국제바칼로레이트 · IB Diploma · Mathematics: Applications and Interpretation · HL

Supported HL focus. First assessment 2021; current through 2028. First-assessment-2029 course is separate.. Remaining guide, assessment and practical requirements retain their recorded holds.

Prerequisites: read the stated quantities and units, use arithmetic and the model conditions below. Each lesson develops its own method before independent transfer.

These are original or explicitly fictional teaching examples, not actual measurements or completed assessed learner investigations.

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

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

Example:

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.

Warn:

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 Applications and Interpretation 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:

A prime number that divides the integer exactly. Choose the relationship, show the method, check its assumptions and interpret the result.

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

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

Example:

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.

Warn:

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 Applications and Interpretation 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:

A factor that performs a percentage change in one multiplication. Choose the relationship, show the method, check its assumptions and interpret the result.

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.

Example:

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.

Warn:

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 Applications and Interpretation 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:

The constant multiplier between consecutive terms of a geometric sequence. Choose the relationship, show the method, check its assumptions and interpret the result.

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.

Example:

The fitted model t=8+2d predicts 18 minutes at d=5 km. If the observed time is 21, the residual is 3 minutes. A ¥1000 deposit at 5% compound annual interest becomes 1000×1.05³=1157.625 after 3 years.

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.

Warn:

A high correlation permits reliable extrapolation to any distance. This claim is false. Explain which definition or assumption it violates.

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

• Current first-assessment-2021 Applications and Interpretation 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:

The observed value minus the value predicted by a fitted model. Choose the relationship, show the method, check its assumptions and interpret the result.

Where do the basis vectors go?

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

Choose the mathematical structure

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

Work through a checked case

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

Example:

For A=[[2,1],[0,3]], det A=6. A(1,2)=(4,6). Its inverse is [[1/2,-1/6],[0,1/3]]. The characteristic equation is (2-λ)(3-λ)=0, so the eigenvalues are 2 and 3.

Test a tempting shortcut

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

Warn:

Every square matrix has an inverse. This claim is false. Explain which definition or assumption it violates.

Interpret a new situation

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

Use this in your course

• Current first-assessment-2021 Applications and Interpretation 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:

A scalar that determines whether a square matrix is invertible and, in two dimensions, its signed area factor. Choose the relationship, show the method, check its assumptions and interpret the result.

What if the root is not real?

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

Choose the mathematical structure

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

Work through a checked case

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

Example:

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.

Warn:

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 Applications and Interpretation 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:

The number obtained by reversing the sign of the imaginary part. Choose the relationship, show the method, check its assumptions and interpret the result.

지원 HL focus. 첫 평가 2021; 현재는 2028까지. 첫 평가-2029 과정은 별도임. Remaining guide, assessment 및 practical requirements는 기록된 holds를 유지함.

사전 요구사항: 명시된 양과 단위를 읽고, 산술 및 아래 모델 조건을 사용하십시오. 각 수업은 독립적 이전 전에 자체 방법을 개발합니다.

이는 실제 측정값이나 완성된 평가된 학습자 조사가 아닌, 원작 또는 명시적으로 허구적인 교육 예시입니다.

허용되는 입력은 무엇인가?

• 제곱근 모델은 특정 입력값에서만 실수 출력을 산출한다. 공식 자체만으로는 완전한 함수를 정의하지 못한다.
• 이 수업에서는 정의역(domain) 을 다룬다: 함수에 허용되는 입력값들의 집합이다.

수학 구조 선택하기

• 정의역과 치역을 구하시오. 역함수를 구하기 전, 먼저 함수가 정의역에서 일대일 correspondance를 만족하는지 확인하십시오. 합성 fg는 g를 먼저 적용한 후 f를 적용하는 것을 의미하며, 중간 출력값은 f에 대한 유효한 입력값이어야 합니다.
• 계산 전에 허용된 입력값과 단위를 명시하라. 방정식은 단순히 계산기 입력 내용을 기록하는 것보다 관계성을 표현해야 한다.

검증된 사례를 풀이하기

• 시작값과 결과를 대조하십시오. 원본 관계에 대입하거나, 적절한 경우 그래프와 수치적 답을 비교하십시오.

