Dice Dilemmas 2019 eBook...

Dice Dilemmas

Mathematics Manipulatives Manual

Dr Paul Swan

Activities to promote mental computation and develop thinking about chance processes

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3 Dice Dilemmas© P. Swan

Teacher Guide 4

Connect Three 6

Three Throw 10

Dice Digits 12

More Than Less Than 15

One Hundred or Bust 16

EeeZee 18

Mary's Game 20

Go for Broke 21

Double Dare 22

Maxi-Min 23

Dice Cricket 24

Duelling Dice 25

Coordinate Dice 26

Product Pairs 28

Oh No 30

Dice Tower 32

Add'n Multiply 33

Area Dice 34

The Great Car Race I 36

The Great Car Race II 38

The Great Car Race III 39

Cross the Road 40

Collector Cards 42

Find the Product 43

Fair Enough? 44

Sum Dice 45

Birth Months 46

Teacher Notes and Answers 47

Contents

© P. Swan4Dice Dilemmas

Games Australian Curriculum Links Mathematics Concepts

Connect Three Yr 2: ACMNA030

Yr 3: ACMNA055

Fluency with the four operations.

Three Throw Yr 3: ACMNA056

Yr 3: ACMNA057

Yr 4: ACMNA075

Yr 4: ACMNA076

Practise in recalling multiplication facts

and computing 2-digit by 1-digit products.

Associative property of multiplication.

Duelling Dice Yr 3: ACMNA055

Yr 6: ACMNA128

Addition of 2-digit numbers or decimal

numbers.

Dice Digits

More Than Less Than

One Hundred or Bust

Yr 3: ACMNA053 Developing place value understanding.

EeeZee Yr 3: ACMNA053

Yr 3: ACMNA057

Repeated addition or multiplication with

numbers 1 - 6.

Mary's Game Yr 3: ACMNA055

Yr 3: ACMNA056

Yr 4: ACMNA075

Fluency with the four operations using

numbers 1 - 6.

Go for Broke

Double Dare

Dice Cricket

Yr 3: ACMNA055 Addition of one and two-digit numbers.

Maxi-Min Yr 3: ACMNA055

Yr 4: ACMNA073

Addition and subtraction of numbers in the

thousands. Place value.

Coordinate Dice Yr 6: ACMMG143 Using the cartesian coordinate system in the

positive quadrant.

Product Pairs Yr 3: ACMNA055

Yr 3: ACMNA056

Yr 3: ACMNA057

Yr 4: ACMNA075

Multiplication of numbers 0 - 9 and addition

of double-digit numbers.

Oh No Yr 3: ACMNA053

Yr 3: ACMNA051

Place value into thousands.

Dice Tower

Add'n Multiply

Yr 2: ACMNA030

Yr 3: ACMNA055

Yr 3: ACMNA056

Yr 4: ACMNA075

Addition and multiplication with numbers

1 - 6.

Teacher Guide


Games Australian Curriculum Links Mathematics Concepts

Area Dice Yr 2: ACMNA031 Using the area model to understand

multiplication, including commutative

property.

The Great Car Race I

The Great Car Race II

The Great Car Race III

Cross the Road

Sum Dice

Yr 4: ACMNA071

Yr 5: ACMSP116

Addition, subtraction and multiplication with

numbers 1 - 6. Probability.

Birth Months

Collector Cards

Yr 5: ACMSP116 Investigating the probability of certain

outcomes occuring from a series of equally

likely events.

Find the Product

Fair Enough?

Yr 4: ACMNA071

Yr 5: ACMSP116

Multiplication or subtraction of numbers 1 - 6.

Odd and even probability numbers.

Dice are a cheap, readily available and versatile resource that may be used to develop a wide range of mathematical ideas and skills including:

• Mental computation (+, –, x and ÷),

• place value and

• chance processes.

The games and activities in this book have been designed to make use of commonly available dice.

The pages have been designed with the busy teacher in mind. At a glance a teacher may see what type of dice and how many are needed for any one activity.

Tetrahedron Hexahedron Octahedron Decahedron Dodecahedron Icosahedron

6 © P. SwanDice Dilemmas

2 - 4 Players Requires 2 Ten or 2 Six-Sided Dice

PurposeFluency with the four operations (+, -, ×, ÷).

Materials• Connect Three gameboard,

• counters,

• 2 ten-sided or 2 six-sided dice.

