Transcript
Page 1: Creativity Maths Booklet

North Somerset

Teaching and Learning Team

Autumn 2010

Dave Gale

Creativity in Maths

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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

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Introduction“To 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 I’m 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.

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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 I’m 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 they’ve reached certain points, ask themselves the question:

“WHAT IF ….?”

Hopefully by reading my reflections on the puzzles, you’ll 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. I’ve referenced original versions where possible but otherwise, I’m 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 it’s self-explanatory:

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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.

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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

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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 = 5

b) y = 4x – 12 between x = 5 and x = 6

c) y = 11x – 54 between x = 6 and x = 7

d) y = 2x + 9 between x = 7 and x = 8

e) y = –2x + 41 between x = 8 and x = 9

f) y = –11x + 122 between x = 9 and x = 10

g) y = –4x + 52 between x = 10 and x = 11

h) y = –2x + 30 between x = 11 and x = 15

i) y = 23 between x = 7 and x = 9

j) y = 12 between x = 6 and x = 10

k) y = 10 between x = 7.5 and x = 8.5

l) y = 8 between x = 5 and x = 11

m) y = 4x – 20 between x = 7 and x = 7.5

n) y = –4x + 44 between x = 8.5 and x = 9

o) y = 4 between x = 6 and x = 10

p) y = 2x – 8 between x = 4 and x = 6

q) y = –2x + 24 between x = 10 and x = 12

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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).

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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 you’ve got them all?

Now change the puzzle in some way. Based on what you’ve found out as you investigated the first puzzle, ask yourself “What if I try ……?”

Investigate your new puzzle and make notes on what you’ve found.

Comparing to the first set of questions above, what is different now and what is the same?

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Caterpillars

Here 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 it

ODD Add one

= 1 Stop

Can you find a caterpillar that is longer than the

one shown?

What have you noticed?

What is the shortest

caterpillar possible?

How long is the longest caterpillar possible?

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Probability Game

What are the possible outcomes? How do you know you’ve 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 if……

How 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?

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What is the chance of winning your new game?

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Hints, Guidance and LinksSome of these hints and guidance are taken from the NRich website. Some are my own lines of thought and findings.

I’ve 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 if…

How 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.)

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Painting Between the LinesOriginal ProblemGo to www.nrich.maths.org/7031Or just search for ‘Painting Between the Lines’ at www.nrich.maths.org

Tracing paper! Or a computer image program with a “layers” facility.

A nice link between art and maths. Lots of very striking pictures are quite simple.

How can you use lines to create 3Dness (think vanishing points)?

Look at steepness of lines and their equation.

(Very) Able learners could use curves.

Shading extension allows discussion of inequalities.

Careful consideration of origin positioning and scale of axes is required.

Give students plenty of time to create their image with lines. Homework?

Once students have created their instructions, another student can try to recreate the image without having seen the original.

All students could be given the same photo/image and challenged to create an abstract version with instructions. Ask your art department to judge which is the best abstract representation of the original.

Draw out thoughts about positive and negative gradients.

These are examples of pictures that were drawn:

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Big Ben Empire State

Painting Between the Lines - examples

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Arc de Triomphe

Painting Between the Lines - examples

Statue of Liberty

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Painting Between the Lines

Draw axes for x between 0 and 14 and y between 0 and 17

1 y= 32 -2x between x= 8 and x= 122 y = 8 between x= 4 and x= 123 x = 7.5 between y= 7 and y= 84 y = 2x - 15 between x= 10 and x= 115 y = 7 between x= 4 and x= 116 y = 2x between x= 4 and x= 87 y = 5 between x= 5 and x= 108 y = 15 - 2x between x= 4 and x= 5

Draw axes for x and y between 0 and 17

1 x = 10 between y= 0and y= 3

2 y = x + 2 between x= 1and x= 5

3 y = 18 - x between x= 11and x= 15

4 x = 6 between y= 0and y= 3

5 y = 24 - x between x= 8and x= 13

6 y = 11 between x= 11and x= 13

7 y = 2x + 1 between x= 3and x= 5

8 y = 7 between x= 11and x= 13

9 y = 11 between x= 3and x= 5

10 y = 3 between x= 1and x= 6

11 y = x + 8 between x= 3and x= 8

12 y = 7 between x= 3and x= 5

13 y = 33 - 2x between x= 11and x= 13

14 y = 3 between x= 10and x= 15

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What have you got?