예시:

f(x)=√(x-2)인 경우 x≥2이며 치역은 y≥0입니다. y=√(x-2)에서 x=y²+2로 정리할 수 있습니다. 따라서 역함수 f inverse(x)=x²+2이며, 이는 x≥0의 범위에서 성립합니다. g(x)=x+3인 경우, fg(1)=f(4)=√2입니다.

유혹적인 단축법을 테스트하기

• 제곱 연산은 부수적 해(extraneous solutions)를 생성할 수 있습니다. 역함수가 존재함을 주장하기 전에 포물선의 정의역을 제한하는 것이 필수적입니다. f 내부의 수평 이동 방향은 그래프의 실제 이동 방향과 반대부호를 가집니다.
• 단축법이 실패하면 그 단축법이 위배하는 가정을 식별하십시오. 요청된 최종 반올림까지 정확한 값을 유지하십시오.

경고:

모든 실수 전체에서의 이차함수는 역함수를 가집니다. 이 주장은 틀렸습니다. 어떤 정의나 가정과 위배되는지 설명하십시오.

새로운 상황을 해석하기

• 역함수 정의역에서 f(f inverse(x))=x인지 확인하십시오. 스킷을 사용하여 수평선이 원본 그래프를 한 번 이상 교차하는지 테스트하십시오.
• 완전한 해답은 수학적 결과를 제시하고 그 의미를 설명해야 합니다. 주어진 맥락 내에서 가능한지 확인하십시오.

이 내용을 과정에 활용하기

• 현재 첫 평가-2021 적용 및 해석 HL. 이는 작성자 개념 지원 자료이며, 모든 학습목표를 검증하기 위해서는 완전한 가이드가 필요합니다.
• 최종 답보다 방법을 먼저 제시하고, 시험지의 계산기 및 공식 규칙을 준수하십시오. 틀린 답을 검토할 때는 첫 번째 유효하지 않은 단계 위치를 찾아보십시오.

핵심:

함수에 허용된 입력의 집합입니다. 관계를 선택하고, 방법을 제시하며, 가정 조건을 확인하고 결과를 해석하십시오.

감쇠 모델이 왜 양의 값을 유지하는가?

• 약물의 농도는 매 시간 동일한 비율로 감소한다. 일정한 값을 빼면 결국 음수 값을 예측하게 될 것이다.
• 본 수업에서는 반감기를 다룬다: 감소하는 양이 초기 값의 절반이 되는 데 걸리는 시간이다.

수학 구조 선택하기

• y=Ae^(kt)에서 k는 비례 상수이다. 양변에 로그를 취하면 ln y=ln A+kt가 된다. 로그함수는 양수 arguments만 허용하며, log(x+y)는 log x+log y와 같지 않다.
• 계산 전에 허용된 입력값과 단위를 명시하라. 방정식은 단순히 계산기 입력 내용을 기록하는 것보다 관계성을 표현해야 한다.

검증된 사례를 풀이하기

• 시작값과 결과를 대조하십시오. 원본 관계에 대입하거나, 적절한 경우 그래프와 수치적 답을 비교하십시오.

예시:

y=80e^(-0.2t)이고 y=40이면 e^(-0.2t)=1/2이 된다. 따라서 t=ln2/0.2≈3.466이다. 3^x=20인 경우, x=ln20/ln3이다. ln y를 t에 대해 그래프를 그리면 지수 모델을 직선화할 수 있다.

유혹적인 단축법을 테스트하기

• 피팅된 지수함수는 모델이지 보장条件是 아니다. 사용 시간 단위와 적용 범위를 명시하시오. k가 음수이면 감쇠를 의미하며, 시작 값이 음수인 경우는 보통 문맥과 모순된다.
• 단축법이 실패하면 그 단축법이 위배하는 가정을 식별하십시오. 요청된 최종 반올림까지 정확한 값을 유지하십시오.

경고:

양수 x, y에 대하여 ln(x+y)=ln x+ln y이다. 이 주장이 틀렸음을 보이며, 어떤 정의나 가정과 모순되는지 설명하시오.

새로운 상황을 해석하기

• 실제 관측치와 모델을 비교하시오. 체계적인 편차는 환경 변화 가능성을 시사할 수 있다. 보고서에서는 단순한 식을 제시하기보다 비례 상수와 초기 값이 어떤 의미를 가지는지 설명하시오.
• 완전한 해답은 수학적 결과를 제시하고 그 의미를 설명해야 합니다. 주어진 맥락 내에서 가능한지 확인하십시오.

