Dr Paul Swan’s
Maths Games
Check Sheets, Recording Sheets and
Teacher’s Notes
Pack A
• Division Decision
• Pitstop
• Space Race - Addition
• Treasure Trove
WRITTEN BY DR PAUL SWAN
www.abacused.com.au
CONTENTS
Division Decision
Check Sheets
Teacher’s Notes
2-7
8,9
Space Race - Addition
Check Sheet
Teacher’s Notes
10
11-13
Pitstop
Teacher’s Notes
14
Treasure Trove
Check Sheet
Teacher’s Notes
15
16,17
1
www.abacused.com.au
DIVISION DECISION
÷2, ÷3, ÷4, ÷5
CHECKSHEET A
No
13
14
17
19
21
22
23
25
27
29
31
33
34
35
37
38
39
41
43
46
47
49
51
53
÷2
6
7
8
9
10
11
11
12
13
14
15
16
17
17
18
19
19
20
21
23
23
24
25
26
r
1
0
1
1
1
0
1
1
1
1
1
1
0
1
1
0
1
1
1
0
1
1
1
1
÷3
4
4
5
6
7
7
7
8
9
9
10
11
11
11
12
12
13
13
14
15
15
16
17
17
r
1
2
2
1
0
1
2
1
0
2
1
0
1
2
1
2
0
2
1
1
2
1
0
2
÷4
3
3
4
4
5
5
5
6
6
7
7
8
8
8
9
9
9
10
10
11
11
12
12
13
2
www.abacused.com.au
r
1
2
1
3
1
2
3
1
3
1
3
1
2
3
1
2
3
1
3
2
3
1
3
1
÷5
2
2
3
3
4
4
4
5
5
5
6
6
6
7
7
7
7
8
8
9
9
9
10
10
r
3
4
2
4
1
2
3
0
2
4
1
3
4
0
2
3
4
1
3
1
2
4
1
3
DIVISION DECISION
÷2, ÷3, ÷4, ÷5
CHECKSHEET B
No
55
57
58
59
61
62
63
65
67
69
71
73
74
77
79
81
82
83
85
86
87
89
91
93
÷2
27
28
29
29
30
31
31
32
33
34
35
36
37
38
39
40
41
41
42
43
43
44
45
46
r
1
1
0
1
1
0
1
1
1
1
1
1
0
1
1
1
0
1
1
0
1
1
1
1
÷3
18
19
19
19
20
20
21
21
22
23
23
24
24
25
26
27
27
27
28
28
29
29
30
31
r
1
0
1
2
1
2
0
2
1
0
2
1
2
2
1
0
1
2
1
2
0
2
1
0
÷4
13
14
14
14
15
15
15
16
16
17
17
18
18
19
19
20
20
20
21
21
21
22
22
23
3
www.abacused.com.au
r
3
1
2
3
1
2
3
1
3
1
3
1
2
1
3
1
2
3
1
2
3
1
3
1
÷5
11
11
11
11
12
12
12
13
13
13
14
14
14
15
15
16
16
16
17
17
17
17
18
18
r
0
2
3
4
1
2
3
0
2
4
1
3
4
2
4
1
2
3
0
1
2
4
1
3
DIVISION DECISION
÷3, ÷4, ÷5, ÷6
CHECKSHEET A
No
13
14
17
19
21
22
23
25
27
29
31
33
34
35
37
38
39
41
43
46
47
49
51
53
÷3
4
4
5
6
7
7
7
8
9
9
10
11
11
11
12
12
13
13
14
15
15
16
17
17
r
1
2
2
1
0
1
2
1
0
2
1
0
1
2
1
2
0
2
1
1
2
1
0
2
÷4
3
3
4
4
5
5
5
6
6
7
7
8
8
8
9
9
9
10
10
11
11
12
12
13
r
1
2
1
3
1
2
3
1
3
1
3
1
2
3
1
2
3
1
3
2
3
1
3
1
÷5
2
2
3
3
4
4
4
5
5
5
6
6
6
7
7
7
7
8
8
9
9
9
10
10
4
www.abacused.com.au
r
3
4
2
4
1
2
3
0
2
4
1
3
4
0
2
3
4
1
3
1
2
4
1
3
÷6
2
2
2
3
3
3
3
4
4
4
5
5
5
5
6
6
6
6
7
7
7
8
8
8
r
1
2
5
1
3
4
5
1
3
5
1
3
4
5
1
2
3
5
1
4
5
1
3
5
DIVISION DECISION
÷3, ÷4, ÷5, ÷6
