Patterns In real lifeSession 05-06

Objectives

To determine some patterns in nature, in geometry and in other patterns in real life.

To appreciate the patterns in real life. To document mathematical patterns seen and available in the locality.

ACTIVITY Group yourself into

eight groups. In each group, there

will be a leader, notetaker, presenter and material manager.

Perform the task and have a happy learning!

Palindrome QuizROUND 1

PalindromeNumber or words that reads the same

forwards and backwards.

Example: Tapat, Bob, 55, 101.

10

Question 1

987654321NOON

When both a clock’s hands are on 12, and the sun is overhead, what time is it?

10

Question 2

987654321PUPThis young dog is a palindrome.

10

Question 3

987654321EYEThis part of the body is a palindrome.

10

Question 4

987654321DADThis member of the family is a palindrome.

10

Question 5

987654321MOMThis member

of the family is also a palindrome.

10

Question 6

987654321KAYAKThis type of

boat is a palindrome.

Question 7

What is the greatest palindromic number less than 50?

44

Question 8

What is the smallest palindromic number greater than 20?

22

Question 9

What is the next palindromic number after the one you have just found?

33

Question 10

What is the palindromic time you might have a cup of tea just before 10.00am.

9:59

Question 11

What was the last palindromic year in the 20th Century?

1991

Question 12

What was the first palindromic year in the 21st Century?

2002

What’s next?ROUND 2

Question 01

Draw the next figure.

Question 02

Draw the next figure.

Question 03

Draw the next figure.

Question 04

Draw the next figure.

Question 05

Determine the next number.

1, 8, 21, 40, 65, 96, 133, ___

176

Search for Mr and Miss Golden PNURound 3

1. Sheet with directions on what lengths to measure and ratios to calculate

2. Table to record fractional and decimal representations of each ratio.

Materials Needed1. Calculators (1 per group)

2. Yard Stick (1 per group)3. Measuring Tape (1 per group)4. Activity Sheets and pencils

Activity Sheets

MethodStep 1: Measure the height (B) and the navel height (N) of each member of your group. Calculate the ratio B/N. Record them in your table.

Step 2: Measure the length (F) of an index finger and the distance (K) from the finger tip to the big knuckle of each member of your group. Calculate the ratio F/K. Record them in your table.

Step 3: Measure the length (L) of a leg and the distance (H) from the hip to the kneecap of everyone in your group. Calculate and record the ratio L/H.

Step 4: Measure the length (A) of an arm and the distance (E) from the finger tips to the elbow of everyone in your group. Calculate and record the ratio A/E.

Step 5: Measure the length (X)of a profile, the top of the head to the level of the bottom of the chin and the length (Y) from the bottom of the ear to the level of the bottom of the chin. Calculate and record the ratio X/Y.

Step 6: Select another pair of lengths (Q and P) on the body that you suspect may be in the golden ratio. Measure those lengths. Calculate the ratio (large to small) and record it.

B

N

A

E

XY

NameExpress each

ratio in its both its

fraction and decimal form.

B/N F/K L/H A/E X/Y

1.

2.

3.

4.

5.

And the winner is …

Third Runner-upSecond Runner-upFirst Runner-upMr and Miss Golden PNU 2015

ANALYSIS Share your insights to

the following questions

Which patterns affect or you appreciate most? Why? What particular steps did you follow to identify the

patterns given in the activities? What other patterns do you know that are present in

your locality? How vital these patterns in your career as future

teachers? What realizations do you have about the

mathematics and the patterns in real life?

ABSTRACTION

Does maths really appear in nature?

In a word, yes.However, unless you know what to look out for, it isn’t

very easy to spot.For example, would you have thought that the breeding of

rabbits could be modelled on a simple number sequence?

But that this same sequence can be used to construct the spiral shape that we see on a sea shell?

In this presentation I aim to show some examples of the many different cases where you can find maths in the real world.

A pretty face?

It is quite obvious that the human face is symmetrical about a vertical axis down the nose.

However, studies have shown that the symmetry of another persons face is a large factor in determining whether or not we find them attractive.

The better the quality of the symmetry, the more mathematically perfect it is and the more aesthetically pleasing we consider it to be.

In short, the better the symmetry of someone's face, the more attractive you should find them!

BEAUTY

PHIBEHOLDER

of the

is in the

BeautyA quality that gives aesthetic pleasureVisual pleasantness of a person, animal, object or scene.Pleasantness of sound.