이 내용을 과정에 활용하기

• 현재 첫 평가-2021 적용 및 해석 HL. 이는 작성자 개념 지원 자료이며, 모든 학습목표를 검증하기 위해서는 완전한 가이드가 필요합니다.
• 최종 답보다 방법을 먼저 제시하고, 시험지의 계산기 및 공식 규칙을 준수하십시오. 틀린 답을 검토할 때는 첫 번째 유효하지 않은 단계 위치를 찾아보십시오.

핵심:

감소하는 양이 초기 값의 절반이 되는 데 걸리는 시간이다. 관계식을 선택하고, 계산 방법을 제시하며, 가정을 검증하고 결과를 해석하시오.

Supported HL focus. First assessment 2021; current through 2028. First-assessment-2029 course is separate.. Remaining guide, assessment and practical requirements retain their recorded holds.

Prerequisites: read the stated quantities and units, use arithmetic and the model conditions below. Each lesson develops its own method before independent transfer.

These are original or explicitly fictional teaching examples, not actual measurements or completed assessed learner investigations.

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

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

Example:

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.

Warn:

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 Applications and Interpretation 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:

The multiplier that relates corresponding lengths in similar shapes. Choose the relationship, show the method, check its assumptions and interpret the result.

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.

Example:

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.

Warn:

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 Applications and Interpretation 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:

The side opposite the right angle in a right-angled triangle. Choose the relationship, show the method, check its assumptions and interpret the result.

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

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

Example:

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.

Warn:

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 Applications and Interpretation 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:

The vector sum representing the combined displacement or force. Choose the relationship, show the method, check its assumptions and interpret the result.

지원 HL focus. 첫 평가 2021; 현재는 2028까지. 첫 평가-2029 과정은 별도임. Remaining guide, assessment 및 practical requirements는 기록된 holds를 유지함.

사전 요구사항: 명시된 양과 단위를 읽고, 산술 및 아래 모델 조건을 사용하십시오. 각 수업은 독립적 이전 전에 자체 방법을 개발합니다.

이는 실제 측정값이나 완성된 평가된 학습자 조사가 아닌, 원작 또는 명시적으로 허구적인 교육 예시입니다.

하나의 평균으로 전체를 알 수 있을까?

• 두 집단의 중앙값은 같으나 산포가 다릅니다. 하나의 요약 통계량으로 위치와 일관성을 모두 설명할 수 없습니다.
• 이 수업에서는 **빈도 밀도(frequency density)**를 다룹니다: 빈도를 계급폭으로 나눈 값으로 히스토그램의 높이에 사용됩니다.

수학 구조 선택하기

• 문맥에 맞는 적절한 평균과 산포를 비교하십시오. 히스토그램은 면적이 빈도를 나타내므로, 높이=빈도/계급폭입니다. 그룹화된 추정치는 구간 내 대표값을 가정합니다.
• 계산 전에 허용된 입력값과 단위를 명시하라. 방정식은 단순히 계산기 입력 내용을 기록하는 것보다 관계성을 표현해야 한다.

검증된 사례를 풀이하기

• 시작값과 결과를 대조하십시오. 원본 관계에 대입하거나, 적절한 경우 그래프와 수치적 답을 비교하십시오.

예시:

10부터 20까지인 계급으로 빈도가 30인 경우 밀도는 30/10=3입니다. 20부터 40까지인 계급으로 빈도가 20인 경우 밀도는 20/20=1입니다. 더 넓은 막대가 더 높은 밀도를 의미하는 것은 아닙니다. 값이 2,4,4,6,9일 때, 중앙값은 4이고 평균은 5입니다.

유혹적인 단축법을 테스트하기

• 가장 높은 히스토그램 막대가 가장 많은 관측치를 포함할 필요는 없습니다. 그룹화된 평균은 추정치일 뿐입니다. 상관관계는 인과성을 증명하지 않으며, 외삽(extrapolation)은 관찰 범위를 넘어inta합니다.
• 단축법이 실패하면 그 단축법이 위배하는 가정을 식별하십시오. 요청된 최종 반올림까지 정확한 값을 유지하십시오.