CHECKSHEET B
No
55
57
58
59
61
62
63
65
67
69
71
73
74
77
79
81
82
83
85
86
87
89
91
93
÷3
18
19
19
19
20
20
21
21
22
23
23
24
24
25
26
27
27
27
28
28
29
29
30
31
r
1
0
1
2
1
2
0
2
1
0
2
1
2
2
1
0
1
2
1
2
0
2
1
0
÷4
13
14
14
14
15
15
15
16
16
17
17
18
18
19
19
20
20
20
21
21
21
22
22
23
r
3
1
2
3
1
2
3
1
3
1
3
1
2
1
3
1
2
3
1
2
3
1
3
1
÷5
11
11
11
11
12
12
12
13
13
13
14
14
14
15
15
16
16
16
17
17
17
17
18
18
5
www.abacused.com.au
r
0
2
3
4
1
2
3
0
2
4
1
3
4
2
4
1
2
3
0
1
2
4
1
3
÷6
9
9
9
9
10
10
10
10
11
11
11
12
12
12
13
13
13
13
14
14
14
14
15
15
r
1
3
4
5
1
2
3
5
1
3
5
1
2
5
1
3
4
5
1
2
3
5
1
3
DIVISION DECISION
÷3, ÷4, ÷5, ÷6, ÷7, ÷8, ÷9
CHECKSHEET A
No
13
÷3 r
4 1
÷4 r
3 1
÷5 r
2 3
÷6 r
2 1
÷7 r
1 6
÷8 r
1 5
÷9 r
1 4
14
4
2
3
2
2
4
2
2
2
0
1
6
1
5
17
5
2
4
1
3
2
2
5
2
3
2
1
1
8
19
6
1
4
3
3
4
3
1
2
5
2
3
2
1
21
7
0
5
1
4
1
3
3
3
0
2
5
2
3
22
7
1
5
2
4
2
3
4
3
1
2
6
2
4
23
7
2
5
3
4
3
3
5
3
2
2
7
2
5
25
8
1
6
1
5
0
4
1
3
4
3
1
2
7
27
9
0
6
3
5
2
4
3
3
6
3
3
3
0
29
9
2
7
1
5
4
4
5
4
1
3
5
3
2
31
10 1
7
3
6
1
5
1
4
3
3
7
3
4
33
11 0
8
1
6
3
5
3
4
5
4
1
3
6
34
11 1
8
2
6
4
5
4
4
6
4
2
3
7
35
11 2
8
3
7
0
5
5
5
0
4
3
3
8
37
12 1
9
1
7
2
6
1
5
2
4
5
4
1
38
12 2
9
2
7
3
6
2
5
3
4
6
4
2
39
13 0
9
3
7
4
6
3
5
4
4
7
4
3
41
13 2
10 1
8
1
6
5
5
6
5
1
4
5
43
14 1
10 3
8
3
7
1
6
1
5
3
4
7
46
15 1
11 2
9
1
7
4
6
4
5
6
5
1
47
15 2
11 3
9
2
7
5
6
5
5
7
5
2
49
16 1
12 1
9
4
8
1
7
0
6
1
5
4
51
17 0
12 3
10 1
8
3
7
2
6
3
5
6
53
17 2
13 1
10 3
8
5
7
4
6
5
5
8
6
www.abacused.com.au
DIVISION DECISION
÷3, ÷4, ÷5, ÷6, ÷7, ÷8, ÷9
CHECKSHEET B
No
55
÷3 r
18 1
÷4 r
13 3
÷5 r
11 0
÷6 r
9 1
÷7 r
7 6
57
19 0
14 1
11 2
9
3
8
1
7
1
6
3
58
19 1
14 2
11 3
9
4
8
2
7
2
6
4
59
19 2
14 3
11 4
9
5
8
3
7
3
6
5
61
20 1
15 1
12 1
10 1
8
5
7
5
6
7
62
20 2
15 2
12 2
10 2
8
6
7
6
6
8
63
21 0
15 3
12 3
10 3
9
0
7
7
7
0
65
21 2
16 1
13 0
10 5
9
2
8
1
7
2
67
22 1
16 3
13 2
11 1
9
4
8
3
7
4
69
23 0
17 1
13 4
11 3
9
6
8
5
7
6
71
23 2
17 3
14 1
11 5
10 1
8
7
7
8
73
24 1
18 1
14 3
12 1
10 3
9
1
8
1
74
24 2
18 2
14 4
12 2
10 4
9
2
8
2
77
25 2
19 1
15 2
12 5
11 0
9
5
8
5
79
26 1
19 3
15 4
13 1
11 2
9
7
8
7
81
27 0
20 1
16 1
13 3
11 4
10 1
9
0
82
27 1
20 2
16 2
13 4
11 5
10 2
9
1
83
27 2
20 3
16 3
13 5
11 6
10 3
9
2
85
28 1
21 1
17 0
14 1
12 1
10 5
9
4
86
