1. Primary 3/4 Mathematics Workshop For Parents 14 April 2012 Endeavour Primary School Mathematics Department 2012

2. Workshop Outline Introduction to Problem-Solving Model Method 3 Different types of Models 4 different Heuristics Format of assessment

3. Problem-solving Approach 1.Understand the Problem (Understand) 2.Devise a Plan (Plan) 3.Carry out the Plan (Do) 4.Review and discuss the solution (Check)

4. Problem-solving Approach 1.Understand the Problem (Understand) Read to understand. If at first not clear, read again. Still don’t get it? Read chunk by chunk. Explain the question in another way. Use visualisation tool – model, timeline, diagrams, table etc.

5. Problem-solving Approach 2.Devise a Plan (Plan) Have I seen a similar or related question before? Do I have a ready plan? Do I have all the data? What data is missing? Can I find the missing data? Can I use a smaller number to try first? Use a heuristics?

6. Problem-solving Approach 3.Carry out the Plan (Do) Are all my steps accurate? Are there traps I need to be alert of? Have I used all the data given? Do my steps make sense?

7. Problem-solving Approach 4.Review and discuss the solution (Check) Does the answer make sense? Did I answer the question? Could this problem be solved in a simpler way?

8. Model Method Draw a diagram

9. Why Model Drawing? Visual representation of details –Majority of our children are visual learners Helps children plan the solution steps for solving the problem –Useful in fractions, ratio & percentage

10. Teaches mathematical language Provides foundation for algebraic understanding Empowers students to think systematically and master more challenging problems Why Model Drawing?

11. Model Drawing does NOT Work in every problem Specify ONE RIGHT model Specify ONE RIGHT operation

12. Concrete-Pictorial-Abstract Approach Concrete – Manipulatives: Base-Ten Blocks Pictorial - Models: 100 30? Abstract – Symbols: 100 – 30 = 70

13. 4 + 2 = 6 Concrete-Pictorial-Abstract Approach

14. Types of Models 1.Part-whole model a) Whole Numbers b) Fractions 2. Comparison Model a) Comparing 2 items b) Comparing 3 items c) Other Comparison Models 3.Before-After Model a) Total unchanged b) Total changed

15. 1. Part-whole Model Find value of unknown part Find value of whole

16. Part-whole Model: Whole Numbers Calvin earns $2000 every month. He pays $300 for food. He also spends $200 on his car, $500 on housing and saves the rest. How much does he save every month? Calvin earns $2000 every month. He pays $300 for food. He also spends $200 on his car, $500 on housing and saves the rest. How much does he save every month?

17. Part-whole Model: Whole Numbers Calvin earns $2000 every month. He pays $300 for food. He also spends $200 on his car, $500 on housing and saves the rest.. How much does he save every month? $300 $200 car $500 food housing ? savings $2000

18. $300 $200 car $500 food housing ? $2000 saving Used$300 + $200 + $500 = $1000 He saves $1000 every month. Savings$2000 - $1000 = $1000

19. How can we check if $1000 is a reasonable answer? What is another way to solve this problem?

20. Part-whole Model: Whole Numbers Qi Ying bought some sweets. She ate half of them and gave 5 sweets to Joy. She had 7 sweets left. How many sweets did Qi Ying buy?

21. Part-whole Model: Whole Numbers Qi Ying bought some sweets. She ate half of them and gave 5 sweets to Joy. She had 7 sweets left. How many sweets did Qi Ying buy? ? Ate 1 unit (half) 5 (Joy) 7 (Left) 1 unit (half)

22. Part-whole Model: Whole Numbers ? Ate 1 unit 5 (Joy) 7 (Left) 1 unit 5 + 7 = 12 Qi Ying bought 24 sweets. 2 units2 × 12 = 24

23. How can we check if ‘24 sweets’ is a reasonable answer? What is another way of representing this problem? ? 5 + 7 × 2 ÷ 2

24. Part-whole Model: Fractions

25. ? girls 24 boys 2 units 24 1 unit 24 ÷ 2 = 12 There are 12 girls.

26. How can we check if the answer is reasonable?

27. Part-whole Model: Fractions ¼ of the fish in an aquarium are goldfish. There are 4 more guppies than goldfish in the aquarium. The remaining 16 fish are carps. How many fish are there in the aquarium?

