creativity maths booklet

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Creativity Maths Booklet

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Creativity in Maths

Table of Contents

Contents.Page 1

Introduction.Page 2

The Problems

Sweets in a Box..Page 4

Painting Between the Lines..Page 5

Four Card Fun..Page 8

Caterpillars..Page 9

Probability Game..Page 10

Hints, Guidance and Links.Page 11

Links to Schemes of Work.Page 22

Inspiration and Discussion Points.Page 25

Links/References.Page 26

IntroductionTo discover the joy of banging your head against a mathematical wall, and then discovering that there may be ways of either going around or over that wall.P43 (Olkin and Schoenfeld, 1994)Real mathematics is both beautiful and creative yet as teachers, we can often find this difficult to convey to students. While the formula for the area of a trapezium can be derived and shown to have a satisfying sense of rightness it is perhaps too simplistic to be considered beautiful. Also, the vast majority of maths that gets taught in school is already well understood by the teacher so the student is unlikely to discover something that their teacher did not already know. This can lead to students believing that maths has all been done already and there is nothing more to be found. It is understandable why students and many non-mathematicians find it hard to equate maths with creativity.This bookletOffers some ideas for allowing students to be creative within the maths classroom but specifically within the maths they try. It is not my intention to have your students making endless pretty display material (although there is always some scope for that), rather that your students should have a chance to explore mathematical ideas and create new problems for themselves.Further, as we all know, any material provided to teachers need to be easily accessible, user friendly, tried and tested, helpfully guided.I would expect you to be able to use the problems/puzzles compiled in this booklet the next time you have a suitable class. You can easily do that by simply displaying or photocopying the initial problem pages and allowing your students to have a go. If this booklet simply introduces you to some new, interesting maths puzzles you can use in the classroom, then Im happy with that. If, however, you find the comments and hints & tips useful and are inspired to look for creative opportunities in your lessons then so much the better.

What is creativity in Maths?Tangrams and origami certainly have their place in allowing the use of mathematics to produce creative objects. There are also numerous maths songs and videos available on the internet but neither of these approaches are what Im aiming for.I want students to be creative in their use of maths when solving puzzles or when creating their own puzzles. It can be quite a creative leap for a student to discover that there is something worth investigating in a puzzle or that a conjecture is worth following up. The overriding theme for this booklet is allowing students to start an investigation and once theyve reached certain points, ask themselves the question:WHAT IF .?Hopefully by reading my reflections on the puzzles, youll see how this develops.Using this bookletI recommend that you look at the problems/puzzles as they are presented and select a class you think they might work with. For at least one of the puzzles, try to be brave and go into the classroom without having worked through the problem first. You can still look at the hints and tips where I provide suggestions for developing the lessons without giving the answers. Allow students the opportunity to see you working through a problem while they do the same. This way, you can model good investigational practice and provide your students the opportunity to develop their mathematical thinking.The majority of the problems in this booklet have been borrowed from other places and are not my creations. Ive referenced original versions where possible but otherwise, Im just bringing to your attention some great puzzles and offering my thoughts and reflections to help you utilize them in class.An important messageI hope its self-explanatory:

Sweets in a BoxProblem | Teachers' Notes | Hint | Solution | Printable page | Stage: 2 Challenge Level:

A sweet manufacturer has decided to design some gift boxes for a new kind of sweet. Each box is to contain 36 sweets placed in lines in a single layer in a geometric shape without gaps or fillers.How many different shaped boxes can you design?

The sweets come in 4 colours, 9 of each colour.Arrange the sweets so that no sweets of the same colour are adjacent to (that is 'next to') each other in any direction. In the diagram below none of the squares marked x can have a red sweet in them.

Arrange the sweets in some of the boxes you have drawn.

Possible extensions:Now try making boxes of 36 sweets in 2, 3 or 4 layers.Can you arrange the sweets, 9 each of 4 colours, so that none of the same colour are on top of each other as well as not adjacent to each other in any direction?See if you can invent a good way of showing your arrangement.Try different numbers of sweets such as 24 or 60 in each box.

