1. June 2012 Understanding is a measure of the quality and quantity of connections that an idea has with existing ideas. (VandeWalle, Pg. 23)

2. 1. Conceptual understanding 2. Procedural fluency 3. Strategic competence 4. Adaptive reasoning 5. Productive disposition Source: Common Core (Pg. 6)

3. Definition: Number sense is a good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms. (Van de Walle, Pg. 119)

4. According to a 2007 Trends in International Mathematics and Science Study, Singapore students are among the best in the world in math achievement. The Singapore model drawing approach bridges the gap between the concrete and abstract models we tend to jump to in the US. Model drawing reinforces students visualization and understanding of math processes. Model drawing can be used effectively to solve 80 percent of problems in all texts.

5. Computation is about students comprehending what they are doing, not following a set of rules. Students need to understand both what to do and why. Students will have a variety of strategies to solve problems. Changes students attitudes toward math and problem solving. Language based learning- think alouds and math talks Can be used as a supplement to an adopted curriculum Singapore Math Video

6. Instruction begins at the concrete level with manipulatives to build understanding of basic concepts and skills. (begin with proportional manipulatives) Then students are introduced to the pictorial stage: model drawing. Students are not introduced to formulaic or algorithmic procedures, the abstract, until they have mastered model-drawing.

8. Discuss. Catch thinking on chart paper. Write a summary paragraph.

9. 1) Teacher modeling and thinking aloud about the strategy 2) Students practice with the teacher 3) Students practice in small groups 4) Students practice in partners 5) Independent practice

10. Purpose: Modeling, communicating, promoting a more efficient strategy, promoting reasoning, moving to a more sophisticated level of thinking. Im thinking… Im wondering… What are you thinking? How did you figure that out? Is there another way? Why did you choose this way? How do you know this answer is correct? What would happen if? Model clear, explicit language about concepts Mathematical thinking and language promote more understanding than memorization or rules

11. 346+475= First, Im going to add the hundreds. That means 300+400=700. Now I will add the tens, four tens (40) plus seven tens (70) equals eleven tens. I can make 110. So now I have 700+100+10. Now I will add the ones and 6+5 makes 11. This is one ten and one 1 so I have 700+100+10+10+1. My answer is 821. Place Value Talk is critical!

12. Purpose: Unstick someone, get help, clarify, promote deeper thinking, make connections

13. Purpose: Promote productive math conversations Example: Practicing questions in multiple- choice format Step 1: Solve individually. Write down your answer. Step 2: Compare. Same or different? Step 3: Explain why you chose that answer.

14. Purpose: Use words that proficient mathematicians would use, make connections Example: End of Year Jeopardy Review Game. 500 point question: What is addition?

15. Look at their work. Do the model, the picture, and the equation match the question and each other? Read or listen to their explanation. Ask a math question. Seek professional help – Ask a student expert, the teacher, or other adult.

16. Place Value Strips Place Value Disks Place Value Chart Number-bond cards Part-whole cards Decimal Tiles Decimal Strips Gratiot Isabella ISD Maniplatives Link

17. Begin with no regrouping Sequence 1. Number bonds 2. Decomposing numbers 3. Left-to-right addition 4. Place value disks and charts 5. Vertical addition 6. Traditional addition

19. Spatial Relationships pattered arrangements One and Two More, One and Two Less counting on and counting back 7 is 1 more than 6 and it is 2 less than 9 Anchoring Numbers to 5 and 10 using 5 and 10 to build on and break from is foundational for working with facts Part-Part-Whole Relationships understand that a number is can be made of 2 or more parts

20. A Pre-Place Value Relationship with 10 11 through 20, think 10 and some more Extending More and Less Relationships i.e. 17 is one less than 18 like 7 is one less than 8 Double and Near-Double Relationships special cases of the part-part-whole construct use pictures

21. Build Up and Down through 10 Break Apart to Find an Unknown Fact

22. Estimation and Measurement More or less than _______? Closer to _____ or _____? About _______. More Connections Add a Unit to Your Number Is It Reasonable? Graphs Make bar graphs and pictographs

23. Big Ideas: 1. Number relationships provide the foundation for strategies that help children remember the basic facts. (i.e. relate to 5, 10, and doubles…) 2. Think addition is the most powerful way to think of subtraction facts. 3. All of the facts are conceptually related. You can figure out new or unknown facts from those you already know. 4. What is mastery? 3 seconds or less

24. 40 7 10 1 11 1111

25. 33+56= Decompose each number by place value. Put the tens together and one ones together. (30+50) + (3+6) 80 + 9 = 89 305063

26. 35 +26 10 1 11 1111 111 1

27. 35 +26 10 1 11 1111 111 1

28. 35 +26 6 1 10 1

29. 36 +28 50 +14 64 Add from left to right.