OrganisationA game for 2 - 4 players.

Rules• The first player rolls the dice for the given game. The player may use one or

more operations on the numbers shown on the dice to produce a total shown on the board. The total is then covered with a counter. The player may not cover a number which was previously covered. If a player is unable to find a total which has not already been covered, he/she must pass the dice to the next player.

• The winner is the person who completes a row of three, either horizontally, vertically or diagonally.

Sample Game• Two Dice Version

If a player throws a 3 and a 6 he/she may cover 18 (6 x 3), 9 (6 + 3), 3 (6 – 3) or 2 (6 ÷ 3). The player should clearly state how they arrived at the answer.

• Three Dice Version If playing the 3 dice version of Connect Three, players may mix the operations used to reach a total. For example if a player rolls three sixes the following totals could be formed: 216 (6 x 6 x 6), 18 (6 + 6 + 6), 42 (6 x 6 + 6), 30 (6 x 6 – 6), 7 (6 ÷ 6 + 6), etc. Remember to watch the order of operations.

Variations• Play the ten-sided dice game or the three six-sided dice game.

• Allow players to remove counters from previously covered numbers if they can make the total using the numbers shown on their dice.

Connect Three 2×or2×

Dice Dilemmas7© P. Swan

Connect Three

1 2 3 4

5 6 7 8

9 10 11 12

15 16 18 20

24 25 30 36

2×

Requires 2 Six-Sided Dice


2×

0 1 2 3 4

5 6 7 8 9

10 11 12 13 14

15 16 17 18 20

21 24 25 27 28

30 32 35 36 40

42 45 48 49 54

56 63 64 72 81

Connect ThreeRequires 2 Ten-Sided Dice


Connect Three

1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16

17 18 19 20 21 22 23 24

25 26 27 28 29 30 31 32

33 34 35 36 37 38 39 40

41 42 44 45 48 50 54 50

60 64 66 72 75 80 90 96

100 108 120 125 144 150 180 216

3×



Three Throw2 - 4 Players Requires 3 Six-Sided Dice

PurposePractise in recalling multiplication facts and computing 2-digit by 1-digit products. Associative property of multiplication.

Materials

• Three Throw gameboard,

• 3 six-sided dice or 1 six-sided die to be rolled 3 times.


Rules• Roll 3 six sided dice and multiply the three numbers together. Alternatively roll a

single die three times.

• Put a counter on the answer square.

• Try to get 3 counters in a line, horizontally, vertically or diagonally.

• If a counter already occupies a number you cannot place another one on it.

Sample Game

Variations• Allow players to remove opponentsʼ counters if they make the total with their

dice.

1 2 3 4 5

6 8 9 10 12

15 16 18 20 24

25 27 30 32 36

40 45 48 50 54

60 64 72 75 80

90 96 100 108 120

125 144 150 180 216

The winner

3×



Three Throw

Three Throw Gameboard

1 2 3 4 5

6 8 9 10 12

15 16 18 20 24

25 27 30 32 36

40 45 48 50 54

60 64 72 75 80

90 96 100 108 120

125 144 150 180 216

3×


Dice DigitsWhole Class Requires 1 Ten-Sided Die

PurposeDeveloping place value understanding.

Materials

• Dice Digits record sheet,

• 1 ten-sided die.

OrganisationA game for the whole class.

Rules• The die is rolled and the number shown placed in the box on the top row of the

record sheet. When the die is rolled a second time the number may be placed in either the first or second box in the second row of record sheet.

• The second row must be completed before moving on to the next row and so on.

• When all spaces are filled the numbers are added.

• The winner is the person with the largest total.

Variations• Try to make the smallest possible total.

• Vary the number of rows/boxes depending on the ability of the children.

• Vary the gameboard.

1×


Requires 1 Ten-Sided Die

Dice Digits 1×


Dice Digits (Variation) 1×



More Than Less Than


Materials• 1 ten-sided die or,

• 1 six-sided die.

OrganisationA game for 2 players.

Rules• Players take turns to roll the die.

• Players enter their numbers on the game board after each throw.

• Repeat until all boxes on the game board are full.

• The player with the largest difference wins the game.

Variations• Change the game board. Increase the size. Introduce decimals.

• Aim for the smallest difference.

Player Two

Player One

2 Players Requires 1 Ten or 1 Six-Sided Die

1×or1×


2 Players or Whole Class


Materials• One Hundred or Bust recording sheet,

• 1 ten-sided or 1 six-sided die.

