1. Handy Problem Solving Strategies for Primary Mathematics Siti Aishah Bte Shukri Rebecca Yeo

2. Something About Us

3. At the end of this session... You should be able to: List the stages of the Polya’s problem solving process Explain when and how to use different bar models in problem solving. Demonstrate how to use the branching method and bar models to understand and solve word problems

4. What kind of problems do students with learning difficulties face in Maths?

5. Common Maths Difficulties faced by students with Dyslexia Sequencing and naming Planning Organizing Language: Comprehending what is being read Handwriting and copying skills Working memory Remembering Maths facts Conceptual, logical, and spatial reasoning So how do we help them? Essential skills needed for maths problem solving

6. We can… Use the same strategies we use to teach students with dyslexia on how to read and write Integrate Visual and Language systems Provide organization and integration of teaching instructions that are multi- sensorial. We know that using pictures to prompt their recall of sounds or words could be helpful as it provides cues and context clues which gives meaning to their learning. Writing down key points/notes aids in working memory issues Underlining/highlighting Provide them a structure and opportunity to walk through the process of problem solving Backtrack and build the building blocks of the essential skills

7. Profile of our Maths students Students with dyslexia who have maths difficulties Primary 1 to Primary 6

8. Teaching Approach Orton-Gillingham Principles Cognitive Language-based Simultaneously Multisensory Structured, Sequential and Cumulative Diagnostic and Prescriptive Emotionally Sound

9. Singapore Mathematics Framework

10. Sample Questions

11. What is a problem? A problem is something that has no easy access to a solution. Dependent on the individual

12. Polya’s Problem Solving Process Understand Devise a Plan Solve Check

13. It’s not enough to just follow the process.. Take note of students’ 1. Cognitive resources 2. Heuristics 3. Monitoring and Control 4. Affect and Beliefs

14. WALK THROUGH THE PROCESS

15. 1. Understand the Problem (UP) Extract clues from the problem  A clue in every sentence! Make sense of the clues Define mathematical terms Restate what the problem is trying to say Use a heuristic (e.g. draw a picture)

16. What are some words frequently associated with these operations?

17. Some math terms (primary level)

18. Example Problem At a party, there were twice as many girls as boys. Each girl was given 3 balloons, and each boy was given 4 balloons. The total number of balloons given to the children was 70. How many girls were there at the party?

19. Strategy: Read sentence by sentence, annotate key terms At a party, there were twice as many girls as boys. Each girl was given 3 balloons, and each boy was given 4 balloons. The total number of balloons given to the children was 70. How many girls were there? 2x (1)

20. 2. Devise a Plan (DP) Choose a problem solving strategy/heuristic Give a representation Make a model Make a list Use equations Make a calculated guess Guess and check Look for patterns Make an assumption Go through the process Act it out Work backwards Before-after Change the problem Restate the problem Simplify the problem Solve part of the problem

21. Why devise a plan? A suitable heuristic can enable you to understand the problem better. It can also allow you to formulate hypotheses about the problem and test if your hypotheses work. It may help you spot patterns and important clues you would not have seen if you had just relied on the problem alone. This stage breaks down the problem further, allowing you more time to decide what to do.

22. Introduction to the Bar Model Visual problem solving heuristic Usually used to help with understanding of the problem by representing it in a pictorial form. Makes visible the part-whole thinking that is key to the learning of primary mathematics. Exposes students to informal algebra by promoting algebraic thinking before they are ready for formal algebra. Integrate Visual and Language systems

23. Bar Model When drawing a bar model, we need to take note of: 1) Size of the bars 2) Position of the bars

24. Part-Whole Model Used when adding or subtracting two or more numbers

25. Part-Whole Model E.g. What is the sum of 12 and 8? Understanding the problem Sum: The whole when two or more numbers are added together. ?

26. Part-Whole Model E.g. What is the sum of 12 and 8? Solve the problem 12 + 8 = 20 Devise plan: Draw a part-whole model Add the parts ?

