Maths Games Workshop: Part Seven: Multiplication in Five Lessons

Maths Games Workshop: Part Seven: Multiplication in Five LessonsAuthor(s): Dave KirkbySource: Mathematics in School, Vol. 16, No. 5 (Nov., 1987), pp. 12-15Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214390 .

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M GA

WORKSHOP Dave Kirkby Part Seven Sheffield Polytechnic

Multiplication in Five Lessons

Many simple games can be devised to help children practise learning multiplication facts (learning their "tables"). Five such games are illustrated here, together with suggestions of ways in which an apparent short activity can be developed into a whole lesson or series of lessons. Many simple games lend themselves to the creation of mathematical activity at various levels.

The dice for these games can be made by writing numbers on blank cubes.

Lesson one - Products Game. Two dice are required; one numbered 1, 2, 3, 4, 5, 6 and the other numbered 4, 5, 6, 7, 8, 9.

Rules 1. Players take turns to roll the two dice, their score being

the product of the two numbers. These scores are recorded and then added.

2. The first player to reach 200 wins the set. 3. The first player to win three sets is the winner.

Play. Pupils play the game in pairs.

Development 0 What is the minimum number of throws required to

reach 100? e.g.

54

54

54

48 TOTAL 108 TOTAL 102

Here are two ways of reaching 100 in two throws. Are there other ways?

What is the minimum number of throws required to reach 100 EXACTLY? e.g.

54

142 1

TOTAL100

[-] n8= 48

-C = 42 10

TOTAL100

Here are two ways of reaching 100 exactly in three throws. Are there other possibilities? How many different scores are possible with one throw? A multiplication table can be constructed.

xl 4 5 6 7 8 9

1 4 5 6 7 8 9 2 8 10 12 14 16 18 3 12 15 18 21 24 27 4 16 20 24 28 32 36 5 20 25 30 35 40 45 6 24 30 36 42 48 54

Which scores can be obtained with one throw in more than one way? e.g.

= 18

= 18

Investigate similar problems with changes in the numbering of the dice. Play the game with variations in the rules. e.g. 1. Try changing the target number of 200. e.g. 2. Play the game by starting with a score of 200 and

subtracting the score each time. Invent variations of the game.

12 Mathematics in School, November 1987



Lesson two --Products and Sums Game. Two dice are required: one numbered 1, 2, 3, 4, 5, 6 and the other numbered 4, 5, 6, 7, 8, 9.

Rule. The rules are the same as for the game in Lesson one except that you add the two numbers indicated, then multiply them. These two results are added for the score. e.g.

=11(3 +8)+24(3x8)=35

Play. Pupils play the game in pairs.

Development " With one throw how can the following scores be ob-

tained - 44, 15, 62, etc.? " Find two different ways of obtaining a one-throw score

of 29.

=11(2+9)+18(2x9)=29

F =] 9(4+5)+20(4x5)=29

0 How many different one-throw scores are possible between 20 and 30?

0 How many different one-throw scores are possible?

A table of one-throw scores can be obtained by combining the entries for an addition table and a multiplication table.

+456789

1 5 6 7 8 9 10 2 6 7 8 9 10 11 3 7 8 9 10 11 12 4 8 9 10 11 12 13 5 9 10 11 12 13 14 6 10 11 12 13 14 15

xl 4 5 6 7 8 9

1 4 5 6 7 8 9 2 8 10 12 14 16 18 3 12 15 18 21 24 27 4 16 20 24 28 32 36 5 20 25 30 35 40 45 6 24 30 36 42 48 54

4 5 6 7 8 9

1 9 11 13 15 17 19 2 14 17 20 23 26 29 3 19 23 27 31 35 39 4 24 29 34 39 44 49 5 29 35 41 47 53 59 6 34 41 48 55 62 69

Which one-throw scores can be obtained in more than one way? Is it possible to obtain a total of 100 exactly in two throws? e.g.

=10(3+7)+21(3x7)=

=15(6+9)+54(6x9)=

TOTAL100

Can this be achieved in other ways? Investigate similar problems with changes in the numbering of the dice. Investigate changes in the scoring. For example, suppose the score is obtained by adding the product to the difference between the two dice. e.g.

[3 7]=4(7-3)+21(3x7)=25

Invent variations of the game.

Lesson three - Design a Board Game. Two dice are required: one numbered 1, 2, 3, 4, 5, 6, and the other numbered 2, 3, 5, 7, 8, 9. A set of counters for each player and a board is also needed.

24 45 7 60 9 42 6

35 14 5 16 54 40 10

30 27 25 21 20 12 32

8 48 18 15 50 28 36

Rules

1. Players take turns to roll the two dice and multiply the scores together.

2. If the product appears on the board then the player places a counter of his own colour on the square containing the product.

