6H were doing number sequences a few weeks ago and they all enjoyed it immensely. Then Miss Hunte played her usual trick of telling the class, without any warning or discussion with me that, as she had to leave the room for a few minutes, “Mr Terry will talk to you.”
If anyone from a university education department is reading this, may I suggest that delivering a lesson, ‘off the cuff’ should be a requisite of any teacher-training course? I based a whole career on it. The very first time I was seen by an OFSTED inspector, he came up to me, as I was about to begin the lesson and asked me where my lesson plan was.
“Up ’ere mate,” I said, tapping my temple.
I decided to introduce 6H to the beauty that is the Fibonacci Sequence. In case you don’t know it, it begins: 0, 1 and every subsequent number is the sum of the previous two numbers. So, it goes,
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …………….
There is a formula to work out, say the 46th number in the sequence but it is much too complicated for here. In case you’re interested, and I know you are, I’ve just worked out the 46th number and it is: 18363111903. The 146th number in the sequence has 31 digits and that took me nearly five minutes to calculate! It really won’t add anything for me to reproduce it, so I won’t.
I told 6H about how it occurs in nature. Any Fibonacci number divided by the preceding number, after the first few, gives 1.618 … an irrational number. The ratio 1:1.618 is known as the ‘Golden Ratio’. Rectangles with length and width in this ratio are called ‘Golden Rectangles’. They are the most pleasing rectangles to the eye and their co-ordinates are used in many renaissance paintings.
The golden ratio occurs naturally in the spiral of sunflower seeds and the coils of snail and nautilus shells. I constructed a coil on the whiteboard using squares 1,1; 1,1; 2,2; 3,3; 5,5; 8,8 and they saw and understood how the spiral developed. You’ve really got to see it to understand it.
Fibonacci, from Pisa, lived in the 13th century. He did not discover the golden ratio. Indian mathematicians and the Egyptians knew of it several thousand years earlier. The Great Pyramid of Gisa was built utilising it in places and The Parthenon was built incorporating it in many aspects and dimensions. It occurs in the Mona Lisa and often in parts of The Last Supper by da Vinci. It is aesthetically pleasing to the eye.
The length of the average human forearm to the length of the hand is 1.618 as is the length of the human face to its width. I’ve been studying old school photographs of rugby teams and found that David’s is 1.284. (Shrek’s is 1.262)
Also in the human body, the ratio of the width of mouth to the width of nose is in the golden ratio, as is the ratio of the distance between the navel and knee to the distance between the knee and the end of the foot. The same sequence can be seen in the leaves of poplar, cherry, apple, plum and oak trees.
Interestingly (in my opinion though not in Caroline’s), paper sizes are not, ’golden rectangles’. The ‘A’ series are in the ratio of 1 : √2. This allows for a sheet to be folded over and over without changing the ratio. That can’t be done with a Golden Rectangle.
6H loved it. Then I made a bit of a mistake. I forgot for a moment that I was talking to 10-year olds and started to tell them about the two rabbits in a field.
Imagine a field with no rabbits in it. (0) Into that field after one month is put one pair of newborn baby rabbits (0, 1). We are counting the number of pairs of rabbits in the field after every month.Rabbits become sexually mature at the age of two months.
“No, they don’t,” interrupted Rozzard. I ignored him.
So, after another month, there is still just one pair in the field (0, 1, 1). One month after that, the female rabbit, now mature, gives birth to one pair, one male and one female and so now there are two pairs (0,1,1,2).
This is when the trouble started: “My rabbit had eight babies. They wouldn’t just have two.”
Another month passes and the original female gives birth to another pair, one male and one female making three pairs (0, 1, 1, 2, 3). After yet another month the first female to be born is mature and she too gives birth to a pair, one male and one female; but so, does the original female and so now there are five pairs in the field (0, 1, 1, 2, 3, 5).
“That’s disgusting! That means the father is their mother’s brother,” said Anique.
“Or their grandfather,” added Rozzard helpfully.
I’d lost them and so I set them homework. I don’t really agree with setting homework for primary age children. I think that at that age they should be out, doing things but I set it to use up some time until Ms Hunte returned.
“Make up a number sequence of at least five numbers, either arithmetic or geometric and I’ll work out the next two numbers in the sequence. There’ll be a prize for anyone who beats me.”
The next time I went back they handed me their homework. Most of them were easy to work out. 1, 2, 4, 7, 11, ____ was typical and I did them all. One of them stumped me. It was Rozzard’s.
This was it: 6, 12, 48, 768, 196608, _____
I studied it for several minutes totally neglecting other things. I was not going to be beaten by Rozzard. It was clearly not an arithmetic sequence but I couldn’t see any apparent logical progression.
“Are you sure that this works Rozzard,” I asked him and then I had a thought. “They’re not just random numbers, are they?”
He was very indignant. “Miss Hunte checked it,” he said grumpily.
Miss Hunte grinned and nodded. She was enjoying my frustration as much as Rozzard.
At last I got it. “Sorry Rozzard, no prize for you,” I said. “6 x 2 = 12. 12 x 2 squared (4) = 48. 48 x 4 squared (16) = 768. 768 x 16 squared (256) = 196608 and so 196608 x 256 squared (65536) is 12 884 901 888. Is that the answer?”
He wasn’t happy. “Yes,” he grumped, “but you got it the wrong way so I still should get a prize.”
I couldn’t devote any more time to it then as I had standards to raise. I brought it home and worked on it during the afternoon and cracked it, I think. Rozzard is going to be upset when I next see him if I’ve done it his way, because Ms Hunte told him that my prize was to be a trip to Disney World in Orlando.
This is how Rozzard did it:
6, 12, 48, 768, 196608, ____________
6² = 36 and 36 ÷ 3 = 12
12² = 144 and 144 ÷ 3 = 48
48² = 2304 and 2304 ÷ 3 = 768
768² = 589824 and 589824 ÷ 3 = 196608
196608² = 38654705664 and 38654705664 ÷ 3 = 12884901888
and so, that is the next number in the sequence.
Fibonacci 146 = 1454489111232772683678306641953
31 digits, so don't bother counting them!
This is a photo of us last Thursday. I’m going to miss them all very much.
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