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Teacher Notes Notes For Teachers Slides 3 to 8: Introduction to the Fibonacci sequence via the traditional “Rabbit Problem”. Slides 9 to 31: An investigation with a printable worksheet exploring some of the properties of the Fibonacci sequence. Slide 32: The well known 64 = 65 puzzle. Slide 33: Fibonacci Numbers in Pascal’s Triangle (From the Pascal’s Triangle presentation). Slides 34 to 36: A brief look at Fibonacci numbers in nature. Slide 37 to 48: These give you three alternative (non-rabbit) ways of introducing/investigating the sequence or re-visiting it at a later date. They are: 1. Paving Stones Problem, 2. Phone Box Problem. 3. Stairs Problem. Slide 49: A couple of Fibonacci Websites that students may wish to visit.

History The Fibonacci Sequence Leonard of Pisa (Fibonacci) 1175 - 1250 Leonardo of Pisa (Fibonacci) was a mathematician of medieval Pisa. He travelled extensively throughout the Africa and the Middle East on business and trade and he observed the numerical systems in use in different cultures. He was influential in introducing the Indo-Arabic system of arithmetic to Europe (which was using the cumbersome Latin System). In his famous book Liber Abaci (book of the calculator) he introduces the ten symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 and the method of place value. 358 x 94 1432 32220 33652 CCCLVIII x XCIV or Which one would you prefer to do? The sequence for which Fibonacci is famous is introduced in Liber Abaci by way of an (idealised) look at the growth in population of rabbits starting with a single male and female.

Rabbits The Fibonacci Sequence A pair of baby rabbits (male and female) are born in a field on 1 st January. They (and all future pairs) always give birth to another mixed pair (male and female) two (calendar) months after birth. Assuming this idealised situation continues, how many pairs of rabbits will there be in the field at the end of the first year? 1 st January 1 st February 1 st March 1 st April 1 st May Fibonacci’s Rabbit Problem Leonard of Pisa (Fibonacci) 1175 - 1250

The Fibonacci Sequence How many pairs of rabbits are in the field after 12 months? Worksheet MonthPairs of RabbitsNumber of Pairs January1 February1 March2 April3 May5 June July August September October November December?

The Fibonacci Sequence MonthPairs of RabbitsNumber of Pairs January1 February1 March2 April3 May5 June8 July13 August21 September34 October55 November89 December144 How many pairs of rabbits are in the field after 12 months?

The Rabbit Problem By considering the special case of the five rabbits in the field in May can you explain the above? There are 144 pairs of rabbits in the field at the end of the year. From the table of results it can be seen that the number of pairs of rabbits in the field in any particular month is equal to the sum of the pairs of rabbits in the field in each of the two preceding months. For Example: MonthNumber of Pairs January1 February1 March2 April3 May5 June8 July13 August21 September34 October55 November89 December144 May April March The number of pairs of rabbits in the field in May is made up of those that were in the field in April (1 month earlier) plus the new bred pairs from each of the fertile pairs in the field in March (2 months earlier). In general, the number of pairs of rabbits in any month = pairs one month earlier + offspring from each pair (all fertile) two months earlier.

The Fibonacci Sequence Leonard of Pisa (Fibonacci) 1175 - 1250 Fibonacci’s sequence contains many interesting and unusual properties. Each term in the sequence is formed by the addition of the two previous terms. Thus F 6 (the 6 th term = 8) is produced by F 5 (the 5 th term = 5) and F 4 (the 4 th term = 3). That is 8 = 5 + 3. So in this case we have F 6 = F 5 + F 4. In general (from the third term onwards) we write: F n = F n-1 + F n-2 F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1

Investigation Sheet Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? 3.Check that every third term is even. Why is this the case? Find an explanation and write it up clearly. 4.What is the distribution of odd and even numbers within the sequence? 5.The sum (F 1 to F 4 ) = 1 + 1 + 2 + 3 = 7 = F 6 – 1. Is this true in general that Sum(F 1 to F n ) = F n + 2 – 1?. Check this for six more cases. 6.F 1 + F 3 + F 5 + F 7 = F 8 ? Investigate. 7.F 2 + F 4 + F 6 + F 8 = F 9 - 1? Investigate. 8.Choose any number from the sequence and square it. Compare the result with pairs of adjacent numbers. Write down the rule that connecting F n 2 and F n – 1 and F n + 1 9.F 3 2 + F 4 2 = F 7 ? Investigate and generalise using algebraic symbols. 10.Compare the ratios of successive terms F 2 /F 1, F 3 /F 2, F 4 /F 3.........to F 16 /F 15. Produce a table of results (to 7 d.p) using either a calculator or spreadsheet. What do you notice? State any conclusions. Worksheet

Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. F 13 233 F 14 377 F 15 610 F 16 987 F 17 1597 F 18 2584 F 19 4181 F 20 6765

Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? No

Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? 3.Check that every third term is even. Why is this the case? Find an explanation and write it up clearly. The 3 rd term is even since it is the sum of the first two odd terms (1, 1) and odd + odd = even. The 4 th term is odd since (1 + 2 = 3), odd + even = odd and similarly the 5 th term is odd since (2 + 3 = 5), even + odd = odd. Since we now have another two adjacent odd terms the sequence repeats: odd, odd, even, odd, odd, even,…..

Investigating Some Properties of The Fibonacci Sequence 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? 3.Check that every third term is even. Why is this the case? Find an explanation and write it up clearly. 4.What is the distribution of odd and even numbers within the sequence? F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 2/3 odd an 1/3 even

Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? 3.Check that every third term is even. Why is this the case? Find an explanation and write it up clearly. 4.What is the distribution of odd and even numbers within the sequence? 5.The sum (F 1 to F 4 ) = 1 + 1 + 2 + 3 = 7 = F 6 – 1. Is this true in general that Sum(F 1 to F n ) = F n + 2 – 1?. Check this for six more cases. 1 + 1 + 2 = 5 - 1 1 + 1 + 2 + 3 + 5 = 13 - 1 1 + 1 + 2 + 3 + 5 + 8 = 21 - 1 1 + 1 + 2 + 3 + 5 + 8 + 13 = 34 - 1 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 = 55 - 1 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 144 -1 These 6 cases are all true so it looks as though it may be true in general.

Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? 3.Check that every third term is even. Why is this the case? Find an explanation and write it up clearly. 4.What is the distribution of odd and even numbers within the sequence? 5.The sum (F 1 to F 4 ) = 1 + 1 + 2 + 3 = 7 = F 6 – 1. Is this true in general that Sum(F 1 to F n ) = F n + 2 – 1?. Check this for six more cases. 6.F 1 + F 3 + F 5 + F 7 = F 8 ? Investigate.

Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? 3.Check that every third term is even. Why is this the case? Find an explanation and write it up clearly. 4.What is the distribution of odd and even numbers within the sequence? 5.The sum (F 1 to F 4 ) = 1 + 1 + 2 + 3 = 7 = F 6 – 1. Is this true in general that Sum(F 1 to F n ) = F n + 2 – 1?. Check this for six more cases. 6.F 1 + F 3 + F 5 + F 7 = F 8 ? Investigate. It appears that the sum of all successive odd terms up to any point in the sequence is equal to the value of the next (even) term in the sequence. We can express this algebraically as: F 1 + F 3 + F 5 +... +F 2n – 1 = F 2n

Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? 3.Check that every third term is even. Why is this the case? Find an explanation and write it up clearly. 4.What is the distribution of odd and even numbers within the sequence? 5.The sum (F 1 to F 4 ) = 1 + 1 + 2 + 3 = 7 = F 6 – 1. Is this true in general that Sum(F 1 to F n ) = F n + 2 – 1?. Check this for six more cases. 6.F 1 + F 3 + F 5 + F 7 = F 8 ? Investigate. 7.F 2 + F 4 + F 6 + F 8 = F 9 - 1? Investigate.

Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? 3.Check that every third term is even. Why is this the case? Find an explanation and write it up clearly. 4.What is the distribution of odd and even numbers within the sequence? 5.The sum (F 1 to F 4 ) = 1 + 1 + 2 + 3 = 7 = F 6 – 1. Is this true in general that Sum(F 1 to F n ) = F n + 2 – 1?. Check this for six more cases. 6.F 1 + F 3 + F 5 + F 7 = F 8 ? Investigate. 7.F 2 + F 4 + F 6 + F 8 = F 9 - 1? Investigate. It appears that the sum of all successive even terms up to any point in the sequence is equal to the value of the next (odd) term in the sequence minus 1. We can express this algebraically as: F 2 + F 4 + F 6 +... +F 2n = F 2n + 1 -1

Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? 3.Check that every third term is even. Why is this the case? Find an explanation and write it up clearly. 4.What is the distribution of odd and even numbers within the sequence? 5.The sum (F 1 to F 4 ) = 1 + 1 + 2 + 3 = 7 = F 6 – 1. Is this true in general that Sum(F 1 to F n ) = F n + 2 – 1?. Check this for six more cases. 6.F 1 + F 3 + F 5 + F 7 = F 8 ? Investigate. 7.F 2 + F 4 + F 6 + F 8 = F 9 - 1? Investigate. 8.Choose any number from the sequence and square it. Compare the result with pairs of adjacent numbers. Write down the rule that connecting F n 2 and F n – 1 and F n + 1 5 2 = 3 x 8 + 1

Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? 3.Check that every third term is even. Why is this the case? Find an explanation and write it up clearly. 4.What is the distribution of odd and even numbers within the sequence? 5.The sum (F 1 to F 4 ) = 1 + 1 + 2 + 3 = 7 = F 6 – 1. Is this true in general that Sum(F 1 to F n ) = F n + 2 – 1?. Check this for six more cases. 6.F 1 + F 3 + F 5 + F 7 = F 8 ? Investigate. 7.F 2 + F 4 + F 6 + F 8 = F 9 - 1? Investigate. 8.Choose any number from the sequence and square it. Compare the result with pairs of adjacent numbers. Write down the rule that connecting F n 2 and F n – 1 and F n + 1 8 2 = 5 x 13 - 1

Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? 3.Check that every third term is even. Why is this the case? Find an explanation and write it up clearly. 4.What is the distribution of odd and even numbers within the sequence? 5.The sum (F 1 to F 4 ) = 1 + 1 + 2 + 3 = 7 = F 6 – 1. Is this true in general that Sum(F 1 to F n ) = F n + 2 – 1?. Check this for six more cases. 6.F 1 + F 3 + F 5 + F 7 = F 8 ? Investigate. 7.F 2 + F 4 + F 6 + F 8 = F 9 - 1? Investigate. 8.Choose any number from the sequence and square it. Compare the result with pairs of adjacent numbers. Write down the rule that connecting F n 2 and F n – 1 and F n + 1 13 2 = 8 x 21 + 1

Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? 3.Check that every third term is even. Why is this the case? Find an explanation and write it up clearly. 4.What is the distribution of odd and even numbers within the sequence? 5.The sum (F 1 to F 4 ) = 1 + 1 + 2 + 3 = 7 = F 6 – 1. Is this true in general that Sum(F 1 to F n ) = F n + 2 – 1?. Check this for six more cases. 6.F 1 + F 3 + F 5 + F 7 = F 8 ? Investigate. 7.F 2 + F 4 + F 6 + F 8 = F 9 - 1? Investigate. 8.Choose any number from the sequence and square it. Compare the result with pairs of adjacent numbers. Write down the rule that connecting F n 2 and F n – 1 and F n + 1 21 2 = 13 x 34 - 1 It seems that F n 2 = F n – 1 x F n + 1 + 1 if n is odd. F n 2 = F n – 1 x F n + 1 - 1 if n is even.

Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? 3.Check that every third term is even. Why is this the case? Find an explanation and write it up clearly. 4.What is the distribution of odd and even numbers within the sequence? 5.The sum (F 1 to F 4 ) = 1 + 1 + 2 + 3 = 7 = F 6 – 1. Is this true in general that Sum(F 1 to F n ) = F n + 2 – 1?. Check this for six more cases. 6.F 1 + F 3 + F 5 + F 7 = F 8 ? Investigate. 7.F 2 + F 4 + F 6 + F 8 = F 9 - 1? Investigate. 8.Choose any number from the sequence and square it. Compare the result with pairs of adjacent numbers. Write down the rule that connecting F n 2 and F n – 1 and F n + 1 9.F 3 2 + F 4 2 = F 7 ? Investigate and generalise using algebraic symbols. 2 2 + 3 2 = 13

Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? 3.Check that every third term is even. Why is this the case? Find an explanation and write it up clearly. 4.What is the distribution of odd and even numbers within the sequence? 5.The sum (F 1 to F 4 ) = 1 + 1 + 2 + 3 = 7 = F 6 – 1. Is this true in general that Sum(F 1 to F n ) = F n + 2 – 1?. Check this for six more cases. 6.F 1 + F 3 + F 5 + F 7 = F 8 ? Investigate. 7.F 2 + F 4 + F 6 + F 8 = F 9 - 1? Investigate. 8.Choose any number from the sequence and square it. Compare the result with pairs of adjacent numbers. Write down the rule that connecting F n 2 and F n – 1 and F n + 1 9.F 3 2 + F 4 2 = F 7 ? Investigate and generalise using algebraic symbols. 3 2 + 5 2 = 34

Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? 3.Check that every third term is even. Why is this the case? Find an explanation and write it up clearly. 4.What is the distribution of odd and even numbers within the sequence? 5.The sum (F 1 to F 4 ) = 1 + 1 + 2 + 3 = 7 = F 6 – 1. Is this true in general that Sum(F 1 to F n ) = F n + 2 – 1?. Check this for six more cases. 6.F 1 + F 3 + F 5 + F 7 = F 8 ? Investigate. 7.F 2 + F 4 + F 6 + F 8 = F 9 - 1? Investigate. 8.Choose any number from the sequence and square it. Compare the result with pairs of adjacent numbers. Write down the rule that connecting F n 2 and F n – 1 and F n + 1 9.F 3 2 + F 4 2 = F 7 ? Investigate and generalise using algebraic symbols. 5 2 + 8 2 = 89 F n 2 + F n + 1 2 = F 2n + 1 It looks as though:

Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 10. Compare the ratios of successive terms F 2 /F 1, F 3 /F 2, F 4 /F 3.........to F 16 /F 15. Produce table of results (to 7 d.p) using either a calculator or spreadsheet. What do you notice? State any conclusions. F 2 /F 1 1 F 3 /F 2 2 F 4 /F 3 1.5 F 5 /F 4 1.6666666…… F 6 /F 5 1.6 F 7 /F 6 1.625 F 8 /F 7 1.6153846……. F 9 /F 8 1.6190476…….. F 10 /F 9 1.6176470…….. F 11 /F 10 1.6186186…….. F 12 /F 11 1.6179775…….. F 13 /F 12 1.6180555……. F 14 /F 13 1.6180257……. F 15 /F 14 1.6180371……. F 16 /F 15 1.6180344……. The ratio of adjacent terms appears to be getting closer to the Golden Ratio (see presentation).

Puzzle 8 3 5 13 5 5 8 8 A Fibonacci Numbers Puzzle: The Extra Square? 65 -1 = 64 ? Try this with other adjacent Fibonacci numbers. See Problem 8 of the investigation on slide 9.

Pascal’s Triangle 1716 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 33 464 5510 6156 20 7 721 35 88 2856285670 9 3684 126 84 36 9 10 45 210 252 210 120 45 120 10 11 55 330 462 330 165 55 11 12 66 495 792924 792 495 220 66 12 13 78 286 715 1287 1716 1287 715 286 78 13 1 238 21 5589 15 34 144 233 377 Fibonacci Numbers in Pascal’s Triangle? Adding the numbers along the shallow diagonals.

In Nature 1 petal white calla lily 2 Petals euphorbia 3 Petals trillium 5 Petals columbine 13 petals black-eyed Susan 21 petals shasta daisy with 34 petals field daisies The number of petals found on many types of flower are (on average) a Fibonacci number. Daisies with 13, 21, 34, 55 or 89 petals are quite common. Photos courtesy of Jill Britton, mathematics instructor, Camosun College in Victoria, British Columbia, Canada. 8 petals bloodroot

Spirals patterns can be observed in the seed heads of daisies and sunflowers, as well as on the surface of pine cones and pineapples. It is very common for pine cones to have two sets of spirals going in different directions. There are commonly 5 spirals in one direction and 8 in the other or 8 and 13. A similar situation occurs with pineapples. Pineapple Pine cone Sun Flower Spiral pairs in a sun flower seed head are often 21/34 34/55 55/89 or 89/144