경고:

히스토그램 막대의 높이는 항상 빈도와 같다는 주장은 부등한 계급폭에도 적용된다고 합니다. 이는 틀린 말입니다. 어떤 정의나 가정을 위배하는지 설명하십시오.

새로운 상황을 해석하기

• 데이터 유형에 적합한 표시법을 선택하십시오. 수치적 비교와 그것이 인구(모집단)에 대한 의미를 제시하되, 표본이 지지하는 정밀도보다 더 많은 것을 추론하지 마십시오.
• 완전한 해답은 수학적 결과를 제시하고 그 의미를 설명해야 합니다. 주어진 맥락 내에서 가능한지 확인하십시오.

이 내용을 과정에 활용하기

• 현재 첫 평가-2021 적용 및 해석 HL. 이는 작성자 개념 지원 자료이며, 모든 학습목표를 검증하기 위해서는 완전한 가이드가 필요합니다.
• 최종 답보다 방법을 먼저 제시하고, 시험지의 계산기 및 공식 규칙을 준수하십시오. 틀린 답을 검토할 때는 첫 번째 유효하지 않은 단계 위치를 찾아보십시오.

핵심:

빈도를 계급폭으로 나눈 값으로 히스토그램의 높이에 사용됩니다. 관계를 선택하고, 방법을 제시하며, 가정을 검토한 후 결과를 해석하십시오.

첫 번째 추출 후 무엇이 달라지는가?

• 주머니에 3개의 빨간색과 2개의 파란색 마ーカー가 들어 있습니다. 같은 색을 두 번 뽑을 때, 두 번째 뽑기의 확률이 변합니다(비교환법).
• 이 수업에서는 조건부 확률을 다룹니다: 표본공간을 특정 조건으로 제한했을 때 사건이 발생할 확률입니다.

수학 구조 선택하기

• 나뭇가지를 따라 곱하고 분리된 나뭇가지를 더합니다. 치환 시 구성은 일정하게 유지됩니다. 조건부 확률은 P(B) > 0일 때 P(A given B)=P(A∩B)/P(B)입니다.
• 계산 전에 허용된 입력값과 단위를 명시하라. 방정식은 단순히 계산기 입력 내용을 기록하는 것보다 관계성을 표현해야 한다.

검증된 사례를 풀이하기

• 시작값과 결과를 대조하십시오. 원본 관계에 대입하거나, 적절한 경우 그래프와 수치적 답을 비교하십시오.

예시:

비교환법일 때, 두 개 모두 빨간색일 확률=3/5×2/4=3/10. 각 하나씩 나올 확률=3/5×2/4+2/5×3/4=3/5. 만약 P(A∩B)=0.12이고 P(B)=0.3라면, 조건부 확률 P(A given B)=0.4입니다.

유혹적인 단축법을 테스트하기

• 서로소 mutually exclusive는 겹침이 없다는 뜻이며, 독립 independent는 한 사건의 발생 여부가 다른 사건의 확률을 바꾸지 않는다는 뜻입니다. 양의 확률을 가진 서로소 사건들은 독립적이지 않습니다.
• 단축법이 실패하면 그 단축법이 위배하는 가정을 식별하십시오. 요청된 최종 반올림까지 정확한 값을 유지하십시오.

경고:

양의 확률을 가진 서로소 사건들은 반드시 독립적이다. 이 주장은 틀렸습니다. 어떤 정의나 가정을 위반하는지 설명하십시오.

새로운 상황을 해석하기

• 양방향 표는 제한된 분모를 명확하게 보여줍니다. 곱 P(A)P(B)를 사용하기 전에 문맥이나 제공된 정보로부터 독립성을 정당화하십시오.
• 완전한 해답은 수학적 결과를 제시하고 그 의미를 설명해야 합니다. 주어진 맥락 내에서 가능한지 확인하십시오.

이 내용을 과정에 활용하기

• 현재 첫 평가-2021 적용 및 해석 HL. 이는 작성자 개념 지원 자료이며, 모든 학습목표를 검증하기 위해서는 완전한 가이드가 필요합니다.
• 최종 답보다 방법을 먼저 제시하고, 시험지의 계산기 및 공식 규칙을 준수하십시오. 틀린 답을 검토할 때는 첫 번째 유효하지 않은 단계 위치를 찾아보십시오.

