Patterns in Times Tables

Activities to try

1. 1

   Shade, fold and decompose the multiplication square

   Why it works: A noticed pattern can be explained from the structure of the operation itself — the table's symmetry is a consequence (and a picture) of commutativity, and 'always even' follows from decomposing into equal addends — so maths patterns are consequences of properties, not coincidences.

   1. Shade a chosen times-table on a multiplication square (e.g. shade every multiple of 5) and describe what you notice about the digits — for 5s, every product ends in 0 or 5.
   2. Fold or reflect the grid across its main diagonal so a cell like 7×8 lands exactly on 8×7; observe that every cell matches its mirror.
   3. State WHY the table is symmetric: because 7×8 and 8×7 are the same product (commutativity).
   4. Pick a 4×n cell and split it into two equal addends (2n + 2n) to show why 4×n is always even.
   5. Use a 'predict the next' prompt: extend a shaded pattern one or two steps, say what you expect, then check against the grid.
2. 2

   Be the pattern detective: tell me WHY

   Why it works: An explanation names a cause grounded in properties; a few examples suggest a pattern, but a structural reason shows it must always hold.

   1. State a pattern you have spotted (e.g. 'multiples of 5 end in 0 or 5', 'the sum of two even numbers is always even').
   2. When a 'mathematician' partner accepts only a structural reason, give the WHY: for 5s, each step adds 5 and two fives make a ten, so the ones digit lands back on 0; for even+even, each even is made of groups of two, so combining them still gives groups of two.
   3. Explain the table's diagonal symmetry using commutativity, and predict whether a pattern will still hold for a large case (e.g. 5×100) and justify without computing it all.
3. 3

   Match the pattern to its reason (check your explanation)

   Why it works: Being able to select or reject explanations — not just spot patterns — checks whether reasoning is structural rather than a restatement.

   1. Given a pattern, choose the correct explanation from options that include a tempting restatement distractor — e.g. 'Why is 4×n always even?' with a restatement ('because it's in the 4 times table') versus the doubling explanation ('because 4×n is 2n + 2n, two equal groups').
   2. Predict the next entry in a shaded pattern on the table and check it.
   3. Decide whether a stated related pattern (e.g. 'odd + odd is always even') is true and identify the structural reason or counter-reason — odd + odd is indeed always even, because each odd is a group of twos plus one, and the two extra ones pair up.