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Painting Between the Lines

Draw axes for x between 0 and 26 and y between 0 and 12

1 y = 12 between x= 3 and x= 162 x = 18 between y= 2 and y= 43 y = 3 between x= 5 and x= 74 y = 3 between x= 19 and x= 215 y = 4 between x= 4 and x= 86 y = 7 between x= 21 and x= 247 x = 22 between y= 2 and y= 48 y = 31 - x between x= 24 and x= 269 x = 26 between y= 2 and y= 5

10 x = 8 between y= 2 and y= 411 y = 5x - 3 between x= 1 and x= 312 y = 2 between x= 8 and x= 1813 x = 4 between y= 2 and y= 414 x = 7 between y= 1 and y= 315 y = 28 - x between x= 16 and x= 2116 y = 2 between x= 1 and x= 417 y = 2 between x= 22 and x= 2618 x = 19 between y= 1 and y= 319 y = 4 between x= 18 and x= 2220 x = 4 between y= 1 and y= 321 y = 1 between x= 5 and x= 722 x = 21 between y= 1 and y= 323 y = 1 between x= 19 and x= 21

What have you got?

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Painting Between the Lines

Draw axes for x between 0 and 17 and y between 0 and 10

1 y = 6 between x= 9 and x= 102 y = 3 between x= 9 and x= 103 x = 13 between y= 2 and y= 34 y = 3 between x= 6 and x= 75 x = 2 between y= 5 and y= 66 y = x - 8 between x= 15 and x= 177 y = x + 6 between x= 1 and x= 38 y = x - 1 between x= 8 and x= 109 x = 7 between y= 5 and y= 6

10 y = 6 between x= 2 and x= 311 y = 1 between x= 1 and x= 1512 x = 3 between y= 5 and y= 613 x = 10 between y= 1 and y= 314 y = 3 between x= 13 and x= 1415 x = 9 between y= 5 and y= 616 x = 14 between y= 5 and y= 617 y = 5 between x= 13 and x= 1418 x = 6 between y= 5 and y= 619 y = 6 between x= 13 and x= 1420 y = 9 between x= 3 and x= 1721 x = 10 between y= 5 and y= 622 x = 14 between y= 2 and y= 323 x = 13 between y= 5 and y= 624 x = 9 between y= 1 and y= 325 y = 5 between x= 6 and x= 726 x = 6 between y= 2 and y= 327 y = 2 between x= 6 and x= 728 y = 3 between x= 2 and x= 329 y = 7 between x= 1 and x= 1530 y = 2 between x= 13 and x= 1431 x = 2 between y= 1 and y= 332 y = 6 between x= 6 and x= 733 y = x - 14 between x= 15 and x= 1734 x = 3 between y= 1 and y= 335 y = 5 between x= 2 and x= 336 y = 5 between x= 9 and x= 1037 x = 7 between y= 2 and y= 3

What have you got?

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Four Card FunOriginal ProblemUnknown but I collected it from Helen Pemberton at Blurton High school.

Use cards with numbers on for weaker students (kinaesthetic).

Encourage simple changes.

Encourage perseverance at the task they’ve set themselves.

Look for patterns in those simple changes to build to more complex changes.

Is the sum 47 + 13 ‘the same as’ 13 + 47 : discuss.

Link to Factorial notation. 4 cards leads to 4! possible sums.

Looking at the ways to change one sum but still keep the same answer:

Link to transformations. Are these sums (partial) reflections of each other?

What happens if you rotate a sum?

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CaterpillarsOriginal ProblemI believe I collected this from my course tutor Dave Miller during my PGCE at Keele.

