AA: course teaching notes
Version: First assessment 2021; current through 2028. First-assessment-2029 course is separate.
These are original course-owned teaching notes. Objective-level exceptions are listed in the review, and lessons are tier labelled.
1 · Exact arithmetic and estimation
Can we pack without leftovers?
- A supplier packs 72 pencils and 90 pens into identical gift bags. How can we avoid leftovers?
- This lesson studies prime factor 质因数: A prime number that divides the integer exactly.
Choose the mathematical structure
- Prime factors reveal shared structure. Use the smallest common prime powers for the HCF and the largest for the LCM. Estimate before calculating; use brackets to preserve the order of operations.
- 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
$$a=\prod p_i^{\alpha_i},\quad b=\prod p_i^{\beta_i},\quad \operatorname{HCF}(a,b)=\prod p_i^{\min(\alpha_i,\beta_i)}$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
72=2^3×3^2 and 90=2×3^2×5. Their HCF is 2×9=18. Make 18 bags with 4 pencils and 5 pens each. Their LCM is 2^3×3^2×5=360.
Test a tempting shortcut
- The HCF divides both numbers; the LCM is a multiple of both. They answer different questions. A decimal estimate is not an exact fraction.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Watch out · Precaución
The HCF of two positive integers is always larger than either integer. This claim is false. Explain which definition or assumption it violates.
Interpret a new situation
- For a non-calculator paper, keep fractions exact and show cancellation. For a calculator paper, enter the full expression and compare with your estimate.
- 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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
A prime number that divides the integer exactly. Choose the relationship, show the method, check its assumptions and interpret the result.
1 · 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
$$a^m a^n=a^{m+n},\quad a^{-n}=\frac1{a^n},\quad a^{m/n}=\left(\sqrt[n]{a}\right)^m$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
0.000072=7.2×10^(-5). Also 16^(3/4)=(16^(1/4))^3=2^3=8. Simplify √72=6√2, then rationalise 1/√2=√2/2.
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.
Watch out · Precaución
For every positive a, a^2+a^3 equals a^5. This claim is false. Explain which definition or assumption it violates.
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
The power to which a base is raised. Choose the relationship, show the method, check its assumptions and interpret the result.
1 · 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
$$u_n=a+(n-1)d,\qquad S_n=\frac{n}{2}\left[2a+(n-1)d\right]$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
For a=5,d=3,n=8, u_8=5+7×3=26 and S_8=8(10+21)/2=124. For a=12,r=1/2, S infinity=12/(1-1/2)=24. For u_(n+1)=2u_n+1 with u_1=1, the next terms are 3,7,15.
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.
Watch out · Precaución
Every geometric series has a finite sum to infinity. This claim is false. Explain which definition or assumption it violates.
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
The constant multiplier between consecutive terms of a geometric sequence. Choose the relationship, show the method, check its assumptions and interpret the result.
1 · Percentages, ratio and proportional reasoning
Can we recover the original price?
- A coat is reduced by 20% to ¥240. The discount applies to the original price, not to the sale price.
- This lesson studies multiplier 乘数: A factor that performs a percentage change in one multiplication.
Choose the mathematical structure
- A p% increase has multiplier 1+p/100; a decrease has multiplier 1-p/100. Reverse a percentage by dividing by the multiplier. In a ratio, first find the total number of parts.
- 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
$$P_{\mathrm{new}}=P_{\mathrm{old}}\left(1+\frac{r}{100}\right)$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
Let the original price be P. The model is sale price=0.8P. Hence P=240/0.8=300. A later 20% increase gives 240×1.2=288, so the two changes do not cancel.
Test a tempting shortcut
- A percentage uses a stated base. Subtracting the percentages loses that base. For compound change, multiply the multipliers; do not add the percentages.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Watch out · Precaución
A 20% decrease followed by a 20% increase restores the starting price. This claim is false. Explain which definition or assumption it violates.
Interpret a new situation
- For direct proportion use y=kx; for inverse proportion use y=k/x. Calculate k from a known pair before using a new value. State what you held constant.
