A good MYP Criterion B investigation gives students something worth noticing and teachers something worth assessing. It needs an accessible starting point, room to choose an approach, a pattern that rewards further exploration, and a reason to explain why a rule works.

Crossing Paths is an original investigation for MYP Years 4-5. Students count uncovered tiles in a square courtyard crossed by two diagonal paths. The diagrams are simple to read, but the generalisation has a useful twist: odd and even grid sizes behave differently.

The free pack below includes a student task, a task-specific clarification, a complete worked solution and a constructed sample response with teacher annotations. It is designed to help with the planning and the marking, including the often difficult distinction between checking examples and justifying a general rule.

Download the free investigation pack

For students: a three-page task with diagrams, investigation prompts and working space.
For teachers: a seven-page guide with planning notes, marking clarification, answers, annotated responses and feedback prompts.

Download the complete pack - 10-page PDF

Free for classroom copying. The sample responses are constructed teaching examples, not real student submissions or IB-scored exemplars.

The student task: how many tiles are uncovered?

Start with a square grid of tiles, with n tiles along each side. One path follows the tiles from the top-left corner to the bottom-right corner. The other follows the opposite diagonal. Any tile belonging to a path is covered; all other tiles are uncovered. A tile shared by both paths is counted once.

Three-by-three, four-by-four, five-by-five and six-by-six grids with both diagonals coloured; the shared central tile is gold in odd grids — Four starting grids. The white tiles are the quantities students investigate. These diagrams count whole tiles, rather than the area of a thin line drawn across a grid.

Investigate the number of uncovered tiles as the whole-number side length changes. Develop a general rule, or rules, test new cases independently, and explain why your conclusion applies to every permitted grid size.

The domain is whole numbers n ≥ 2. Students can draw, count by rows, organise a table, use algebra, or design a spreadsheet. The student sheet includes an optional results table, but learners can adapt it or use another representation.

For teachers, this matters: the task should give students room to make investigative decisions. If a worksheet prescribes every calculation and announces how to split the data before anyone explores, it offers less evidence of independent mathematical thinking.

Why this pattern produces useful Criterion B evidence

Students can begin by counting. As they generate results, they need to decide how to organise them and whether their first conjecture accounts for every size. A rule that works for one group of grids may fail for another. That creates a natural reason to revise a claim.

The mathematical demand comes from linking a conditional rule to the diagram. It is not enough to identify a familiar-looking list of numbers. Students need to account for what changes when the two diagonals share a tile.

The IB mathematics subject brief identifies investigating patterns as Criterion B. This resource supplies an original classroom task; assessment decisions should use the applicable descriptors in your school’s current licensed mathematics guide.

A practical two-lesson sequence

Launch the arrangement. Allow about 10 minutes to establish what a path tile is and what students are counting. Clarify the diagram without demonstrating a complete counting algorithm.
Allow individual exploration. Give students around 30 minutes to choose a strategy, generate cases and record a conjecture. Ask neutral questions such as “How did you count?” or “Does that cover every grid size?”
Record a provisional rule. Use the final 10 minutes of the first lesson for students to explain what they think is happening and identify anything they still need to check.
Verify and justify. In the second lesson, students test fresh cases, develop a general explanation and review their conclusion. The pack suggests n = 10 and n = 11 so that both types of grid are checked.

Students need square numbers, substitution and some confidence with variables. A calculator or spreadsheet is optional. For classes that need more access support, use large-print diagrams or physical tiles. Record substantial hints, especially if they supply the key structural insight.

Task-specific clarification: make the evidence concrete

A useful clarification translates expectations into this investigation without turning them into a checklist of features that automatically earn marks. The teacher guide separates three kinds of evidence:

Evidence to discuss while marking Crossing Paths
Focus	Look for
Investigative choices	A chosen approach, useful cases, organised findings and a visible response to an incomplete conjecture.
Generalisation	Defined variables, a rule covering each relevant case, and clear conditions for when it applies.
Validity	New cases checked independently, followed by a structural explanation of the rule.

Use the complete response for a best-fit judgment. A neat table, an algebraic formula or an extra page of working does not determine the level by itself. Keep the evidence of investigation distinct from a separately planned assessment of communication.

Teacher notes: the worked solution

This section contains the key insight. Each diagonal has n tiles. In an even-sized grid, the diagonals share no tile, so the paths cover 2n tiles. In an odd-sized grid, they share the central tile, so the number of distinct path tiles is 2n − 1.

Subtracting the path tiles from the total gives:

Even n: U = n² − 2n = n(n − 2).
Odd n: U = n² − 2n + 1 = (n − 1)².

Teacher diagram comparing a six-by-six grid with no shared diagonal tile and a seven-by-seven grid with one shared gold centre tile, alongside the two general rules — The extra 1 in the odd-grid rule corrects the double-counting of the central tile.

The worked guide includes the full results table, equivalent forms of the rules, a position-based proof of the overlap, and independent row counts for the two new cases. It also solves the optional extension: finding every permitted grid size with exactly 120 uncovered tiles.

Verification is not the same as justification

For n = 10, the rule predicts 80 uncovered tiles. Counting eight uncovered tiles in each of ten rows independently confirms that prediction. For n = 11, ten rows contain nine uncovered tiles and the central row contains ten, giving 100.

Those checks provide evidence for the conjecture. They do not, by themselves, establish that every later case will work. A general justification explains why each diagonal contains n tiles and precisely when a shared tile must occur.

This distinction is useful when marking. A response may develop correct rules and test them successfully while still lacking a reason they apply to every whole-number side length. Feedback such as “Explain what each term counts in any grid” is more helpful than simply asking for more examples.

Use the annotated response for a marking conversation

The guide contains a complete constructed model response, with teacher commentary beside each stage: choosing a method, refining an idea, stating the rules, checking new cases, explaining the overlap and limiting the conclusion to the arrangement investigated.

First page of the constructed student response, with annotations explaining the evidence of method choice, discovery, generalisation and independent verification — An excerpt from the annotated model. The full response continues with a general justification in the teacher PDF.

A contrasting extract gives correct rules and checks but claims they must always work because two examples were successful. Comparing the two helps a department discuss the evidence behind adjacent achievement bands. The suggested judgments are illustrative, not official IB marks or guaranteed outcomes.

For a short moderation record, note the proposed level, the locations of the evidence, support given, the reason for the adjacent-band decision and one useful next step.

Adapt the task thoughtfully

Because the answers are publicly available, this pack works particularly well as a formative investigation, a department planning resource or a marking workshop. For summative use, adapt the arrangement and check the new mathematics fully, or use conditions that establish the learner’s independent authorship. A changed diagram can require a different rule and a different clarification.

Download the student sheet separately when sharing with a class, and keep the teacher answers for the discussion afterwards.

Download Crossing Paths: the complete investigation pack

For the wider assessment picture, read our guide to MYP Mathematics Criteria A, B, C and D. For year-group planning, explore the MYP Year 4 and MYP Year 5 curriculum resources.

Original task, diagrams, clarification and constructed responses: Full Marks Education. This resource is not an IB-issued assessment or an endorsed mark scheme. See the IB assessment overview for the role of teacher-designed school assessments.

MYP Criterion B Investigation: Free Task, Worked Solution and Marking Guide