Encourage competition. “I’ve seen an 8 long caterpillar here. Anyone beaten that?”

Between the class, have they tried them all?

Are there good/poor starting numbers?

If you’ve tried 97, what other numbers have you effectively tested?

Do they really all end in 4, 2, 1?

What if you work in reverse? Start at the tail.

Student: “Can I start with 100?”

Teacher: “You tell me.”

What if you start with 3.5?

What if you start with minus 4?

Is zero odd, even, both or neither?

What is an even number?

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Caterpillar

Examples

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Probability GameOriginal ProblemTo the best of my knowledge, I came up with this idea independently but I’m sure there are other versions.

Relate to fun fair games. They are often rigged so that the person running the stall has much more of a chance of winning than may at first be thought.

Depending on the year group, make use of tree diagrams or just experimental outcomes.

For experimental games, use multilink cubes.

Try going back to the original game and state the same rules/win condition but cover up the beads in the box. Are people happy to play or do they want to know what’s in the box first?

The game needs to be appealing to punters.

If linking to tree diagrams, encourage them to think how their game choices will affect their diagram and its complexity.

May have to boycott some ideas. For example, 100 red, 100 green, 100 yellow, 1 black. Pick out two, if they match you win. Beads are kept out for the subsequent games (without replacement) for the remainder of the day. If you pick the Black bead, then you win some big prize. Lovely idea but too complex!

Here are some games/ideas my students came up with:

10 red, 10 purple, 10 yellow, 10 black.

Pick 3. Win if:

PRY (any order)

PPP

Black = instant lose

5 red and 10 white plastic cases.

Pick 3.

RRR = open the cases, take the gobstoppers inside.

Include a rainbow ball. Picking this means you

double your win.

50 yellow, 20 blue, 10 purple.

Pick 2.

Matching wins an amount determined

by the colour.

£1 to play. 3 bags. One is chosen at random.

One contains 1 gold, 1 silver, 1 bronze and 7 black.

Other two contain 10 black.

Pick one bead.

Gold = £10, Silver = £5, Bronze = £3.

Picking a black ball means you get squirted with

water!

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Links to Schemes of WorkProblems that have been used in KS3 can often be revisited in KS4.

Levels and Grades are approximate. The investigative nature of the tasks probably raises the grades shown by one or two levels.

Key Stage 3Problem SMTP ref KS3 link Level

Sweets in a Box

Y7. SSM1Y7. Alg3Y8. SSM1Y9. SSM1

Y9. Alg4

Use 2D representations to visualise 3D shapes.Recognise and use factors.Investigate in a range of contexts: shape and space.Explore connections in mathematics across a range of contexts: shape and space.Use the prime factor decomposition of a number.

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6Painting Between the Lines

Y7. Alg3

Y7. Alg3Y8. Alg3

Y9. Alg4

Generate coordinate pairs that satisfy a simple linear rule; plot the graphs of simple linear functions where y is given in terms of x.Recognise straight line graphs parallel to the x or y axis.Recognise that equations of the form y=mx+c correspond to straight-line graphs.Given values for m and c, find the gradient of lines given by equations of the form y=mx+c

4

46

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Four Card Fun

Y7. Num1

Y8. N/A1

Use standard column procedure to add and subtract whole numbers.Add, subtract integers

3

3

Caterpillars Y7. Num1 Understand negative numbers as positions on a number line.

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Probability Game

Y7. HD1

Y7. HD3

Y8. HD1

Y8. HD1

Y8. HD1

Y9. HD2

Collect data from a simple experiment and record in a frequency table; estimate probabilities based on this dataCompare experimental and theoretical probabilities in simple cases.Find and record all possible mutually exclusive outcomes for two successive events in a systematic way.Understand that repeating experiments may produce different outcomesUnderstand that increasing the number of trials leads to a better estimate of probability.Use the vocabulary of probability

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5

6

5

5

KS3 cont

All Y7 Represent problems mathematically.Make correct use of symbols.

