IB MYP Mathematics: Criteria A and B by year: knowing, understanding and patterns (MYP Mathematics)

This guide covers Criterion A (knowing and understanding) and Criterion B (investigating patterns) as set out in the IB Middle Years Programme Mathematics guide, first assessment 2022. It follows the descriptor sets for Years 1, 3 and 5. The guide has no tiers; where it names standard and extended mathematics, they differ in content, and the criteria are the same. Test yourself with the practice questions and use the revision notes for quick recall. The subject hub and printable checklist show the wider course. If you are not sure where to start, take a free 10-minute diagnostic.

For the other two criteria and a general overview, see the four criteria in practice. This guide goes deeper on A and B and compares the years side by side.

What this unit covers

Criterion	Strands	Maximum	What it rewards
A: Knowing and understanding	i select, ii apply, iii solve	8	Choosing suitable mathematics, using it well, reaching correct solutions
B: Investigating patterns	i techniques, ii describe, iii verify, justify or prove	8	Discovering a pattern, stating a rule, testing and justifying it

Each criterion has levels 1 to 8 in four bands: 1-2 (limited), 3-4 (adequate), 5-6 (substantial) and 7-8 (excellent). A level of 0 means no descriptor is reached. Teachers make a best-fit judgement across the strands.

Criterion A: knowing and understanding

The three strands describe one chain of thought.

• i. Select. You decide which mathematics a problem needs. This is the part that cannot be done for you.
• ii. Apply. You carry out the chosen method successfully.
• iii. Solve correctly. Your solutions are right, across a variety of contexts.

How the bands progress

The bands change the kind of problem you can handle, not the skills listed.

Band	Problems you handle	Situations
1-2	simple	familiar
3-4	more complex	familiar
5-6	challenging	familiar
7-8	challenging	familiar and unfamiliar

Two points stand out. First, the jump from 5-6 to 7-8 is about unfamiliar situations: you must choose the right mathematics when the question does not look like one you have practised. Second, strand iii in every band says you “generally” solve correctly in a variety of contexts, so range of context matters as well as accuracy.

How it differs by year

The guide prints the Criterion A descriptors with the same wording for Years 1, 3 and 5. What changes is the mathematics available to you. A challenging problem in Year 1 draws on the Year 1 framework (for example ratio and area). A challenging problem in Year 5 draws on algebra, Pythagoras’ theorem, trigonometry or exponential models (the guide’s framework places these in years 4 and 5). The ceiling stays level 8, and the problems rise to meet it.

Worked example 1 (Year 1 style, familiar)

A pack of 24 pencils costs 6.60. How much would 40 pencils cost at the same rate?

1. Select: the cost scales with the number of pencils, so use a unit rate.
2. Apply: 6.60 ÷ 24 = 0.275 per pencil.
3. Solve: 0.275 × 40 = 11.00.
4. Check by proportion: 40 is 5/3 of 24, and 6.60 × 5/3 = 11.00. The two routes agree.

Worked example 2 (Year 3 style, an unfamiliar framing)

A rectangular pen is 4 m longer than it is wide, and its fence is 36 m long. Gravel 1 m wide will be laid along the inside of the fence all the way round. How much of the pen is left for grass?

1. Select: the question never mentions an equation, but the unknown width and the perimeter call for forming one. Let the width be w m.
2. Apply: 2(w + w + 4) = 36, so 4w + 8 = 36 and w = 7. The pen is 7 m by 11 m.
3. The grass area is a smaller rectangle, 2 m less in each direction: 5 m by 9 m = 45 m².
4. Check: the pen is 7 × 11 = 77 m², so the gravel covers 77 − 45 = 32 m². Counting the strips directly, two full-length strips of 11 m and two end strips of 5 m give 11 + 11 + 5 + 5 = 32 m², which agrees.
5. Answer in context: 45 m² of grass.

Spotting that the grass rectangle loses 1 m on each side, 2 m in total per direction, is strand i in action.

Worked example 3 (Year 5 style, an exponential model)

A culture of 500 bacteria grows by 12% each hour. After how many whole hours does it first exceed 2000?

1. Select: repeated percentage growth is exponential, so N = 500 × 1.12^t.
2. Apply: solve 500 × 1.12^t > 2000, so 1.12^t > 4.
3. Take logarithms: t > ln 4 ÷ ln 1.12 = 12.23. (The guide lists logarithms under extended mathematics, so a standard-level solution can test whole values of t instead, as in step 4.)
4. Check: at t = 12, N = 500 × 1.12^12 = 1948 (below 2000). At t = 13, N = 2182 (above).
5. Answer: 13 hours. The check shows why 12.23 must be rounded up, not down.

Criterion B: investigating patterns

An investigation is open. You are given a situation, not a method. The guide says tasks that tell you which technique to use are too guided, which limits the level you can earn (see the table below). Teachers should still give enough direction for everyone to begin.

The strands by year

Strand	Year 1	Year 3	Year 5
i	use problem-solving techniques to recognise patterns	choose and use techniques to discover complex patterns	choose and use techniques to discover complex patterns
ii	state relationships or general rules that fit the findings	relationships and/or general rules	general rules
iii	check the pattern works for other examples	verify and justify relationships and/or rules	prove, or verify and justify, general rules

The progression is easy to read. Year 1 asks you to recognise and test. Year 3 asks for complexity and justification. Year 5 requires general rules and gives credit for proof.

