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

This guide is aligned to the IB Mathematics guide for the Middle Years Programme (for use from September 2020/January 2021, first assessment 2022). It covers Table 2, the twelve related concepts for mathematics, and the guide’s glossary meanings of them. It also covers branch links and unit planning. The guide sets no separate list of related concepts for standard and extended mathematics, so everything here applies across the course. For a quick recall version, use the revision notes. To test yourself, use the practice set. The course hub is at MYP Mathematics and the printable checklist lists every topic. Try the free 10-minute diagnostics to see where you stand.

Why related concepts matter

The MYP frames learning with key concepts and related concepts. For mathematics the guide names three key concepts: form, logic and relationships. Related concepts are narrower. They belong to the discipline and let a class look at a key concept in more detail. The guide says a related concept can arise from what a unit is about, or from how mathematics is done, meaning its features and processes. It also says teachers may choose related concepts beyond Table 2, including ones from other subject groups.

A statement of inquiry joins a key concept, related concepts and a global context. The guide names six global contexts, such as fairness and development, and scientific and technical innovation.

The twelve concepts at a glance

Related concept	Meaning in the guide’s glossary (paraphrased)
Approximation	A value or representation that comes near to the true one without being exact
Change	A difference in how large something is, how much there is, or how it acts
Equivalence	Being exactly the same in value, so that one can stand in for the other; used for statements, quantities and expressions
Generalization	A broad statement built from particular cases
Models	Real-life events shown through expressions, equations or graphs
Patterns	Collections of 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 something 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 that leads to a conclusion that is true and accurate, or to a sensible interpretation of results

Which branch each concept links to

For each branch, the guide lists related concepts that could be used within it. The lists are suggestions, not limits.

Related concept	Numerical and abstract reasoning	Thinking with models	Spatial reasoning	Reasoning with data
Approximation	yes			yes
Change		yes	yes	yes
Equivalence	yes	yes	yes	yes
Generalization	yes			yes
Models			yes	yes
Patterns		yes	yes	yes
Quantity	yes	yes	yes	yes
Representation		yes	yes	yes
Simplification	yes	yes		yes
Space			yes
Systems	yes	yes	yes	yes
Validity	yes			yes

Equivalence, quantity and systems appear under all four branches. Space appears under spatial reasoning only. The list for thinking with models does not name “models”, a reminder that teachers may bring in any concept that fits a unit.

Worked examples, concept by concept

Each example below shows the concept doing real work in a calculation or argument. They suit the range from MYP 1-3 to later MYP years.

Approximation

Estimate 4.83 × 19.7, then compare with the exact value. Round each factor to the nearest whole number that is easy to use: 4.83 → 5 and 19.7 → 20. Estimate: 5 × 20 = 100. Exact: 4.83 × 19.7 = 95.151. Error of the estimate: 100 − 95.151 = 4.849, about 5.1% of the exact value. Both factors were rounded up, so the estimate is too high. The direction of an error is often as useful as its size.

Context decides the rounding. A floor measures 4.8 m by 3.9 m and is tiled with square tiles of side 0.5 m. Floor area = 4.8 × 3.9 = 18.72 m². Each tile covers 0.25 m². Tiles needed = 18.72 ÷ 0.25 = 74.88. You cannot buy 0.88 of a tile, so round up to 75. Buying 74 would leave a gap, so the situation sets the rounding rule.

Change

A seedling grows from 12 cm to 27 cm in 5 days. Change in height = 27 − 12 = 15 cm. Rate of change = 15 ÷ 5 = 3 cm per day. Percentage change = 15 ÷ 12 × 100 = 125%. These describe one change three ways: its size, its speed and its size relative to the start. A linear model predicts 12 + 3 × 10 = 42 cm at day 10, but whether growth stays steady is a question of validity.

Equivalence

Show that 2(x + 3) + x and 3x + 6 are equivalent, and that (x + 3)² and x² + 9 are not. Expand and collect: 2(x + 3) + x = 2x + 6 + x = 3x + 6, so they are equivalent. A test value supports this but does not prove it: at x = 5 both give 21. For the second pair, a single value is enough to show they differ. At x = 1, (1 + 3)² = 16 but 1² + 9 = 10. Equivalence also covers fractions, decimals and percentages: 18/24 = 3/4 = 0.75 = 75%.

Generalization

A pattern of tiles has 5, 8, 11 and 14 tiles in figures 1 to 4. Find a general rule. The terms rise by 3 each time, so the rule involves 3n. At n = 1, 3n gives 3, and we need 5, so add 2. Rule: tiles = 3n + 2. Check against the data: n = 2 gives 8, n = 3 gives 11, n = 4 gives 14. Prediction: figure 10 has 3 × 10 + 2 = 32 tiles. A generalization from few cases can be wrong. The terms 2, 4, 8 suggest doubling (next term 16), yet n² − n + 2 also gives 2, 4, 8 and then 14. Test a rule on a new case and explain why it works.

Models

A taxi company charges a fixed 4 plus 1.5 per kilometre. Model the cost C for d km. C = 4 + 1.5d. Cost for 12 km: 4 + 1.5 × 12 = 22. Largest whole number of kilometres affordable with 30: 4 + 1.5d ≤ 30, so d ≤ 26 ÷ 1.5 = 17.33, giving 17 km (cost 29.5). The model ignores waiting time, night rates and rounding of fares. Saying what a model leaves out is part of using it well.