27. Comparison Model Usually used when comparing two or more numbers Can be used to add two or more quantities 20 8

28. Words used for comparison More than Less than Longer than Shorter than Heavier than Lighter than Taller than Shorter than

29. Comparison Model E.g. Jolin has 20 sweets. Sam has 8 sweets. How many more sweets does Jolin have than Sam? Jolin Sam 20 8 Key information? What do we want to find?

30. Comparison Model E.g. Jolin has 20 sweets. Sam has 8 sweets. How many more sweets does Jolin have than Sam? 8 Jolin Sam 20 Devise a plan: Implement plan (Solve): Draw a comparison model 20 – 8 = 12 ? Jolin’s amount – Sam’s amount

31. Unitary model Used when the question involves equal parts. Each unit can represent a quantity more than 1. 77777

32. Unitary model A sum of money was shared among Annie, Benny and Charlie. Annie received of the money. Benny received of the money. Charlie received the remaining $250. How much was the sum of money shared? Annie BennyCharlie $250

33. Let’s do a recap What are the 3 types of bar models? When do we use it?

34. Back to the example problem At a party, there were twice as many girls as boys. Each girl was given 3 balloons, and each boy was given 4 balloons. The total number of balloons given to the children was 70. How many girls were there at the party? Which heuristic would you use?

35. Strategy: Draw a unitary comparison model At a party, there were twice as many girls as boys. Each girl was given 3 balloons, and each boy was given 4 balloons. The total number of balloons given to the children was 70. How many girls were there? How many girls were there at the party? 33 4 3 4 70 Girls Boys ? 3

36. 3. Implement the Plan (IP) At this stage, students decide what steps to take to solve the question based on the work they have done thus far.

37. 3. Implement the plan (IP) Strategy: Group the students into equal groups (based on bar model) Therefore, number of girls = 14. If 1 group  10 balloons, 70 ÷ 10 = 7 groups Girls  7 groups x 2 girls = 14 girls 1 group

38. 4. Check (CP) Strategy: Work backwards. 1 group  10 balloons If there are 14 girls, 14 ÷ 2 = 7 boys 14 x 3 = 42 balloons 7 x 4 = 28 balloons 42 + 28 = 70!

39. Strategy: Use a different method (Draw a picture + Make a table + look for a pattern) 4. Check (CP) GirlsBoysTotal PaxBalloonsPaxBalloons 22 x 3 = 611 x 4 = 46 + 4 = 10 44 x 3 = 1222 x 4 = 812 + 8 = 20 66 x 3 = 1833 x 4 = 1218 + 12 = 30 1414 x 3 = 4277 x 4 = 2842 + 28 = 70 Therefore, number of girls = 14.

40. Guided Practice 1 Megan is paid $4 per hour for working on weekdays. She is paid twice as much per hour for working on weekends. If Megan works 8 hours each day during the weekdays and 5 hours on Saturday, how much does she earn each week?

41. Possible Solution Megan is paid $4 per hour for working on weekdays. She is paid twice as much per hour for working on weekends. If Megan works 8 hours each day during the weekdays and 5 hours on Saturday, how much does she earn each week? UP WeekdaysWeekends Examples Monday, Tuesday, Wednesday, Thursday, Friday Amount ($)4 per hour Saturday, Sunday 4 x 2 = 8 per hour Working hours 8 hours each day5 hours on Saturday Week

42. Possible Solution Megan is paid $4 per hour for working on weekdays. She is paid twice as much per hour for working on weekends. If Megan works 8 hours each day during the weekdays and 5 hours on Saturday, how much does she earn each week? DP: Draw a unitary model (by hour) Weekdays Weekends $4 $8 Work 8 hours each day on weekdays Work 5 hours on Saturday (weekend)