21 34 2 3 5 8 13 1 Click to view Distribution of seed heads in a sunflower

Paving Stones 10 ft 2 ft 1 ft 2 ft Paving stones (2 ft x 1 ft) are to be laid so that they completely cover an area of ground measuring 10 ft x 2 ft as shown above. The stones can be laid either vertically or horizontally. Investigate the different ways of laying the stones and find the total number of different arrangements that can be used to cover the ground in question. 1 ft lengths 1 Way 2 ft lengths 2 Ways 3ft lengths Length (ft)Ways 11 22 33 4 5 6 7 8 9 10? 3 Ways Use square paper to help continue the investigation and tabulate your results. We will start by first considering the number of arrangements for smaller lengths of land that are 2 ft wide.

Paving Stones (Results) 10 ft 2 ft 1 ft 2 ft Length (ft)Ways 11 22 33 45 58 613 721 834 955 1089 There are 89 different ways to cover the 10 ft length of ground. From the table of results it can be seen that (from the 3 ft length onwards) the number of stone arrangements for any particular length is equal to the sum of the arrangement for the two previous lengths. For example: F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 The table of results corresponds to the numbers in the famous Fibonacci sequence (starting with F 2 ). You may have met this sequence before and the general term is given by: F n = F n-1 + F n-2

Paving Stones (Results) 10 ft 2 ft 1 ft 2 ft Length (ft)Ways 11 22 33 45 58 613 721 834 955 1089 There are 89 different ways to cover the 10 ft length of ground. From the table of results it can be seen that (from the 3 ft length onwards) the number of stone arrangements for any particular length is equal to the sum of the arrangement for the two previous lengths. For example: By considering the special case of the five 4 ft length arrangements can you explain why the above happens? 4ft arrangements 3ft arrangements 2ft arrangements The five 4ft arrangements are made by adding 1 ft lengths to the three 3 ft arrangements and 2 ft lengths to the two 2 ft arrangements. This same situation will apply in all cases and so is true in general.

Phone Box Investigate the following situation: You have a large quantity of 10p and 20p coins only and you wish to use them to make calls of different values (10p, 20p, 30p, 40p etc) up to the value of £1 from a local telephone box. Coins inserted in different orders are regarded as different ways of paying for the calls. Paying for Telephone Calls For example a 30p call be made in 3 different ways by inserting coins in the following order. 10, 10, 10 20, 10 10, 20 Cost of CallWays 10p 20p 30p3 40p 50p 60p 70p 80p 90p £1?

Paying for Telephone Calls (Results) Cost of CallWays 10p1 20p2 30p3 40p5 50p8 60p13 70p21 80p34 90p55 £189 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 There are 89 different ways to pay for a £1 call. From the table of results it can be seen that the number of ways to pay for any particular call is equal to the sum of the ways to pay for the two preceding calls. For example: The table of results corresponds to the numbers in the famous Fibonacci sequence. You may have met this sequence before and the general term is given by: F n = F n-1 + F n-2

Paying for Telephone Calls (Results) Cost of CallWays 10p1 20p2 30p3 40p5 50p8 60p13 70p21 80p34 90p55 £189 From the table of results it can be seen that the number of ways to pay for any particular call is equal to the sum of the ways to pay for the two preceding calls. For example: By considering the special case of the eight ways to make a 50p call can you explain the above? 10,10,1020,1030p call  10,20 40p call  10,10,10,10 10,10,2010,20,10 20,10,10 50p call  10,10,10,10,1010,10,20,1010,20,10,10 20,20 20,10,10,1020,20,10 10,10,10,2010,20,2020,10,20

Paying for Telephone Calls (Results) Cost of CallWays 10p1 20p2 30p3 40p5 50p8 60p13 70p21 80p34 90p55 £189 From the table of results it can be seen that the number of ways to pay for any particular call is equal to the sum of the ways to pay for the two preceding calls. For example: 10,10,1020,1030p call  10,20 40p call  10,10,10,10 10,10,2010,20,10 20,10,10 50p call  10,10,10,10,1010,10,20,1010,20,10,10 20,20 20,10,10,1020,20,10 10,10,10,2010,20,2020,10,20 The eight 50p ways are made by adding 10p to each of the five 40p ways and 20p to each of the three 30p ways. (8 = 5 + 3) This same situation will apply in all cases and so is true in general. By considering the special case of the eight ways to make a 50p call can you explain the above?

