Unit 1 · Topic 1: Number and Algebra
The Complete IB Mathematics: Analysis and Approaches Guide · IB Mathematics: Analysis and ApproachesUnit 1 · Topic 1: Number and Algebra
- The Complete IB Mathematics: Analysis and Approaches Guide
- IB Mathematics: Analysis and Approaches
- 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. Sequences, binomial coefficients, logarithms, complex numbers, and proof all reward a visible chain of exact statements. Arithmetic sequence and geometric sequence describe different recurrences; convergence belongs only to a geometric series whose ratio has magnitude below one.
- How IB Mathematics: Analysis and Approaches assesses this International Baccalaureate publishes no per-unit percentage for IB Mathematics: Analysis and Approaches; 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 term rule or finite-sum expression, an exact algebraic statement with its domain, a complex number in Cartesian or polar form
- Key skills Arithmetic and Geometric Sequences and Series, Sigma Notation, and Financial Applications, Scientific Notation, Laws of Exponents and Logarithms, and Exponential Equations, The Binomial Theorem for Natural-Number Powers, Simple Deductive Proof and Identities, Counting Principles, Permutations and Combinations, and the Extended Binomial Theorem (HL), Partial Fractions and Systems of Linear Equations (HL), Complex Numbers: Cartesian, Polar and Euler Forms, De Moivre's Theorem, and Roots (HL), Proof by Induction, Contradiction, and Counterexample (HL)
- How to study for Unit 1 This page turns a term rule or finite-sum expression, an exact algebraic statement with its domain, a complex number in Cartesian or polar form into one route: Name the sequence, expansion, logarithmic form, complex representation, or proof method that matches the prompt..
- The organizing decision Choose the exact algebraic structure before calculating, and keep the index, domain, or proof claim visible.
What IB Mathematics: Analysis and Approaches Unit 1 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Arithmetic and Geometric Sequences and Series, Sigma Notation, and Financial Applications
arithmetic sequence; geometric series; finite sum02Scientific Notation, Laws of Exponents and Logarithms, and Exponential Equations
exponential equation; logarithm domain; exact form03The Binomial Theorem for Natural-Number Powers
general binomial term; coefficient; requested power04Simple Deductive Proof and Identities
deductive proof; identity; counterexample05Counting Principles, Permutations and Combinations, and the Extended Binomial Theorem (HL)
counting principle; permutation; combination06Partial Fractions and Systems of Linear Equations (HL)
partial fractions; linear system; exact algebra07Complex Numbers: Cartesian, Polar and Euler Forms, De Moivre's Theorem, and Roots (HL)
modulus; argument; Cartesian form08Proof by Induction, Contradiction, and Counterexample (HL)
induction hypothesis; contradiction; quantified conclusionInternational Baccalaureate does not publish a per-unit weighting for this exam.
Official unit name and weighting: IB subject guide / subject brief.
Patterns, exact forms, and proof
Connect the published share to the unit model
Sequences, binomial coefficients, logarithms, complex numbers, and proof all reward a visible chain of exact statements. Arithmetic sequence and geometric sequence describe different recurrences; convergence belongs only to a geometric series whose ratio has magnitude below one.
A binomial expansion begins with a general term, while an exponential equation begins with a domain and a choice of exact transformation. Complex number work uses modulus and argument, and polynomial proof uses factor, remainder, induction, contradiction, or a counterexample.
On a no-calculator paper, preserve exact form and show the law used. With a calculator, record the equation entered, the numerical result, and a domain or precision check.
The decision that organizes this unit
Define the system and choose the route before calculating
Choose the exact algebraic structure before calculating, and keep the index, domain, or proof claim visible.
Name the sequence, expansion, logarithmic form, complex representation, or proof method that matches the prompt.
Mechanism route and repair branches
Relationships to preserve
- Arithmetic sequences have a constant difference, while geometric sequences have a constant ratio.
- A binomial coefficient follows from the selected power in the general term.
- Complex roots share a modulus and are separated by equal changes in argument.