How the bands progress (Years 3 and 5)

Band	Strand i	Strand ii	Strand iii
1-2	with teacher support, discover simple patterns	state predictions	not demonstrated at this level
3-4	apply techniques to simple patterns	suggest rules consistent with findings	not demonstrated at this level
5-6	select and apply techniques; complex patterns	describe rules consistent with findings	verify the rule (Year 5: verify its validity)
7-8	as 5-6	describe rules consistent with correct findings	verify and justify (Year 5: prove, or verify and justify)

Year 1 follows the same shape with gentler wording: the 3-4 band is about recognising patterns and suggesting how they work, 5-6 adds relationships or rules and testing on another example, and 7-8 requires correct patterns and findings.

Strand iii first appears at 5-6. Without verification, a rule cannot reach that band.

Caps and traps

• A fully guided task caps the level: 6 in Years 1 and 2, and 4 from Year 3 up.
• From Year 3 up, a rule that is consistent with incorrect findings can earn at most 6, provided the rule is of equivalent complexity.
• The guide gives non-exhaustive examples at Year 5. Simple patterns are linear, or quadratic as ax² or ax² + c. Complex patterns include other quadratics, cubics, quartics, exponentials and rational forms f(x)/g(x).

A method for any investigation

1. Gather data. Build the first six or so cases from a diagram or by calculation, and record them in a table.
2. Look for structure. Test differences, second differences, ratios and links to squares or powers.
3. State a conjecture in words, then as an algebraic rule.
4. Test on a new case you did not use to find the rule.
5. Justify by explaining why the structure of the situation forces the rule, or prove it algebraically.

Worked example 4 (Year 1 style): the tile border

A square pool with n tiles along each side is surrounded by a single row of border tiles, corners included. How many border tiles are needed?

Pool side n	1	2	3	4	5
Border tiles	8	12	16	20	24

The border grows by 4 each time, so the rule begins “multiply n by 4”. When n = 1 the border is 8, so add 4: border = 4n + 4.

Verify with a new case. For n = 7 the rule gives 32. Drawing the pool, each side of the border has 9 tiles, with 4 corners shared: 4 × 9 − 4 = 32. The rule works.

Worked example 5 (Year 3 style): diagonals of a polygon

How many diagonals does an n-sided polygon have?

Sides n	3	4	5	6	7	8
Diagonals d	0	2	5	9	14	20

First differences: 2, 3, 4, 5, 6. Second differences: all 1. A constant second difference points to a quadratic with leading term n²/2.

Try d = n²/2 + bn + c. Using n = 3 (d = 0) and n = 4 (d = 2): 4.5 + 3b + c = 0 and 8 + 4b + c = 2. Subtracting gives 3.5 + b = 2, so b = −1.5 and c = 0. Conjecture: d = n(n − 3)/2.

Test on a new case, n = 9: 9 × 6 ÷ 2 = 27, and the table’s pattern continues 20, 27. For justification, each vertex joins to n − 3 others (not itself or its two neighbours). That gives n(n − 3) joins, and each diagonal is counted from both ends, so divide by 2. The structure forces the rule, which is the justification the 7-8 band asks for.

Worked example 6 (Year 5 style): a proof

Evaluate n(n + 1)(n + 2)(n + 3) + 1 for n = 1, 2, 3.

n	Product + 1	Square root
1	25	5
2	121	11
3	361	19

The results are perfect squares. The roots 5, 11, 19 have differences 6 and 8, so the second difference is 2: try n² + 3n + 1. Conjecture: the expression equals (n² + 3n + 1)².

Test: n = 4 gives 4 × 5 × 6 × 7 + 1 = 841 = 29², and 16 + 12 + 1 = 29.

Proof: pair the outer and inner factors. Let m = n² + 3n. Then n(n + 3) = m and (n + 1)(n + 2) = m + 2. The expression is m(m + 2) + 1 = m² + 2m + 1 = (m + 1)² = (n² + 3n + 1)² for every n. This is a general proof, not a check of cases.

Common errors

• Treating the Criterion A descriptors as different for each year. The wording is the same; the problems get harder.
• Choosing a method because it was used in the last question, not because the problem needs it.
• Stopping at a correct answer in Criterion A without checking it against the context, such as rounding 12.23 hours down.
• Presenting a pattern rule with no table, so there is no evidence of how it was found.
• Fitting a rule to two data points only. Any two points suit many rules.
• Treating “it works for n = 4” as a proof. In Year 5, testing is verification; proof needs algebra or a structural argument.
• Following a worksheet that dictates the method and then expecting level 7-8 in Criterion B.

Next steps

• Try the practice questions.
• Revise with the revision notes.
• Work through the reasoning with data guide for Criterion A practice in statistics.
• Check your progress with the free diagnostics.

Official syllabus

International Baccalaureate Organization, Middle Years Programme Mathematics guide, from 2020, first assessment 2022. Criterion A and Criterion B descriptors for Years 1, 3 and 5 are paraphrased from this guide.