43. Possible Solution Megan is paid $4 per hour for working on weekdays. She is paid twice as much per hour for working on weekends. If Megan works 8 hours each day during the weekdays and 5 hours on Saturday, how much does she earn each week? Weekdays Weekends $4 $8 IP: Break into smaller cases Step 1 - Case 1: Solve for weekdays first 1 day  8 hours 5 days  8 x 5 = 40 hours Pay for 5 days  40 x $4 = $160 Work 8 hours each day on weekdays Work 5 hours on Saturday (weekend)

44. Possible Solution Megan is paid $4 per hour for working on weekdays. She is paid twice as much per hour for working on weekends. If Megan works 8 hours each day during the weekdays and 5 hours on Saturday, how much does she earn each week? Weekdays Weekends $4 $8 IP: Break into smaller cases Step 2 - Case 2: Solve for Saturday 1 hour  $8 5 hours  $8 x 5 = $40 Step 3 – Add the amount for the cases $160 + $40 = $200 Work 8 hours each day on weekdays Work 5 hours on Saturday (weekend)

45. Branching Method An alternative visual problem solving heuristic Use mostly for fraction questions with remainders Usually taught after the bar model methods Provides a clearer visual representation meaning for multi-level fraction problems as compared to bar models

46. Example Problem Roy baked some muffins. of them were chocolate muffins and the rest were blueberry muffins. After giving away 12 blueberry muffins, Roy had 9 blueberry muffins left. How many muffins did he bake at first?

47. Clues: There are chocolate muffins and blueberry muffins. 5/8 of the muffins were chocolate muffins. After giving away 12 blueberry muffins, Roy had 9 blueberry muffins left. Find: Number of muffins he baked at first. Roy baked some muffins. of them were chocolate muffins and the rest were blueberry muffins. After giving away 12 blueberry muffins, Roy had 9 blueberry muffins left. How many muffins did he bake at first?

48. 2. Devise a Plan + 3. Solve Strategy: Draw a bar model + Use equations We can see that 3 out of 8 of the muffins are blueberry muffins which is made up of 12 that were given away as well as 9 that were left. Hence, 3 units  12 + 9 = 21 1 unit  21 ÷ 3 = 7 8 units  7 X 8 = 56

49. 2. Devise a Plan + 3. Solve Strategy: Draw a branching model + Use equations So if 5/8 are chocolate muffins, it means the rest are blueberry which is 3/8. (1-5/8 = 3/8) These blueberry muffins then are made up of the 12 that were given away and 9 that were left. (12 + 9 = 21) Hence, 3 units  12 + 9 = 21 1 unit  21 ÷ 3 = 7 (Total)8 units  7 X 8 = 56 In one whole (total), there are 8 out of 8 units.

50. 4. Check Strategy: Work backwards If there are 56 muffins in total and 21 blueberry muffins, it means there are 56 – 21 = 35 chocolate muffins Since 1 unit  7 chocolate muffins 5 units  7 X 5 = 35 chocolate muffins.

51. GUIDED PRACTICE 2 Sarah had some money. She spent of it on a Prada bag. After spending $225 of the remainder on a pair of shoes, she had $275 left. How much money did she have at first?

52. Strategy: Draw a branching model + Use equations Remainder: $225 + $275 = $500 Hence, 2 units  $500 1 unit  $500 ÷ 2 = $250 (Total)5 units  $250 X 5 = $1250

53. Now’s your turn!

54. Practice Question A teacher has 100 happy face stickers in three colours: red, yellow and blue. of the stickers are red. The number of yellow stickers is twice the number of red stickers. a)What fraction of the stickers is blue? b) How many of the stickers are blue?

56. Siti.aishah@das.org.sg rebecca@das.org.sg

57. References Ministry of Education (2007). Primary Mathematics Teaching and Learning Syllabus. Singapore. Polya, G. (2004). How to solve it: A new aspect of mathematical method (Expanded Princeton Science Library Edition). United States of America: Princeton University Press. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition and sense making in mathematics. In D. Grouws (Ed.). Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). New York: Macmillan. Yeap, B. H. (2014). Teaching to Mastery Mathematics - Bar Modeling: A Problem- solving Tool. Singapore: Marshall Cavendish Education.