핵심:

표본공간을 특정 조건으로 제한했을 때 사건의 확률입니다. 관계식을 선택하고, 계산 방법을 제시하며, 해당 가정을 검증한 뒤 결과를 해석하십시오.

어떤 요인이 횟수를 예측 가능하게 하는가?

• 품질 검사원이 불량품을 세는 수치는 무작위이지만, 모델은 그 예상 범위를 설명할 수 있습니다.
• 이 수업에서는 기대값 expected value를 다룹니다: 무작위 변수의 확률 가중 평균입니다.

수학 구조 선택하기

• 이항 모델은 고정된 n, 독립적 실험, 두 가지 결과, 일정한 p를 필요로 합니다. E(X)=np이고 Var(X)=np(1-p)입니다. 정규 모델에는 z=(x-μ)/σ을 사용합니다. 푸아송 모델은 일정한 속도와 적절한 독립성 가정을 가진 계수들을 묘사합니다.
• 계산 전에 허용된 입력값과 단위를 명시하라. 방정식은 단순히 계산기 입력 내용을 기록하는 것보다 관계성을 표현해야 한다.

검증된 사례를 풀이하기

• 시작값과 결과를 대조하십시오. 원본 관계에 대입하거나, 적절한 경우 그래프와 수치적 답을 비교하십시오.

예시:

X가 이항분포 binomial(5,0.2)일 때, P(X=0)=0.8^5=0.32768이며, E(X)=1, Var(X)=0.8입니다. μ=100, σ=15인 정규량에서 값 130의 z는 ⟨2⟩입니다. 시간당 푸아송 평균이 3이면 2시간 동안의 평균은 ⟨6⟩입니다.

유혹적인 단축법을 테스트하기

• 모든 세기(count)가 이항 분포를 따르는 것은 아닙니다. p의 변화나 종속성은 이를 무효화할 수 있습니다. 연속 변수의 경우 특정 한 값에 대한 확률은 0입니다. 이산 분포를 정규 분포로 근사할 때는 연성 보정(continuity correction)이 중요합니다.
• 단축법이 실패하면 그 단축법이 위배하는 가정을 식별하십시오. 요청된 최종 반올림까지 정확한 값을 유지하십시오.

경고:

모든 성공의 세기는 종속성에 상관없이 이항 분포를 따른다는 주장은 틀렸습니다. 어떤 정의나 가정을 위반했는지 설명하십시오.

새로운 상황을 해석하기

• 계산기 분포 함수를 사용하기 전에 사건을 부등식으로 작성하십시오. P(X<k), P(X≤k), 그리고 꼬리의 보완을 구분하십시오. 맥락에서 가정을 명시하십시오.
• 완전한 해답은 수학적 결과를 제시하고 그 의미를 설명해야 합니다. 주어진 맥락 내에서 가능한지 확인하십시오.

이 내용을 과정에 활용하기

• 현재 첫 평가-2021 적용 및 해석 HL. 이는 작성자 개념 지원 자료이며, 모든 학습목표를 검증하기 위해서는 완전한 가이드가 필요합니다.
• 최종 답보다 방법을 먼저 제시하고, 시험지의 계산기 및 공식 규칙을 준수하십시오. 틀린 답을 검토할 때는 첫 번째 유효하지 않은 단계 위치를 찾아보십시오.

핵심:

확률 변수의 확률 가중 평균입니다. 관계식을 선택하고, 방법을 제시하며, 그 가정을 확인하고 결과를 해석하십시오.

우연으로 결과를 설명할 수 있는가?

• 한 공장이 품목 중 불량품 비율이 단 10%에 불과하다고 주장합니다. 표본에는 더 많은 불량이 포함되어 있으나, 우연만으로 일부 차이를 설명할 수 있습니다.
• 이번 수업에서는 유의수준(significance level) 을 다룹니다: 기저 가설을 기각하기 위해 선택한 확률 기준치입니다.

수학 구조 선택하기

• 결과 전 Population 모수를 사용하여 H₀과 H₁을 명시하십시오. H₀ 하에 적절한 꼬리 확률을 계산하십시오. 증거가 지정된 유의 규칙을 충족하면 H₀을 기각하되, 그렇지 않으면 충분한 증거가 없다고 명시하십시오.
• 계산 전에 허용된 입력값과 단위를 명시하라. 방정식은 단순히 계산기 입력 내용을 기록하는 것보다 관계성을 표현해야 한다.

