1, the integral of ax^n is ax^(n+1)/(n+1)+C. A definite integral is signed accumulation. Split at sign changes when total area or distance is 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

• The integral of 1/x is ln|x|+C, not the power rule with n=-1. Signed area can be zero even when the total enclosed area is positive.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Identify what the integral means and include the correct units. A rate measured per second integrates to the underlying quantity, not to another rate. Differentiate an antiderivative to check it.
• 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 mathematics; official unit P2. 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.

P2 · Root finding and numerical integration

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 equal strip width h, the trapezium estimate is h/2 times the sum of the two endpoint heights plus twice the internal heights.
• 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 equal strip widths and include internal heights twice. Curvature decides whether the estimate is above or below the exact area; being increasing alone does not.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• P2 introduces the trapezium rule. Newton–Raphson and other root-finding methods belong to other named units; they are not part of this P2 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 mathematics; official unit P2. 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.

P3 · Domains, inverses and composition

Which inputs are allowed?

• A square-root model returns a real output only for some inputs. Its formula alone does not specify a complete function.
• This lesson studies domain 定义域: The set of allowed inputs to a function.

Choose the mathematical structure

• State the domain and range. For an inverse, first ensure the function is one-to-one on its domain. Composition fg means apply g first, then f; the intermediate output must be an allowed input to f.
• 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

• Squaring can introduce extraneous solutions. Restricting a parabola's domain is essential before claiming an inverse. A horizontal translation inside f has the opposite sign to the graph's movement.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Check f(f inverse(x))=x on the inverse domain. Use a sketch to test whether a horizontal line meets the original graph more than once.
• 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 mathematics; official unit P3. 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.

P3 · Radians, identities and trigonometric equations

How far does the rim travel?

• A rotating wheel sweeps an arc. Radians let its angle connect directly to the distance travelled along the rim.
• This lesson studies radian 弧度: The angle subtended by an arc whose length equals the radius.

Choose the mathematical structure

• For θ in radians, arc length s=rθ and sector area A=r²θ/2. Use sin²θ+cos²θ=1 and angle identities to simplify or solve. List every solution in the stated interval.
• 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 calculator's inverse sine gives one principal value. It does not give every solution. Dividing by sin x can discard solutions where sin x=0; check that case first.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Choose an identity that reduces the equation to one trigonometric function. Keep exact familiar angles where possible, and check each solution against the original equation.
• 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 mathematics; official unit P3. 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.

P3 · Exponentials, logarithms and modelling

Why does a decay model stay positive?

• A medicine concentration falls by the same percentage each hour. A constant subtraction would eventually predict a negative amount.
• This lesson studies half-life 半衰期: The time for a decaying quantity to fall to half its initial value.

Choose the mathematical structure

• For y=Ae^(kt), k is a proportional rate. Taking logs gives ln y=ln A+kt. Logarithms require positive arguments, and log(x+y) is not log x+log y.
• 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 fitted exponential is a model, not a guarantee. Specify the time units and range of use. A negative k describes decay; a negative starting amount usually contradicts the context.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Compare actual observations with the model. Systematic departures may indicate changing conditions. In a report, explain what the rate and initial value mean, rather than giving a bare equation.
• 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 mathematics; official unit P3. 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.

P3 · 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 mathematics; official unit P3. 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.

P3 · Integrals, area and accumulation

How much change has accumulated?

• A velocity graph tells us motion at an instant. How can we recover the displacement accumulated over time?
• This lesson studies antiderivative 原函数: A function whose derivative equals the given integrand.

Choose the mathematical structure

• For n≠-1, the integral of ax^n is ax^(n+1)/(n+1)+C. A definite integral is signed accumulation. Split at sign changes when total area or distance is 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

• The integral of 1/x is ln|x|+C, not the power rule with n=-1. Signed area can be zero even when the total enclosed area is positive.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Identify what the integral means and include the correct units. A rate measured per second integrates to the underlying quantity, not to another rate. Differentiate an antiderivative to check it.
• 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 mathematics; official unit P3. 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.

P3 · Fixed-point iteration

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

• Rearrange the equation as x=g(x), choose a starting value and iterate. A limit L must satisfy L=g(L). Near a fixed point, |g prime(L)|<1 provides a local convergence check; a sign bracket can validate the reported rounded root.
• 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

• Different rearrangements can have different convergence behaviour. Check the iterates, domain and original equation; a square-root update cannot reach a negative fixed point.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• P3 uses equation rearrangement and numerical iteration. This lesson excludes Newton and numerical integration, which belong to other units.
• 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 mathematics; official unit P3. 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.

P4 · 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 mathematics; official unit P4. 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.

P4 · Equations, identities and rearrangement

When do two plans cost the same?

• Two mobile plans cost 20+3x and 44+x yuan for x GB. When do they cost the same?
• This lesson studies identity 恒等式: An equality that holds for every allowed value of its variable.

Choose the mathematical structure

• An equation asks which inputs satisfy an equality; an identity holds for all allowed inputs. Preserve equality by applying the same operation to both sides. State restrictions before dividing by a variable.
• 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

• Cancelling a term is not the same as cancelling a factor. In (x²+2x)/x, factor the numerator and retain x≠0. Check a rearrangement by substitution.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• Set up the equation from units and the meaning of the unknown. A negative or fractional solution may be algebraically correct but impossible for a count.
• 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 mathematics; official unit P4. 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.

P4 · Binomial expansion and valid approximations

How does a small change affect a power?

• A small measurement change affects a power of a quantity. An expansion can show the size of first and second effects.
• This lesson studies binomial coefficient 二项式系数: The number of ways to choose a specified number of objects from a set.

Choose the mathematical structure

• For positive integer n, (a+b)^n is a finite binomial expansion. For noninteger n, expand (1+x)^n as 1+nx+n(n-1)x²/2+... with |x|<1. Factor out constants before using this form.
• 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 expansion in 2x requires |2x|<1 for the infinite series, not merely |x|<1. Positive-integer expansions are finite and do not have that convergence restriction.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Interpret a new situation

• State the range of validity alongside an approximation. Retain enough terms to support the requested accuracy, and distinguish a coefficient from the whole term containing x.
• 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 mathematics; official unit P4. 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.

P4 · 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 mathematics; official unit P4. 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.

P4 · 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 mathematics; official unit P4. 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.

P4 · 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 mathematics; official unit P4. 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.

P4 · 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

• Separate the variables in dy/dx=ky, integrate both sides and apply an initial condition. Include any equilibrium solution excluded when dividing by y.
• 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

• This unit uses first-order separation and initial conditions. Euler numerical integration and second-order complementary functions are excluded from 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 mathematics; official unit P4. 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.

P4 · 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 mathematics; official unit P4. 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 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 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 · ⁨momento⁩ 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 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 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 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 · ⁨momento⁩ 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 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 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 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.

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 · ⁨densidade de frequência⁩ 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 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 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 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 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 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 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 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 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.

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 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 · ⁨flutuar⁩ 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 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 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.