Representations to read
- a term rule or finite-sum expression
- an exact algebraic statement with its domain
- a complex number in Cartesian or polar form
Branches to reject
- using the term number as the exponent without checking the starting index
- turning an exact result into a decimal too early
- checking examples without completing the general proof
| Key concept | Why it's hard | What scores |
|---|---|---|
| Sequence model | Indexing and convergence can be mixed up | Names the model, term or sum rule, index, and units |
| Algebraic expansion | The requested power must be selected before arithmetic | Shows a general term and isolates the required exponent |
| Proof and complex form | A numerical pattern is not a general argument | States every branch or closes the logical implication |
How IB Mathematics: Analysis and Approaches 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 |
|---|---|---|
| Arithmetic and Geometric Sequences and Series, Sigma Notation, and Financial Applications | arithmetic sequence; geometric series; finite sum | using the term number as the exponent without checking the starting index |
| Scientific Notation, Laws of Exponents and Logarithms, and Exponential Equations | exponential equation; logarithm domain; exact form | turning an exact result into a decimal too early |
| The Binomial Theorem for Natural-Number Powers | general binomial term; coefficient; requested power | checking examples without completing the general proof |
| Simple Deductive Proof and Identities | deductive proof; identity; counterexample | using the term number as the exponent without checking the starting index |
| Counting Principles, Permutations and Combinations, and the Extended Binomial Theorem (HL) | counting principle; permutation; combination | turning an exact result into a decimal too early |
| Partial Fractions and Systems of Linear Equations (HL) | partial fractions; linear system; exact algebra | checking examples without completing the general proof |
| Complex Numbers: Cartesian, Polar and Euler Forms, De Moivre's Theorem, and Roots (HL) | modulus; argument; Cartesian form | using the term number as the exponent without checking the starting index |
| Proof by Induction, Contradiction, and Counterexample (HL) | induction hypothesis; contradiction; quantified conclusion | turning an exact result into a decimal too early |
Model a geometric saving plan
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Practice 1: Use the correct geometric index and sum
A student follows a monthly saving plan.
A student deposits $120 in month 1 and increases each later deposit by 5%. Find (a) the month-8 deposit and (b) the total deposited through month 8.
Show your working.
Give monetary answers to the nearest cent.
A. Month 8: 120(1.05)⁷; total: 120((1.05)⁸−1)/(1.05−1).
✅ Correct! 😊 Correct. The first expression is the eighth term because counting begins at exponent 0, and the second sums all eight geometric terms.
B. Month 8: 120(1.05)⁸; total: 120((1.05)⁷−1)/(1.05−1).
❌ Not quite 😭 Both indices are shifted. The eighth term uses exponent 7, while a total through month 8 contains eight terms.
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C. Month 8: 120+8(0.05); total: 8(120+0.05).
❌ Not quite 😭 A percentage increase creates a constant ratio, not a constant difference. These arithmetic expressions do not model the deposits.
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D. Month 8: 120(1.05)⁷; total: 8·120(1.05)⁷.
❌ Not quite 😭 The eighth-deposit expression is correct, but multiplying it by eight assumes every deposit equals the last one. The total needs the geometric-sum rule.
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Full explanation
- Step 1The first term is 120 and each term is multiplied by 1.05, so the sequence is geometric.
- Step 2The eighth term is 120(1.05)⁸⁻¹=120(1.05)⁷=168.852… .
- Step 3The finite total is 120((1.05)⁸−1)/(1.05−1)=1145.892… .
- Step 4To the nearest cent, the two answers are $168.85 and $1,145.89.
Practice 2: Select a binomial term and preserve logarithmic form
Answer both independent algebraic parts.
(a) Find the coefficient of x⁴ in (3+x)⁶. (b) Solve log₂y=6, stating the required domain condition.
Give exact answers.
Show your working.
A. Set k=4 in C(6,k)3⁶⁻ᵏxᵏ, and rewrite log₂y=6 as y=2⁶ with y>0.
✅ Correct! 😊 Correct. The chosen general term contains x⁴, and exponential form gives the positive solution y=64.
B. Set k=2 because 6−k=4, and rewrite log₂y=6 as y=12.
❌ Not quite 😭 The power of x is k, so k must be 4. A logarithm is reversed by exponentiation, not by multiplying its base and value.
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C. Set k=4 in C(6,k)3ᵏx⁶⁻ᵏ, and rewrite the logarithm as y=6².