OrganisationA game for 2 players or the whole class.

Rules• Each player takes a turn to roll the die.

• Write the number rolled in either the tens or the ones place on the record sheet. For example, a three may represent either 3 or 30.

• After 7 throws the numbers are totalled and player closest (but below) 100 wins.

• If you go over 100 you “bust”.

Variations• Begin with 100 and subtract the values on each line of the record sheet. The

player whose total is closest to zero wins.

• One player rolls the die and all players must place that number in either the ones or the tens place on the board.

• Roll the die six times and fill in all six spaces on the”Six Roll Board”. Then find the total.

• Alter the total.

• Play One Thousand or Bust using a H.T.O (Hundreds, Tens, Ones) board.

One Hundred or BustRequires 1 Ten or 1 Six-Sided Die

1×or1×


Requires 1 Ten or 1 Six-Sided Die

Six Roll Boards

One Thousand or Bust

One Hundred or Bust

Tens Ones Tens OnesTens Ones Tens Ones

Tens Ones Tens OnesTens Ones

Hundreds Tens Ones Hundreds Tens Ones

1×or1×


2 - 4 Players Requires 5 Six-Sided Dice

PurposeRepeated addition or multiplication with numbers 1 - 6.

EeeZee is similar to the commercial game Yahtzee, but simpler. Once children have mastered this game they may like to move on to Yahtzee.

Materials• EeeZee score sheet,

• 5 six-sided dice.


Rules• The aim of the game is to have the highest total after all six numbers on the

recording sheet have been used. In order to do this players roll the five dice and hope that the same number appears on several of the dice. The more dice showing the same number the better. A player may choose to separate some of the dice and roll the others a second time in an attempt to improve his/her result. A score is then determined. Play alternates until each player has had six turns. The score is then totalled.

• For example, if on your first throw the numbers 1, 2, 3, 3, and 6 appear, the two dice showing 3 might be separated from the others and the remaining dice rolled a second time in the hope that more threes turn up. It is possible, of course, that the remaining three dice may all end up showing numbers other than three, or perhaps the same number eg 4 in which case the player would choose to mark three 4ʼs on the record sheet. A score of 12 would then be registered. The number four may not be used again during the game.

Sample Game• First throw: 5, 5, 2, 1, 4. The player decides to throw

the last three dice again and keep the two fives.

• On the next throw he/she throws a 6, 4 and another 6. The five dice now show: 5, 5, 6, 4, 6. Either the 5ʼs, 6ʼs or 4 may now be entered onto the score sheet to produce a total of either 10, 12 or 4.

• The player must make a decision as to which of the three scores to record, keeping in mind the aim of the game. In this case they decided to use the two fives in case later on he/she throws more than two sixes.

Number How Many Score

1

2

3

4

5 2 10

6

Total

EeeZee 5×



EeeZee


1

2

3

4

5

6

Total


1

2

3

4

5

6

Total


1

2

3

4

5

6

Total


1

2

3

4

5

6

Total


1

2

3

4

5

6

Total


1

2

3

4

5

6

Total

5×


2 Players or Whole Class Requires 2 Six-Sided Dice

PurposeFluency with the four operations using numbers 1 - 6.

Materials• 2 sets of cards numbered 0 - 12,



Rules• Each player sets his/her cards out in front of themselves. It is a good idea to place

them in order.

• Players take it in turns to throw the 2 dice. Players use any of the operations (+, -, ×, ÷ ) to make a number from 0 to 12. For example, if the values shown on the dice are 4 and 2, the player may turn any one of the following cards over: 8, (4 x 2) or 6, (4 + 2) or 2 (4 – 2, 4 ÷ 2).

• As a player finds an answer, they turn down that card in front of themself, providing it has not already been used. The first to have all their cards turned over, wins. Alternatively the player with the most cards turned over after a set time is declared the winner.

0 1 2 3 4 5 6

7 8 9 10 11 12

0 1 2 3 4 5 6

7 8 9 10 11 12

Mary's Game 2×


2 Players or Whole Class Requires 1 Six-Sided Die

PurposeAddition of one and two-digit numbers.

Materials• 1 six-sided die.


Rules• Each player in turn throws the die, as often as he/she likes, and keeps adding to

their score. If, however, the die shows 6, the score for the whole turn becomes zero.