Subjectivity of BeautyPerception of each personRace, culture, era

No unification or generalization“Beauty is in the eye of the Beholder”

The Divine Proportion The Golden Ratio

PHI

Mathematics and the Golden Ratio

Ratio = 1.618… = Phi = ø

Fibonacci NumbersEach number is phi times the last 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…Found in nature

Shasta Daisy Field Daisy Passion Flower

Columbine BloodrootWhite Calla Lily Euphorbia

Black-eyed SusanTrillium

Golden RectangleLeonardo Da VinciMona Lisa Vitruvius

Leonardo Da Vinci

The Parthenon

Golden Pentagon

Raphael Crucifixion

The Sacrament of the Last SupperSalvador Dali

Golden Spiral

Nautilus ShellCapitulum: SunflowerPine Cones

The Milky Way Galaxy

DNADNA

PHI MASKDr Stephen Marquardt

Roman: Lucilla Verus 164 A.D.

Egyptian: Queen Nefertiti 1400 BC

Raphael: The Small Cowper, Madonna 1505 A.D

Greta Garbo: 1931

Marylin Monroe: 1957

Beehive basics

A beehive is made up of many hexagons packed together.

Why hexagons? Not squares or triangles?

Hexagons fit most closely together without any gaps, so they are an ideal shape to maximise the available space.

Rabbit multiplicationThe breeding of rabbits is a very effective

way of demonstrating the Fibonacci sequence.

The Fibonacci sequence is a sequence of numbers formed by adding together the 2 previous numbers.

The Fibonacci sequence starts as-0, 1, 1, 2, 3, 5, 8, 13, 21, 34.So how is this relevant to rabbits breeding?Lets suppose a newly-born pair of rabbits,

one male and one female, are put in a field. Rabbits are able to mate at the age of one month, so that at the end of its second month of life a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male and one female) every month.

What would happen?

So what happens?1. We start off with 1 pair of rabbits in the

field.2. At the end of the first month the original

pair mate but there is still one only 1 pair. 3. At the end of the second month the female

produces a new pair, so now there are 2 pairs of rabbits in the field.

4. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.

5. At the end of the fourth month, the original female has produced yet another new pair and the female born two months ago produces her first pair, making 5 pairs.

Noticed the sequence yet?Over the course of 5 months the number of

pairs in the field has gone 1, 1, 2, 3, 5. The Fibonacci sequence!

More FibonacciThe Fibonacci sequence can also be used in another, more

visual, way.This is the process of creating Fibonacci rectangles.1. Start with two small squares of size 1 next to each other.

On top of both of these draw a square of size 22. We can now draw a new square - touching both a unit

square and the latest square of side 2 - so having sides 3 units long

3. Then another touching both the 2-square and the 3-square (which has sides of 5 units).

4. We can continue adding squares around the picture, each new square having a side which is as long as the sum of the last two square's sides.

This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, we call the Fibonacci Rectangles.

This is only the first 7 numbers in the Fibonacci sequence.What would happen if we carried on a lot longer?

So what happens?

As we go on, the squares begin to form a certain pattern. If we draw a line through the corner of each square we start to get a spiral shape.

The same spiral shape that we can see on this sea shell!

Fibonacci flowers?

The Fibonacci sequence previously mentioned appears in other cases.

The ratio of consecutive numbers in the Fibonacci sequence approaches a number known as the golden ratio, or phi (1.618033989). Phi is often found in nature. A Golden Spiral formed in a manner similar to the Fibonacci spiral can be found by tracing the seeds of a sunflower from the centre outwards.

More Fibonacci flowers?On many plants, the number of petals is a

Fibonacci number:3 petals: lily, iris 5 petals: buttercup, wild rose, larkspur 8 petals: delphiniums 13 petals: ragwort, corn marigold, cineraria, some daisies 21 petals: aster, black-eyed susan, chicory 34 petals: plantain, pyrethrum 55, 89 petals: michaelmas daisies, the asteraceae family. Ever wondered why its so difficult to find a 4 leaf clover? Or even a plant of any kind with 4 petals? Few plants have 4 petals, and 4 is not a Fibonacci number.

Coincidence?

Natural fractals?Fractals don’t appear in nature as such,

but they are another clear example of the way maths and nature can be linked together.

A fractal is a geometric pattern that is repeated at every scale and so cannot be represented by classical geometry.

