PATTERN and RELATIONSHIPS

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Felix Preston
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PATTERN and RELATIONSHIPS Patterns are all around us outside in both the natural and built environments. They come in many guises: Number patterns are part of the joy and wonder of maths.

1 PATTERN and RELATIONSHIPS Patterns are all around us outside in both the natural and built environments. They come in many guises: Number patterns are part of the joy and wonder of maths. Forms such as the Fibonacci sequence are link numbers with shape and form in nature. Patterns help us understand order both of numbers and objects, e.g. 1 st, 2 nd, 3 rd, etc. Logic patterns made different attributes including shapes, colours, sizes and lines. Logic patterns naturally work as part of different art activities Word patterns because all languages have patterns within them based upon their lettering or symbols. Rhymes are lovely examples of word patterns. There are intrinsic links between mark making and pattern work. Patterns also provide many ideas for developing descriptive vocabulary Patterns which are heard or felt such as music and dance activities Within some curricula, the focus is primarily on number patterns and relationships which in many ways limits possibilities. What is a pattern? Children explore pattern in different ways, from the casual arranging of shapes, toys and artefacts to making large complicated patterns. Allow children time to create and re-create simple patterns before moving onto more complex ones. Let them comment, question and enjoy the simplicity and complexity of patterns. Making patterns Let children make patterns for each other to continue using natural materials found outside. Another option is to get children to copy each other s pattern. This can be surprisingly challenging if you request that size and similarity matters. Snow is a wonderful medium for making patterns. Use this as a creative opportunity where children make patterns that tell a story, a bit like how trackers follow trails. Juliet Robertson, Creative STAR Learning Ltd,

2 Leaf Logic Doing this logic activity with leaves adds an additional level of challenge in the leaves are not homogenous in size, shape or colour so a continuous discussion is needed as to how to make decisions as to what makes a leaf big or small or a particular colour. Ask each child to gather a few leaves from under a tree. It works best if the leaves have several different colours. Maple trees are especially good for this. Everyone needs to be sitting in a circle around a large sheet. The challenge of the group is to see if everyone can contribute a leaf to the logic line. If you are playing this in a windy place then put stones on top of the leaves to stop them blowing away. This is how it works: The first person puts a leaf down in the middle of the sheet. The next person puts a leaf beside it. One attribute is changed. The leaf is still a big, maple leaf only this time the colour has changed to green. The third person puts a leaf down. This time the leaf is still a green, maple. The attribute that has been changed is size. This leaf is small. The fourth person puts down another leaf. Here the one attribute that has changed is the colour. We've gone back to yellow. The activity continues until no more leaves can be placed in a line. Like with dominoes you can work either end of the line. Once children have got the hang of this activity, it is easy to introduce Carroll Diagrams, which involve sorting objects according to defined attributes. Strategy Games In the majority world countries, many cultural activities take place outdoors. Thus a strategy game is more likely to be played outside using stones and holes or circles drawn in the ground. This is a tradition which has happened for thousands of years. Strategy games have a universal appeal. They are often known by different names and have slightly different rules in different countries. PLEASE SEE THE SEPARATE HANDOUT FOR STRATEGY GAMES Make number sequences with gathered objects from outside The children can be challenged to create a number sequence each and then have to swap with a friend and work out each other s pattern. These can get quite sophisticated over time. Secret Codes Children enjoy making up numerical codes and sequences for others to crack. This blog post explains how this can be taken a step further when outdoors: Juliet Robertson, Creative STAR Learning Ltd,

3 Cows and Bulls This game is the predecessor to Mastermind a classic Seventies strategy game. It can be played as a whole class prior to children working in smaller groups of pairs. This version uses natural materials whereas the traditional approach was to use numbers which is also a useful alternative. The aim is to break the code - that is to work out the pattern of the hidden natural materials Materials White cloth or long box At least 6 different types of natural materials in groups of 6, e.g. 6 stones, 6 sticks, 6 leaves, 6 shells, 6 bark chips, 6 conkers Create a line of 4 stones or shells. Do not let the children see this pattern. Put the pattern in a box or hide it somehow. On top of the white cloth, the children take turns or work in small groups to take turns to put a line of 4 stones or shells onto the cloth facing you and the hidden pattern. For every object correctly placed, then you say it is a bull. For every correct object but in the wrong place, then this is a cow. The games continues until a group or individual has worked out the code. You may want painted pebbles or similar to help children remember the bulls and cows in each line of objects. The game is played in a line of 4 with 6 different types of natural materials can the children work out the number of possibilities? There are 6x6x6x6 = 1296 possible combinations if one allows for duplicate use of natural materials, e.g. stones are used more than once in a line. The other challenge is to try and crack the code in six turns or less. For younger children or when starting out, begin with a line of 3 objects. Fibonacci in Woodlands A Mathematical Investigation Fibonacci lived from He was the son of an Italian merchant. He developed a passion for numbers and discovered the following sequence that can be observed in leaf arrangements, flower segments, pine cones, etc.: 1, 1, 2, 3, 5, 8, 13, 21 If you look at a pine cone you can see that the scales of the cone form regular spirals some go to the left and some to the right. If you count the numbers of scales at each level, you will find that they follow the Fibonacci sequence. Many plants produce new branches in quantities that are based upon Fibonacci numbers. Juliet Robertson, Creative STAR Learning Ltd,