• For example, if you throw a 2 and then a 4, your score so far is 6. You can stop then, with a score of 6 and allow your opponent to throw or you can choose to throw again. If you roll again and roll a 2, your score so far is 8 but, if you roll a 6 on the next throw, your total for the round is zero. If you allow your opponent to throw then you keep your current score and add to it next time you throw.

• The first to reach 100 wins.

Variations• Use a single eight, ten, twelve or twenty sided die.

• Change the penalty number from 6 to another number. Does it matter if a different penalty number is chosen?

• This game can be played with the whole class by asking players to stand and then sit down to accumulate their points. If they remain standing and the penalty number turns up, then they lose all their points.

• Discuss various strategies for deciding when to stop.

Go for Broke 1×


2 Players Requires 2 Six-Sided Dice


Materials• 2 six-sided dice.


Rules• The object of the game is to score as many points over five rounds as possible.

• One player rolls two dice and records the total of the two dice. The player may choose to continue accumulating points but if a one comes up he/she loses all of his/her points for that round. Once the player chooses to pass the dice to the other player then he/she is allowed to keep all of the points accumulated from that round, even if a one turns up in the next round.

• If double one turns up then all of the points accumulated in previous rounds are also lost.

Variations• Instead of playing five rounds players may like to aim for a total such as 100.

Round Number Player One Player Two

Round 1

Round 2

Round 3

Round 4

Round 5

Total

Double Dare 2×



PurposeAddition and subtraction of numbers in the thousands. Place Value.

Materials• Pencil and paper,

• 4 six-sided dice,

• a calculator (optional).


Rules• Each player begins with 10 000 points.

• The four dice are thrown and the four digits shown are used to produce a number by arranging them in any order. This four-digit number is then added to the playerʼs total.

• Next time the player has a turn, the new four-digit number is subtracted from the total.

• Play continues in this fashion until one player exceeds 50 000.

Sample Game

Variations• Alter the finishing value.

• Use 2 or 3 dice.

• Use 8 or 10 sided dice.

• The player who exceeds the finishing value is the loser.

Maxi-Min 4×

(It is a good idea to form the largest possible number when adding.)

(It is a good idea to form the lowest possible number when subtracting.)

Player One Player Two

56 4

5

23 1

2

64 5

6

45 6

4

45 6

4

23 1

2

45 6

4

45 6

4

31 2

3

45 6

4

31 2

3

56 4

5

12 3

1

64 5

6

31 2

3

64 5

6

10000+ 5 43 115431

-224613185

10000+643315431

- 4 45 611977

1st Throw

2nd Throw

1st Throw

2nd Throw


2 Players Requires 1 Six-Sided Die




Rules• Each player writes the numbers 1 to 11 onto a score sheet. One player “bats”, and

the other “bowls”.

• The player “batting” rolls a die and scores the number of runs equal to the value displayed by the die, unless the result is a five. A five is considered as an appeal for a wicket and the “bowler” is given the opportunity of rolling the die and determining if and how the player “batting” is out. If the die turns up:

• When a batter is out their score is tallied. The player continues batting until 10 team members are dismissed.

• When the first playerʼs team has been dismissed, the batting and bowling roles are reversed.

• The ‘Winner’ is the player whose team scores the most runs.

Variations• Play limited overs “one day cricket” or "twenty-twenty cricket" (20 rolls each), i.e.

restrict the number of rolls.

Dice Cricket 1×

Number Rolled Outcome

1 The batter is out, hit wicket.

2 The batter is out, bowled.

3 The batter is out, caught.

4 The batter is out, lbw.

5 The batter is not out.

6 The batter is run out.


Duelling DiceSmall Groups Requires 1 Ten-Sided Die

PurposeAddition of 2-digit numbers or decimal numbers.

Materials


OrganisationA game for small groups.

Rules• Each player rolls a ten sided dice several times in an attempt to make two-digit

numbers that add to make a number as close as possible to one hundred.

• When a single cell is left on the player's board, that player has three rolls to complete the game. After each roll a decision is made whether to use the number or not. The player can lock in a number after any roll but cannot go back to a previous roll once a new roll is made.

Variations• The boards may easily be adapted to focus on decimals, eg:

Different target numbers may be chosen, eg 99.

6 82

Player One's Board

7 12

Player Two's Board

(Add to make one)

1×


2 Players Requires 2 Differently Coloured Six-Sided Dice

PurposeUsing the cartesian coordinate system in the positive quadrant.

