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Times Table Patterns

Calculation · Year 2–Year 6

Join each number to its multiple and watch the pattern appear.

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About the table patterns

Ten numbers round a circle, and a line from each one to where its multiple lands. Three sevens is twenty-one, so three joins one. Four sevens is twenty-eight, so four joins eight. Draw all ten and the sevens leave two four-sided shapes crossing in the middle of the circle, and no two tables leave the same thing behind.

What the chords show on ten points is the LAST DIGITS of the table, which is why the patterns repeat and why some tables give closed shapes and others give lines. It also explains the oddest thing on the board: the threes and the sevens draw exactly the same picture. Three sevens is twenty-one, which ends in one, so multiplying by seven undoes multiplying by three and the same pairs of numbers end up joined, just the other way round.

Four tables do something a class will want explaining. The fives send every even number to nought and every odd one to five, which is a picture of what the five times table ends in. The tens send everything to nought. The nines join every number to its mirror image, so they draw a ladder of flat lines. And the sixes leave every even number exactly where it is and send every odd one straight across the circle.

The elevens do the strangest thing of all: they draw nothing. Eleven of something ends in the digit it started with, so every number lands back on itself. Those are drawn as a ring round the number rather than as a line with no length, and the board says so instead of leaving a class looking at an empty circle.

The times table is listed beside the circle and fills in as the chords are drawn, so the arithmetic and the picture are on the board together. Press Next to take it one chord at a time and ask where the next one will go before you press.

Then turn the points up. The same arithmetic on two hundred points draws smooth curves instead of shapes: the twos give a cardioid with one cusp, the threes give two cusps and the fours give three. Nothing has changed about the maths, which is the whole point of it.

How to use it on your whiteboard

  1. 1Tap a table down the left hand side. The pattern redraws for that table.
  2. 2Press Next to draw one chord at a time, All to draw the lot, or None to clear the circle and start again.
  3. 3Read the caption: it says what the chord that just appeared is doing, and the times table beside the circle fills in with it.
  4. 4Change Points for a finer pattern. Ten is the last digits of the table; two hundred is a curve.
  5. 5Tap Fullscreen to fill the board. Share copies a link with this table and this pattern on it.

Questions

What are the lines actually showing?
Each number is joined to its multiple in that table, wrapped round the circle. On ten points that is the last digit: three sevens is twenty-one, so three joins one. On more points it is the remainder after dividing by however many points there are, which is the same rule.
Why does the six times table look broken?
It is not. On ten points, two sixes is twelve, so two joins two and stays where it is, and four, six, eight and nought do the same. The odd numbers all go straight across the circle. So that pattern is five rings and five lines through the middle, and the board says which numbers stay.
Why does the eleven times table draw nothing?
Because eleven of something ends in the same digit it started with: three elevens is thirty-three, seven elevens is seventy-seven. Every number lands back on itself, so there is nothing to join it to. The board says so rather than leaving a class looking at an empty circle.
Which table gives the best pattern?
On ten points the threes and the sevens give the same pair of crossing quadrilaterals, the fives give two fans and the nines give a ladder of flat lines. On two hundred points the twos give a cardioid with a single cusp, the threes give two cusps and the fours give three, which is a pattern in itself worth asking a class about.
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