IB MYP Mathematics: Key concepts in MYP mathematics: form, logic and relationships (MYP Mathematics)

This study guide is for IB Middle Years Programme (MYP) Mathematics. It follows the IB Mathematics guide (for use from September 2020 or January 2021, first assessment 2022) and covers its section on key concepts: the full list of MYP key concepts, the three that mathematics contributes (form, logic and relationships), example statements of inquiry, and factual, conceptual and debatable inquiry questions. The guide describes a mathematics course and an extended mathematics course; this topic applies to both, because key concepts frame units rather than adding content.

Related pages: revision notes, practice questions, the MYP Mathematics hub, the printable checklist, the syllabus guide and the free diagnostics.

What this topic covers

Part	What you should be able to do
Table 1: the 16 key concepts	Name them and say which three come from mathematics
Form, logic, relationships	Explain each in mathematical terms and recognise it in a task
Statements of inquiry (Table 3)	Read one and pick out the key concept, related concepts and global context
Inquiry questions (Table 4)	Tell factual, conceptual and debatable questions apart and write your own
Links to other subjects	Name key concepts from other subjects that connect to each branch

Why key concepts matter

A concept is a big idea that stays true across topics and places. MYP units use concepts so that you learn why mathematics works, not only how to run a method. Key concepts help connect courses inside mathematics and link mathematics with other subject groups.

Key concepts sit beside related concepts, which belong to mathematics itself, and a global context, which says why the inquiry matters.

Table 1: the 16 key concepts

Across the MYP the key concepts are: aesthetics, change, communication, communities, connections, creativity, culture, development, form, global interactions, identity, logic, perspective, relationships, systems, and time, place and space.

Mathematics contributes three: form, logic and relationships. The other thirteen can still appear in mathematics units, and the framework chapter links most of them to branches (see below).

Form

Across the MYP, form covers an object’s shape and inner structure: how it is organised, what it essentially is and how it appears.

In mathematics, form means that the structure and shape of an object are told apart by its properties. A square, a rhombus and a rectangle are different forms because their sides, angles and diagonals have different properties. The guide adds that form lets you appreciate the aesthetic side of mathematical constructs.

Form questions ask: what is this object, and which properties decide it? Typical places: classifying shapes, the shape of a graph, the structure of an expression.

Worked example 1: deciding a form from properties

Points A(0, 0), B(4, 1), C(5, 5) and D(1, 4) are joined in order. Decide which form ABCD has.

1. Vectors for opposite sides: AB = (4, 1) and DC = (4, 1). They are equal, so AB and DC are parallel and the same length.
2. BC = (1, 4) and AD = (1, 4). The other pair is also parallel and equal, so ABCD is a parallelogram.
3. Compare adjacent sides: AB² = 4² + 1² = 17 and AD² = 1² + 4² = 17. Adjacent sides are equal, so ABCD is a rhombus.
4. Test for a right angle: AB · AD = 4×1 + 1×4 = 8. This is not 0, so the angle at A is not 90°, and ABCD is not a square.
5. Diagonals: AC · BD = (5)(−3) + (5)(3) = 0, so the diagonals are perpendicular, as expected for a rhombus.

ABCD is a rhombus but not a square. The form is decided by a chain of properties, not by how the sketch looks.

Logic

Across the MYP, logic means reasoning by agreed principles so that you can construct arguments and arrive at conclusions. In mathematics you apply it when deciding things about numbers, shapes and variables, and it lets you explain why a conclusion is valid. The guide warns that this is not the same as the branch of mathematics called symbolic logic.

The glossary supports this with related terms: justification (valid reasons that explain why a rule works), lines of reasoning (steps that follow on from one another), proof (logical steps to a required result) and test (trying a rule on different values to see if it holds). Testing is useful, but a test is not a proof.

Worked example 2: from testing to justifying

Claim: when you square two consecutive whole numbers and subtract the smaller square from the larger, the result is always odd.

Test. 4² − 3² = 16 − 9 = 7, 6² − 5² = 36 − 25 = 11, 10² − 9² = 100 − 81 = 19. Each is odd. This supports the claim but cannot cover every pair.

Justify.

1. Call the smaller number n, a whole number. The next one is n + 1.
2. The difference of the squares is (n + 1)² − n² = n² + 2n + 1 − n² = 2n + 1.
3. 2n is even for every whole n, so 2n + 1 is odd.
4. The result is also the sum of the two numbers, n + (n + 1).

The claim is true for every pair. The algebra covers all cases at once, which a list of examples never can.

Relationships

Across the MYP, relationships are the links among objects, properties, ideas and people, including our links with the world. When a relationship changes, there are consequences, some small and some far-reaching.

In mathematics, relationships link quantities, properties or concepts, and you can write them as models, rules or statements. They let you explore the patterns you find around you. The glossary also says that repeated features of a pattern can be described as relationships or general rules.

The guide’s forms of representation include words, formulae, tables, charts, graphs, diagrams and models: any way of showing mathematical information. One relationship can be shown in several of these, and moving between them is a skill in itself.

Worked example 3: one relationship, four representations

A taxi charges a fixed 3.50 dollars plus 1.20 dollars for each kilometre.

• Words: the fare is the fixed charge plus the distance charge.
• Rule: F = 3.5 + 1.2d, where d is the distance in km and F is the fare in dollars.
• Table:

d (km)	0	10	20
F (dollars)	3.50	15.50	27.50

• Graph: a straight line with intercept 3.5 on the F axis and gradient 1.2.

