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Notes for Teachers

This activity is based upon Non-Transitive Dice, and is an excellent exploration into some seemingly complex probability.. The three dice version has been around for a while, but with different numbers on the dice. The version here is so it fits with the 5 dice version. (If you have a three dice set, the probabilities in each case are the same, it is just the numbers on the dice that need changing in the Tree Diagrams).

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Notes for Teachers






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Slide1

Notes for Teachers

This activity is based upon Non-Transitive Dice, and is an excellent exploration into some seemingly complex probability.

The three dice version has been around for a while, but with different numbers on the dice. The version here is so it fits with the 5 dice version. (If you have a three dice set, the probabilities in each case are the same, it is just the numbers on the dice that need changing in the Tree Diagrams)

It is best done using the Non-Transitive Dice, which you can buy from

http://mathsgear.co.uk/collections/dice/products/non-transitive-grime-dice

You could also make the dice as a starter activity, and a recap on nets (just use different coloured card, and remember to put the correct numbers on each die).

The slides talk the students through what they need to do, and I have put some comments on ideas for questions and practicalities in the notes box.

The Grime dice (5 dice set) were discovered by James Grime of the University of Cambridge, and his video description and article can be found at

http://grime.s3-website-eu-west-1.amazonaws.com/

This slideshow is an attempt at a teacher friendly, usable in the classroom, way of presenting this information

.

The

spreadsheet

calculates all the probabilities and allows users to change the values on the dice.

There is another great way to introduce

Non-Transitive dice at

http://

nrich.maths.org/7541

For more interactive resources, visit my website at

http://www.interactive-maths.com/

Slide2

Dice Games

In your pairs, you are going to play a game.

You each have a coloured die, and you are going to both throw your die.

The player with the highest score wins that round.

Play 10 rounds.

Who is winning overall?

Play a further 90 rounds (100 in total).

Is the game fair?Slide3

What did we discover?

We saw that

RED

beats BLUE.

How did RED and BLUE compete?

We saw that BLUE beats GREEN.

How did BLUE and GREEN compete?

What do we expect in the RED vs GREEN games?

We expect that since

RED

beats

BLUE

and

BLUE

beats

GREEN

, then RED will beat GREEN.

This is called a Transitive Property – the win is transferred through the blue!

Numbers are transitive: if 5 > 3 and 3 > 1, then 5 > 1!Slide4

We see that

GREEN

beats RED.

What actually happened in the RED and GREEN games?

BEATS

BEATS

BEATS

Non-Transitive DiceSlide5

Why is it that this happens?

Let’s take a look at the probabilities!

First we need to know what numbers are on each die.

4, 4, 4, 4, 4, 9

2, 2, 2, 7, 7, 7

0, 5, 5, 5, 5, 5

Now we can use our knowledge of probabilities to calculate the probability in each battle.

We shall use a tree diagram to consider the multiple outcomes.Slide6

4, 4, 4, 4, 4, 9

2, 2, 2, 7, 7, 7

0, 5, 5, 5, 5, 5

RED vs

BLUE

RED

4

9

BLUE

2

7

 

 

2

7

 

 

 

 

 

 

 

 

So

RED

wins over

BLUE

with probability

 Slide7

4, 4, 4, 4, 4, 9

2, 2, 2, 7, 7, 7

0, 5, 5, 5, 5, 5

Use the values on the three die to make two further Tree Diagrams to show that the Dice are indeed Non-Transitive.Slide8

4, 4, 4, 4, 4, 9

2, 2, 2, 7, 7, 7

0, 5, 5, 5, 5, 5

BLUE vs

GREEN

BLUE

2

7

GREEN

0

5

 

 

0

5

 

 

 

 

 

 

 

 

So

BLUE

wins over

GREEN

with probability

 Slide9

4, 4, 4, 4, 4, 9

2, 2, 2, 7, 7, 7

0, 5, 5, 5, 5, 5

GREEN vs

RED

GREEN

0

5

RED

4

9

 

 

4

9

 

 

 

 

 

 

 

 

So

GREEN

wins over

RED

with probability

 Slide10

Pair up with somebody with the same colour die as you.

Now make a group of 4 by joining another pair (there should be two dice of two

different

colours in your group).

We are going to play the game again, but taking the total of the same coloured dice.

Play 100 rounds as before, and keep track of how many rounds each colour wins.Slide11

What did we discover this time?

We saw that

BLUE

beats RED.

How did RED and BLUE compete?

We saw that GREEN beats BLUE.

How did BLUE and GREEN compete?

We saw that RED beats GREEN.

How did GREEN and RED compete?

This is the opposite to what happened with only one die of each colour!!!Slide12

With two dice, the rules are a little bit different!

