1. Welcome Highland Ranch and Midland! December 10, 2014 Facilitator: Andrea Barraugh Please make a nametag – Large, Dark Letters

2. Introducing Division while Laying the Groundwork for Fractions “Sharing Cookies”

3. Table Talk How are fractions and division related?

5. Focus Practices 3. Construct viable arguments and critique the reasoning of others. 5. Model with mathematics 6. Attend to precision. 7. Look for and make use of structure.

6. Sharing Cookies LAUNCH

16. What if there is one more person at the door? (12 cookies with 13 people) How many cookies will each person get? More or less than 1 whole cookie? More or less than ½ cookie?

20. Sequencing the Social Interactions 1.Independent 2.Partner Exploration 3.Table Sharing 4.Whole Group Sharing

21. Exploration Choices Share paper cookies with 4 children.  6 cookies with 4 children  5 cookies with 4 children  4 cookies with 4 children  3 cookies with 4 children  2 cookies with 4 children  1 cookies with 4 children Share paper cookies using these scenarios:  4 cookies with 4 children  5 cookies with 2 children  2 cookies with 4 children  7 cookies with 6 children  4 cookies with 5 children  13 cookies with 10 children  12 cookies with 20 children Choose a sequence of sharing situations to explore. Model each situation using the cookies provided or with another visual model. What observations can you make about equal shares?

22. Communication Choices Cookie Sharing Scenario Communication Choices Cookie Sharing Conjecture Problem: Model (glued and labeled paper cookies): Explanation: Answer: Conjecture: If ______________ then _____________________________. 3 Supporting Examples with Visual Models

23. Math Summary What did you learn about the relationship between division and fractions?

24. Professional Reflections In which practices were you engaged during this “sharing” experience? How might you use this experience with your students?

25. 3 rd Grade Highlighted CC “Division as Sharing” Standard Interpret quotient of 56 ÷ 8: – 56 cookies shared by 8 people – how many cookies each? (number of objects) – 56 cookies, each person gets 8 cookies – how many people? (number of shares)

26. 3 rd Grade CC Fraction Standards Understand Fractions: Simple Equivalent Fractions: 1/2 = 2/42/3 = 4/6 Whole Numbers as Fractions: 3/3 = 1 4/1 = 4 Compare Two Fractions with Same Numerator or Same Denominator: 2/3 > 1/3 2/3 > 2/4 1/4 3/4 1/42/4 3/4 4/4 0/4 0 1 1/4

27. Highlighted 4 th Grade Fraction Standard CCSS.MATH.CONTENT.4.NF.B.3.A Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

28. Highlighted 5 th Grade CC Fraction Standard Apply and extend previous understandings of multiplication and division. CCSS.MATH.CONTENT.5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. – For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. – If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

29. Making Sense of Fractions Changing the Whole Modeling Operations

30. Fraction Operations Standards What are the fraction operations standards for your grade level? How do they progress across 3-5?

31. Pattern Blocks Investigate the pattern blocks by looking for mathematical relationships.

32. 32 Fraction Exploration Changing the Whole = 1 = 1/2 = 1/6 = 1/3 = 1 = ___

33. 33 Challenges = 1 = = 1 =

34. Modeling Operations with Fractions

35. Fraction Problems (Solve each problem with pencil and paper.) 1/3 + 1/3 = 2/3 + 1/2 2/3 – 1/2 1/2 x 1/3 1/2 ÷ 1/3 35

36. How can you model the same problems with pattern blocks? 1/3 + 1/3 = 2/3 + 1/2 2/3 – 1/2 1/2 x 1/3 1/2 ÷ 1/3 36

37. Multiplication as Grouping Connecting to Whole Numbers 3 x 4 = 3 groups of 4 stars = 4 + 4 + 4 = 12

38. Using Pattern Blocks and Language to Connect Whole Number and Fraction Multiplication 3 x 1/6 = 5 x ½ = 1/3 x 6 = 3 x 2/3 =

39. Connecting to Grouping Whole Numbers: Multiplying a Fraction by a Fraction 1/2 x 2/3 1/2 groups of 2/3 1/2 of 2/3

40. Try These... ½ x ½ =2/3 x 1/2 = ½ x 1/3 =3/2 x 2/3 = ½ x 2/3 = 1/3 x ½ =

41. Division with Fractions How can interpretation, language, and modeling support our understanding? 20 ÷ 4 =  Twenty divided into 4 groups  How many in each group?  How many groups of 4 are in 20?  How many 4s are in 20? 2 ÷ ½ = Two divided into groups of ½, how many ½s? How many 1/2s are in two? How many are in ? 1/2 ÷ 1/3 = How many one thirds are in one half?

42. More Opportunities to Grapple Practice 5/6 ÷ 2/6 2/3 ÷ 1/6 1/6 ÷ 3/6 1 2/3 ÷ 1/2 Challenges 1/6 + 1/2 =5/6 + 1/3 = 5/6 – 1/2 =5/6 – 2/3 = 1/2 x 5/6 =5/6 x 1/2 = 1/3 ÷ 2/6 = 1 ÷ 1/6 = 42

43. Processing the Experience What practices were you engaged with as you explored these mathematical ideas? What new insights do you have into operations with fractions? What new insights do you have into teaching fractions?

44. Multiplication Statement Whole Numbers (Always true, Never True, Sometimes True) Fractions (Always true, Never True, Sometimes True) 1. Multiplication is the same as repeated addition. 2. Times means “groups of.” 3. A multiplication problem can be shown as a rectangle. 4. You can reverse the order of the factors and the product stays the same. 5.You can break numbers apart to make multiplying easier. 6. When you multiply two numbers, the product is larger than the factors.

45. Division Statement Whole Numbers (Always true, Never True, Sometimes True) Fractions (Always true, Never True, Sometimes True) 1. You can solve a division problem by subtracting. 2. To divide two numbers, a ÷ b, you can think, “How many bs are in a?” 3. You can check a division problem by multiplying. 4. The division sign (÷) means “into groups of.” 5. The quotient tells “how many groups” there are. 6. You can break the dividend apart to making dividing easier.

46. Division Statement Whole Numbers (Always true, Never True, Sometimes True) Fractions (Always true, Never True, Sometimes True) 7. Remainders can be represented as whole numbers or fractions. 8. If you divide a number by itself, the answer is one. 9. If you divide a number by one, the answer is the number itself. 10. You can reverse the order of the dividend and the divisor, and the quotient stays the same.