28. ¼ Part-whole Model: Fractions ¼ of the fish in an aquarium are goldfish. There are 4 more guppies than goldfish in the aquarium. The remaining 16 fish are carps. How many fish are there in the aquarium? ¼ 16 carps ¼ goldfish guppies ? ¼ 2 units 4 + 16 4 units 2 × 20 = 40 There are 40 fish. = 20 4

29. How can we check if the answer is reasonable?

30. 2. Comparison Model Find total sum given between difference and value of an item Find value of an item given difference and sum

31. Comparison Model: 2 items Sven collected 3426 stamps. He collected 841 fewer stamps than Jerome. How many stamps did they collect?

32. Comparison Model: 2 items Sven collected 3426 stamps. He collected 841 fewer stamps than Jerome. How many stamps did they collect? Who has more? 3426 Sven 841 fewer ? Jerome Whose bar should be longer? ?

33. Jerome 3426 + 841 = 4267 They collected 7693 stamps. Total 3426 + 4267 = 7693 3426 Sven 841 fewer ? Jerome ?

34. How can we check if ‘7693 stamps’ is a reasonable answer? What is another way to solve this problem?

35. Comparison Model: 2 items

36. Smaller ¼ ? Larger

37. 2 units 1 unit Smaller ¼ ? Larger

38. How can we check if the answer is reasonable?

39. Comparison Model: 3 items Kyle, Siti and Alice have a total of 290 stickers. Kyle has twice as many stickers as Siti. Alice has 50 stickers more than Siti. How many stickers does Alice have?

40. Comparison Model: 3 items Kyle, Siti and Alice have a total of 290 stickers. Kyle has twice as many stickers as Siti. Alice has 50 stickers more than Siti. How many stickers does Alice have? Siti Kyle Alice 290 50 Note how ‘50’ is represented.

41. Alice has 110 stickers. 4 units 290 – 50 = 240 1 unit 240 ÷ 4 = 60 Siti Kyle Alice 290 50 Let Siti have x stickers. Kyle 2x Alice x + 50 4x + 50 = 290 4x = 240 x = 60 60 + 50 = 110 Alice 60 + 50 = 110

42. Comparison Model: 3 items Kyle, Siti and Alice have a total of 270 stickers. Kyle has thrice as many stickers as Siti. Alice has half as many stickers as Siti. How many stickers does Siti have?

43. Comparison Model: 3 items Kyle, Siti and Alice have a total of 270 stickers. Kyle has thrice as many stickers as Siti. Alice has half as many stickers as Siti. How many stickers does Siti have? Siti Kyle Alice 270

44. SitiKyle Alice 270 1 unit 2 units Siti has 60 stickers. 270 ÷ 9 30 x 2 = 60 2709 units = 30

45. How can we check if the answer is reasonable?

46. Other Comparison Models 2 files and 3 pens cost $18 altogether. A file costs 3 times as much as a pen. Find the cost of 1 file.

47. Other Comparison Models 2 files and 3 pens cost $18 altogether. A file costs 3 times as much as a pen. Find the cost of 1 file. Pens Files

48. Other Comparison Models 2 files and 3 pens cost $18 altogether. A file costs 3 times as much as a pen. Find the cost of 1 file. Pens Files $18 ?

49. 2 files and 3 pens cost $18 altogether. A file costs 3 times as much as a pen. Find the cost of 1 file. Pens Files $18 9 units 1 unit 1 file costs $6. $18 $18 ÷ 9 = $2 3 units$2 x 3= $6 ?

50. Other Comparison Models 2 crystal vases and 3 plates cost $161. The cost of 1 plate is half the cost of 1 vase. What is the cost of 1 vase?

51. Other Comparison Models 2 crystal vases and 3 plates cost $161. The cost of 1 plate is half the cost of 1 vase. What is the cost of 1 vase? Vases Plates

52. Other Comparison Models 2 crystal vases and 3 plates cost $161. The cost of 1 plate is half the cost of 1 vase. What is the cost of 1 vase? Vases Plates $161 ?

53. 2 crystal vases and 3 plates cost $161. The cost of 1 plate is half the cost of 1 vase. What is the cost of 1 vase? Vases Plates $161 7 units 1 unit 1 vase costs $46. $161 $161 ÷ 7 = $23 2 units$23 x 2= $46 ?