Materials• Coordinate Dice gameboard,

• 2 differently coloured six-sided dice. The playing board may be used for six to twelve sided dice.


Rules• One colour dice is used to represent the value on the horizontal axis and a

different colour dice is used to represent the value on the vertical axis.

• The first player rolls the dice and circles the coordinate in one colour.

• The second player rolls the two dice and circles that point in a different colour.

• If a point is already occupied then a player cannot go and must pass the dice to the other player.

• The first player to circle three points in a row either vertically, horizontally or diagonally is the winner.

Variations• Use different dice. The playing board may be used for six to twelve sided dice.

Coordinate Dice 2×

Player One

Player Two

0 0 1

Y A

xis

6-Si

ded

Die

X Axis 6-Sided Die2 3 4 5 6

1

2

3

4

5

6

The winner


2×Coordinate Dice

00 1 2 3 4 5 6 7 8 9 10 11 12

3

4

5

6

7

8

9

10

11

12

1

2

_ D

ice

_ Dice

Requires 2 Differently Coloured Six-Sided Dice


2 Players Requires 1 Ten or 1 Six-Sided Die

PurposeMultiplication of numbers 0 - 9 and addition of double-digit numbers.

Materials• Product Pairs gameboard,

• 1 ten-sided die or 1 six-sided die.


Rules• Players take turns to throw the die and place the number somewhere on their

grid.

• After four throws each, the grid should be full.

• Players then choose which pairs to multiply.

• The pairs are multiplied and then added.

• The winner is the player who makes the largest total.

Variations• Aim for the largest/smallest, even/odd number etc.

• Use different types of boards. (Note that on some boards diagonal pairings may not work.)

6

Player One's Board Player Two's Board

Product Pairs 1×or1×

3 6

1 9

3 6

1 931

69

189

27

===

××

Total

36

19

35457

===

××

Total

31

96

276

33

===

××

Total

3 6

1 9


1×or1×Product Pairs

Product Triples

Requires 1 Ten or 1 Six-Sided Die


2 Players Requires 1 Ten-Sided Die

PurposePlace value into thousands.

Materials• Oh No gameboard,



Rules• Each player rolls the die and places the number shown in either the thousands,

hundreds, tens or ones column of his/her own record sheet or on his/her opponentʼs record sheet.

• On which sheet a player chooses to put the number will depend on the target set for the game. For example the target might be: - the largest number - the smallest number - the largest/smallest, even/odd number - the number nearest …

• If the target was the largest number then large digits would be recorded on your own gameboard in the thousands or hundreds column, while you would put digits like 0, 1 and 2 on your opponentʼs record sheet in the thousands and hundreds column. Strategies will vary according to the target. If the target was to produce an odd number then you would try to place an odd digit in the ones column on your own recording sheet and an even digit in the ones place on your opponentʼs recording sheet.

• A point is awarded to the winner of each round. Games continue for five rounds. The player with the most points is the winner.

Variations• Vary the target.

• Use 3-digit numbers.

• Use decimals.

• Change the recording sheet. For example play three rounds and then add the values to see if the total hits the target.

Oh No 1×


Oh No 1×


TH H T O

1

2

3

4

5

Player One:

TH H T O

1

2

3

4

5

Player One:

TH H T O

1

2

3

4

5

Player One:

1000 100 10 1

1

2

3

Player One:

1000 100 10 1

1

2

3

Player Two:

TH H T O

1

2

3

4

5

Player Two:

TH H T O

1

2

3

4

5

Player Two:

TH H T O

1

2

3

4

5

Player Two:


Whole Class Requires 6 Differently Coloured Six-Sided Dice

PurposeAddition and multiplication with numbers 1 - 6.

Materials• A set of coloured cubes from which at least ten may be selected,

• 6 differently coloured six-sided dice that match the different coloured cubes.


Rules• Ask each member of the class to build a tower using ten cubes and at

least two colours.

• Roll the six dice to determine a value for each cube colour.

• Each player then works out the value of his/her tower by multiplying the number of cubes of each colour by the value shown on the related die.

