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Dr Paul Swan’s Maths Games

Check Sheets, Recording Sheets andTeacher’s Notes

Pack A

Division Decision• Pitstop• Space Race - Addition• Treasure Trove•

WRITTEN BY DR PAUL SWAN

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CONTENTS

Division Decision Check Sheets 2-7 Teacher’s Notes 8,9

Space Race - Addition Check Sheet 10 Teacher’s Notes 11-13

Pitstop Teacher’s Notes 14

Treasure Trove Check Sheet 15 Teacher’s Notes 16,17

1

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DIVISION DECISION÷2, ÷3, ÷4, ÷5

CHECKSHEET A

No ÷2 r ÷3 r ÷4 r ÷5 r13 6 1 4 1 3 1 2 314 7 0 4 2 3 2 2 417 8 1 5 2 4 1 3 219 9 1 6 1 4 3 3 421 10 1 7 0 5 1 4 122 11 0 7 1 5 2 4 223 11 1 7 2 5 3 4 325 12 1 8 1 6 1 5 027 13 1 9 0 6 3 5 229 14 1 9 2 7 1 5 431 15 1 10 1 7 3 6 133 16 1 11 0 8 1 6 334 17 0 11 1 8 2 6 435 17 1 11 2 8 3 7 037 18 1 12 1 9 1 7 238 19 0 12 2 9 2 7 339 19 1 13 0 9 3 7 441 20 1 13 2 10 1 8 143 21 1 14 1 10 3 8 346 23 0 15 1 11 2 9 147 23 1 15 2 11 3 9 249 24 1 16 1 12 1 9 451 25 1 17 0 12 3 10 153 26 1 17 2 13 1 10 3

2

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DIVISION DECISION÷2, ÷3, ÷4, ÷5

CHECKSHEET B

No ÷2 r ÷3 r ÷4 r ÷5 r55 27 1 18 1 13 3 11 057 28 1 19 0 14 1 11 258 29 0 19 1 14 2 11 359 29 1 19 2 14 3 11 461 30 1 20 1 15 1 12 162 31 0 20 2 15 2 12 263 31 1 21 0 15 3 12 365 32 1 21 2 16 1 13 067 33 1 22 1 16 3 13 269 34 1 23 0 17 1 13 471 35 1 23 2 17 3 14 173 36 1 24 1 18 1 14 374 37 0 24 2 18 2 14 477 38 1 25 2 19 1 15 279 39 1 26 1 19 3 15 481 40 1 27 0 20 1 16 182 41 0 27 1 20 2 16 283 41 1 27 2 20 3 16 385 42 1 28 1 21 1 17 086 43 0 28 2 21 2 17 187 43 1 29 0 21 3 17 289 44 1 29 2 22 1 17 491 45 1 30 1 22 3 18 193 46 1 31 0 23 1 18 3

3

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DIVISION DECISION÷3, ÷4, ÷5, ÷6

CHECKSHEET A

No ÷3 r ÷4 r ÷5 r ÷6 r13 4 1 3 1 2 3 2 114 4 2 3 2 2 4 2 217 5 2 4 1 3 2 2 519 6 1 4 3 3 4 3 121 7 0 5 1 4 1 3 322 7 1 5 2 4 2 3 423 7 2 5 3 4 3 3 525 8 1 6 1 5 0 4 127 9 0 6 3 5 2 4 329 9 2 7 1 5 4 4 531 10 1 7 3 6 1 5 133 11 0 8 1 6 3 5 334 11 1 8 2 6 4 5 435 11 2 8 3 7 0 5 537 12 1 9 1 7 2 6 138 12 2 9 2 7 3 6 239 13 0 9 3 7 4 6 341 13 2 10 1 8 1 6 543 14 1 10 3 8 3 7 146 15 1 11 2 9 1 7 447 15 2 11 3 9 2 7 549 16 1 12 1 9 4 8 151 17 0 12 3 10 1 8 353 17 2 13 1 10 3 8 5

4

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DIVISION DECISION÷3, ÷4, ÷5, ÷6