Painting Between the LinesProblem | Teachers' Notes | Hint | Solution | Printable page | Stage: 3 and 4 Challenge Level:

In abstract or computer generated art, a real object is often represented by a simplified set of shapes, lines or curves.

Take a look at the picture below:

(Photo: Przemyslaw "Blueshade" Idzkiewicz)

The picture could be represented in an abstract way using straight lines:

If you wanted to describe mathematically the line segments used to draw this picture, all you would need to do is decide where to put the axes and then work out the equations of all the straight lines.

Here is a set of instructions to create an abstract representation of another well known landmark (download as a Word or PDF file).A Straight-Line Landmark

Draw a set of axes with x from 0 to 16 and y from 0 to 25.

Now draw the following straight lines between the given values of x:

a) y = 2x 2 between x = 1 and x = 5b) y = 4x 12 between x = 5 and x = 6c) y = 11x 54 between x = 6 and x = 7d) y = 2x + 9 between x = 7 and x = 8e) y = 2x + 41 between x = 8 and x = 9f) y = 11x + 122 between x = 9 and x = 10g) y = 4x + 52 between x = 10 and x = 11h) y = 2x + 30 between x = 11 and x = 15i) y = 23 between x = 7 and x = 9j) y = 12 between x = 6 and x = 10k) y = 10 between x = 7.5 and x = 8.5l) y = 8 between x = 5 and x = 11m) y = 4x 20 between x = 7 and x = 7.5n) y = 4x + 44 between x = 8.5 and x = 9o) y = 4 between x = 6 and x = 10p) y = 2x 8 between x = 4 and x = 6q) y = 2x + 24 between x = 10 and x = 12

What have you got?

Painting Between the Lines - continuedNow try this for yourself - choose an image, perhaps a photograph of a famous location, or a famous painting.

Think how it might be broken down into shapes or segments of lines.

Can you create a set of instructions for your picture?

Once you have created your instructions, give them to a friend to follow.

Possible extension:Can you devise a way to describe mathematically each region so that you can give instructions for colouring your image?

Please send us any of your creations (together with your instructions).

Four Card FunUsing the cards:

Create as many sums as you can that look like this:

How many different sums can you find?How many different answers do you get?What is the biggest?What is the smallest?How could you check that youve got them all?

Now change the puzzle in some way. Based on what youve found out as you investigated the first puzzle, ask yourself What if I try ?Investigate your new puzzle and make notes on what youve found.Comparing to the first set of questions above, what is different now and what is the same?

CaterpillarsHere is a six-segment mathematical caterpillar:

You pick the starting number to go in his head. This must be under 100.To get the next numbers, you follow these simple rules:EVEN Halve itODDAdd one= 1StopHow long is the longest caterpillar possible?Can you find a caterpillar that is longer than the one shown?

What have you noticed?What is the shortest caterpillar possible?

Probability Game

What are the possible outcomes? How do you know youve found them all? Can you find a systematic way of writing them down? What is the probability of winning this game? Would you play it?

How could you change this game?What ifHow many beads?How many colours and how many of each colour?How many get picked out?What are the win conditions?What else could you change?What is the chance of winning your new game?

Hints, Guidance and LinksSome of these hints and guidance are taken from the NRich website. Some are my own lines of thought and findings. Ive deliberately avoided giving you the answers (where they exist).Sweets in a BoxOriginal ProblemGo to www.nrich.maths.org/84Or just search for Sweets in a Box at www.nrich.maths.org

Are there any simplifying assumptions necessary (size/uniformity of sweet)?Counters could be provided for lower ability (kinaesthetic).Have you got all the possible rectangular designs?Encourage non-rectangular shapes. Put different sorts of paper out to use: Isometric, Square dotty, blank.How could the puzzle be changed? What ifHow many ways can it be coloured? Will you method work for other layouts?Are there any amounts of colours that make the task impossible? (ie can it be done with 3 colours, 5 colours?)Possible link to the four colour theorem.What if the sweets are triangular?Different tray sizes: Link to Least Common Multiple and Prime Factor Decomposition.Could they make the tray? (Nets, Max Box problem.)

Painting Between the LinesOriginal ProblemGo