IB MYP Mathematics: The twelve related concepts and their meanings (MYP Mathematics) – Revision Notes

These notes are aligned to the IB Mathematics guide for the Middle Years Programme (for use from September 2020/January 2021, first assessment 2022). They condense Table 2, the twelve related concepts, with the glossary meanings and the branch links. For full explanations and longer worked examples, read the study guide. When you are ready to test yourself, use the practice set. The course hub is MYP Mathematics, the printable checklist tracks your progress, and the free 10-minute diagnostics show what to revise first. The guide gives one list of related concepts for the whole course, with no split between standard and extended mathematics.

Key facts to learn

• The key concepts the guide gives to mathematics are form, logic and relationships.
• Related concepts belong to the subject. They let you look at a key concept more closely and can arise from a unit’s topic or from the way mathematics is done.
• There are twelve in Table 2. Teachers may add others, including from other subject groups.
• A statement of inquiry combines a key concept, related concepts and a global context.
• The four branches are numerical and abstract reasoning, thinking with models, spatial reasoning and reasoning with data.

Glossary meanings in one line each

Concept	Remember it as
Approximation	Near the true value, but not exact
Change	A difference in size, quantity or behaviour
Equivalence	Same in value, so one can replace the other (statements, quantities, expressions)
Generalization	A broad claim built from particular cases
Models	Real-life events shown through expressions, equations or graphs
Patterns	Numbers or objects arranged by a set order or rule
Quantity	How much there is: a count or a measure
Representation	The form in which something is shown
Simplification	Rewriting in an easier form
Space	The geometric dimensions within which an object is described
Systems	Sets of parts that connect to and depend on one another
Validity	Sound, logical mathematics giving a true and accurate conclusion or a sensible interpretation of results

Branch links at a glance

Branch	Related concepts the guide lists for it
Numerical and abstract reasoning	approximation, equivalence, generalization, quantity, simplification, systems, validity
Thinking with models	change, equivalence, patterns, quantity, representation, simplification, systems
Spatial reasoning	change, equivalence, models, patterns, quantity, representation, space, systems
Reasoning with data	all twelve except space: validity, systems, simplification, representation, quantity, patterns, models, generalization, equivalence, change, approximation

Memory hooks:

• Equivalence, quantity and systems appear in all four branches.
• Space appears in spatial reasoning only.
• Reasoning with data has the longest list: eleven of the twelve, everything except space.
• Approximation, generalization and validity sit in numerical reasoning and reasoning with data.
• Models is not named under thinking with models, but a teacher may still use it there.

Methods in steps

Approximating and checking

1. Round each number to one that is easy to use.
2. Calculate with the rounded numbers.
3. Work out the exact value if needed, then find the error: estimate minus exact.
4. Say whether the estimate is too high or too low, and decide whether the context needs rounding up or down.

Describing a change

1. Change = final value − starting value.
2. Rate = change ÷ time (or ÷ the other quantity).
3. Percentage change = change ÷ starting value × 100.

Showing equivalence

1. Expand and collect terms on both sides, or convert to the same form.
2. If the forms match, they are equivalent.
3. To show they are not equivalent, one counterexample is enough.
4. A matching test value supports equivalence but does not prove it.

Generalizing from a pattern

1. Tabulate terms and find differences.
2. Propose a rule, such as an + b.
3. Test it on every given term and on a new one.
4. Explain why it works, if you can.

Using a model

1. Define the variables and units.
2. Write the rule (expression, equation or graph).
3. Calculate, then say what the model leaves out.
4. Judge whether the answer is valid in the situation.

Worked reminders

Approximation. 6.1 × 29.8 ≈ 6 × 30 = 180. Exact: 181.78. The estimate is 1.78 too low.

Change. A price rises from 40 to 46: change 6, percentage change 6 ÷ 40 × 100 = 15%.

Equivalence. 4(x − 1) + 2x = 4x − 4 + 2x = 6x − 4. At x = 3 both sides give 14.

Generalization. Terms 4, 9, 14, 19 rise by 5, so the rule is 5n − 1. The 10th term is 49.

Quantity. 3.5 m³ is 3.5 × 1000 = 3500 litres.

Simplification. (10p²q) / (15pq²) = 2p / (3q), after dividing by 5, p and q.

Space. At scale 1:30 a real length of 4.5 m (450 cm) is 450 ÷ 30 = 15 cm on the model. Lengths scale by k, areas by k², volumes by k³.

