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

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – the IB holds copyright in its own papers. Use these alongside the official past papers available through your school or the IB store.

These questions cover the key concepts section of the IB Middle Years Programme Mathematics guide (for use from September 2020 or January 2021, first assessment 2022): the key concepts form, logic and relationships, statements of inquiry, and factual, conceptual and debatable inquiry questions. The topic applies to the guide’s mathematics and extended mathematics courses. The mark totals are for self-marking only.

Related pages: the study guide, the revision notes, the MYP Mathematics hub and the printable checklist.

Questions

1. State the three MYP key concepts that mathematics contributes. [2]

2. Explain what the key concept form means in MYP mathematics, and give one example of two forms that are told apart by their properties. [3]

3. Classify each question as factual, conceptual or debatable.

(a) What is the area of a circle with radius 6 cm? [1] (b) Why does multiplying a positive number by a fraction smaller than 1 give a smaller number? [1] (c) Should a school judge a class’s progress by one average alone? [1]

4. The points P(1, 1), Q(5, 3), R(4, 5) and S(0, 3) are joined in order. Show that PQRS is a rectangle but not a square. [4]

5. Claim: the sum of five consecutive integers is always a multiple of 5.

(a) Test the claim using the integers 4, 5, 6, 7 and 8. [1] (b) Justify the claim for every set of five consecutive integers. [2] (c) Explain why testing five more sets would still not justify it. [1]

6. A mobile plan costs 15 dollars per month plus 0.08 dollars for each minute of calls. The monthly cost is C dollars for m minutes, so C = 15 + 0.08m.

(a) Find the cost for 250 minutes. [1] (b) Find the number of minutes for a monthly cost of 47 dollars. [2] (c) State the gradient of the graph of C against m and explain what it means. [1] (d) State one restriction on the values of m for which the rule is valid. [1]

7. A unit studies solving linear equations by balancing both sides. Write one statement of inquiry for it. Name the key concept, one related concept from the mathematics list and a global context. [3]

8. The guide links key concepts from other subjects to the spatial reasoning branch.

(a) Name two of these key concepts. [2] (b) For each one, give a spatial reasoning topic it can be linked with. [2]

9. A cube has side length 3 cm. It is enlarged by scale factor 4.

(a) Find the surface area and volume of the original cube. [2] (b) Find the surface area and volume of the enlarged cube. [2] (c) Find the factor by which each increased, and state the general rule for scale factor k. [2]

10. A student investigates the sum of four consecutive integers, starting at n.

(a) Complete a table of the sum for n = 1, 2, 3 and 4. [2] (b) Write a rule for the sum in terms of n, in simplest form. [2] (c) Show that the sum is never a multiple of 4. [2] (d) Part (b) mainly uses one mathematics key concept and part (c) mainly uses another. Name each and explain your choices. [2]

11. A bag holds 3 red and 2 blue counters. Two counters are drawn one after the other.

(a) Find P(red) on the first draw, and P(both red) if the first counter is replaced before the second draw. [2] (b) Find P(both red) if the first counter is not replaced. [2] (c) Use the product rule to decide in which case the two draws are independent. [2]

Answers

1. Form [1], logic and relationships [1]. Form, logic, relationships. [2] Examiner insight: Naming related concepts such as patterns or validity instead of key concepts is an easy way to lose credit, so check the three against Table 1 of the guide.

2. Form is about the underlying structure and shape of something being told apart by its properties [1]. Example: a rectangle and a rhombus [1]. A rectangle has four right angles, while a rhombus has four equal sides, so each is recognised by a different property [1]. [3] Examiner insight: A good answer names the properties that separate the two forms; “they look different” does not use the idea of form.

3. (a) Factual: it has one calculable answer, 36π cm² [1]. (b) Conceptual: it asks for an idea about multiplication that applies to all such numbers [1]. (c) Debatable: there is no single right answer and you must weigh evidence [1]. [3] Examiner insight: If you add a one-phrase reason to each label, you show the sorting is deliberate rather than guessed.

4. PQ = (4, 2) and SR = (4, 2), so PQ and SR are parallel and equal [1]. PS = (−1, 2), so PQ · PS = 4(−1) + 2(2) = 0 [1]. The angle at P is 90° and PQRS is a parallelogram, so it is a rectangle [1]. PQ² = 20 and PS² = 5, so adjacent sides are unequal, and it is not a square [1]. [4] Examiner insight: A “show that” answer gains credit for every step of the reasoning, so state both the parallelogram property and the right angle before naming the rectangle.