54. 3. Before and After Model Total unchanged Total changed

55. Before and After (total unchanged) Alan Ben 558

56. Alan and Ben had 558 cards altogether. Alan gave of his cards to Ben. After that, Ben had twice the number of cards as Alan. How many cards did Ben have at first? Alan Ben 558 9 units 1 unit Ben had 310 cards at first. 558 558 ÷ 9 = 62 5 units62 x 5= 310 ?

57. Before and After (Total Changed) Alice and Betty had the same amount of money each. After Alice spent $120 and Betty spent $45, Betty had twice as much money as Alice. How much money did each girl have at first? Alice Betty

58. Before and After (Total Changed) Alice Betty 1 unit $45 Alice and Betty had the same amount of money each. After Alice spent $120 and Betty spent $45, Betty had twice as much money as Alice. How much money did each girl have at first?

59. Alice and Betty had the same amount of money each. After Alice spent $120 and Betty spent $45, Betty had twice as much money as Alice. How much money did each girl have at first? Alice Betty 1 unit $45 ? 1 unit Each girl had $195 at first. $120 - $45= $75 $75 + $120= $195

60. Alice and Betty had the same amount of money each. After Alice spent $120 and Betty spent $45, Betty had twice as much money as Alice. How much money did each girl have at first? Alice Betty 1 unit $45 ? 1 unit Each girl had $195 at first. $120 - $45= $75 $150 + $45= $195 2 units$75 x 2 = $150

61. Before and After (Total Changed) There was an equal number of male and female passengers in a train at first. After 193 male passengers and 46 female passengers alighted, there were 4 times as many female passengers as male passengers left in the train. How many male passengers were in the train at first? Male Female

62. Before and After (Total Changed) There was an equal number of male and female passengers in a train at first. After 193 male passengers and 46 female passengers alighted, there were 4 times as many female passengers as male passengers left in the train. How many male passengers were in the train at first? Male Female 1 unit 461 unit

63. Before and After (Total Changed) There was an equal number of male and female passengers in a train at first. After 193 male passengers and 46 female passengers alighted, there were 4 times as many female passengers as male passengers left in the train. How many male passengers were in the train at first? Male Female 1 unit 461 unit ?

64. Male Female 1 unit 46 3 units There were 242 male passengers at first. 193 - 46 = 147 147 ÷ 3 = 49 1 unit There was an equal number of male and female passengers in a train at first. After 193 male passengers and 46 female passengers alighted, there were 4 times as many female passengers as male passengers left in the train. How many male passengers were in the train at first? ? 1 unit 49 + 193 = 242

65. Other Heuristics 1.Work Backwards 2. Guess and Check 3.Make a Systematic List 4.Make a Table Is model drawing the only method? No!

66. Work Backwards Find the missing number. ? 108 54 50 - 4 ÷ 2x 3 + 4 x 2 ÷ 3 108 ÷ 3 = 36 The missing number is 36. 50 + 4 = 54 54 x 2 = 108

67. Work Backwards A train carrying some passengers left Station A. At Station B, 7 passengers boarded. At Station C, half of the passengers alighted. At Station D, 8 passengers alighted. As the train left Station D, there were 28 passengers on the train. How many passengers were on the train when it left Station A? 28 ÷2 + 7 - 8 ? A B C D

68. A train carrying some passengers left Station A. At Station B, 7 passengers boarded. At Station C, half of the passengers alighted. At Station D, 8 passengers alighted. As the train left Station D, there were 28 passengers on the train. How many passengers were on the train when it left Station A? 65 passengers were on the train when it left Station A. 28 + 8 = 36 72 36 28 ÷2 + 7 - 8 ? 36 x 2 = 72 72 – 7 = 65 A B C D + 8 x 2 - 7

69. Work Backwards John took 50 minutes to wash his car and another 1 h 40 min to polish it. He finished washing and polishing his car at 2 pm. At what time did he start washing his car? 2pm + 50 min + 1h 40 min

70. John took 50 minutes to wash his car and another 1 h 40 min to polish it. He finished washing and polishing his car at 2 pm. At what time did he start washing his car? He started washing his car at 11.30 am. ?12.20pm 2pm - 1h 40 min - 50 min 12.20 pm 11.30 am - 50 min 2 pm 1 pm 12.20 pm - 40 min - 1 h + 50 min + 1h 40 min

71. Guess and Check (1) At a park, there are 25 bicycles and tricycles. These vehicles have a total of 55 wheels. If there are more bicycles than tricycles, how many bicycles are there at the park?