검증된 사례를 풀이하기

• 시작값과 결과를 대조하십시오. 원본 관계에 대입하거나, 적절한 경우 그래프와 수치적 답을 비교하십시오.

예시:

H₀:p=0.1, H₁:p>0.1이며 n=10일 때, 적어도 3개의 불량을 관측할 확률은 ⟨1⟩-(0.9^10+10×0.1×0.9^9+45×0.1²×0.9^8)≈0.070191입니다. 5% 수준에서 이는 불량률이 10%를 초과한다는 불충분한 증거입니다.

유혹적인 단축법을 테스트하기

• H₀을 기각하지 않는 것이 H₀이 참임을 증명하는 것은 아닙니다. 작은 결과가 나오는 꼬리가 아닌, H₁에 기반한 꼬리를 선택하십시오. 통계적 유의성은 효과의 실제 크기를 측정하지 않습니다.
• 단축법이 실패하면 그 단축법이 위배하는 가정을 식별하십시오. 요청된 최종 반올림까지 정확한 값을 유지하십시오.

경고:

귀무가설을 기각하지 않는 것은 그것이 참임을 증명합니다. 이 주장은 틀렸습니다. 이 주장이 위배하는 정의 또는 가정을 설명하십시오.

새로운 상황을 해석하기

• Population와 원래 주장에 대한 문장으로 마무리하십시오. 모델의 가정을 명시하고, 표집 절차가 이를 지지하는지 고려하십시오.
• 완전한 해답은 수학적 결과를 제시하고 그 의미를 설명해야 합니다. 주어진 맥락 내에서 가능한지 확인하십시오.

이 내용을 과정에 활용하기

• 현재 첫 평가-2021 적용 및 해석 HL. 이는 작성자 개념 지원 자료이며, 모든 학습목표를 검증하기 위해서는 완전한 가이드가 필요합니다.
• 최종 답보다 방법을 먼저 제시하고, 시험지의 계산기 및 공식 규칙을 준수하십시오. 틀린 답을 검토할 때는 첫 번째 유효하지 않은 단계 위치를 찾아보십시오.

핵심:

영가설을 기각하기 위해 선택한 확률 임계값입니다. 관계성을 선택하고, 방법을 제시하며, 가정 조건을 확인하고 결과를 해석하십시오.

표본 평균의 불확실성은 얼마나 되는가?

• 두 샘플은 서로 다른 평균을给出합니다. 하나의 샘플이 모집단과 같다는 약속이 아닌, 샘플링 불확실에 대한 statement가 필요합니다.
• 이 수업에서는 신뢰구간을 다룹니다: 특정 장기적 비율로 모수(population parameter)를 포함하도록 설계된 절차를 통해 계산된 구간입니다.

수학 구조 선택하기

• σ가 알려진 정규 분포의 평균에 대해, 95% 신뢰구간은 표본 평균±1.96σ/√n입니다. 카이제곱분포는 적절한 자유도를 사용하여 관측치와 기대치의 차이 제곱을 합산한 후 기대치로 나눈 값인 ∑(관측-기대)²/기대로 계산합니다. 데이터와 가정에 맞는 검정을 선택하십시오.
• 계산 전에 허용된 입력값과 단위를 명시하라. 방정식은 단순히 계산기 입력 내용을 기록하는 것보다 관계성을 표현해야 한다.

검증된 사례를 풀이하기

• 시작값과 결과를 대조하십시오. 원본 관계에 대입하거나, 적절한 경우 그래프와 수치적 답을 비교하십시오.

예시:

평균=50, σ=10, n=100일 때 표준 오차=1이며, 95% 신뢰 구간은 [48.04,51.96]입니다. 관측 빈도 30과 20, 기대 빈도 25과 25에 대해 카이제곱=25/25+25/25=2이며, 자유도와 기각 임계값을 결정하기 전입니다.

유혹적인 단축법을 테스트하기

• 한 번의 95% 신뢰 절차가 고정된 모수가 해당 구간에 들어올 확률이 95%임을 의미하지 않습니다. 기대치와 추정된 매개변수는 카이제곱 검정의 유효성과 자유도에 영향을 미칩니다.
• 단축법이 실패하면 그 단축법이 위배하는 가정을 식별하십시오. 요청된 최종 반올림까지 정확한 값을 유지하십시오.