- 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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
A factor that performs a percentage change in one multiplication. Choose the relationship, show the method, check its assumptions and interpret the result.
1 · Complex numbers and roots
What if the root is not real?
- The equation x²+1=0 has no real roots. Extending the number system lets us represent and calculate with both roots.
- This lesson studies complex conjugate 共轭复数: The number obtained by reversing the sign of the imaginary part.
Choose the mathematical structure
- Write z=a+bi with i²=-1. The conjugate is a-bi and z times its conjugate=a²+b². The modulus is √(a²+b²); the argument needs the correct quadrant. Conjugate roots occur for polynomials with real coefficients.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Work through a checked case
$$z=a+bi,\qquad |z|=\sqrt{a^2+b^2},\qquad z\overline z=a^2+b^2$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
For z=3+4i, |z|=5 and z conjugate=3-4i. The reciprocal is (3-4i)/25. For x²-6x+13=0, the roots are 3±2i. Multiplying by the conjugate makes a complex denominator real.
Test a tempting shortcut
- The square root of a sum is not the sum of the square roots. An inverse tangent alone may return the wrong quadrant. Do not confuse modulus with squared modulus.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Watch out · Precaución
For a+bi, its modulus is always a+b. This claim is false. Explain which definition or assumption it violates.
Interpret a new situation
- Draw the real axis horizontally and imaginary axis vertically. For products, moduli multiply and arguments add; powers extend this geometric pattern through de Moivre's theorem.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Use this in your course
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
The number obtained by reversing the sign of the imaginary part. Choose the relationship, show the method, check its assumptions and interpret the result.
1 · 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
$$(1+x)^n=1+nx+\frac{n(n-1)}{2}x^2+\cdots$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
(1+2x)^5=1+10x+40x²+80x³+80x⁴+32x⁵. For (1+x)^(-1), the first three terms are 1-x+x²; at x=0.1 this gives 0.91 versus the exact 1/1.1≈0.909091.
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.
Watch out · Precaución
Every binomial expansion is a finite polynomial, including powers that are not positive integers. This claim is false. Explain which definition or assumption it violates.
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
The number of ways to choose a specified number of objects from a set. Choose the relationship, show the method, check its assumptions and interpret the result.
1 · Series, finite differences and Taylor expansions
Can a polynomial replace an exponential nearby?
- Near a known point, a complicated function can be replaced by a polynomial approximation. The approximation needs a centre and a valid scale.
- This lesson studies Taylor polynomial 泰勒多项式: A polynomial formed from a function's derivatives at a chosen centre.
Choose the mathematical structure
- About x=a, f(x)=f(a)+f prime(a)(x-a)+f double prime(a)(x-a)²/2!+... when the expansion is valid. At a=0 it is a Maclaurin expansion. Finite sums may also simplify by cancellation or standard sum formulae.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Work through a checked case
$$f(x)=f(0)+f^{\prime}(0)x+\frac{f^{\prime\prime}(0)}{2!}x^2+\cdots$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
The first three terms of e^x are 1+x+x²/2. At x=0.1 this gives 1.105, close to e^0.1≈1.105170. For the sum of k² from k=1 to 5, n(n+1)(2n+1)/6 gives 5×6×11/6=55.
Test a tempting shortcut
- The factorial belongs in the denominator of every Taylor coefficient. The expansion centre is not always zero. An approximation close to its centre can be poor far away.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Watch out · Precaución
A Taylor polynomial is always exact for the original function at every input. This claim is false. Explain which definition or assumption it violates.
Interpret a new situation
- For a telescoping series, write several terms and identify both surviving ends. For a Taylor approximation, state its order and compare with a known value or a remainder estimate where required.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Use this in your course
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
A polynomial formed from a function's derivatives at a chosen centre. Choose the relationship, show the method, check its assumptions and interpret the result.