28 2
21 2
17 1
14 2
12 2
10 6
9
5
87
29 0
21 3
17 2
14 3
12 3
10 7
9
6
89
29 2
22 1
17 4
14 5
12 5
11 1
9
8
91
30 1
22 3
18 1
15 1
13 0
11 3
10 1
93
31 0
23 1
18 3
15 3
13 2
11 5
10 3
7
www.abacused.com.au
÷8 r
6 7
÷9 r
6 1
DIVISION DECISION
TEACHER’S NOTES 1/2
Aims
Students will perform divisions involving single-digit divisors.
Prior Knowledge
Students will require a level of fluency with basic multiplication and associated division facts.
Students will require an understanding of the division operation
Prior to playing this game students will require experience with the basic ideas behind division, such as
sharing and grouping. Rather than use the ‘goes into’ or ‘guzinta’ as many students call it, focus on the
sharing notion of division. Division should be linked to multiplication.
Language
Remainder: The number remaining after a division of whole numbers. When dividing 12 by 5, the
remainder is 2. Note that some students mistakenly think that 5 r 2 is the same as 5•2, when really it
means 2 2/5 or 5•4
Division Strategies
The early numbers on the board relate to the basic multiplication facts. When learning basic multiplication
facts the associated division facts should be emphasised. For example a child who knows the fact 7 x 6
= 42, should be explicitly taught the associated facts, 6 x 7 = 42, 42 ÷ 7 = 6 and 42 ÷ 6 = 7. Links to the
associated fraction may also be made, 1/6 of 42 is 7 and 1/7 of 42 is 6.
A three-part flashcard may be used to emphasise the links between multiplication and division.
6
48
8
Links may be made between all three parts, by covering individual sections of the card. For example,
covering 48 emphasises the multiplication fact 6 eights. Covering either the 6 or 8 emphasises the
associated division facts 48 ÷ 6 = 8, 48 ÷ 8 = 6, 6 x ? = 48, 8 x ? = 48.
Once division moves beyond the basic facts, students can partition numbers into know facts in order
to perform a division. For example, 79 ÷ 6 may be broken into three separate calculations: 60 ÷ 6 = 10,
18 ÷ 6 = 3 and then one remains. The key to adopting this strategy is to break the original number into
multiples of the divisor. In this case the divisor was 6 and so the dividend (79) was broken into 60 and 18,
leaving a remainder of one.