For a 14 km trip: F = 3.5 + 1.2 × 14 = 3.5 + 16.8 = 20.30 dollars.

Work backwards. A fare of 30.20 dollars gives 30.2 = 3.5 + 1.2d, so 1.2d = 26.7 and d = 26.7 ÷ 1.2 = 22.25 km.

A negative distance has no meaning, so the rule applies for d ≥ 0 only.

Example statements of inquiry (Table 3)

A statement of inquiry joins a key concept, related concepts and a global context into one sentence. The guide’s table gives seven examples, summarised here.

Idea in the statement	Key concept	Related concepts	Global context	Possible study
Structure design depends on how finite resources are used responsibly	Form	Space, quantity	Fairness and development	Geometry and trigonometry: volume
Logic helps judge whether measured findings are valid	Logic	Validity	Orientation in space and time	Parallel lines and transversals
Models of relationships improve decisions	Relationships	Models, representation	Identities and relationships	Quadratic functions
Patterns in forms and space can fuel creativity	Form	Patterns, space	Personal and cultural expression	Transformations
A logical process that simplifies patterns helps us understand the world	Logic	Patterns, simplification, models	Scientific and technical innovation	Projectile motion
Relationships and the changes that affect them help us see human impact on environmental systems	Relationships	Systems, change	Globalization and sustainability	Exponentials and logarithms
Natural patterns described as relationships lead to principles and discoveries	Relationships	Patterns	Scientific and technical innovation	Line of best fit

Read these as models of the format. Teachers can choose other related concepts and global contexts.

How to read a statement of inquiry

1. The big idea names the key concept.
2. The supporting ideas are the related concepts, such as validity or models.
3. Why it matters gives the global context.

Inquiry questions (Table 4)

Inquiry questions come from the statement of inquiry and steer the unit. There are three types.

Type	Purpose	Example (adapted from the guide)
Factual	Remember facts and topics	How do the slopes of two perpendicular lines compare?
Conceptual	Analyse big ideas	Why can approximation be useful?
Debatable	Evaluate perspectives and develop theories	How big is infinity?

The guide’s other examples include factual questions on volume versus area and on independent events, conceptual questions on what a “solution” of a function means, and debatable questions on whether order or chaos is more natural.

A quick test: a factual question has an answer you can look up or calculate. A conceptual question has an answer that explains an idea and applies across many cases. A debatable question has no single right answer, and you must weigh evidence and argue a position.

Worked example 4: answering the factual questions

Gradients of perpendicular lines. A line through (0, 1) and (3, 3) has gradient (3 − 1)/(3 − 0) = 2/3. A line perpendicular to it has gradient −3/2. Check: (2/3) × (−3/2) = −1. The product of the gradients of perpendicular lines is −1.

Volume and area. Take a cube of side 2 cm. Surface area = 6 × 2² = 24 cm². Volume = 2³ = 8 cm³. Enlarge the side by scale factor 3, giving side 6 cm. Surface area = 6 × 6² = 216 cm², which is 216 ÷ 24 = 9 = 3² times larger. Volume = 6³ = 216 cm³, which is 216 ÷ 8 = 27 = 3³ times larger. Area scales with the square of the scale factor and volume with the cube, so they are different kinds of quantity.

Independence. Flip a fair coin and roll a fair die. P(head) = 1/2 and P(6) = 1/6. P(head and 6) = 1/12, and 1/2 × 1/6 = 1/12. The product rule holds, so the events are independent.

Writing your own questions

Start from a statement of inquiry such as “Changes in relationships show us how people affect natural systems”.

• Factual: What is the value of 1.05ⁿ after 3 years of 5% growth?
• Conceptual: How does a constant percentage change differ from a constant amount of change?
• Debatable: Should long-term plans for resources be based on models that we know are simplified?

Key concepts from other subjects

The framework chapter links each branch of mathematics to key concepts from other MYP subjects, with example topics.

Branch	Key concepts from other subjects, with example topics
Numerical and abstract reasoning	Change (ratios, number bases); communication (number lines, units); connections (number bases, sequences, Venn diagrams); development (sequences, prime numbers); identity (sets, factors); systems (sets, number systems)
Thinking with models	Aesthetics (patterns, sequences, graphs); change (algebraic expressions, transformations); connections (patterns, functions and graphs); systems (functions, series); time, place and space (functions, equations)
Spatial reasoning	Aesthetics (shapes, transformations); change (identities, transformations); communities (angle and triangle properties); creativity (transformations, similarity and congruency); identity (unit circle, identities); perspective (coordinate geometry, similarity and congruency); time, place and space (three-dimensional coordinates, transformations)
Reasoning with data	Communication (representation, probability of events); communities (samples, populations); connections (successive trials, central tendency); development (successive trials, population sampling); global interactions (population sampling, representations); systems (probability of events, conditional probability)

The guide also lists related concepts from mathematics for each branch, drawn from Table 2: approximation, change, equivalence, generalization, models, patterns, quantity, representation, simplification, space, systems and validity.

Common errors

• Writing “symmetry” or “number” as a key concept. The 16 in Table 1 are the only key concepts listed there. Related concepts such as patterns and equivalence are different.
• Treating form as “what a shape looks like”. Name the properties that decide it.
• Offering five tested examples as a justification. A test checks a rule; justification explains why it works.
• Confusing logic in the MYP with symbolic logic. The guide separates them.

Official syllabus

IB Middle Years Programme Mathematics guide (for use from September 2020 or January 2021), first assessment 2022, published by the International Baccalaureate Organization. Next: revision notes and practice questions.