2 · 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
$$f(g(x))=(f\circ g)(x),\qquad f^{-1}(f(x))=x$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
For f(x)=√(x-2), x≥2 and the range is y≥0. From y=√(x-2), x=y²+2. Thus f inverse(x)=x²+2 with x≥0. For g(x)=x+3, fg(1)=f(4)=√2.
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.
Watch out · Precaución
Every quadratic function on all real numbers has an inverse function. This claim is false. Explain which definition or assumption it violates.
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
The set of allowed inputs to a function. Choose the relationship, show the method, check its assumptions and interpret the result.
2 · 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
$$y=Ae^{kt},\qquad \ln y=\ln A+kt$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
If y=80e^(-0.2t), y=40 gives e^(-0.2t)=1/2. Hence t=ln2/0.2≈3.466. For 3^x=20, x=ln20/ln3. Plotting ln y against t linearises this exponential model.
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.
Watch out · Precaución
For positive x and y, ln(x+y)=ln x+ln y. This claim is false. Explain which definition or assumption it violates.
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
The time for a decaying quantity to fall to half its initial value. Choose the relationship, show the method, check its assumptions and interpret the result.
3 · Angle reasoning, similarity and mensuration
Does volume scale like length?
- A model has lengths one third of the real object. How much smaller are its area and volume?
- This lesson studies scale factor 相似比: The multiplier that relates corresponding lengths in similar shapes.
Choose the mathematical structure
- For similar shapes with length scale factor k, areas scale by k² and volumes by k³. State angle reasons explicitly. A circle's tangent is perpendicular to the radius at the contact point.
- 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
$$\frac{A_2}{A_1}=k^2,\qquad \frac{V_2}{V_1}=k^3$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
If model-to-real length factor is 3, a model area of 12 cm² gives 12×3²=108 cm² and a model volume of 8 cm³ gives 8×3³=216 cm³. A cylinder with r=3,h=5 has volume πr²h=45π.
Test a tempting shortcut
- Equal angles alone establish similarity, not equal size. Use corresponding lengths in the same order. Convert linear units before calculating area or volume, or square/cube the conversion factor correctly.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Watch out · Precaución
Doubling every length of a solid doubles its volume. This claim is false. Explain which definition or assumption it violates.
Interpret a new situation
- A geometric proof should name the relevant theorem, identify the equal angle or ratio, and draw the conclusion. A scale drawing is evidence only when the task permits measurement.
- 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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
The multiplier that relates corresponding lengths in similar shapes. Choose the relationship, show the method, check its assumptions and interpret the result.
3 · 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
$$a^2+b^2=c^2,\qquad \tan\theta=\frac{\mathrm{opposite}}{\mathrm{adjacent}}$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
The ladder length is c=√(3²+4²)=5 m. Its angle to the ground satisfies tanθ=4/3, so θ≈53.1°. With two sides 6 and 8 enclosing 60°, c²=6²+8²-2×6×8 cos60°=52.
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.
Watch out · Precaución
Pythagoras applies to every triangle, including triangles without a right angle. This claim is false. Explain which definition or assumption it violates.
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
The side opposite the right angle in a right-angled triangle. Choose the relationship, show the method, check its assumptions and interpret the result.
3 · 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
$$s=r\theta,\qquad A=\frac12 r^2\theta,\qquad \sin^2\theta+\cos^2\theta=1$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
For r=4 and θ=π/3, s=4π/3 and A=8π/3. Solving sin x=1/2 on 0≤x<2π gives x=π/6 and 5π/6. For sin2x=1/2, solve first on 0≤2x<4π before dividing by 2.
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.
Watch out · Precaución
The principal inverse sine value is always the only solution in a full turn. This claim is false. Explain which definition or assumption it violates.
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
The angle subtended by an arc whose length equals the radius. Choose the relationship, show the method, check its assumptions and interpret the result.
3 · 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. A displacement from A to B is OB-OA. Parallel vectors are scalar multiples. A vector has magnitude and direction; a distance is a scalar.