72. Guess and Check (1) At a park, there are 25 bicycles and tricycles. These vehicles have a total of 55 wheels. If there are more bicycles than tricycles, how many bicycles are there at the park?

73. Guess and Check (1) At a park, there are 25 bicycles and tricycles. These vehicles have a total of 55 wheels. If there are more bicycles than tricycles, how many bicycles are there at the park? Conditions stated in the question: 1)Total vehicles: 25 2)Total wheels: 55 3)More bicycles than tricycles.

74. At a park, there are 25 bicycles and tricycles. These vehicles have a total of 55 wheels. If there are more bicycles than tricycle, how many bicycles are there at the park? GuessNo. of bicycles (more) No. of bicycle wheels (no. x 2) No. of tricycles (fewer) No. of tricycle wheels (no. x 3) Total number of wheels (55) Check

75. At a park, there are 25 bicycles and tricycles. These vehicles have a total of 55 wheels. If there are more bicycles than tricycle, how many bicycles are there at the park? GuessNo. of bicycles (more) No. of bicycle wheels (no. x 2) No. of tricycles (fewer) No. of tricycle wheels (no. x 3) Total number of wheels (55) Check 11515 x 2 = 30 25 ‒ 15 = 10 10 x 3 = 30 30 + 30 = 60 

76. At a park, there are 25 bicycles and tricycles. These vehicles have a total of 55 wheels. If there are more bicycles than tricycle, how many bicycles are there at the park? GuessNo. of bicycles (more) No. of bicycle wheels (no. x 2) No. of tricycles (fewer) No. of tricycle wheels (no. x 3) Total number of wheels (55) Check 11515 x 2 = 30 25 ‒ 15 = 10 10 x 3 = 30 30 + 30 = 60  21717 x 2 = 34 25 ‒ 17 = 8 8 x 3 = 24 34 + 24 = 58 

77. At a park, there are 25 bicycles and tricycles. These vehicles have a total of 55 wheels. If there are more bicycles than tricycle, how many bicycles are there at the park? GuessNo. of bicycles (more) No. of bicycle wheels (no. x 2) No. of tricycles (fewer) No. of tricycle wheels (no. x 3) Total number of wheels (55) Check 11515 x 2 = 30 25 ‒ 15 = 10 10 x 3 = 30 30 + 30 = 60  21717 x 2 = 34 25 ‒ 17 = 8 8 x 3 = 24 34 + 24 = 58  32020 x 2 = 40 25 – 20 = 5 5 x 3 = 15 40 + 15 = 55  There are 20 bicycles at the park.

78. At a park, there are 25 bicycles and tricycles. These vehicles have a total of 55 wheels. If there are more bicycles than tricycle, how many bicycles are there at the park? Method 1: Guess and Check Method 2: Supposition Suppose all the vehicles are bicycles, the number of wheels But there are 55 wheels altogether. 55 ‒ 50 = 5 extra wheels Each tricycle has 1 wheel more than a bicycle, 5 ÷ 1 = 5 There are 5 tricycles. 25 ‒ 5 = 20 There are 20 bicycles at the park. 2 x 25 = 50

79. Guess and Check (2) Sue has 23 coins. Some are 10¢ coins and the others are 20¢ coins. She has more 10¢ coins than 20¢ coins. The total value of the coins is $3.40. How many 20¢ coins are there?

80. Guess and Check (2) Sue has 23 coins. Some are 10¢ coins and the others are 20¢ coins. She has more 10¢ coins than 20¢ coins. The total value of the coins is $3.40. How many 20¢ coins are there?

