Unit 1 · Topic 1: Number and Algebra
The Complete IB Mathematics: Applications and Interpretation Guide · IB Mathematics: Applications and InterpretationUnit 1 · Topic 1: Number and Algebra
- The Complete IB Mathematics: Applications and Interpretation Guide
- IB Mathematics: Applications and Interpretation
- 5 sections
International Baccalaureate does not publish a per-unit weighting for this exam. Exam structure (official): SL: Paper 1 (90 min, 40%) · Paper 2 (90 min, 40%) · Internal Assessment (20%); HL: Paper 1 (120 min, 30%) · Paper 2 (120 min, 30%) · Paper 3 (60 min, 2 questions, 20%) · Internal Assessment (20%). Be ready to carry the same unit skills and representations into a complete answer on every paper. This topic develops approximation, sequences, exponentials, and matrix modeling. The official subject brief names the topic but does not publish a topic-by-topic examination percentage, so no percentage is claimed here on the “Number and Algebra: choose, calculate, interpret” page.
- How IB Mathematics: Applications and Interpretation assesses this International Baccalaureate publishes no per-unit percentage for IB Mathematics: Applications and Interpretation; Paper 1 and Paper 2 each account for 40% of the SL total and 30% of the HL total. · SL: Paper 1 (90 min, 40%) · Paper 2 (90 min, 40%) · Internal Assessment (20%) | HL: Paper 1 (120 min, 30%) · Paper 2 (120 min, 30%) · Paper 3 (60 min, 2 questions, 20%) · Internal Assessment (20%) · show the model with a labeled sequence or recurrence, an exponential model with its domain and units, a matrix with a labeled state vector
- Key skills Scientific Notation, Rounding, Bounds, and Percentage Error, Arithmetic and Geometric Sequences and Series, Compound Interest, Depreciation, Amortization, and Annuities, Exponents, Logarithms, and Solving Equations and Systems with Technology, Laws of Logarithms, Rational Exponents, and Infinite Geometric Series (HL), Complex Numbers: Cartesian, Polar, and Exponential Forms (HL), Matrices: Algebra, Determinants, Inverses, and Solving Systems (HL), Eigenvalues, Eigenvectors, and Diagonalization (HL)
- How to study for Unit 1 This page turns a labeled sequence or recurrence, an exponential model with its domain and units, a matrix with a labeled state vector into one route: Name each quantity and unit, then identify whether the change is additive or multiplicative before calculating..
- The organizing decision Decide whether a situation calls for approximation, additive change, multiplicative change, a recurrence, or a matrix model.
What IB Mathematics: Applications and Interpretation Unit 1 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Scientific Notation, Rounding, Bounds, and Percentage Error
relative error; geometric sequence; state vector02Arithmetic and Geometric Sequences and Series
relative error; geometric sequence; state vector03Compound Interest, Depreciation, Amortization, and Annuities
relative error; geometric sequence; state vector04Exponents, Logarithms, and Solving Equations and Systems with Technology
relative error; geometric sequence; state vector05Laws of Logarithms, Rational Exponents, and Infinite Geometric Series (HL)
relative error; geometric sequence; state vector06Complex Numbers: Cartesian, Polar, and Exponential Forms (HL)
relative error; geometric sequence; state vector07Matrices: Algebra, Determinants, Inverses, and Solving Systems (HL)
relative error; geometric sequence; state vector08Eigenvalues, Eigenvectors, and Diagonalization (HL)
relative error; geometric sequence; state vectorInternational Baccalaureate does not publish a per-unit weighting for this exam.
Official unit name and weighting: IB subject guide / subject brief.
Number and Algebra: choose, calculate, interpret
Connect the published share to the unit model
This topic develops approximation, sequences, exponentials, and matrix modeling. The official subject brief names the topic but does not publish a topic-by-topic examination percentage, so no percentage is claimed here on the “Number and Algebra: choose, calculate, interpret” page.
Vocabulary in use: relative error, arithmetic sequence, geometric sequence, exponential model, recurrence, matrix, state vector, precision. Each term is applied on these pages and collected again in the glossary with a checked page pointer on the “Number and Algebra: choose, calculate, interpret” page.
A strong number-and-algebra solution labels quantities and units, distinguishes additive from multiplicative change, retains calculator precision, and tests the result against the modeled setting. The tank, membership, reserve, and commuter situations in this topic are independently written illustrations.
The decision that organizes this unit
Define the system and choose the route before calculating
Decide whether a situation calls for approximation, additive change, multiplicative change, a recurrence, or a matrix model.
Name each quantity and unit, then identify whether the change is additive or multiplicative before calculating.
Mechanism route and repair branches
Relationships to preserve
- Relative error compares an absolute deviation with a reference magnitude.
- Arithmetic sequences use a constant difference, while geometric sequences use a constant ratio.
- A recurrence gives the next state from the current state, and a transition matrix updates all labeled states together.