- 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
$$\overrightarrow{AB}=\mathbf b-\mathbf a,\qquad |\mathbf v|=\sqrt{v_x^2+v_y^2}$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
With a=(4,1) and b=(1,3), a+b=(5,4). Their scalar product is a·b=4×1+1×3=7. The magnitude of a is √17. For A=(1,2), B=(5,5), AB=(4,3) and |AB|=5.
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.
Watch out · Precaución
AB and BA always have the same components. This claim is false. Explain which definition or assumption it violates.
Interpret a new situation
- For a translation, every point uses the same displacement. For advanced line problems, distinguish the position vector from a direction vector and solve parameter equations consistently.
- 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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
The vector sum representing the combined displacement or force. Choose the relationship, show the method, check its assumptions and interpret the result.
4 · 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 · densidad de frecuencia 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
$$\overline x=\frac{\sum x_i}{n},\qquad \mathrm{density}=\frac{\mathrm{frequency}}{\mathrm{class\ width}}$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
A class from 10 to 20 with frequency 30 has density 30/10=3. A class from 20 to 40 with frequency 20 has density 20/20=1. Its wider bar must not be mistaken for a larger density. For values 2,4,4,6,9, the median is 4 and mean is 5.
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.
Watch out · Precaución
A histogram bar's height always equals its frequency, even with unequal class widths. This claim is false. Explain which definition or assumption it violates.
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
Frequency divided by class width, used as histogram height. Choose the relationship, show the method, check its assumptions and interpret the result.
4 · 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
$$P(A\mid B)=\frac{P(A\cap B)}{P(B)},\qquad P(B)>0$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
Without replacement, P(two red)=3/5×2/4=3/10. P(one of each)=3/5×2/4+2/5×3/4=3/5. If P(A∩B)=0.12 and P(B)=0.3, P(A given B)=0.4.
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.
Watch out · Precaución
Mutually exclusive events with positive probabilities must be independent. This claim is false. Explain which definition or assumption it violates.
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
The probability of an event after restricting the sample space to a stated condition. Choose the relationship, show the method, check its assumptions and interpret the result.
4 · 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
$$P(X=k)=\binom nk p^k(1-p)^{n-k},\qquad X\sim B(n,p)$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
For X binomial(5,0.2), P(X=0)=0.8^5=0.32768, E(X)=1 and Var(X)=0.8. For a normal quantity with μ=100,σ=15, the value 130 has z=2. A Poisson mean of 3 per hour gives mean 6 over two hours.
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.
Watch out · Precaución
Every count of successes has a binomial distribution regardless of dependence. This claim is false. Explain which definition or assumption it violates.
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
The probability-weighted mean of a random variable. Choose the relationship, show the method, check its assumptions and interpret the result.
5 · 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
$$\frac{d}{dx}(ax^n)=anx^{n-1},\qquad f^{\prime}(x)=0$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
For y=x³-3x, dy/dx=3x²-3. At x=1, the gradient is 0 and y=-2. The second derivative is 6x, positive at x=1, so this is a local minimum. At x=-1, y=2 and the second derivative is negative, giving a local maximum.
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.
Watch out · Precaución
Every point with zero derivative is a local maximum or minimum. This claim is false. Explain which definition or assumption it violates.
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
The instantaneous rate of change, also the gradient of a tangent. Choose the relationship, show the method, check its assumptions and interpret the result.
5 · 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
$$\int ax^n\,dx=\frac{ax^{n+1}}{n+1}+C\quad(n\ne-1)$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
For v(t)=3t²-3 over 0≤t≤2, displacement=[t³-3t]_0^2=2. Since v changes sign at t=1, distance=-[t³-3t]_0^1+[t³-3t]_1^2=2+4=6. The constants cancel only for a definite integral.
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.
Watch out · Precaución
A definite integral always equals the total positive area, even when the graph crosses the axis. This claim is false. Explain which definition or assumption it violates.
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
A function whose derivative equals the given integrand. Choose the relationship, show the method, check its assumptions and interpret the result.