8
www.abacused.com.au
DIVISION DECISION
TEACHER’S NOTES 2/2
Assessment
After playing Division Decision several times students may be given similar examples to solve. These may
be given in the context of playing the game.
Problem 1
“Imagine you were playing Division Decision and you landed on 79. Which number(s) would you prefer to
spin in order to move the furthest?” Explain.
Problem 2
“Imagine you were playing Division Decision and were using the ÷3, ÷ 4, ÷ 5, ÷ 6, ÷ 7, ÷ 8, ÷9 spinner
and you landed on 79. If you wanted to move forward one which number(s) would you want to spin up?
Explain.
Differentiating the Curriculum
This game may be played at three different levels, hence the inclusion of three spinners (variations)
•
•
•
÷2, ÷3, ÷4, ÷5
÷3, ÷4, ÷5, ÷6
÷3, ÷4, ÷5, ÷6, ÷7, ÷8, ÷9
This means that different groups of students may be playing the same game but at a level appropriate to
their ability.
Support for Weaker Students
Students who require support may be given a small multiplication chart. This will help them with the basic
multiplication facts. They will still need to make the links between multiplication and division and will
need to partition dividends that are beyond the basic facts. The CD Maths Basic Facts: Multiplication and
Division includes many support materials.
Extending Students
Ask the students to consider the structure of the board. Some numbers are missing from the board. Why?
Students may note that some numbers have several factors, eg 6 x 6 is 36 and 4 x 9 are 36, which means
when 36 is divided by two, four, six or nine there won’t be a remainder. Other numbers (prime numbers)
such as 37 have only one and itself as factors. This means that when dividing by two, three, four etc there
will always be a remainder.
9
www.abacused.com.au
SPACE RACE
ADDITION
CHECK SHEET
Planet +1
Credits
1+1
2
2+1
3
3+1
4
4+1
5
5+1
6
6+1
7
7+1
8
8+1
9
9+1
10
2+2
4
3+2
5
4+2
6
5+2
7
6+2
8
7+2
9
8+2
10
9+2
11
2+3
5
3+3
6
4+3
7
5+3
8
6+3
9
7+3
10
8+3
11
9+3
12
1+?=10
9
2+?=10
8
3+?=10
7
4+?=10
6
5+?=10
5
6+?=10
4
7+?=10
3
8+?=10
2
9+?=10
1
1+1
2
2+2
4
3+3
6
4+4
8
5+5
10
6+6
12
7+7
14
8+8
16
9+9
18
2+3
5
3+4
7
4+5
9
5+6
11
6+7
13
7+8
15
8+9
17
9+10
19
9+2
11
9+3
12
9+4
13
9+5
14
9+6
15
9+7
16
9+8
17
9+9
18
Planet +2
Credits
1+2
3
Planet +3
Credits
1+3
4
Make to Ten
Credits
Double
Credits
Near Double
Credits
1+2
3
Planet 9+
Credits
9+1
10
10
www.abacused.com.au
SPACE RACE - ADDITION
TEACHER’S NOTES 1/3
Aims
This game is designed to focus on specific mental strategies for adding single digit numbers.
Prior Knowledge
The students will need to be able to count.
The students need to understand that 3 + 6 is the same as 6 + 3 (commutative law)
Extra Materials
If players are expected to keep a running record of the game they need to use an appropriate table.
Assessment
Players will need to write down their space credits (score) for each turn and keep a cumulative total. The
player with the highest total at the end of the game wins. The Admiral (checker) should use the check
sheet to make sure that each calculation is correct. Teachers may collect the students recording to double
check.
As students are playing the game the teacher may observe which planets seem to cause the most
difficulties for students. For example, if the speed of calculation slows considerably or students make more
mistakes when orbiting the ‘Make to Ten’ planet it is likely that some explicit teaching of this strategy or
extra practice with these facts may be required.
When observing students play, listen for or watch for students who are over reliant on counting in ones.