• For example a student might build a tower using 6 red cubes, 3 green cubes and 1 blue cube. The dice corresponding to the cube colours are then thrown.The following values might come up: red die = 5, green die = 4 and blue die = 2. Other coloured dice are thrown at the same time as other players may have used cubes of different colours. The total value of the tower is found by multiplying the number of cubes of a particular colour by the corresponding value shown on the dice and adding the values eg (6 x 5) + (3 x 4) + (1 x 2) = 44. The total for this tower is 44.

• The winner is the player whose tower is worth the most.

Variations• Change the number of cubes used to make the tower.

• Alter the number of colours allowed.

• Use different types of dice i.e. 4, 8, 10 and 12-sided.

Dice Tower 6×



PurposeAddition and multiplication with numbers 1 - 6.

Materials• 4 six-sided dice.


Rules• The aim of the game is to produce the largest total by adding the values on pairs

of dice and then multiplying.

• The four dice are rolled and players race to produce the largest value.

• For example, if the dice show 2, 3, 4 and 6 they may be paired in the following ways.

• The player who finds the highest value first wins.

Variations• Aim for the smallest total.

• Add the numbers showing on three dice together and multiply by the number on the fourth.

• Use different types of dice i.e. 4, 8, 10 and 12-sided.

222

346

568

===

+++

433

664

1097

===

+++

568

505456

1097

===

×××

and and and

8 × 7 = 56 is the combination that produces the winning total.

Add'n Multiply 4×



PurposeUsing the area model to understand multiplication, including commutative property.

Materials• Two Area Dice gameboards,

• coloured pencils,



Rules• One player rolls the dice and colours in a rectangle on his/her playing board

according to the dimensions shown on the dice. If the numbers 2 and 4 are shown on the dice then the player may shade in a 2 x 4 or a 4 x 2 rectangle on his/her board.

• The second player then rolls the dice and shades a rectangle on his/her board.

• If a player cannot draw a rectangle in the space available then he/she misses a turn.

• The first player to cover the entire board wins.

Variations• Change the grid sizes. Shade parts of the grid.

• Use different dice. Make sure the dimensions of the grid match the type of dice being used.

• Both players use the same roll of the dice to form a rectangle on their respective grids.

• Both players use one board. The winner is the last player to shade a rectangle on the board.

Area Dice 1×


Area Dice 1×

Requires 1 Six-Sided Die


Small Groups Requires 2 Six-Sided Dice

PurposeAddition with numbers 1 - 6. Probability.

Materials• 11 counters,


OrganisationA game for small groups.

Rules• Place counters on each of the squares numbered 2 – 12 to represent race cars.

• Throw 2 dice and add the values to see which counter moves forward one square.

Questions• Which car do you think will win?

• Play the game several times keeping a note of the winning cars. Is it a fair race? Explain.

The Great Car Race I 2×


76

8

5

9

4

10

3

11

2

12

FINISH

LINE

The Great Car Race I 2×




PurposeSubtraction with numbers 1 - 6. Probability.




Rules• One player places his/her counter on track A and the

other on B.

• Roll two dice for each move. Subtract the smaller value shown on the dice from the larger value to determine who moves.

Variations• Game 1

Car A moves one space if the difference between the numbers on each die is 0, 1 or 2. Car B moves one space if the difference is 3, 4 or 5. Who do you think will win?

• Game 2 Car A moves one space if the difference is 0, 2 or 4. Car B moves one space if the difference is 1, 3,or 5. Who do you think will win?

• Game 3 Car A moves 3 spaces if the difference is zero. Car B moves one space if the difference is not zero. Who do you think will win?

Questions• Do you think the games are fair/unfair? Why?

• Play each game a number of times and compare your results with your predictions.

The Great Car Race II 2×

A B

Start

Finish Line


2 - 6 Players Requires 2 Six-Sided Dice





Rules• Each player places his/her counter on one of the starting squares (START).

• Then any player rolls 2 dice and adds the numbers shown together. The player who has this total at the front of his/her car moves one square towards the finish.

• The winner is the first to reach the Finish Line.

Questions• Play this several times and keep a record of which players win.

• Do you think it is a fair race? Explain.

The Great Car Race III 2×

2 or 8

Sta

rt

Fini

sh3 or 9

4 or 10

5 or 11

6 or 12

7 only



Cross the Road 2×


Materials• Cross the Road gameboard,

• 22 counters (11 one colour and another 11 another colour),



Rules• Each player places one counter on each of the numbers on their side of the

board. Each counter represents a person trying to cross the road.