CHECKSHEET B

No ÷3 r ÷4 r ÷5 r ÷6 r55 18 1 13 3 11 0 9 157 19 0 14 1 11 2 9 358 19 1 14 2 11 3 9 459 19 2 14 3 11 4 9 561 20 1 15 1 12 1 10 162 20 2 15 2 12 2 10 263 21 0 15 3 12 3 10 365 21 2 16 1 13 0 10 567 22 1 16 3 13 2 11 169 23 0 17 1 13 4 11 371 23 2 17 3 14 1 11 573 24 1 18 1 14 3 12 174 24 2 18 2 14 4 12 277 25 2 19 1 15 2 12 579 26 1 19 3 15 4 13 181 27 0 20 1 16 1 13 382 27 1 20 2 16 2 13 483 27 2 20 3 16 3 13 585 28 1 21 1 17 0 14 186 28 2 21 2 17 1 14 287 29 0 21 3 17 2 14 389 29 2 22 1 17 4 14 591 30 1 22 3 18 1 15 193 31 0 23 1 18 3 15 3

5

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DIVISION DECISION÷3, ÷4, ÷5, ÷6, ÷7, ÷8, ÷9

CHECKSHEET A

No ÷3 r ÷4 r ÷5 r ÷6 r ÷7 r ÷8 r ÷9 r13 4 1 3 1 2 3 2 1 1 6 1 5 1 414 4 2 3 2 2 4 2 2 2 0 1 6 1 517 5 2 4 1 3 2 2 5 2 3 2 1 1 819 6 1 4 3 3 4 3 1 2 5 2 3 2 121 7 0 5 1 4 1 3 3 3 0 2 5 2 322 7 1 5 2 4 2 3 4 3 1 2 6 2 423 7 2 5 3 4 3 3 5 3 2 2 7 2 525 8 1 6 1 5 0 4 1 3 4 3 1 2 727 9 0 6 3 5 2 4 3 3 6 3 3 3 029 9 2 7 1 5 4 4 5 4 1 3 5 3 231 10 1 7 3 6 1 5 1 4 3 3 7 3 433 11 0 8 1 6 3 5 3 4 5 4 1 3 634 11 1 8 2 6 4 5 4 4 6 4 2 3 735 11 2 8 3 7 0 5 5 5 0 4 3 3 837 12 1 9 1 7 2 6 1 5 2 4 5 4 138 12 2 9 2 7 3 6 2 5 3 4 6 4 239 13 0 9 3 7 4 6 3 5 4 4 7 4 341 13 2 10 1 8 1 6 5 5 6 5 1 4 543 14 1 10 3 8 3 7 1 6 1 5 3 4 746 15 1 11 2 9 1 7 4 6 4 5 6 5 147 15 2 11 3 9 2 7 5 6 5 5 7 5 249 16 1 12 1 9 4 8 1 7 0 6 1 5 451 17 0 12 3 10 1 8 3 7 2 6 3 5 653 17 2 13 1 10 3 8 5 7 4 6 5 5 8

6

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DIVISION DECISION÷3, ÷4, ÷5, ÷6, ÷7, ÷8, ÷9

CHECKSHEET B

No ÷3 r ÷4 r ÷5 r ÷6 r ÷7 r ÷8 r ÷9 r55 18 1 13 3 11 0 9 1 7 6 6 7 6 157 19 0 14 1 11 2 9 3 8 1 7 1 6 358 19 1 14 2 11 3 9 4 8 2 7 2 6 459 19 2 14 3 11 4 9 5 8 3 7 3 6 561 20 1 15 1 12 1 10 1 8 5 7 5 6 762 20 2 15 2 12 2 10 2 8 6 7 6 6 863 21 0 15 3 12 3 10 3 9 0 7 7 7 065 21 2 16 1 13 0 10 5 9 2 8 1 7 267 22 1 16 3 13 2 11 1 9 4 8 3 7 469 23 0 17 1 13 4 11 3 9 6 8 5 7 671 23 2 17 3 14 1 11 5 10 1 8 7 7 873 24 1 18 1 14 3 12 1 10 3 9 1 8 174 24 2 18 2 14 4 12 2 10 4 9 2 8 277 25 2 19 1 15 2 12 5 11 0 9 5 8 579 26 1 19 3 15 4 13 1 11 2 9 7 8 781 27 0 20 1 16 1 13 3 11 4 10 1 9 082 27 1 20 2 16 2 13 4 11 5 10 2 9 183 27 2 20 3 16 3 13 5 11 6 10 3 9 285 28 1 21 1 17 0 14 1 12 1 10 5 9 486 28 2 21 2 17 1 14 2 12 2 10 6 9 587 29 0 21 3 17 2 14 3 12 3 10 7 9 689 29 2 22 1 17 4 14 5 12 5 11 1 9 891 30 1 22 3 18 1 15 1 13 0 11 3 10 193 31 0 23 1 18 3 15 3 13 2 11 5 10 3