For example, if a student was orbiting the 9 + planet and landed on 6, watch for students who use their
fingers to count on , 10 ,11, 12, 13, 14, 15, or students who sub vocalise or tap their fingers. Students who
do this often mis-count or lose count.You may notice that their answers are typically one more or one
less than the actual answer
Background (Language/Terminology)
The planets are presented in order of difficultly and highlight the five mental strategies for single-digit
addition (that is, the basic addition facts).
1.
2.
3.
4.
5.
Count on from the larger 1, 2 or 3
Make to ten
Doubles
Near doubles
Bridge ten
11
www.abacused.com.au
SPACE RACE - ADDITION
TEACHER’S NOTES 2/3
Mental Computation Strategies
Count on from the larger 1, 2, or 3
The first three planets focus on adding 1, 2 or 3. Encourage all students to count on from the larger
number. When adding 3 and 6 a student should turn the calculation around (use the commutative
property) 6 + 3 and count 7, 8, 9. Watch for students who count 6, 7, 8. Eventually students need to
develop fluency and learn that 6 and 3 is 9.
Watch (listen) for:
•
students who begin the count at one each time, 1, 2, 3, 4, 5, 6, 7, 8, 9
•
students who start the count at the wrong place ie 6, 7, 8 – instead of 7, 8, 9.
Make to Ten – Visit ‘Make to Ten’ planet
The students will need to learn the following addition facts
1 + 9 = 10
2 + 8 = 10
3 + 7 = 10
4 + 6 = 10
5 + 5 = 10
6 + 4 = 10
7 + 3 = 10
8 + 2 = 10
9 + 1 =10
If the facts are recorded in an organised manner students are more likely to notice patterns such as the
commutative property of addition, that is 6 + 4 = 10 and 4 + 6 = 10.
In order to orbit the Make to ten planet. Cuisenaire rods may be used to model part part whole
understanding.
WHOLE
PART
PART
The commutative property 7 + 3 = 3 + 7 may be highlighted.
10
7
3
10
3
7
12
www.abacused.com.au
SPACE RACE - ADDITION
TEACHER’S NOTES 3/3
The part part whole model may be used to highlight links between addition and subtraction, that is,
10 – 7 = 3
10 – 3 = 7
7 + _ = 10
10
7
3 + _ = 10
10
3
Doubling:Visit the Doubling Planet
A student who lands on 3 would double it to make 6 space credits.
Cutting up grid paper often helps students learn to double. For example, 3 + 3 or double 3 is 6.
Ten frames may be used in a similar fashion to help students focus on doubling. For example if four
counters of the same colour are placed on a ten frame then another four counters of a different counters
may be placed on the ten frame to show that double four is eight.
Near Doubles:Visit the Near Double Planet
At the start of the game a decision needs to be made to either:
•
Double and add one, or
•
Double and subtract one.
For example, if the rule to double and add one is chosen then a student who lands on 4 would score 9
space credits (4 + 4 + 1)
Students could create a table t0 record this information for each number rolled.
Number
4
7
Double -1
7
13
Double
8
14
Double +1
9
15
13
www.abacused.com.au
PITSTOP
TEACHER’S NOTES
Aim
To park all your cars (counters) in pit stops.
Prior Knowledge
Students will need to know the addition facts that make 10.
Students will need to know complementary numbers, that is numbers that add together to form a multiple
of ten, eg 46 and another 4 make 50.
Students will need to know how to add and subtract 9 and 11 (Adjust and Compensate)
Language
Encourage students to discuss the various strategies they use to add/subtract nine or eleven. For example,
when subtracting 9 a student might subtract 10 and then move one forward.
Encourage students to explain any links they might notice between addition and subtraction. For example,
6 + 4 = 10, 4 + 6 = 10, 10 – 6 = 4, 10 – 4 = 6, 6 + ? = 10, 4 + ? = 10.
Assessment
When students are adding or subtracting 9 or 11 watch whether the students count in ones, move
directly to the correct spot or use a strategy such as move ten and then adjust.