Climbing Stairs Climbing The Stairs Investigate the following situation: You are going to climb a set of stairs consisting of ten steps. You are allowed to take either a single or a double step as you climb to the top. In how many different ways can you climb to the top? Three of the possible ways are shown.

Climbing The Stairs Investigate the following situation: You are going to climb a set of stairs consisting of ten steps. You are allowed to take either a single or a double step as you climb to the top. In how many different ways can you climb to the top? Three of the possible ways are shown. StepWays 1 2 33 4 5 6 7 8 9 10? Start the investigation by considering smaller stepped stairs first and record your results in a table as shown below. Use squared paper to sketch the steps. 3 stepped stairs 3 ways 1, 1, 1 1, 2 2, 1

Climbing The Stairs (Results) StepWays 11 22 33 45 58 613 721 834 955 1089 There are 89 different ways to get to the top of the stairs. From the table of results it can be seen that the number of ways to get to a particular step is equal to the sum of the ways to get to each of the two preceding steps. For example: The table of results corresponds to the numbers in the famous Fibonacci sequence (starting with F 2 ). You may have met this sequence before and the general term is given by: F n = F n-1 + F n-2 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1

Climbing The Stairs Results) StepWays 11 22 33 45 58 613 721 834 955 1089 There are 89 different ways to get to the top of the stairs. From the table of results it can be seen that the number of ways to get to a particular step is equal to the sum of the ways to get to each of the two preceding steps. For example: By considering the special case of the five ways to get to step 4 can you explain the above? 1, 1, 1 1, 2 2, 1 1, 1, 1, 1, 11, 2, 1 2, 1, 1 1, 1, 2 2,2 2,

Climbing The Stairs Results) StepWays 11 22 33 45 58 613 721 834 955 1089 There are 89 different ways to get to the top of the stairs. From the table of results it can be seen that the number of ways to get to a particular step is equal to the sum of the ways to get to each of the two preceding steps. For example: By considering the special case of the five ways to get to step 4 can you explain the above? 1, 1, 1 1, 2 2, 1 1, 2, 1, 1, 1, 11, 2, 1 2, 1, 1 1, 1, 2 2,2 The five ways to get to step 4 must comprise of the three ways of getting to step 3 with an extra single step added, plus the two ways to get to step 2 with a double step added. This same situation will apply in all cases and so is true in general.

To learn more about the Fibonacci numbers in nature visit http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm To learn much more about the Fibonacci numbers in general visit http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

Investigation Worksheet Investigating Some Properties of The Fibonacci Sequence F 12 144 F 11 89 F 10 55 F9F9 34 F8F8 21 F7F7 13 F6F6 8 F5F5 5 F4F4 3 F3F3 2 F2F2 1 F1F1 1 1.Try not to look at the numbers above. Produce a neat table of all numbers to F 20. 2.Do odd an even numbers appear in equal numbers in the sequence? 3.Check that every third term is even. Why is this the case? Find an explanation and write it up clearly. 4.What is the distribution of odd and even numbers within the sequence? 5.The sum (F 1 to F 4 ) = 1 + 1 + 2 + 3 = 7 = F 6 – 1. Is this true in general that Sum(F 1 to F n ) = F n + 2 – 1?. Check this for six more cases. 6.F 1 + F 3 + F 5 + F 7 = F 8 ? Investigate. 7.F 2 + F 4 + F 6 + F 8 = F 9 - 1? Investigate. 8.Choose any number from the sequence and square it. Compare the result with pairs of adjacent numbers. Write down the rule that connecting F n 2 and F n – 1 and F n + 1 9.F 3 2 + F 4 2 = F 7 ? Investigate and generalise using algebraic symbols. 10.Compare the ratios of successive terms F 2 /F 1, F 3 /F 2, F 4 /F 3.........to F 16 /F 15. Produce a table of results (to 7 d.p) using either a calculator or spreadsheet. What do you notice? State any conclusions. Worksheet

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