A famous example is the Koch curve (shown on the right).

Stage 1 is to draw a straight line.All stages afterwards are constructed by

rubbing out the middle of a line and drawing 2 more diagonal lines in its place (resembling a triangle).

So what would happen if we carried this fractal on for many more stages?

So what would happen?

We would end up with the famous Koch snowflake.

By just repeating a simple pattern, you can create a snowflake, yet another example of how maths and nature can share a connection.

Are there any more fractals that create natural images?

Yes! Over the next few slides are some of the most impressive natural fractals that have been discovered.

Fractal trees

Fractal ferns

Fractal spiral

Figurate Numbers

Formulas for Triangular, Square, and Pentagonal NumbersFor any natural number n,

( 1)the th triangular number is given by ,2n

n nn T

2the th square number is given by , andnn S n

(3 1)the th pentagonal number is given by .2

nn nn P

Example: Figurate NumbersUse a formula to find the sixth pentagonal number

Solution (3 1)

2nn nP

with n = 6:6

6[6(3) 1] 51.2

P

Use the formula

What is a Tessellation?

A tessellation is any repeating pattern of interlocking shapes.

The word tessellation comes from the Latin Tessella, which was a small Square stone or tile used in ancient Roman mosaics.  Tiles and Mosaics are common synonyms for tessellations. 

Some shapes, or polygons, will tessellate and others will not.  As for the regular polygons, tessellations can easily be created using squares, equilateral triangles and hexagons.

A Roman floor mosaic

Tessellations can be simple…

Tessellations can also be very complex…

M. C. Escher: Tessellation Master

Escher produced '8 Heads' in 1922 - a hint of things to come.

His inspiration…Escher took a boat trip to Spain and went to the Alhambra, an extravagant palace full of pattern.

There, he copied many of the tiling patterns. '8 Heads' - 1922

Alhambra CastleAlhambra consists of palacesbuilt by several rulers, each had his own castle.

One of the most well known example of Muslim architecture.

The Alhambra Palace is a famous example of Moorish architecture.

Islamic art does not usually use representations of people, but usesgeometric patterns.

The idea behind several of the buildings of Alhambra was to create a Paradise on earth.

Escher copied many of the designs he saw a Alhambra, adding his own flair

M. C. Escher 4 Motifs 1950

Design for Wood Intarsia Panel for Leiden Town Hall, 1940

Tessellation transitions

by M. C. Escher

Realism & Tessellations Combined

Sometimes, Escher would combine realism and tessellations.

Reptiles is an example of this combination.

'Reptiles' - 1943

Metamorphosis I, 1937

by M. C. Escher

Realism & Tessellation Combined

Cycle, 1938

by M. C. Escher

Realism & Tessellation Combined

Day and Night, 1938 by M. C. Escher

Realism & Tessellation Combined

Escher’s Last Tessellation

His last tessellation was a solution to a puzzle sent to him by Roger Penrose, the mathematician. Escher solved it and, true to form, changed the angular wood blocks into rounded 'ghosts'.

Penrose 'Ghosts' - 1971

How do you create a successful tessellation?

Begin with a simple geometric shape - the square

Change the shape of one side

Copy this line on the opposite side

Rotate the line and repeat it on the remaining edges

Erase the original shape

Add lines to the inside of the shapes to turn them into pictures.

Add color to enhance your picture.

By repeating your shape you create a tessellated picture

How do you create a more complex tessellation?

Draw a line that separates the two hidden shapes you have found.

Add a few lines that bring out your hidden shapes.

Make four versions of each shape, each version with more detail

The most detailed shape can be changed quite a bit

Make four versions of each shape with more detail

The most detailed shape can be changed quite a bit

Color all of one type of shape the same basic color scheme

Line up the simplest shape with the most complex along the bottom

Add the next row in the same way

Completed Tessellation

Tanong ko lang: Bakit rectangle ang karamihan hugis ng

kulungan ng baboy? Bakit bilog ang takip ng manhole? Bakit bilog ang hugis ng gulong?

See problem solving book in the boarding

APPLICATION Prepare an AV

presentation/documentation of the mathematical concepts in the environment and its relevance to everyday living. Use either powerpoint presentation or moviemaker. Plan for your concept in presenting the output. Use music, graphics, texts, and other audio visual devices or materials. Submit the requirement after two weeks.

FLEXIBLE LEARNING ACTIVITY