# Blog

Thinking outside the box - stimulating family projects
By David Hanson
01 Mar

David Hanson has over 40 years' experience of teaching, has been a member of the ISEB Mathematics exam setting team and has written numerous titles for Galore Park, including the Mathematics for Common Entrance 13 Plus revision series.

Have you ever wondered what to do with your child that
• encourages lateral thinking
• stimulates creativity
• stretches the imagination
• reinforces ideas in a core subject
• can be done anywhere, any time
• is fun
• costs nothing?
Here is one idea for a family project that satisfies all of the above and, at the same time, lends itself to participation by
• the whole family
• groups of families

Six is a special number
Introduction

Six is a rather special number for several reasons, including:

• Hexagons feature in honeycombs.

• Quartz crystals occur as terminated hexagonal prisms.

• If a regular hexagon is circumscribed (a circle is drawn passing through all six vertices), the radius of the circle is the same as the length of side of the hexagon.

• Seven identical circular coins can be arranged as shown here.

• 1 x 2 x 3 = 6 and 1 + 2 + 3 = 6

• Apart from 2 and 3, all prime numbers are either one less or one more than a multiple of six.

Base six in history

Throughout the ages, many different bases have been used in everyday mathematics, and some are still used but, in the twenty-first century, base ten (the decimal system) is the base in general use. It is widely accepted that this is because we have ten digits on our two hands although many people argue that base twelve (the duodecimal system) would have been better – not least because twelve has more factors than ten.

The Sumerian civilisation, more than four thousand years ago, developed a sexagesimal (base sixty) number system which used the two bases six and ten alternately.

A sexagesimal abacus would look something like this.

The units spike has spaces for 9 counters (base ten), the next spike – the tens spike has space for 5 counters (base six), the sixtys spike has space for 9 counters (base ten) and so on.

The sexagesimal number 1025 is the same as 625 in base ten (1 six hundred + 0 sixtys + 2 tens + 5 units).

In this chapter, base ten numbers written in figures will be bold italic, as with 625 above.

The use of this system continued in Babylonia and spread through Mesopotamia to be used by every nation in the Mediterranean which used the Babylonian measurement units. We still use base sixty in measurement of time and angle and sixes feature in many places – egg boxes, balls in a cricket over, and so on.

How base six works

There are alternative names (including senary) which could be used, but we will stick to the simple names base six and heximal.

It could be considered that base six is the most logical base to be used by humans because we have five digits on each hand. Study the photographs below showing ‘finger counting’ of the numbers three (3), four (4) and five (5).

The next counting numbers are 10 (one six and no units), 11 (one six and one unit – which is 7 in base ten) and so on.

Counting in this way, we could count up to 55 in base six, which is 35 (thirty-five) in base ten.

A base six abacus looks like this. It has spaces for 5 counters on each spike.

The spikes from right to left show the numbers of

• Units            1s                                                         (1s base ten)

• Sixes            10s                                                       (6s base ten)

• Sixes²          100s             we’ll call these Trixes!       (36s base ten)

In base ten           36                            6                 1

We will read the heximal number 213 as “two trix and one-six three”.
This is equivalent to the decimal number 81 (2 x 36 + 1 x 6 + 3 x 1).

The next spike to the left would be

• Sixes³          1000s           we’ll call these Quixes!      (216s base ten)

Working in base six

From now on, unless indicated otherwise, all numbers are in base six (heximal). Any numbers in base ten (decimal) will be bold and italic, for example 35

A base six multiplication table looks like this.

 1 2 3 4 5 10 2 4 10 12 14 20 3 10 13 20 23 30 4 12 20 24 32 40 5 14 23 32 41 50 10 20 30 40 50 100

In base six some common fractions are relatively easy to deal with.

Proper fraction                      Heximal fraction
one half                  ½                 0.3               here the point is a heximal
one third                 1/3               0.2               point, not a decimal point!
one quarter             ¼                 0.13
one sixth                 1/10              0.1
two thirds                2/3               0.4
three quarters         ¾                 0.52
one fifth                  1/5               0.111111… a recurring heximal fraction

In base six

• ½ + 1/3 = 5/10                  0.5

• ½ – 1/3 = 1/10                  0.1

• ½ x 1/3 = 1/10                  0.1

• ½ ÷1/3 = 1½                     1.3

This abacus shows the heximal number 4.23 (four Units, two sixths and three trixths).

The addition of three trixths (0.03) would make the number 4.3 (four and a half!)

The first 23 prime numbers in base six are

2, 3, 5, 11, 15, 21, 25, 31, 35, 45, 51, 101, 105, 111, 115, ….

Apart from 2 and 3, all primes have the units digit 1 or 5
Note that not all numbers with a units digit of 1 or 5 are prime!

A few suggestions
You might like to consider

• making up a short test of basic skills where all of the numbers are in base six

• writing a short story book where all numbers (including page numbers of course) are in base six

• re-writing a song (for example ‘The twelve days of Christmas’) so that all of the numbers are in base six

• studying number sequences (such as the square numbers, triangle numbers or Fibonacci numbers) in base six

• creating a ‘world’ where base six is the generally adopted base for all number work, including measurements

• writing a BBC BASIC program, similar to the one below, which will convert base six numbers to decimal numbers

10 REM BASE TEN TO BASE SIX CONVERSION
20 PRINT "Enter an integer less than 500"
30 INPUT N
40 IF N > 499 OR N <> INT(N) THEN GOTO 20
50 LET D = N DIV 216
60 LET R = N MOD 216
70 LET C = R DIV 36
80 LET S = R MOD 36
90 LET B = S DIV 6
100 LET A = S MOD 6
110 PRINT "In base six, the number is ",D;C;B;A
120 GOTO 20

Anyway, now it is over to you. Use your imagination and see what you can discover. Remember to keep a record of your ideas and discoveries.

Once you've discovered the magic of number six it might be time to delve deeper into your maths revision with one of our stimulating and rigorous textbooks! Our range has been developed with high achievers in mind, with books to stretch even the keenest maths student.

Mathematics for Common Entrance 13 Plus One and Two books cover Common Entrance Levels 1 and 2 and introduce Level 3 for extension work. The content covered exceeds National Curriculum requirements at KS3 and lays the foundations for success at 13 Plus. Book One can be used for Year 7, and Book Two for Year 8. Book Three (Extension) is ideal for Level 3 candidates, and those taking scholarship exams.

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