30. 38 +85 123

31. Begin with no regrouping Sequence 1. Number Bonds 2. Place Value Disks and Charts 3. Traditional Subtraction

33. 86-8 10 11111 1

34. 86-8 10 11111 1 1111 1111 1 1

35. 86-8 70 + 8 = 78 86-8=78 10 11111 1 1111 1111 1 1

36. 54 - 28 26

37. Before memorizing multiplication facts, students must first understand the concept of multiplication----they must know it is repeated addition with special attention to place value Stages 1. Number bonds 2. Place value disks and charts 3. The distributive property 4. Area model 5. Traditional multiplication

38. 16 4 16 16 or 4 4 4 16 16 2 8 1 16 Vocab: Factor x Factor=Product

39. 141x3 100 10 1 1 1

40. 141x3 100 10 1 1 1 100

41. 141x3 4 2 3 141x 3=423 100 10 1 1 1 100

42. 45x3 (40x3) + (5x3) 120 + 15 = 135

43. 6x33 30 3 6 180 18 6x33=180+18 180+18=198

44. 15x26 20 6 10 200 60 5 100 30 15x26=(200+100)+(60+30) 300 + 90 =390 15x26=390

46. Begin by introducing division as repeated subtraction Use number bonds to demonstrate the inverse relationship of multiplication and division Sequence for teaching 1. Number bonds 2. Place value disks and charts 3. The distributive property 4. Partial quotient division 5. Traditional long division 6. Short division

47. 9 27 ? What should the second factor be? Vocab: Dividend ÷ Divisor=Quotient

48. 64÷5 Step 1 Build with disks 10 1111

49. 64÷5 Step 2: The divisor tells us how many groups we need: Draw 5 rows 1 2 3 4 5 10 1111

50. 64÷5 Step 3: Begin with the tens. Do we have enough tens to put one in every row? Yes 1 2 3 4 5 10 1111

51. 64÷5 Step 4: If there are any large disks left, trade them for an equivalent value of smaller disks 1 2 3 4 5 10 1111 11111 11111

52. 64÷5 Step 5:Divide the ones equally among the groups 1 2 3 4 5 1 2 10+2=12 64÷5=12 r 4 10 1111 1 1 1 1 1 1 1 11 1

53. 345÷3= 345=300+40+5 unfriendly 345=300+30+15 (300÷3) + (30÷3) + (15÷3)= 100 + 10 + 5 = 115

54. 4 504 -400 100 104 -100 25 4 - 4 1 0 100+25+1=126

55. 157 3 472 -3 17 -15 22 -21 1

56. 169 2 3 4 676

57. Drawing simple visual models to represent word problems. Steps 1. Read the entire problem. 2. Rewrite the question in sentence form, leaving a space for the answer. 3. Determine who and/or what is involved in the problem. 4. Draw unit bar(s). 5. Chunk the problem, adjust the unit bars, and fill in the question marks. 6. Correctly compute and solve the problem. 7. Write the answer in the sentence, and make sure the answer makes sense.

58. Janet picked 3 daisies and 2 sunflowers from her garden. How many total flowers did Janet pick from her garden? Work problem on your Part-Part Whole Model Handout

59. Scrooge had 17 pennies in his piggy bank. He also had 8 dimes. How many total coins did Scrooge have in his piggy bank? Work problem on your Part-Part Whole Model Handout

60. Liz earned $500 as her weekly pay. She paid $413 to cover her bills for the week. How much money did she have left to spend? Work problem on your Part-Part Whole Model Handout

61. A movie theater has 1,250 seats. If 756 people attended todays matinee, how many seats will there be left in the movie theater? Work problem on your Part-Part Whole Model Handout