Representations to read
- a labeled sequence or recurrence
- an exponential model with its domain and units
- a matrix with a labeled state vector
Branches to reject
- using the full rounding interval instead of half of it as the maximum absolute error
- confusing a single sequence term with the sum of several terms
- writing a matrix answer without preserving the order of its labeled states
| Key concept | Why it's hard | What scores |
|---|---|---|
| relative error | Distinguishing relative error from arithmetic sequence requires their definitions and units. | Define the quantity measured by relative error and attach its units before calculation. |
| geometric sequence | A calculator screen does not show why geometric sequence is the suitable model feature. | Record the relation used for geometric sequence and one identifying intermediate value. |
| matrix | A computed matrix value still needs a domain and scale check. | Check the restrictions, precision, and contextual meaning of the matrix result. |
How IB Mathematics: Applications and Interpretation assesses Topic 1: Number and Algebra
What a complete response must make visible
Match the task to evidence that a reader can audit, then check the most likely reasoning failure before finalizing the response.
| Task | Evidence to show | Hurdle |
|---|---|---|
| Scientific Notation, Rounding, Bounds, and Percentage Error | relative error; geometric sequence; state vector | using the full rounding interval instead of half of it as the maximum absolute error |
| Arithmetic and Geometric Sequences and Series | relative error; geometric sequence; state vector | confusing a single sequence term with the sum of several terms |
| Compound Interest, Depreciation, Amortization, and Annuities | relative error; geometric sequence; state vector | writing a matrix answer without preserving the order of its labeled states |
| Exponents, Logarithms, and Solving Equations and Systems with Technology | relative error; geometric sequence; state vector | using the full rounding interval instead of half of it as the maximum absolute error |
| Laws of Logarithms, Rational Exponents, and Infinite Geometric Series (HL) | relative error; geometric sequence; state vector | confusing a single sequence term with the sum of several terms |
| Complex Numbers: Cartesian, Polar, and Exponential Forms (HL) | relative error; geometric sequence; state vector | writing a matrix answer without preserving the order of its labeled states |
| Matrices: Algebra, Determinants, Inverses, and Solving Systems (HL) | relative error; geometric sequence; state vector | using the full rounding interval instead of half of it as the maximum absolute error |
| Eigenvalues, Eigenvectors, and Diagonalization (HL) | relative error; geometric sequence; state vector | confusing a single sequence term with the sum of several terms |
Measure error before trusting a model
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Practice 1: Measure a rounding error
A community garden records the volume of a water tank as 820 L, correct to the nearest 10 L.
Find the greatest possible relative error and express it as a percentage.
Show your working.
A. 10 ÷ 820 × 100 = 1.22% because the rounding interval is 10 L wide.
❌ Not quite 😭 Nearest-10-L rounding gives a maximum deviation of half the rounding increment, not the full 10 L interval.
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B. 5 ÷ 815 × 100 = 0.613% because 815 L is the lower bound.
❌ Not quite 😭 The 5 L maximum error is correct, but relative error here uses the recorded reference value 820 L, not an interval bound.
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C. 5 ÷ 820 × 100 ≈ 0.610%, with bounds 815 L and 825 L.
✅ Correct! 😊 Correct. The bounds verify a 5 L maximum deviation, which is divided by the recorded 820 L and converted to a percentage.
D. 5 ÷ 820 = 0.00610%, with bounds 815 L and 825 L.
❌ Not quite 😭 The ratio is correct, but it has been labeled as a percentage without multiplying by 100.
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Full explanation
- Step 1Rounding to the nearest 10 L gives the interval from 815 L to 825 L.
- Step 2The greatest possible absolute error is half the rounding increment: 10 ÷ 2 = 5 L.
- Step 3Relative error = 5 ÷ 820 = 0.00610 using the recorded value as reference.
- Step 4Convert the ratio to a percentage: 0.00610 × 100 ≈ 0.610%.
Practice 2: Build a reserve recurrence
A reserve begins with 1500 dollars. At the end of each year, 4% interest is applied and then 300 dollars is deposited.
Write a recurrence for the balance Bₙ and find the balance after two yearly updates.
Show each update.
Give the final balance to the nearest cent.
A. B₀ = 1500, Bₙ₊₁ = 1.04Bₙ + 300; B₂ = 2232.40 dollars.
✅ Correct! 😊 Correct. Each current balance is multiplied by 1.04 before the fixed deposit is added, giving two traceable updates.
B. B₀ = 1500, Bₙ₊₁ = 1.04(Bₙ + 300); B₂ = 2256.96 dollars.
❌ Not quite 😭 This applies interest to each new deposit immediately, contrary to the stated order that interest comes before the deposit.
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C. B₀ = 1500, Bₙ₊₁ = Bₙ + 0.04 + 300; B₂ = 2100.08 dollars.
❌ Not quite 😭 A four-percent increase is multiplicative, not an addition of 0.04 dollars. This recurrence does not model percentage growth.