3. The game continues until all the board has been filled. 4A. The player who has used up most counters wins.

OR 4B. Score ONE POINT for covering up a number and an

EXTRA POINT for every counter next to the one placed on the board. The player with the most points wins. OR

4C. Players add up the scores covered by their counters and the one with the largest total wins.

Play. Pupils play the game in pairs trying variations in Rule 4.

Development 0 Design a board for a similar game which uses one dice

numbered 3, 4, 5, 6, 7, 8 and the other numbered 1, 2, 3, 7,8, 9.

The different possible products are obtained from the multiplication table.

xl 3 4 5 6 7 8

1 3 4 5 6 7 8

2 6 8 10 12 14 16 3 9 12 15 18 21 24

4 28 35 42 49 56

5 32 40 48 ( 64

6 27 36 45 54 63 72

Mathematics in School, November 1987 13



If the duplicate products are eliminated, then 30 different products remain. These can be arranged randomly on a 5 x 6 board.

48 5 16 42 10 9

18 45 28 40 4 64

8 72 3 21 27 35

49 14 32 12 54 7

6 36 15 63 56 24

Design boards for games using different dice. Invent variations in the rules.

Lesson four-- Product Grid Game. The game can be played with a whole class, or a small group. One dice numbered 1, 2, 3, 4, 5, 6 is required. Each player draws this grid.

24 36 48 SCORE

30

20

6

SCORE TOTAL

Rules 1. The teacher throws the dice and announces the number

showing. 2. Each player then places that number in one of the nine

available spaces. 3. This is repeated until all the nine available boxes have

been filled. 4. When a number is written in a box it cannot be changed. 5. Score ONE POINT for each time a row or column

heading is the product of the three numbers in that row or column. e.g.

24 36 48 SCORE

30 3 5 2 1

20 4 5 2 0

6 2 3 1 1

SCORE 1 0 0 3 TOTAL

Play. Play several games.

Development 0 Given that the dice showed 3, 5, 2, 4, 5, 2, 2, 3, 1, what is

the best possible score? 0 Place any numbers in the range 1 to 6 in order to score

maximum.

e.g.

24 36 48 SCORE

30 3 2 5 1

20 4 1 5 1

6 2 3 1 1

SCORE 1 0 0 4 TOTAL

Is it possible to improve on this score of 4? Investigate using numbers in the range 1 to 9. Try with different headings. e.g.

12 15 18 SCORE

16

8

10

SCORE TOTAL

Investigate arrangements which will score a maximum of 6 points. e.g.

30 18 16 SCORE

24 2 3 4 1

10 5 1 2 1

36 3 6 2 1

SCORE 1 1 1 6 TOTAL

Lesson five - Multiples Game. The game is played in a similar way to the game described in Lesson four. Each player draws this grid.

SCORE

SCORE TOTAL

The dice numbered 1, 2, 3, 4, 5, 6 is thrown four times and the resulting numbers are placed, in turn, in the four available boxes.

Rules 1. When all four boxes have been filled, this then produces

four two-digit numbers - two reading across the two rows, and two reading down the two columns.

14 Mathematics in School, November 1987



2. Points are awarded according to the largest divisor less than ten

If the number is a multiple of I - SCORE 1 POINT If the number is a multiple of 2 - SCORE 2 POINTS If the number is a multiple of 3 - SCORE 3 POINTS If the number is a multiple of 4 - SCORE 4 POINTS If the number is a multiple of 5 - SCORE 5 POINTS If the number is a multiple of 6 - SCORE 6 POINTS If the number is a multiple of 7 - SCORE 7 POINTS If the number is a multiple of 8 - SCORE 8 POINTS If the number is a multiple of 9 - SCORE 9 POINTS

So, for example, 21 is a multiple of 3 and a multiple of 7, hence SCORE 7 POINTS. e.g.

SCORE

3 5 7

7 2 9

SCORE 1 4 21 TOTAL

Play. Play several rounds.

Development 0 What is the maximum possible score for different sets of

four throws, e.g. 3, 4, 2, 6; 2, 1, 5, 4; 8, 6, 9, 1, etc.? 0 Suppose one can choose any four digits from five, e.g. 1,

5, 3, 4, 2. 0 Which four throws can produce total scores greater than

30? 0 Investigate for changes in the numbering of the dice. 0 Try variations in the scoring system. For example,

SCORE 1 POINT for each number that is either a SQUARE number, or a PRIME NUMBER. Otherwise, SCORE 0 POINTS. e.g.

SCORE

3 6 1

7 2 0

SCORE 1 0 2 TOTAL

References

Kirkby, D. Maths Games in the Classroom No. 6 "Multiplication Dice Games", Eigen Publications 1983. Kirkby, D. Maths Games in the Classroom No. 7 "Grid Games", Eigen Publications 1983.

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Maths Games Workshop: Part Seven: Multiplication in Five Lessons