• Players take turns to throw the two dice and add the numbers thrown. If the number matches the position where there is a counter, then that person may cross the road.

• The winner is the first player to have all of their counters cross the road.

Questions• Which number or numbers (people) tend(s) to be the first/last to cross the road?

• Try to explain why.

Variations• Allow students to place more than one counter on a particular number.


Cross the Road 2×

2 123 114 105 96 87 78 69 510 411 312 2



Whole Class Requires 1 Six-Sided Die

PurposeInvestigating the probability of certain outcomes occuring from a series of equally likely events.



Activity• Each pack of Breakfast Bickies contains a collector card. There are six cards to

collect to make a set.

• What is the least number of packets that you would need to collect all six?

• How many packets would you expect to buy to collect all six cards?

• Instead of buying actual packs of cereal we can model, or simulate the situation using a six sided die. Each number on the die may be used to represent one of the collector cards. Throw the die and record which numbers come up. When each number has come up stop throwing the die and count the number of throws made. This represents the number of packets required to collect all six cards.

• Collect data from the whole class and find the average number of packets that would be have to be bought to collect all six cards.

Variations• Use eight and ten-sided dice to simulate collecting a set of eight or ten cards.

Card 1 Card 2 Card 3 Card 4 Card 5 Card 6

The last card (number) was number 2.

Collector Cards 1×



Find the Product 1×

PurposeMultiplication or subtraction of numbers 1 - 6. Odd and even. Probability.



ActivityJeremy and Leighland were playing a game with two dice.

One player would roll the dice and then multiply the two numbers displayed on the top. If the answer produced an odd number Leighland would get a point. If the answer was even Jeremy would get a point. After playing for a while Leighland complained that the game was unfair.

• Play the game yourself with a friend until one player reaches ten points. Play the game again several times.

• Do you think the game is fair? Do some numbers occur more often than others?

• The products for all throws of the dice can be found by completing a multiplication square.

• Shade in all of the odd products with one colour and then use another colour to shade all the even products.

• Are there more even or more odd products shaded?

• If you have a choice of even or odd products, which one would you choose? Why?

• Design a point system to make the game fairer.

× 1 2 3 4 5 6

1

2 4

3 12

4

5 15

6



PurposeSubtraction of numbers 1 - 6. Probability.




Rules• Place the counters in the start position and then roll two dice.

• Work out the difference between the numbers shown on the dice. One player moves to the left when a difference of 0, 1 or 2 is thrown and the other player moves to the right if a difference of 3, 4 or 5 is thrown.

• Play the game several times to see who wins.

Questions• Is this game fair?

All the differences that may be found by subtracting the values on the six-sided dice can be shown by completing the following chart.

• Shade all of the 0, 1 and 2 differences.

• Compare these with the non-shaded numbers.

• Explain whether you think the game is fair.

• Suggest some ways the game might be made fairer.

- 1 2 3 4 5 6

1 0 1 2

2 1 0 1

3 2 1 0

4 3 0

5 0

6 0

Fair Enough? 2×

StartWin Win

3, 4 or 5 move one space this way.0, 1 or 2 move one space this way.Player One Player Two



Sum Dice 2×


Materials• Sum Dice gameboards,

• counters,



Rules• In the game Sum Dice players are given boards

like those shown above with twelve numbers printed on them. Two dice are thrown and the values shown on each are added. If the number is shown on the board it is covered. The winner is the first player to cover his/her board.

• Try playing the game with a friend using the two boards shown above.

You probably noticed that the game was very unfair because the number 1 never comes up and the number seven comes up quite often.

All the different sums for two dice may be shown on a chart. Complete this chart to find out which sums occur most often.

Questions• Which number comes up most often?

• Which number(s) come up least often?

Challenge• Create two boards, each containing a different mix of numbers, that are fair to

both players.

1 1 1 1

1 1 1 1

1 1 1 1

7 7 7 7

7 7 7 7

7 7 7 7

+ 1 2 3 4 5 6

1 2 3 4

2 4 5

3 4

4 8

5

6


Groups of 5 or Whole Class Requires 1 Twelve-Sided Die

Birth Months 1×

PurposeInvestigating the probability of certain outcomes occuring from a series of equally likely events.

Materials• 1 twelve-sided die.

OrganisationA game for groups of 5 or the whole class.

Activity• Five people meet at a party. What is the chance that at least one pair will share a

common birth month?