5. (a) 4 + 5 + 6 + 7 + 8 = 30, which is 5 × 6 [1]. (b) n + (n + 1) + (n + 2) + (n + 3) + (n + 4) = 5n + 10 [1] = 5(n + 2), a multiple of 5 for every whole n [1]. (c) Examples can never cover all integers, whereas the algebra covers every case [1]. [4] Examiner insight: Using letters for the general case is the step that turns a test into a justification; a longer list of examples tends to earn less.

6. (a) C = 15 + 0.08 × 250 = 35 dollars [1]. (b) 47 = 15 + 0.08m, so 0.08m = 32 [1], and m = 400 minutes [1]. (c) The gradient is 0.08, the cost in dollars for each extra minute of calls [1]. (d) m ≥ 0, because time cannot be negative (a sensible upper limit is also acceptable) [1]. [5] Examiner insight: Interpreting the gradient in the context of the problem, with units, is what separates a relationship from a bare number.

7. Example: “Balancing both sides of an equation shows how equivalent forms help us solve problems that matter in daily life.” [1] Key concept: relationships (or logic, if the unit’s focus is the reasoning steps) [1]. Related concept: equivalence; global context: scientific and technical innovation or identities and relationships (any sensible choice from the six) [1]. [3] Examiner insight: Several answers are acceptable; credit tends to follow whether the sentence holds a big idea and links it to a context rather than only naming a topic.

8. (a) Any two of: aesthetics, change, communities, creativity, identity, perspective, time, place and space, for example perspective [1] and communities [1]. (b) A matching topic for each: perspective with coordinate geometry or similarity and congruency [1]; communities with angle properties or triangle properties [1]. Other valid pairs are creativity with transformations and identity with the unit circle. [4] Examiner insight: Keep the link accurate: the topic you give should be one that the guide pairs with the concept you named.

9. (a) Surface area = 6 × 3² = 54 cm² [1]; volume = 3³ = 27 cm³ [1]. (b) New side = 12 cm. Surface area = 6 × 12² = 864 cm² [1]; volume = 12³ = 1728 cm³ [1]. (c) Area: 864 ÷ 54 = 16 = 4², and volume: 1728 ÷ 27 = 64 = 4³ [1]. In general, area scales by k² and volume by k³ [1]. [6] Examiner insight: Show the side length of the enlarged cube before squaring or cubing, so that one slip does not hide the rest of your working.

10. (a) n = 1: 10; n = 2: 14; n = 3: 18; n = 4: 22 all four values correct [1] with a method shown, such as adding the four integers [1]. (b) n + (n + 1) + (n + 2) + (n + 3) [1] = 4n + 6 [1]. (c) 4n + 6 = 4(n + 1) + 2 [1], so the remainder on division by 4 is always 2 and the sum is never a multiple of 4 [1]. (d) Part (b): relationships, because it expresses the connection between n and the sum as a rule [1]. Part (c): logic, because it gives a justification that works for all n [1]. [8] Examiner insight: In a “show that” argument the final sentence should link the algebra back to the claim, saying why the remainder of 2 rules out a multiple of 4.

11. (a) P(red) = 3/5 [1]. With replacement, P(both red) = 3/5 × 3/5 = 9/25 [1]. (b) P(both red) = 3/5 × 2/4 [1] = 6/20 = 3/10 [1]. (c) With replacement the second draw has the same probability, 3/5, whatever happened first, so P(R then R) = P(R) × P(R) and the draws are independent [1]. Without replacement 3/10 ≠ 9/25, so the draws are not independent [1]. [6] Examiner insight: Naming the product rule and comparing the two values is stronger than saying only that “the numbers change”.

Where marks are usually lost

• Giving related concepts, such as patterns or validity, when a key concept is asked for.
• Describing form by appearance rather than by properties.
• Treating several worked examples as if they proved a general rule.
• Leaving out the reason when asked to classify an inquiry question.
• Stating a statement of inquiry with no global context.
• Forgetting that area scales by k² and volume by k³.
• Giving a rule or model without saying where it is valid.
• Writing answers to a “show that” without every step in a clear line of reasoning.

Next steps

• Re-read the study guide and the revision notes.
• Use the MYP Mathematics hub and printable checklist to see what else to cover.
• Take the free 10-minute diagnostics to find weak spots.
• Book a free trial class.

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.