8.3.14 golden ratio activity. 8.3.14 video on golden ratio

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Golden Ratio Activity

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Video on Golden Ratio

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Measurements• a = Top-of-head to chin = ………cm

• b = Top-of-head to pupil = ……… cm

• c = Pupil to nosetip = ……… cm

• d = Pupil to lip = ……… cm

• e = Width of nose = ……… cm

• f = Outside distance between eyes = ……… cm

• g = Width of head = ……… cm

• h = Hairline to pupil = ……… cm

• i = Nosetip to chin = ……… cm

• j = Lips to chin = ……… cm

• k = Length of lips = ……… cm

• l = Nosetip to lips = ……… cm

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Ratios

• Now, find the following ratios:

• a/g = ……… cm

• b/d = ……… cm

• i/j = ……… cm

• i/c = ……… cm

• e/l = ……… cm

• f/h = ……… cm

• k/e = ……… cm

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Unit 14: The golden ratio

Lesson 1: The golden ratio

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The objectives of this lesson are:

• to solve more complex problems by breaking them into smaller steps or tasks, choosing and using efficient techniques for calculation, graphical representation and resources, including ICT

• to round decimals to the nearest whole number or to one or two decimal places

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The objectives of this lesson are:

• to consolidate understanding of the relationship between ratio and proportion; reduce a ratio to its simplest form, including a ratio expressed in different units, recognising links with fraction notation

• to enter numbers and interpret the (calculator) display in different contexts (fractions, decimals).

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Fibonacci’s problem

A certain man put a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair from which the second month on becomes productive?

Fibonacci, Liber abbaci, p283–284

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Suppose that the pair of rabbits are put in the place surrounded by a wall on 1 January.

Fibonacci’s problem

How many rabbits will there be a month later, on 1 February?

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How many rabbits will there be a month later, on 1 March?

Fibonacci’s problem

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How many rabbits will there be a month later, on 1 April?

Fibonacci’s problem

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How many rabbits will there be a month later, on 1 May?

Fibonacci’s problem

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How many rabbits will there be a month later, on 1 June?

Fibonacci’s problem

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How many rabbits will there be a month later, on 1 July?

Fibonacci’s problem

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How many rabbits will there be a month later, on 1 August?

Fibonacci’s problem

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The row for 1 August is incomplete.

Fibonacci’s problem

Where should the new pairs be?

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Which pairs are A’s children?

Fibonacci’s problem

Which pairs are B’s children?

How are pair C11 related to pair A?

B, C, D, E, F, G

B1, B2, B3, B4

They are A’s great-grandchildren.

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How many pairs of rabbits are there on the first day of each month?

Fibonacci’s problem

1, 1, 2, 3, 5, 8, 13, 21

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1, 1, 2, 3, 5, 8, 13, 21

Fibonacci’s problem

What will the next number in the sequence be?

34 (13 + 21)

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This sequence of numbers is known as the Fibonacci sequence.

Fibonacci’s problem

Here are the first twenty terms:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765

1, 1, 2, 3, 5, 8, 13, 21, 34,...

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Ratio of pairs of successive terms

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765

What is the ratio of:

• the 2nd to the 1st term?• the 3rd to the 2nd term?• the 4th to the 3rd term?• the 5th to the 4th term?

1 : 12 : 13 : 25 : 3

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Ratio of pairs of successive terms

Express the first four ratios as decimal fractions.

2nd term ÷ 1st term =3rd term ÷ 2nd term =4th term ÷ 3rd term =5th term ÷ 4th term =

121.51.6666...

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765

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Continue the sequence of ratios, writing down all the digits shown on your calculator.

Ratio of pairs of successive terms

1, 2, 1.5, 1.66667, 1.6, 1.625, 1.61538, 1.61905, 1.61765, 1.61818, 1.61798, 1.61806, 1.61803, 1.61804, 1.61803, 1.61803, 1.61803...

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765

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Plenary

We'll now plot on a graph the first fifteen ratios of successive terms of the Fibonacci sequence.

1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987

2.0

1.9

1.8

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

ratio

: te

rm d

ivid

ed b

y pr

evio

us t

erm

term of Fibonacci sequence

Ø, the golden ratio

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The ratio is oscillating around 1.618 033 988 749 9 …

This number is known as the golden ratio and is usually denoted by Ø.

It is an irrational number (meaning that its digits go on forever without repeating).

Plenary

___2

1 + 5The exact value of Ø is