• Model this situation by forming groups of five in the class and collecting some data. Another way:

• Another way to simulate this situation is by rolling a 12-sided die. Each number on the die may be used to represent a month in the year.

• Roll the twelve-sided die five times and record how often the same number (month) comes up.

• Repeat this experiment ten times.

• Record your results.

• Find the average.

• Write a brief report on your findings.


The Great Car Race I page 36• All cars should have an equal chance of winning the race

although the board looks highly unfair favouring cars 2

and 12. A chart [Figure 1.] showing the chances of two dice

adding to a particular total reveals why this is the case.

• Clearly seven is the most common outcome when two

dice are rolled and the values added. Totals of 6 and 8 are

equally likely as are 5 and 9 or 3 and 10 or 4 and 11 or 2 and

12.

The Great Car Race II page 38• Game 1

The following chart [Figure 2.] lists possible outcomes for

tossing two dice and subtracting the values. Examining the

chart reveals the chance of winning Game 1. The chance of

the difference being 0, 1 or 2 is 24/36 or 2/3. The chance of the

difference being 3, 4 or 5 is 12/36 or 1/3. Therefore car A has

twice as much chance of winning as car B.

• Game 2

The chance of the difference being 0, 2 or 4 is 18/36 or 1/2. The

chance of the difference being 1, 3 or 5 is also 18/36 or 1/2. Both

cars have an equal chance of winning.

• Game 3

The chance of the difference being 0 is 6/36 or 1/6. The

chance of the difference not being 0 is 30/36 or 5/6. While

the point system allows Car A to move three spaces if a

difference of zero is thrown, this does not fully make up for

the advantage that Car B has. A fairer point system would

allow Car A to move five spaces if a difference of zero is

rolled.

The Great Car Race III page 39• Examining the dice addition chart [Figure 1.] shows why

this game is fair. There are 6 chances out of 36 possible

outcomes that result in a 2 or 8. Likewise there is 6/36 of a

chance that the total will be 3 or 9 and 1/6 of a chance that

the total will be 4 or 10. In fact all outcomes are equally

likely.

Teacher Notes and Answers

- 1 2 3 4 5 6

1 0 1 2 3 4 5

2 1 0 1 2 3 4

3 2 1 0 1 2 3

4 3 2 1 0 1 2

5 4 3 2 1 0 1

6 5 4 3 2 1 0

+ 1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

Figure 1.

Figure 2.


Cross the Road page 40• Placing counters on position 7 gives the best chance for

crossing the road, as seen by examining [Figure 1.]. Placing

counters on 2 or 12 gives the least chance of crossing the

road.

Collector Cards page 42• Theoretically it should take 15 (14.7 or 6/1 + 6/2 + 6/3 + 6/4 + 6/5 +

6/6) packets of cereal to collect a full set of toys. Of course it

may only take six packets to get a full set.

Find the Product page 43• The completed multiplication square [Figure 3.] shows

that 9/36 or 1/4 of the numbers are odd and 27/36 or 3/4 of the

numbers are even. The game is therefore unfair. Amee

should get three points every time an odd number is rolled

to make the game fair.

Fair Enough page 44• The completed subtraction chart [Figure 2.] indicates that

there are 12/36 or 1/3 of a chance of obtaining a difference

of a 3,4 or 5. The game is unfair. The player rolling 0, 1 or 2

has twice as much chance of winning as the other player.

To make the game fairer the board could be altered so

that the 0, 1 , 2 player has to travel twice as far to win.

Alternatively player two could be allowed to move two

spaces every time 3, 4 or 5 turns up.

Sum Dice page 45• The addition chart [Figure 1.] shows that seven comes up

most often and two and twelve come up least often. There

are many different boards [see Figure 4.] that may be

created that give each player an equal chance of winning.

Birth Months page 46• The chance of a pair of people in a group of five sharing

the same birth month is 1 – 12/12 x 11/12 x 10/12 x 9/12 x 8/12, or

approximately 62%.

× 1 2 3 4 5 6

1 1 2 3 4 5 6

2 2 4 6 8 10 12

3 3 6 9 12 15 18

4 4 8 12 16 20 24

5 5 10 15 20 25 30

6 6 12 18 24 30 36

Figure 3.

2 3 4 5

7 7 7 7

6 5 4 3

12 11 10 9

7 7 7 7

8 9 10 11

Figure 4.

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Dice Dilemmas

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