Degrees of understanding

23rd April 2004 at 01:00
Q) Can you tell me why are there 360 degrees in a circle?

I always thought it is because 360 is a nice number to chop up as it has so many factors, and that it must be related to time, as the subdivisions are minutes and seconds. Your question made me think there must be a deeper underlying reason.

A) Most theories relate to astronomy. The sun takes about 360 days to complete a circuit of the sky (actually 365.25 days), the lunar year is about 354.37 days; so 360 could be a compromise. The lunar month is 29.5 days long.

The Mesopotamians (Sumerians, Akkadians and Babylonians in the area that is now southern Iraq) based their number system on 60. This may have been for commercial reasons (60 has lots of factors so can be easily divided into fractions). The Sumerian calendar from 2400BC divides the year into 12 months of 30 days each. The Ancient Egyptians contributed the degree symbol.

"The basis of angular measure for the mathematicians of Babylon was the angle at each of the corners of an equilateral triangle. They did not have decimal fractions and thus found it difficult to deal with remainders when doing division. So they agreed to divide the corner of an equilateral triangle into 60 degrees, because 60 could be divided by 2, 3, 4, 5 and 6 without remainder. Each degree was divided into 60 minutes and each minute into 60 seconds. If the angles at the corners of six equilateral triangles are placed together they form the angle formed by a complete circle. It is for this reason that there are six times 60 degrees of arc in the complete circle." (www.physlink.com EducationAskExpertsae373.cfm)

When you put those six equilateral triangles together, there are 12 angles at the circumference of the circumscribed circle - the number of months in a year?

Q) Do you know any 'tricks' or hints that would help people remember their times tables?

A) If you are working with "tricks" or hints for finding or recalling multiplication facts, it is essential to encourage the learner to investigate why the "trick" works.

This helps the learner to recall a fact and to understand the underlying maths. When I researched issues associated with learning multiplication tables, pupils reported finding the eight-times table the most difficult to learn. At that time, acquiring fluency was based on "rote" learning, which does not encourage interaction with numbers.

The eights are double the fours, which in turn are double the twos. 7 X 8 is an easy one to remember when the answer is written first - five six seven eight!

56=7x8

In the five times table, all the answers associated with the even numbers end in zero and with odd-numbers in five.

1 X 5 = 5

2 X 5 = 10

3 X 5 = 15

4 X 5 = 20

5 X 5 = 25

6 X 5 = 30

7 X 5 = 35

8 X 5 = 40

9 X 5 = 45

10 X 5 = 50

11 X 5 = 55

12 X 5 = 60

The five times table is of course half the tens, which helps in finding the answer when the learner isn't sure.

The fact of the even numbers ending in zero is really helpful for finding the answers to the even multipliers when looking at the six times table. A neat little 'trick' as the answer ends in the multiplier as can be seen in the table below. Multiply 6 by 8 and the answer ends in 8!

4 X 6 = 24

6 X 6 = 36

8 X 6 = 48

Even better when you also notice that the first digit of the answer to the even multipliers is half of the multiplier. So when multiplying by 4 the beginning of the answer is found by halving the four to give 2 (which of course is 20)

4 X 6 = (4 divided by 2)4

4 X 6 = 24

Clever isn't it? Why does it work? Because to find the answer in the six times table an easy mental method is to use the answer to the five times table plus one lot of each multiplier.

5 X 8 = 40

1 X 8 = 8

6 X 8 = 48

The pattern that helps in developing fluency of the nine times table (without reverting to the hands) can be found under hints at www.perfect-times.co.uk and pupils can then use this to practice their nines for free on the site.

Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.

www.nesta.org.uk

Email your questions to Mathagony Aunt at teacher@tes.co.uk Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX

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