AMC Y5-6 Advanced Course
For Years 5–6 with secure foundations. Develop upper-primary problem-solving through the Australian AMC pathway.
First 3 starter lessons or units free with an accountSee curriculum & course details →Register & try 3 free →AJMO Patterns and Parity: Find Structure Instead of Adding Every Case
The short answer
A useful pattern explains how the objects are built, not just what the first numbers happen to be. In a coloured multiplication grid, grouping odd and even indices can replace a long addition. In a growing model, separate the first object from what each new object adds.
What makes a pattern useful?
A list of values can suggest a relationship, but the construction must explain it. The official sample materials include a multiplication-grid problem where the colouring separates index groups. Our smaller model below makes that structure visible without reproducing the original problem.
Write the rule for a cell before summing it. In this example, row i and column j contain i×j. A green cell has i and j both odd or both even. It is not enough to say that “half the cells are green”: differently placed cells have different values.
How can a coloured grid turn into two products?
The odd row indices sum to 1+3 = 4, and the odd column indices also sum to 4. Their green-cell total is (1+3)(1+3) = 16. The even indices sum to 2+4 = 6, so the even/even contribution is 36. The whole green total is 52.
Why does multiplication work? Expanding (1+3)(1+3) gives 1×1 + 1×3 + 3×1 + 3×3: exactly the four odd/odd cells, each once. The same reasoning covers the even/even cells. It provides a completeness check, rather than a shortcut whose meaning is hidden.
The total of all sixteen cells is (1+2+3+4)² = 100, leaving 48 in the other colour. This independent check helps catch a colouring reversal. On a 5×5 version, the odd indices sum to 9 and the even indices sum to 6, giving 9² + 6² = 117 in the same-parity colour.
How do we build an equation from a growing shape?
The shared-side condition justifies the rule. A ring or a cluster can introduce extra shared sides, so the same equation may no longer apply. Ask the child to explain what is new at each step, not merely to extend 6, 11, 16 by guessing another increase.
How can we tell when a pattern argument is incomplete?
Checking three small cases does not by itself prove a rule for every case. Explain why the construction continues unchanged, or show how the general expression accounts for every object. Keep track of a first-step exception, a boundary row or a repeated intersection.
Try a changed colouring rule or a different overlap arrangement. If the old expression fails, identify the assumption that changed. This is a valuable part of practice: it teaches when a method can be used, rather than only how to calculate with it.
What should a parent look for?
Look for a clear statement of the rule, a justified grouping, and a check against a small example. These habits connect pattern work to equations and later written reasoning. Use the related AMC curriculum to practise the required foundations, then return to mixed problems where the topic is not named in advance.
Give practice a clear learning pathway
Explore the complete curriculum, then start the free lessons in your own learning account. These existing AMC and AIMO courses develop related skills; they are not dedicated AJMO courses or official competition entry.
AMC Y7-8 Advanced Course
For Years 7–8 ready to extend beyond foundations. Build geometry, number-theory and counting techniques.
First 3 starter lessons or units free with an accountSee curriculum & course details →Register & try 3 free →AIMO Preparation Course
For students ready for deeper problems and written reasoning. Inspect the level before choosing the stretch.
First 3 starter lessons or units free with an accountSee curriculum & course details →Register & try 3 free →Questions parents ask
Why is the green total not half the total of the grid?
The colours may have equal numbers of cells while the cell values differ. Group the actual row and column indices rather than averaging by cell count.
What does parity mean in this example?
It means whether a row or column index is odd or even. The selected cells use indices with matching parity.
Can we prove a growing pattern by checking three values?
Those checks provide evidence, but the construction must explain why the same rule continues and accounts for every object.
When does the hexagon stick formula change?
It changes if hexagons share additional sides or overlap differently. The stated formula applies to the straight row with exactly one new shared side at each step.