62. Joe has $45. Tom has $28. How much more money does Joe have than Tom? Work problem on your Comparison Model Handout

63. Thomas makes $35 more dollars a week than Bobby. Bobby makes $82 dollars a week. How much money does Thomas make? Work problem on your Comparison Model Handout

64. When planting her flower garden, Tanya placed her flowers in 3 rows. She put 8 flowers in each row. How many flowers did she plant in her garden? Work problem on your Part-Part Whole Model Handout

65. Ann saved $45 dollars in nine weeks. She saved the same amount each week. How much did she save each week? Work problem on your Part-Part Whole Model Handout

66. The movie theater has 12 rows of chairs with 36 chairs in each row. How many chairs are in the movie theater? If there are 250 kids in our school, can we all go to the movie at the same time? Work problem on blank piece of paper.

67. There are 8 white flowers. There are 3 times as many red flowers. How many red flowers are there? Work problem on your Comparison Model Handout

68. There are 36 red flowers. There are 4 times as many red flowers as white flowers. How many white flowers are there? Work problem on your Comparison Model Handout

69. Joe buys 27 toys. 2/3 of them are trucks. How many of them are not trucks. Solve on a scrap sheet of paper.

70. Susie spent 2/5 of her money of a purse. The purse cost $15. How much money did she have before she bought the purse? Solve on a scrap sheet of paper.

71. From a 5 th grade Singapore text. Mrs. Chen made some tarts. She sold 3/5 of them in the morning and ¼ of the remainder in the afternoon. Is she sold 200 more tarts in the morning than in the afternoon, how many tarts did she make? Solve on a scrap sheet of paper.

72. In a class of 35 students, the ratio of girls to boys is 3:4. How many more boys than girls are there? Solve on scrap paper.

73. There are 30 dogs, cats, and hamsters altogether at a pet store. The ratio of dogs to cats to hamsters is 5:3:2. How many dogs and cats are there at the pet store? Solve on scrap paper.

74. There are 30 dogs, cats, and hamsters altogether at a pet store. The ratio of dogs to cats to hamsters is 5:3:2. How many dogs and cats are there at the pet store? Solve on scrap paper.

75. There were 250 people at a concert. Of these, 40% were children and the rest were adults. How many adults were at the concert? Solve on scrap paper.

76. Of the 60 students in the third grade, we know that 60% are girls. We also know that 50% of the girls have blue eyes and 25% of the boys have blue eyes. How many of the students in third grade have blue eyes? Solve on scrap paper.

77. We know that ¾ of a sum of money is $72. What is the sum of money? Solve on scrap paper.

78. Susie bought 26 treats for her 2 cats, Joe and Moe. She gave Joe 4 more treats than Moe. How many treats did each cat receive? Solve on scrap paper.

79. Purpose: Concept development, communicating ideas Activity: Work with a partner or in a small group. Choose one of the problems that we wrapped a story around. 1- Build a Model 2- Draw a Picture 3- Write an Equation 4- Write your answer in a complete sentence. 5- Explain.

80. Purpose: Provide context, aid concept development, make real-world connections Activity: Article: Round-Robin Story TellingRound-Robin Story Telling

81. Author(s): Joseph Martinez and Nancy Martinez Source: Instructor (1990). 109.6 (Mar. 2000): p70. Round-robin storytelling works well as a math activity with elementary-school students. Provide each small group with a story beginning such as: Jon and Michelle make paper hats. Their standard hat is paper with a tissue tassel. It costs $.30 to make and sells for $1.25. Within each group, students add to the story For example: They also make a fancy hat that sells for $2.50. The last student in the group wraps up the story and poses the problem: Jon and Michelle sell 60 standard hats, and 40 fancy hats. How much profit will they make? Each group works to solve its problem. The groups then swap stories.

82. Purpose: Assure progress in terms of academic achievement Activity:

83. Why Before How: Singapore Math Computation Strategies: Jana Hazekamp Building Number Sense: Catherine Jones Kuhns The Singapore Model Method for Learning Mathematics: Mastery of Education Singapore The Parent Connection for Singapore Math: Sandra Chen 8-Step Model Drawing: Singapores Best Problem-Solving MATH Strategies: Bob Hogan & Char Forsten Step-by-Step Model Drawing: Char Forsten