❌ Not quite 😭 This general term assigns the powers to the wrong binomial terms. The base 2 must be raised to 6, rather than reversing those roles.
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D. Expand only the first four terms, and allow y=±64 because logarithms have positive and negative branches.
❌ Not quite 😭 The x⁴ term may not appear in a partial expansion that stops too early. For real logarithms the input y must be positive, so there is no negative branch.
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Full explanation
- Step 1The general term in (3+x)⁶ is C(6,k)3⁶⁻ᵏxᵏ.
- Step 2Set k=4: the coefficient is C(6,4)3²=15·9=135.
- Step 3Rewrite log₂y=6 in exponential form as y=2⁶=64.
- Step 4The logarithm requires y>0, and y=64 satisfies this condition.
Key terms for Unit 1: Topic 1: Number and Algebra
Models, uses, and boundaries
- Geometric sequence model
- The nth term of a geometric sequence is the first term multiplied by the common ratio raised to one less than n. Use it when consecutive terms change by a constant ratio.
- Monthly deposit
- The eighth deposit is 120 multiplied by 1.05 to the seventh power. Use it to calculate the month-eight term in the savings example.
- Binomial coefficient
- The required binomial coefficient is five choose three multiplied by two squared, which equals 40. Use it after selecting the term containing x cubed.
- Complex cube roots
- The three cube roots have modulus 2 and arguments separated by two pi over three. Use it to generate every root rather than only the positive real root.
- Induction step
- The next induction case equals the assumed case plus three times k times k plus one. Use it to show that the next expression remains divisible by three.
IB Mathematics: Analysis and Approaches Unit 1 FAQ
How much of IB Mathematics: Analysis and Approaches 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 the sequence, expansion, logarithmic form, complex representation, or proof method that matches the prompt.
Which relationships should I preserve?
Arithmetic sequences have a constant difference, while geometric sequences have a constant ratio. A binomial coefficient follows from the selected power in the general term. Complex roots share a modulus and are separated by equal changes in argument.
Which representations should I practice?
Practice moving among a term rule or finite-sum expression, an exact algebraic statement with its domain, a complex number in Cartesian or polar form.
What error should I check before submitting an answer?
Check for using the term number as the exponent without checking the starting index; turning an exact result into a decimal too early; checking examples without completing the general proof.
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.
- Model a geometric saving plan
- Extract a coefficient, then solve a logarithm
- Find all cube roots in the complex plane
- Prove a polynomial divisibility claim
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
Related IB Mathematics: Analysis and Approaches unit guides
IB Mathematics: Analysis and Approaches Exam Guide & Review
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International Baccalaureate does not publish a per-unit weighting for this exam.03Topic 3: Geometry and Trigonometry
International Baccalaureate does not publish a per-unit weighting for this exam.04Topic 4: Statistics and Probability
International Baccalaureate does not publish a per-unit weighting for this exam.05Topic 5: Calculus
International Baccalaureate does not publish a per-unit weighting for this exam.How to study IB Mathematics: Analysis and Approaches Unit 1
Start with the organizing decision
Before solving, restate the decision in operational terms: Choose the exact algebraic structure before calculating, and keep the index, domain, or proof claim visible.. Your first written move should be to Name the sequence, expansion, logarithmic form, complex representation, or proof method that matches the prompt..
Practice the same idea in several representations
Rotate through a term rule or finite-sum expression, an exact algebraic statement with its domain, a complex number in Cartesian or polar form. Use each representation to practice Arithmetic and Geometric Sequences and Series, Sigma Notation, and Financial Applications, Scientific Notation, Laws of Exponents and Logarithms, and Exponential Equations, The Binomial Theorem for Natural-Number Powers, Simple Deductive Proof and Identities, Counting Principles, Permutations and Combinations, and the Extended Binomial Theorem (HL), Partial Fractions and Systems of Linear Equations (HL), Complex Numbers: Cartesian, Polar and Euler Forms, De Moivre's Theorem, and Roots (HL), Proof by Induction, Contradiction, and Counterexample (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 term number as the exponent without checking the starting index; turning an exact result into a decimal too early; checking examples without completing the general proof. 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: Analysis and Approaches to see every unit and plan your revision.