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Y8

Y9

Understand the significance of a counter-example.Solve more demanding problems and investigate in a range of contexts: number.Identify the necessary information to solve a problem.Use correct notation.Use logical argument to establish the truth of a statement.Suggest extensions to problems, conjecture and generalise; identify exceptional cases or counter-examples.Generate fuller solutions to increasingly demanding problems.Present a concise, reasoned argument, using symbols, diagrams and related explanatory text.

Notes:

SMTP = Sample Medium Term Plan

Alg = algebra

Num = Number

HD = Handling data

SSM = Shape, Space and Measure

N/ A = Number/algebra

Where a phrase is repeated in a later part of the same year group’s plan, I’ve only included the first instance. Eg, if something is listed in Alg1 and Alg3, I’ve just listed Alg1.

Key Stage 4 – AQA GCSE

Grades shown are approximate.

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In the numbering below (taken from the AQA specification):

N = Number and Algebra

G = Geometry

S = Statistics

The numbers reference specific parts of the specification. If there is an ‘h’ after the numbers, it is a higher tier skill.

Problem KS4 links Grades

Sweets in a Box

N1.6 Concepts of factors, multiples, HCF, LCM, PFDG1.4 Recall the properties … of quadrilateralsG2.4 Use 2D representations of 3D shapes

CE to CE/D

Painting Between the Lines

N6.3 Use the conventions for coordinates in the plane and plot points in all four quadrantsN6.4 Recognise and plot equations that correspond to straight-line graphs in the coordinate plane, including finding their gradientsN6.5h Understand the roles that m and c play in y=mx+c

F

D

C/B

Four Card Fun

N1.3 Understand and use number operations G upwards

Caterpillars No explicit GCSE links

Probability Game

S5.1 Understand and use vocabulary of probabilityS5.3 List all outcomes of two successive events in a systematic wayS5.5h Know when to add or multiply probabilitiesS5.6h Use tree diagrams to represent outcomes of compound events recognising when events are independentS5.7 Compare experimental and theoretical probabilities

F/GE

CA*

CAll Functional elements:

RepresentingAnalysingInterpreting

These problems would be considered AO3 (or above!)

Quality of Written Communication (QWC)Clearly there is significant need for QWC within these problems.

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Inspiration and Discussion PointsJust some ideas:

to see what your classes come up with as a solution/approach, to capture some of my thoughts through this process.

What is creativity?

Teacher A: Recognising that there’s a question to be asked and choosing a line of enquiry is creative.

Teacher B: No. That’s just what you do in maths.

Teacher A: Exactly! Maths is creative.

What if you have a variety of balls available?

How do you run a test to decide which is ‘best’?

What would a mathematical cartoon look like?

Could my students use the word conjecture more often?

Have I got wall space where I could put problems that we keep coming back to?

Jo has three numbers which she adds in pairs. When she does that, she gets 11, 17 and 22.

What did she start with?

How would I solve this?

How would my students solve this?

How would my faculty solve this?

Whose method is ‘best’?

Do my students ever see me struggling/persevering with maths?

When asked to rate themselves on “I feel confident that I can tackle new problems”, University maths students ranked themselves among the lowest.

Am I allowing my students an opportunity to try their own ways of solving problems or do I always steer them down my preferred methods?

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Links/referencesBooks

Further Mathematical Diversions, Martin Gardiner

How Long is a Piece of String? Rob Eastaway and Jeremy Wyndham

How many Socks make a Pair? Rob Eastaway

Mathematical Puzzles and Diversions, Martin Gardiner

Professor Stewart’s Cabinet of Mathematical Curiosities, Ian Stewart

Professor Stewart’s Horde of Mathematical Treasures, Ian Stewart

Why do Buses Come in Threes? Rob Eastaway and Jeremy Wyndham

Websites

www.nrich.maths.org Nrich maths resources

www.rsscse.co.uk Royal Statistical Society

www.tes.co.uk/resources Times Educational Supplement - resources


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