4 Introducing Fibonacci to young children Get the children to gather some loose material whatever is readily available in the wood, e.g. cones or sticks. As a group, layout the material in the Fibonacci sequence on a light coloured cloth so that the children can see the pattern and write down the numbers beside this, e.g. with sticks: 1 I 1 I 2 II 3 III 5 IIIII 8 IIIIIIII 13 IIIIIIIIIIIIII It s unlikely that the children will understand the pattern. However, you can demonstrate how it is created by moving the sticks. Finish up with the story of Mr Fibonacci and how he used pine cones to practise counting 1,1,2,3,5,8,13,21,34, etc (demonstrate this with a pine cone). This was a problem for him. For example when he went to buy food in a shop he always counted out the wrong amounts. If his lemons cost 10 lire, he couldn t count the number 10 so he always gave 13 coins. Everybody laughed at him and thought he was very silly. Over time, he grew more and more unhappy. One day a little girl who had just learned to count realised his problem. When she saw Fibonacci using a pine cone to count, she gave him a daisy and showed him how to pull the petals off and count like everybody else. So Mr Fibonacci was very happy but to this day we are very pleased about the way he counted because he showed the world one of the cleverest number patterns of all! Make your own Fibonacci pattern Collect cones, flowers, stones, leaves or other loose material and try and arrange to create a Fibonacci pattern of your own. Which materials work best for this? Does it depend upon shape, size, weight or another factor? Think about how this can be followed up with an art activity (indoors or out) that uses the Fibonacci pattern as an inspiration. Flower petal challenge Do the number of petals on a flower match the numbers in the Fibonacci sequence? Decide as a class how you will work this out. Juliet Robertson, Creative STAR Learning Ltd,

5 Fibonacci woodland poems The beginning of the Fibonacci sequence can be used to create Haiku-like poetry or stories based upon syllables in each line: 1 Trees 1 in 2 the woods 3 standing tall 5 waving their green leaves 8 catching and filtering sunlight Fibonacci Rabbits Population control! The original problem that Fibonacci investigated (in the year 1202) was about how fast rabbits could breed in ideal circumstances. Suppose a newly-born pair of rabbits, one male, 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 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, one female) every month from the second month on. The puzzle that Fibonacci posed was... How many pairs will there be in one year? At the end of the first month, they mate, but there is still one only 1 pair. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs. Fibonacci and the Golden Ratio The golden ratio is another story for another day. It is hugely significant in nature (e.g. spirals), artwork, building design and is inextricably linked to the Fibonacci sequence: 1 / 1 = 1, 2 / 1 = 2, 34 / 21 = / 2 = 1 5, 5 / 3 = , 8 / 5 = 1 6, 13 / 8 = 1 625, 21 / 13 = If you continue, eventually the pattern settles into the Golden Number of approximately This is a great extension into an investigative project around the Golden Ratio. Have a look at this YouTube video and be inspired Juliet Robertson, Creative STAR Learning Ltd,

6 The Tower of Hanoi The Tower of Hanoi is a maths puzzle that is traditionally completed on rods with wooden discs. However, it is possible to do this using different sizes and colours of leaves. The first job is to create a "base". This can be drawn in forest litter with a stick. Alternatively, sticks, stones, cones or any other material to hand can be used to make the three squares. Next find three leaves of different sizes. These go in the left hand square. The aim of the puzzle is to move all the leaves into another square so they end up in the same order with the largest leaf on the bottom and the smallest leaf on top. There are some rules to follow: Firstly only one leaf may be moved at a time. You may only move the top leaf on a pile. It must be moved to one of the other squares. No leaf may be placed on top of a smaller leaf. With just three leaves this puzzle is straightforward. The more leaves in your pile, the more challenging the problem becomes. With three leaves, it takes seven moves to complete the puzzle. With four leaves, it takes fifteen moves. With five leaves it takes thirty-one moves. Can you work out the pattern? For getting into the deep maths, have a look at the Wiki page Children can use leaves on the ground by themselves or in pairs. However using tyres or large blocks of different sizes, spaced further apart, turns this into a much larger group problem solver especially if the team is timed to see how quickly they can complete the puzzle. Fractals in Nature Fractals also work well and can be introduced through using sticks to explain the rules of selfsimilarity in patterns. These blog posts explain fractals in more detail: Juliet Robertson, Creative STAR Learning Ltd,

Grade 6 Math Circles February 21/22, Patterns

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles February 21/22, 2017 Patterns Tower of Hanoi The Tower of Hanoi is a puzzle with