5 · 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
$$\frac{d}{dx}f(g(x))=f^{\prime}(g(x))g^{\prime}(x)$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
For y=(3x+1)^4, y prime=4(3x+1)^3×3=12(3x+1)^3. At x=0, the gradient is 12. For x²+y²=25, differentiate implicitly: 2x+2y y prime=0, so y prime=-x/y when y≠0.
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.
Watch out · Precaución
The derivative of u(x)v(x) is always u prime times v prime. This claim is false. Explain which definition or assumption it violates.
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
The rule that multiplies the outer derivative by the inner derivative for a composite function. Choose the relationship, show the method, check its assumptions and interpret the result.
5 · 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
$$\int u v^{\prime}\,dx=uv-\int u^{\prime}v\,dx$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
For integral xe^x dx, take u=x and v prime=e^x. Then the integral is xe^x-e^x+C. For integral 2x/(x²+1) dx, substitute w=x²+1, dw=2x dx; the answer is ln(x²+1)+C.
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.
Watch out · Precaución
The integral of a product equals the product of its separate integrals. This claim is false. Explain which definition or assumption it violates.
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
An integration method based on the derivative of a product. Choose the relationship, show the method, check its assumptions and interpret the result.
5 · Differential equations and numerical solutions
What does the starting value decide?
- A growing population's rate is proportional to its current size. The rate equation describes a whole family until an initial population is supplied.
- This lesson studies initial condition 初始条件: A specified value that selects a particular solution from a family.
Choose the mathematical structure
- For dy/dx=ky, separate variables: dy/y=k dx, giving y=Ae^(kx). Use an initial condition to find A. Euler's method takes y next=y+h f(x,y), with a chosen step h.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Work through a checked case
$$\frac{dy}{dx}=ky,\qquad y=Ae^{kx}$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
With dy/dx=0.5y and y(0)=2, y=2e^(0.5x). Euler with h=0.2 gives y(0.2)≈2+0.2×1=2.2 and y(0.4)≈2.2+0.2×1.1=2.42. The exact second value is about 2.4428.
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.
Watch out · Precaución
Every step size gives exactly the same Euler approximation. This claim is false. Explain which definition or assumption it violates.
Interpret a new situation
- For second-order linear equations, combine the complementary function with an appropriate particular integral, then apply the required initial conditions. This is Further Pure scope; check the named unit.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Use this in your course
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
A specified value that selects a particular solution from a family. Choose the relationship, show the method, check its assumptions and interpret the result.
IA · Planning a mathematical exploration
What makes an investigation mathematical?
- A student wants to predict queue waiting time at the school canteen. A useful exploration needs a question, data and mathematics the student can explain.
- This lesson studies assumption 假设: A condition adopted to make a mathematical model possible and whose effect should be evaluated.
Choose the mathematical structure
- Frame a focused question with measurable variables and a feasible method. Keep a dated record of decisions, sources and mathematical reasoning. Use technology to support analysis, and explain every important output in your own words.
- 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
$$\widehat t=1+0.2n,\qquad e=t-\widehat t$$
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Example · Ejemplo
A pilot records mean waiting times of 3,5,7 minutes when 10,20,30 people are ahead. A first model is t=1+0.2n. At n=25 it predicts 6 minutes. An actual wait of 8 gives residual 2; investigate varying service rates rather than hiding the mismatch.
Test a tempting shortcut
- Do not copy a sample exploration, invent data or manufacture personal engagement. A prediction is not evidence until it is compared with observations. Consent and privacy matter when collecting school data.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Watch out · Precaución
The school-supervised mathematical exploration can be replaced by a timed written mock exam. This claim is false. Explain which definition or assumption it violates.
Interpret a new situation
- The school supervises topic approval, checkpoints, authenticity and assessment. For the current 2021 course use the current school's guide and criteria; the 2029 inquiry criteria belong to a different assessment version.
- 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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
Key idea · Idea clave
A condition adopted to make a mathematical model possible and whose effect should be evaluated. Choose the relationship, show the method, check its assumptions and interpret the result.