경고:

강한 상관관계만으로 인과관계가 성립함을 주장하는 것은 틀렸습니다. 이 주장은 어떤 정의나 가정 위반을 설명하십시오.

새로운 상황을 해석하기

• 불확실성을 맥락에서 보고하십시오. 회귀 분석에서는 잔차(residuals)를 점검하고 부당한 외삽(extrapolation)을 피하며, fitting된 관계와 인과 설명을 구분하십시오. S3 및 IB AI HL은 서로 다른 검증된 scopes 경계를 요구합니다.
• 완전한 해답은 수학적 결과를 제시하고 그 의미를 설명해야 합니다. 주어진 맥락 내에서 가능한지 확인하십시오.

이 내용을 과정에 활용하기

• 현재 첫 평가-2021 적용 및 해석 HL. 이는 작성자 개념 지원 자료이며, 모든 학습목표를 검증하기 위해서는 완전한 가이드가 필요합니다.
• 최종 답보다 방법을 먼저 제시하고, 시험지의 계산기 및 공식 규칙을 준수하십시오. 틀린 답을 검토할 때는 첫 번째 유효하지 않은 단계 위치를 찾아보십시오.

핵심:

특정 장기적 비율로 모수를 포함하도록 설계된 절차를 통해 계산된 구간입니다. 관계를 선택하고 방법을 제시하며, 가정을 점검하고 결과를 해석하십시오.

Supported HL focus. First assessment 2021; current through 2028. First-assessment-2029 course is separate.. Remaining guide, assessment and practical requirements retain their recorded holds.

Prerequisites: read the stated quantities and units, use arithmetic and the model conditions below. Each lesson develops its own method before independent transfer.

These are original or explicitly fictional teaching examples, not actual measurements or completed assessed learner investigations.

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.

Example:

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.

Warn:

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 Applications and Interpretation 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:

The instantaneous rate of change, also the gradient of a tangent. Choose the relationship, show the method, check its assumptions and interpret the result.

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

• A continuous function with opposite signs at two endpoints has a root between them. Newton's method uses x next=x-f(x)/f prime(x), with a nonzero derivative. The trapezium rule approximates a definite integral using endpoint 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.

Example:

For f(x)=x²-2 and x₀=1.5, Newton gives x₁=1.5-(2.25-2)/3=1.4166667. With y=x² on [0,2] and two equal strips, h=1 and trapezium area=(1/2)(0+2×1+4)=3; the exact area is 8/3.

Test a tempting shortcut

• A sign change across a discontinuity does not prove a root. Iteration can diverge or cycle. The trapezium rule's overestimate or underestimate depends on curvature, not just whether the function increases.
• When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Warn:

Every sign change proves a root, including one across a discontinuity. This claim is false. Explain which definition or assumption it violates.

Interpret a new situation

• Give a stopping criterion and report sensible accuracy. Confirm the approximated root by a sign bracket around the stated rounded answer; explain any failure of the chosen iteration.
• 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 Applications and Interpretation 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:

A repeated update in which each new approximation is calculated from the previous one. Choose the relationship, show the method, check its assumptions and interpret the result.

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.

Example:

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.

Warn:

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 Applications and Interpretation 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:

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.

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.

Example:

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.

Warn:

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 Applications and Interpretation 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:

An integration method based on the derivative of a product. Choose the relationship, show the method, check its assumptions and interpret the result.

What does the starting value decide?

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

Choose the mathematical structure

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

Work through a checked case

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

Example:

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.

Warn:

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 Applications and Interpretation 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:

A specified value that selects a particular solution from a family. Choose the relationship, show the method, check its assumptions and interpret the result.

Supported HL focus. First assessment 2021; current through 2028. First-assessment-2029 course is separate.. Remaining guide, assessment and practical requirements retain their recorded holds.

Prerequisites: read the stated quantities and units, use arithmetic and the model conditions below. Each lesson develops its own method before independent transfer.

These are original or explicitly fictional teaching examples, not actual measurements or completed assessed learner investigations.

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

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

Example:

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.

Warn:

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 Applications and Interpretation 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:

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.