Patterns

Find the rule for 1, 3, 6, 10, 15. First differences: 2, 3, 4, 5. Second differences: 1, 1, 1. A constant second difference points to a quadratic rule with leading term n²/2. Try n(n + 1)/2: n = 1 gives 1, n = 2 gives 3, n = 3 gives 6, n = 4 gives 10, n = 5 gives 15. All match. The 20th term is 20 × 21 ÷ 2 = 210.

Quantity

A tank measures 2.5 m by 1.2 m by 0.8 m. How many litres does it hold when full? Volume = 2.5 × 1.2 × 0.8 = 2.4 m³. One cubic metre is 1000 litres, so the tank holds 2400 litres. A measured quantity comes with units. The number 2.4 means little without “m³”, and the conversion is part of the mathematics.

Representation

Show y = 2x − 1 in four ways. Words: double the input, then subtract one. Table: x = 0, 1, 2, 3 gives y = −1, 1, 3, 5. Graph: a straight line with gradient 2 that crosses the y-axis at −1. Equation: y = 2x − 1. The table gives values quickly, the graph shows the steady rise, and the equation allows exact calculation. The guide’s glossary names several forms of mathematical representation, including graphs, tables, diagrams, charts, formulae, words and models.

Simplification

Simplify (6x²y) / (9xy²). Divide top and bottom by the common factor 3: (2x²y) / (3xy²). Cancel one x and one y: 2x / (3y). Check with x = 3, y = 2: the original is 108/108 = 1 and the simplified form is 6/6 = 1. Simplification also applies to models. Treating a round pond as a cylinder of radius 1.5 m and depth 0.8 m gives V = π × 1.5² × 0.8 ≈ 5.65 m³, which is easier to calculate than the real shape.

Space

A model of a shed is built to scale 1:40. The model is 15 cm by 10 cm by 6 cm. Real lengths: 15 × 40 = 600 cm = 6 m; 10 × 40 = 400 cm = 4 m; 6 × 40 = 240 cm = 2.4 m. Model volume = 15 × 10 × 6 = 900 cm³. Real volume = 6 × 4 × 2.4 = 57.6 m³. The volume scale factor is 40³ = 64 000, and 900 × 64 000 = 57 600 000 cm³ = 57.6 m³. Lengths scale by 40, areas by 40², volumes by 40³. Working in one, two or three dimensions is working with space.

Systems

Convert between binary and decimal. 1101 in base 2 = 1 × 8 + 1 × 4 + 0 × 2 + 1 × 1 = 13. To write 45 in base 2, take the largest powers of two: 32 + 8 + 4 + 1, so the digits for 32, 16, 8, 4, 2, 1 are 1, 0, 1, 1, 0, 1. Answer: 101101. A number system is a group of interrelated elements with rules for combining them. The guide also lists sets, number systems, probability of events and conditional probability as places where the key concept of systems applies, in the branch descriptions.

Validity

A rectangle has length (x + 2) m and width x m. Its area is 35 m². Find x. (suited to later MYP years) x(x + 2) = 35, so x² + 2x − 35 = 0, which factorises as (x + 7)(x − 5) = 0. Solutions: x = 5 or x = −7. A width of −7 m is not valid in this situation, so x = 5: the rectangle is 7 m by 5 m, and 7 × 5 = 35 confirms it. Validity also applies to conclusions from data. Suppose ice-cream sales and sunburn cases rise together across a summer. The correlation is real, but concluding that ice cream causes sunburn is not valid. Hot sunny weather is a likely common cause. The guide names causation versus correlation under reasoning with data.

Using the concepts to plan a unit

Here is an original unit, not one of the guide’s own.

• Key concept: relationships.
• Related concepts: models, quantity, representation, validity.
• Global context: fairness and development.
• Statement of inquiry: Models that represent how quantities relate can help communities share a limited resource fairly, if their conclusions are valid.
• Possible study: algebra, linear functions, using a water-tariff table.
• Inquiry questions. Factual: what does 12 m³ cost under a fixed charge plus a unit rate? Conceptual: why is a simplifying model still useful? Debatable: should heavy users pay a higher rate per unit?

Notice how each concept does a job. Quantity gives the units, representation gives the table and graph, the model gives the equation, and validity asks whether the equation still holds for very large or very small use.

Links to the assessment criteria

Criterion B (investigating patterns) fits patterns and generalization: you find a rule and test it. Criterion D (applying mathematics in real-life contexts) fits models and validity: you build an equation and judge whether it makes sense. See criteria in practice and reasoning with data for more.

Common errors

• Using an everyday meaning instead of the glossary sense, such as treating “equivalence” as only “equal”.
• Forgetting that an approximation is close, and that its error tells you how close.
• Calling a rule a generalization when it was fitted to two cases and never tested.
• Assuming each concept belongs to exactly one branch. Most appear in several.
• Stating that a model is “correct” instead of saying where it holds and what it ignores.
• Leaving a solution without checking it against the context.

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.