Mental mathematics

Numbers have more depth than simple stepwise counting. Explore patterns to plumb the depths.

Train your brain in sequences. We are surrounded by them all the time. Mental dexterity to jump around inside these sequences will make it less of a chore to work things out.

The first page on this spreadsheet is for Base 10, the decimal system. It shows various times tables (multiplication tables), highlighted by colour. Look at them and familiarise yourself with the feeling of the patterns, and also with the stepwise changes, and also with the numbers that they typically end with. Note that my times tables' lines start with 1 and end on 10.

Concept:

Rows of numbers should start on 1, not 0.

The tens are not the start of a row, but rather are the end, the completion.

Concept:

A full row is from 1 to 10. Think of this as a SET instead of as a 10.

Thinking this way will help you in shifting between different bases.

Looking for patterns

4× Table

For example, in the 4 times table:

  • all ending numbers are even

  • a sets contain either three — ending in 0, 4 and 8 — or two — ending in 2 and 6

  • all the sets containing three are odd tens: 12-16-20

  • all the sets containing two are even tens: 24-28

  • thus, a heuristic emerges:

    • even sets end in 4, 8

    • odd sets end in 2, 6, 0

FOR 4× TABLE: a division heuristic emerges — expand to read & try examples

44 ÷ 4

  • all even sets contain 2 of the numerator

  • all odd sets contain 3 of the numerator

  • (that is equivalent to 5 in every 2 sets [20])

  • 44 / 4 = 40r4 = 4 sets & 4 units

  • 1st set [2] + 2nd set [3] + 3rd set [2] + 4th set [3] = 2 + 3 + 2 + 3 = 10

  • r4 : 5th set contains 2 (44, 48) ; 44 is the 1st so add 1

  • 10 + 1 = 11

  • 44 / 4 = 11

Heuristic for 4× table

  • 44 is in the 5th set

  • There are 4 complete sets

  • 2 sets (20) contains 5 of the numerator

  • 4 sets ÷ 2 sets = 2

  • 2 × 5 = 10

  • the whole value is 44

  • 4 is the first item in an evenly-numbered set

  • the 40s is evenly-numbered (4X) but is the 5th set

  • thus, the 5th set contains one of the numerator: +1

  • 2 × 5 = 10 → 10 + 1 = 11

  • 44 ÷ 4 = 11

Now test the heuristic with another, higher, value:

176 ÷ 4

  • 176 rounded up is 180: 176 is in the 18th set

  • There are 17 complete sets

  • 17 sets ÷ 2 sets = 8 (5 nums) and 1 even set (2 nums)

  • 8 × 5 = 40 ; 1 × 2 = 2 ; 40 + 2 = 42

  • 18th set is oddly-numbered, 17X, so has 3 of numerator: 2, 6, 0

  • 176 is the 2nd: +2

  • (8 × 5) + (1 × 2) + 2

  • 40 + 2 + 2 = 44

  • 176 ÷ 4 = 44 ✅

An additional heuristic detail can be added to the model:

  • First set is even set, first set has 2 nums

  • Therefore, half a double-set has 2 nums

  • Therefore,

    • 1½ sets has 5+2 = 7

    • 2½ sets has 10+2 = 12

    • etc.

  • 17 sets ÷ 2 sets = 8½ sets = 8×5 + 2 = 42

  • Odd numbers:

    • round down to even number,

    • half it,

    • times by 5,

    • add 2.

What patterns can you see?

What heuristics can you derive?

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