81. Guess No. of 10¢ coins (more) Value of 10¢ coins (no. x 10¢) No. of 20¢ coins (fewer) Value of 20¢ coins (no. x 20¢) Total value ($3.40) Check

82. Sue has 23 coins. Some are 10¢ coins and the others are 20¢ coins. She has more 10¢ coins than 20¢ coins. The total value of the coins is $3.40. How many 20¢ coins are there? Guess No. of 10¢ coins (more) Value of 10¢ coins (no. x 10¢) No. of 20¢ coins (fewer) Value of 20¢ coins (no. x 20¢) Total value ($3.40) Check 120$23$0.60$2.60×

83. Sue has 23 coins. Some are 10¢ coins and the others are 20¢ coins. She has more 10¢ coins than 20¢ coins. The total value of the coins is $3.40. How many 20¢ coins are there? Guess No. of 10¢ coins (more) Value of 10¢ coins (no. x 10¢) No. of 20¢ coins (fewer) Value of 20¢ coins (no. x 20¢) Total value ($3.40) Check 120$23$0.60$2.60× 213$1.3010$2$3.30×

84. Sue has 23 coins. Some are 10¢ coins and the others are 20¢ coins. She has more 10¢ coins than 20¢ coins. The total value of the coins is $3.40. How many 20¢ coins are there? Guess No. of 10¢ coins (more) Value of 10¢ coins (no. x 10¢) No. of 20¢ coins (fewer) Value of 20¢ coins (no. x 20¢) Total value ($3.40) Check 120$23$0.60$2.60× 213$1.3010$2$3.30× 312$1.2011$2.20$3.40  There are 11 20¢ coins.

85. Make a Systematic List Mr John has some stickers. If he gives each child 5 stickers, he will have 5 stickers left. If he gives each child 6 stickers instead, he will have 3 stickers short. How many stickers does he have?

86. Make a Systematic List Mr John has some stickers. If he gives each child 5 stickers, he will have 5 stickers left. If he gives each child 6 stickers instead, he will have 3 stickers short. How many stickers does he have?

87. 55 left 63 short

88. Mr John has some stickers. If he gives each child 5 stickers, he will have 5 stickers left. If he gives each child 6 stickers instead, he will have 3 stickers short. How many stickers does he have? 55 left 55 + 5 = 10 10 15 20 25 30 35 40 45 50 63 short

89. Mr John has some stickers. If he gives each child 5 stickers, he will have 5 stickers left. If he gives each child 6 stickers instead, he will have 3 stickers short. How many stickers does he have? 55 left 55 + 5 = 10 1015 20 25 30 35 40 45 50 55 63 short

90. Mr John has some stickers. If he gives each child 5 stickers, he will have 5 stickers left. If he gives each child 6 stickers instead, he will have 3 stickers short. How many stickers does he have? 55 left 55 + 5 = 10 1015 20 25 30 35 40 45 50 55 63 short 66 ‒ 3 = 3 12 18 24 30 36 42 48 54 60

91. Mr John has some stickers. If he gives each child 5 stickers, he will have 5 stickers left. If he gives each child 6 stickers instead, he will have 3 stickers short. How many stickers does he have? 55 left 55 + 5 = 10 1015 20 25 30 35 40 45 50 55 63 short 66 ‒ 3 = 3 129 1815 2421 3027 3633 4239 4845 5451 6057

92. Mr John has some stickers. If he gives each child 5 stickers, he will have 5 stickers left. If he gives each child 6 stickers instead, he will have 3 stickers short. How many stickers does he have? 55 left 55 + 5 = 10 1015 20 25 30 35 40 45 50 55 63 short 66 ‒ 3 = 3 129 1815 2421 3027 3633 4239 4845 5451 6057 Mr John has 45 stickers.

93. Make a Table Benny, Cindy, David and Evelyn give picture cards to one another. Benny gives Cindy 19 cards. Cindy gives David 15 cards. Evelyn gives David 3 cards but David returns them to Evelyn. David gives Benny 12 cards. Who has fewer picture cards in the end than before?