BEATS

BEATS

BEATS

Let’s have a look at the probabilities again!Slide13

4, 4, 4, 4, 4, 9

2, 2, 2, 7, 7, 7

0, 5, 5, 5, 5, 5

RED vs

BLUE(two dice)

8

18

4

14

 

 

4

14

 

 

 

 

 

13

9

9

 

 

 

4

14

 

 

9

 

 

 

 

 

 

 

 

 

So

BLUE

wins over

RED

with probability

 Slide14

4, 4, 4, 4, 4, 9

2, 2, 2, 7, 7, 7

0, 5, 5, 5, 5, 5

BLUE vs

GREEN(two dice)

4

14

0

10

 

 

0

10

 

 

 

 

 

9

5

5

 

 

 

0

10

 

 

5

 

 

 

 

 

 

 

 

 

So

GREEN

wins over

BLUE

with probability

 Slide15

4, 4, 4, 4, 4, 9

2, 2, 2, 7, 7, 7

0, 5, 5, 5, 5, 5

GREEN vs

RED(two dice)

0

10

8

18

 

 

8

18

 

 

 

 

 

5

13

13

 

 

 

8

18

 

 

13

 

 

 

 

 

 

 

 

 

So

RED

wins over

GREEN

with probability

 Slide16

SUMMARY

BEATS

BEATS

BEATS

One Die

BEATS

BEATS

BEATS

Two Dice

Remember the word lengths get bigger:

RED (3) -> BLUE (4) -> GREEN (5)

How to Use this Game

Place the three dice out, and get a friend to play. Ask them to choose a die to use, and you then pick the one which will beat it. Role the dice 20 times, and you should win.

Once they think they have worked it out, agree to take the die first. When they pick a die, if you are to win, leave it be, but if you are to lose say that you want to “double the stakes” with a second die each. This reverts the order!

4, 4, 4, 4, 4, 9

2, 2, 2, 7, 7, 7

0, 5, 5, 5, 5, 5Slide17

This is a set of 5 Non-Transitive Dice

What do you notice about the dice?

The 3 dice set is included within the 5 dice set.

The numbers 0-9 appear on exactly 1 die.

4, 4, 4, 4, 4, 9

2, 2, 2, 7, 7, 70, 5, 5, 5, 5, 5

3, 3, 3, 3, 8, 8

1, 1, 6, 6, 6, 6This set of dice are called Grime Dice, after their discoverer, James Grime at the University of CambridgeSlide18

As with the 3 dice set, we can work out the probabilities in each pairing.

How many different ways could we pair up the different coloured dice?

RED with each of

BLUE, OLIVE, YELLOW and MAGENTA

BLUE with each of OLIVE,

YELLOW and MAGENTA

OLIVE with each of YELLOW and MAGENTA

YELLOW with MAGENTA

4

3

2

1

We use

OLIVE

and

MAGENTA

instead of green and purple for a good reason we shall see!!!.

So there are 10 possible pairings!

We need to look at all of them!Slide19

Each pair has been given a colour pair to look at. Use a tree diagram to calculate the probabilities involved, and which colour will win.

We already know three:

RED

> BLUE with probability

 

BLUE > OLIVE with probability

 

OLIVE > RED

with probability  Slide20

 

 

 

 

 

 

 

 

 

 

And now for the full list of all the probabilities………

What do you notice?

There are 2 chains that work for the 5 dice

How do the names relate to the chains?

Colour names get longer

Colour names are alphabetical

How do the probabilities compare?Slide21

BEATS

BEATS

BEATS

BEATS

BEATS

BEATS

BEATS

BEATS

BEATS

BEATSSlide22

Notice that we can make several sets of 3 Non-Transitive dice by following paths on this graph.

Each of these 5 subsets of dice will produce a valid set of 3 Non-Transitive Dice.

They are obtained by taking 3 consecutive dice in the Word Length list.Slide23

We can also make sets of 4 Non-Transitive Dice!

Each of these 5 subsets of dice will produce a valid set of

4

Non-Transitive Dice.

They are obtained by taking

4

consecutive dice in the

Alphabetical

list.Slide24

Combine two pairs to make a group of 4 people, with 10 dice!

In your group, investigate what happens in the different combinations available when each pair has 2 dice (of the same colour).

You can use a mixture of experimental probabilities and theoretical probabilities.Slide25

 

 

 

 

 

 

 

 

 

 

And now for the full list of all the probabilities………

What do you notice?

The Word Length Chain is reversed as expected.

The Alphabetical Chain is in the same order

Colour names get longer

Colour names are alphabetical

How do the probabilities compare?Slide26

This line is 50:50 either waySlide27

SUMMARY

How to Use this Game

Place the three dice out, and get a friend to play. Ask them to choose a die to use, and you then pick the one which will beat it. Role the dice 20 times, and you should win.

Once they think they have worked it out, agree to take the die first. When they pick a die, if you are to win, leave it be, but if you are to lose say that you want to “double the stakes” with a second die each. This reverts the order!

One Die

Two Dice

4, 4, 4, 4, 4, 9

2, 2, 2, 7, 7, 7

0, 5, 5, 5, 5, 5

3, 3, 3, 3, 8, 8

1, 1, 6, 6, 6, 6

Word Length

AlphabeticalSlide28

In your groups you are going to create a poster on Non-Transitive Dice.

Colour

Title

Background Info

Some of the Maths

Challenges

PresentationSuccinct

LayoutSlide29

A Special Game

We can now use the set of 10 dice to play two players at once, and improve our chance of beating both of them

Invite two opponents to pick a die each, but do NOT say whether you are playing with one die or two.

If you opponents pick two dice that are next to each other on the alphabetical list (not next to each other around the circle), then play the one die game, and use the diagram to choose the die that will beat both most of the time.

If you opponents pick two dice that are next to each other on the word length list (next

to each other around the circle), then play the two dice game, and use the diagram to choose the die that will beat both most of the time.