Systems. 1011 in base 2 is 8 + 0 + 2 + 1 = 11.

Models and patterns. The sequence 3, 6, 9, 12 gives the model T = 3n. For n = 7, T = 21.

Representation. The rule y = 3 − x gives the table x = 0, 1, 2, 3 and y = 3, 2, 1, 0. The graph is a straight line sloping down with gradient −1.

Validity. x² + 2x − 24 = 0 gives x = 4 or x = −6 for a length. Reject −6 because a length cannot be negative.

Must-know distinctions

Pair	Difference
Equivalence and equality of numbers	Equivalence covers statements and expressions too, such as 2(x + 3) and 2x + 6
Approximation and error	The approximation is the value used. The error tells you how far from exact it is
Pattern and generalization	A pattern is the ordered set. A generalization is the rule you state about it
Model and representation	A model depicts a real-life event. Representation is the form it is shown in, such as a table or graph
Simplification and approximation	Simplifying an expression keeps its value exactly. Approximating changes the value slightly
Quantity and number	A quantity is an amount, usually with units
Validity and correctness	Valid means the reasoning is sound and the result makes sense in context

Command terms that go with the concepts

The guide’s command terms tell you what a question wants. In your own words:

• Estimate: obtain an approximate value.
• Show: give the steps of a calculation or derivation.
• Describe: give a detailed account of a pattern or process.
• Explain: give reasons or causes.
• Justify: give valid reasons or evidence for a conclusion.
• Comment: give a judgment on a given result.

A question on validity often says “justify” or “comment”, so state your reason and refer to the context.

Linking concepts to a unit

To plan or describe a unit, pick a key concept, add related concepts, then add a global context. Example: key concept relationships; related concepts patterns and generalization; global context scientific and technical innovation; possible study statistics with scatter graphs. A statement could be: “Describing patterns in measurements allows us to make general rules that can be tested.”

Link to the criteria as well. Criterion B (investigating patterns) leans on patterns and generalization. Criterion D (applying mathematics in real-life contexts) leans on models and validity. See criteria in practice for more.

Quick self-test

1. Which branch is space linked with in the guide?
2. Name the three related concepts the guide lists for all four branches.
3. Estimate 6.1 × 29.8 by rounding to whole numbers, then find the error against the exact value.
4. A price rises from 40 to 46. Find the percentage change.
5. Are 4(x − 1) + 2x and 6x − 4 equivalent? Show how you know.
6. Find the rule for 4, 9, 14, 19 and give the 10th term.
7. Convert 3.5 m³ to litres.
8. Simplify (10p²q) / (15pq²).
9. Write 1011 (base 2) in base 10.
10. A cost model is C = 5 + 2n. Find C when n = 8, and name one thing the model might ignore.

Answers

1. Spatial reasoning.
2. Equivalence, quantity and systems.
3. 6 × 30 = 180. Exact 181.78. Error = 1.78, so the estimate is too low.
4. 6 ÷ 40 × 100 = 15%.
5. Yes. 4(x − 1) + 2x = 4x − 4 + 2x = 6x − 4.
6. Rise by 5 each time, so 5n − 1. The 10th term is 5 × 10 − 1 = 49.
7. 3.5 × 1000 = 3500 litres.
8. 2p / (3q).
9. 8 + 0 + 2 + 1 = 11.
10. C = 5 + 2 × 8 = 21. It may ignore discounts for bulk buying, extra charges or limits on n (any sensible point).

Where credit is usually lost

• Giving an everyday meaning of a concept when the question asks for its meaning in mathematics.
• Quoting a generalization that has been checked against only the given terms.
• Treating a matching test value as a proof of equivalence.
• Leaving out units, or forgetting to convert them, when working with a quantity.
• Scaling a volume by k instead of k³, or an area by k instead of k².
• Rounding to the nearest value when the context needs rounding up or down.
• Accepting a negative length or an impossible count because it solves the equation.
• Describing a model as perfect, without saying what it ignores.
• Claiming a link between two variables is causal because they are correlated.

Official syllabus

IB Mathematics guide, Middle Years Programme (for use from September 2020/January 2021; first assessment 2022), published by the International Baccalaureate Organization. Sections used: Table 2 (related concepts in mathematics), the branch descriptions with their links to MYP concepts, and the appendix glossary of related concepts.