94. Benny, Cindy, David and Evelyn give picture cards to one another. Benny gives Cindy 19 cards. Cindy gives David 15 cards. Evelyn gives David 3 cards but David returns them to Evelyn. David gives Benny 12 cards. Who has fewer picture cards in the end than before? ReceivesGivesResult Benny Cindy David Evelyn

95. Benny, Cindy, David and Evelyn give picture cards to one another. Benny gives Cindy 19 cards. Cindy gives David 15 cards. Evelyn gives David 3 cards but David returns them to Evelyn. David gives Benny 12 cards. Who has fewer picture cards in the end than before? ReceivesGivesResult Benny19 Cindy19 David Evelyn

96. Benny, Cindy, David and Evelyn give picture cards to one another. Benny gives Cindy 19 cards. Cindy gives David 15 cards. Evelyn gives David 3 cards but David returns them to Evelyn. David gives Benny 12 cards. Who has fewer picture cards in the end than before? ReceivesGivesResult Benny19 Cindy1915 David15 Evelyn

97. Benny, Cindy, David and Evelyn give picture cards to one another. Benny gives Cindy 19 cards. Cindy gives David 15 cards. Evelyn gives David 3 cards but David returns them to Evelyn. David gives Benny 12 cards. Who has fewer picture cards in the end than before? ReceivesGivesResult Benny19 Cindy1915 David15 + 3 Evelyn3

98. Benny, Cindy, David and Evelyn give picture cards to one another. Benny gives Cindy 19 cards. Cindy gives David 15 cards. Evelyn gives David 3 cards but David returns them to Evelyn. David gives Benny 12 cards. Who has fewer picture cards in the end than before? ReceivesGivesResult Benny19 Cindy1915 David15 + 33 Evelyn33

99. Benny, Cindy, David and Evelyn give picture cards to one another. Benny gives Cindy 19 cards. Cindy gives David 15 cards. Evelyn gives David 3 cards but David returns them to Evelyn. David gives Benny 12 cards. Who has fewer picture cards in the end than before? ReceivesGivesResult Benny1219 Cindy1915 David15 + 33 + 12 Evelyn33

100. Benny, Cindy, David and Evelyn give picture cards to one another. Benny gives Cindy 19 cards. Cindy gives David 15 cards. Evelyn gives David 3 cards but David returns them to Evelyn. David gives Benny 12 cards. Who has fewer picture cards in the end than before? ReceivesGivesResult Benny1219Gives more Cindy1915Receives more David15 + 3 = 183 + 12 = 15Receives more Evelyn33No change Benny has fewer picture cards than before.

101. Make a Table (2) In a game, two dice are thrown and the two numbers shown are multiplied to give a score. What whole number less than 10 cannot be a score of this game?

102. Make a Table (2) In a game, two dice are thrown and the two numbers shown are multiplied to give a score. What whole numbers less than 10 cannot be a score of this game?

103. In a game, two dice are thrown and the two numbers shown are multiplied to give a score. What whole number less than 10 cannot be a score of this game? X123456 1 2 3 4 5 6

104. In a game, two dice are thrown and the two numbers shown are multiplied to give a score. What whole number less than 10 cannot be a score of this game? X123456 1 2 3 4 515 6

105. In a game, two dice are thrown and the two numbers shown are multiplied to give a score. What whole number less than 10 cannot be a score of this game? X123456 1123456 224681012 3369 1518 44812162024 551015202530 661218243036

106. In a game, two dice are thrown and the two numbers shown are multiplied to give a score. What whole number less than 10 cannot be a score of this game? X123456 1123456 224681012 3369 1518 44812162024 551015202530 661218243036 The score cannot be 7.

107. Format of Math Paper

108. P5/P6 Math Exam Paper Format P2 – P4P5 - P6 MCQ20 Qns – 40 marks 15 Qns – 20 marks SAQ20 Qns – 40 marks 15 Qns – 20 marks Word Problems 5 Qns – 20 marks 18 Qns – 60 marks

109. P5/P6 Math Exam Paper Format Paper 1 - MCQ and SAQ Paper 2 - a combination of 2, 3, 4 and 5 marks word problems Paper 1 to be completed in 50 minutes without calculator Paper 2 to be completed in 100 minutes with calculator

110. Challenges due to Paper format Paper 1 to be completed within 50 minutes (30 questions – less than 2 minutes per question) Paper 2 – focuses on thinking skills as well as heuristics Culture shock in P5 for pupils

111. Changes to P3 and P4 Format 2012 – P3 and P4 SA2 Papers Section C total marks changed from 20 to 30. 2013 – P4 SA1 and SA2 Papers Section C total marks changed from 30 to 40. Heuristics and thinking skills come into play more. Concept and syllabus becomes basic skills.

112. Thank You