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D. B₁ = 1500, Bₙ₊₁ = 1.04Bₙ + 300; B₂ = 1860.00 dollars.
❌ Not quite 😭 Labeling the initial balance B₁ makes B₂ only one update later. The requested two updates require starting from B₀.
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Full explanation
- Step 1Let B₀ = 1500 represent the balance before any yearly update.
- Step 2Interest first multiplies the current balance by 1.04, then the deposit adds 300: Bₙ₊₁ = 1.04Bₙ + 300.
- Step 3B₁ = 1.04(1500) + 300 = 1860 and B₂ = 1.04(1860) + 300 = 2232.40.
- Step 4Thus the reserve contains 2232.40 dollars after two updates.
Key terms for Unit 1: Topic 1: Number and Algebra
Models, uses, and boundaries
- Compare Relative Error
- Relative error compares absolute deviation with a reference magnitude; convert the resulting ratio to a percentage. Use this relation when the named quantities and conditions in the question match the calculation described here.
- Measure the Rounding Bound
- Show 5 ÷ 820 = 0.00610 before converting the ratio to 0.610%; use the 815 L and 825 L bounds to verify the 5 L maximum absolute error. Use this relation when the named quantities and conditions in the question match the calculation described here.
- Model Repeated Percentage Change
- Write M(n) = 480(1.06)^n and evaluate M(4) = 605.998; check that the repeated multiplier is 1.06 and that a membership count is whole. Use this relation when the named quantities and conditions in the question match the calculation described here.
- Update a Recurrence
- Apply B(n+1) = 1.03B(n) + 250 to obtain 1486, 1780.58, and 2083.9974; verify that interest is applied before the $250 deposit at every update. Use this relation when the named quantities and conditions in the question match the calculation described here.
- Apply a Transition Matrix
- Calculate 0.80(180) + 0.30(120) and 0.20(180) + 0.70(120); confirm that the next-state entries 180 and 120 still total 300 commuters. Use this relation when the named quantities and conditions in the question match the calculation described here.
IB Mathematics: Applications and Interpretation Unit 1 FAQ
How much of IB Mathematics: Applications and Interpretation does Unit 1 carry?
International Baccalaureate does not publish a per-unit weighting for this exam.
What is the first move on a Topic 1: Number and Algebra problem?
Name each quantity and unit, then identify whether the change is additive or multiplicative before calculating.
Which relationships should I preserve?
Relative error compares an absolute deviation with a reference magnitude. Arithmetic sequences use a constant difference, while geometric sequences use a constant ratio. A recurrence gives the next state from the current state, and a transition matrix updates all labeled states together.
Which representations should I practice?
Practice moving among a labeled sequence or recurrence, an exponential model with its domain and units, a matrix with a labeled state vector.
What error should I check before submitting an answer?
Check for using the full rounding interval instead of half of it as the maximum absolute error; confusing a single sequence term with the sum of several terms; writing a matrix answer without preserving the order of its labeled states.
Evidence workshop
Continue from the free model into complete practice
The full unit guide continues with the chapter’s worked examples, figures, scoring tables, and answer checks.
- Measure error before trusting a model
- Project repeated percentage change
- Check and answer: Measure error before trusting a model
- Build and check a recurrence
- Use a transition matrix with labeled states
- Check the result and complete the answer
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
Related IB Mathematics: Applications and Interpretation unit guides
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International Baccalaureate does not publish a per-unit weighting for this exam.How to study IB Mathematics: Applications and Interpretation Unit 1
Start with the organizing decision
Before solving, restate the decision in operational terms: Decide whether a situation calls for approximation, additive change, multiplicative change, a recurrence, or a matrix model.. Your first written move should be to Name each quantity and unit, then identify whether the change is additive or multiplicative before calculating..
Practice the same idea in several representations
Rotate through a labeled sequence or recurrence, an exponential model with its domain and units, a matrix with a labeled state vector. Use each representation to practice Scientific Notation, Rounding, Bounds, and Percentage Error, Arithmetic and Geometric Sequences and Series, Compound Interest, Depreciation, Amortization, and Annuities, Exponents, Logarithms, and Solving Equations and Systems with Technology, Laws of Logarithms, Rational Exponents, and Infinite Geometric Series (HL), Complex Numbers: Cartesian, Polar, and Exponential Forms (HL), Matrices: Algebra, Determinants, Inverses, and Solving Systems (HL), Eigenvalues, Eigenvectors, and Diagonalization (HL), and explain what stays invariant when the surface form changes.
Turn each error into a repair check
After every attempt, audit the response for using the full rounding interval instead of half of it as the maximum absolute error; confusing a single sequence term with the sum of several terms; writing a matrix answer without preserving the order of its labeled states. Then redo only the first step that made the reasoning diverge, keeping units, direction, and model conditions visible.
Confirm current course details in the official IB subject guide / subject brief for the 2027 administration.
Where to go next
Return to the IB Mathematics: Applications and Interpretation to see every unit and plan your revision.