When students roll a number note whether they hesitate or choose the best move. For example, if a
player has counters on 37 and 26 and 65 and rolls a 3 watch whether the student immediately chooses to
move the counter that is on 37 or hesitates.
Encourage students to verbalise each move they make.
Variations
•
Use four counters.
•
Use two dice (add or subtract the two numbers)
•
Allow players to move two counters on any move. That is, the number shown on the dice may be split into two numbers and one car is moved forward one part of the total and the other car, the other part.
14
www.abacused.com.au
TREASURE TROVE
CHECK SHEET
Prime and Composite numbers:
Note: 1 is neither prime nor composite.
Prime Numbers (1-62)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61
Composite Numbers (1-62)
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42,
44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62
Multiples of 3 - 9
Multiples of 3 (1-62)
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60
Multiples of 4 (1-62)
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60
Multiples of 5 (1-62)
5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60
Multiples of 6 (1-62)
6, 12, 18, 24, 30, 36, 42, 48, 54, 60
Multiples of 7 (1-62)
7, 14, 21, 28, 35, 42, 49, 56
Multiples of 8 (1-62)
8, 16, 24, 32, 40, 48, 56
Multiples of 9 (1-62)
9, 18, 27, 36, 45, 54
15
www.abacused.com.au
TREASURE TROVE
TEACHER’S NOTES 1/2
Aim
To collect as much treasure as possible, that is to land on a particular multiple as often as you can.
Prior Knowledge
Prior to playing Treasure Trove students will need some experience with multiples. Prepare students by
counting in multiples, eg 2, 4, 6, 8, 10, 12, … 62. Games such as Multiple Madness, see p. 36 – 27, Swan, P.
(2003). Dice Dazzlers. Will help prepare the students to play this game.
These numbers may be written on the board and the students asked to describe any patterns they
notice, eg with even numbers the units digit is 0, 2, 4, 6, 8 and repeats. The students are then ready to play
Treasure Trove involving multiples of two.
A similar approach may be applied to other multiples. A 0, 5, 0, 5 pattern will become apparent with the
multiples of five
To prepare students to play Treasure Trove: multiples of three, ask the students to count in threes; 3, 6, 9,
12, 15, 18 … 60
Encourage students to look for patterns.You may need to help them spot the digit-sum pattern by asking
them to add the digits.
Multiples of Three
3
6
9
12
15
18
21
24
27
Add the Digits
3
6
9
1+2
1+5
1+8
2+1
2+4
2+7
Digit Sum
3
6
9
3
6
9
3
6
9
The 3, 6, 9 pattern soon becomes obvious. Likewise a digit sum of nine becomes apparent in the multiples
of nine.
If Treasure Trove: Prime is played then students will need to be taught to recognise prime and composite
numbers.
16
www.abacused.com.au
TREASURE TROVE
TEACHER’S NOTES 2/2
Language
A prime number is a counting number with exactly two factors, eg 2, 3, 5, 7, 11, 13, …. Note that 1 is NOT
a prime number as it has only one factor. A list of prime numbers from 1 – 62 is provided on the check
sheet that accompanies this game.
Assessment
One player is designated the Captain of the ship and uses the check sheet that comes with the game to
monitor all the moves during a game.
After playing the game a few times and becoming familiar with a certain multiple students could be shown
a shaded number track and ask the students to pick the odd ones out, or given a list of multiples and
asked to pick out the numbers which are NOT multiples. For example, which of these numbers are NOT
multiples of 4:
•
8, 11, 24, 30, 36, 48.
Variations
•
Change to ten sided dice.
•
Change the amount of treasure that needs to be collected (for a shorter or longer game)
•
Hide treasure under two multiples, eg 3 and 4. A double treasure would then be hidden under 12, 24, 36 etc
17
www.abacused.com.au
OTHER GAMES BY
DR PAUL SWAN
Pack B
Coin Collector
Fraction Action
Get In Shape
Stop The Clock
www.abacused.com.au