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DIVISION DECISIONTEACHER’S NOTES 1/2

AimsStudents will perform divisions involving single-digit divisors.

Prior KnowledgeStudents 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.

LanguageRemainder: 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 StrategiesThe 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.

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.

486

8

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DIVISION DECISIONTEACHER’S NOTES 2/2

AssessmentAfter 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 CurriculumThis 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 StudentsStudents 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 StudentsAsk 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

Planet +11+1 2+1 3+1 4+1 5+1 6+1 7+1 8+1 9+1

Credits 2 3 4 5 6 7 8 9 10

Planet +21+2 2+2 3+2 4+2 5+2 6+2 7+2 8+2 9+2

Credits 3 4 5 6 7 8 9 10 11

Planet +31+3 2+3 3+3 4+3 5+3 6+3 7+3 8+3 9+3

Credits 4 5 6 7 8 9 10 11 12

Make to Ten1+?=10 2+?=10 3+?=10 4+?=10 5+?=10 6+?=10 7+?=10 8+?=10 9+?=10

Credits 9 8 7 6 5 4 3 2 1

Double1+1 2+2 3+3 4+4 5+5 6+6 7+7 8+8 9+9

Credits 2 4 6 8 10 12 14 16 18

Near Double1+2 2+3 3+4 4+5 5+6 6+7 7+8 8+9 9+10

Credits 3 5 7 9 11 13 15 17 19

Planet 9+9+1 9+2 9+3 9+4 9+5 9+6 9+7 9+8 9+9

Credits 10 11 12 13 14 15 16 17 18

SPACE RACEADDITION

CHECK SHEET

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SPACE RACE - ADDITIONTEACHER’S NOTES 1/3

AimsThis game is designed to focus on specific mental strategies for adding single digit numbers.

Prior KnowledgeThe 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 MaterialsIf players are expected to keep a running record of the game they need to use an appropriate table.

AssessmentPlayers 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. Count on from the larger 1, 2 or 32. Make to ten3. Doubles4. Near doubles5. Bridge ten

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SPACE RACE - ADDITIONTEACHER’S NOTES 2/3

Mental Computation Strategies

Count on from the larger 1, 2, or 3The 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’ planetThe students will need to learn the following addition facts

1 + 9 = 102 + 8 = 103 + 7 = 104 + 6 = 105 + 5 = 106 + 4 = 107 + 3 = 108 + 2 = 109 + 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.

The commutative property 7 + 3 = 3 + 7 may be highlighted.

WHOLE

PART PART

10

3 7

10

7 3

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SPACE RACE - ADDITIONTEACHER’S NOTES 3/3

The part part whole model may be used to highlight links between addition and subtraction, that is,

10 – 7 = 310 – 3 = 77 + _ = 10

3 + _ = 10

Doubling: Visit the Doubling PlanetA 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 PlanetAt 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.

10

7

10

3

Number Double -1 Double Double +14 7 8 97 13 14 15

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PITSTOPTEACHER’S NOTES

AimTo park all your cars (counters) in pit stops.

Prior KnowledgeStudents 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)

LanguageEncourage 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.

AssessmentWhen 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.

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TREASURE TROVECHECK SHEET

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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 - 9Multiples 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

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TREASURE TROVETEACHER’S NOTES 1/2

AimTo 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.

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.

Multiples of Three Add the Digits Digit Sum3 3 36 6 69 9 912 1+2 315 1+5 618 1+8 921 2+1 324 2+4 627 2+7 9

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TREASURE TROVETEACHER’S NOTES 2/2

LanguageA 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.

AssessmentOne 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

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OTHER GAMES BYDR PAUL SWAN

Pack B

Coin CollectorFraction Action

Get In ShapeStop The Clock