Transitioning to the Common Core State Standards – Mathematics

1. Transitioning to the Common Core State Standards – Mathematics Pam Hutchison pam.ucdmp@gmail.com

2. Please fill in the lines: • First Name ________Last Name__________ • Primary Email______Alternate Email_______ • . • . • . • . • School____________District______________

3. AGENDA • Fractions • Fractions on a Number Line • Naming and Locating • Fractions, Whole Numbers and Mixed Numbers • Comparing • Equivalent • Assessing Fractions • Stoplighting the Standards

4. Making Math Visible

5. Spending Spree • David spent of his money on a game. Then he spent of his remaining money on a book. If he has $20 left, how much money did he have at first?

6. Fractions

7. Fraction Concepts • Four children share six brownies so that each child receives a fair share. How many brownies (or parts of brownies) will each child receive?

8. Fraction Concepts • Six children share four brownies so that each child receives a fair share. What portion of each brownie will each child receive?

9. Fractions • NF.1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

10. Fraction Concepts

11. Illustrative Mathematics • The importance of the unit or whole • Naming the whole for the fraction • Implication for instruction

12. So what is the definition of a fraction?

13. Definition of Fraction: • Start with a unit, 1, and split it into ___ equal pieces. • Each piece represent 1/___ of the unit. • When we name the fraction__/__, we are talking about ___ of those 1/___ size pieces .

14. Fraction Concepts

15. Fractions Task - NC Mr. Rogers started building a deck on the back of his house. So far, he finished ¼ of the deck. The fraction of the completed deck is below. • Draw 2 pictures of what the completed deck might look like. Use numbers and words to explain how you created your picture.

16. Fraction Task - NC Martha is making a scarf for her sister. Each day she knits 1/6 of a scarf. • What fraction of the scarf will be complete after three days? • What fraction of the scarf will be complete after six days? • How can you use a number line to prove that your answers are correct?

17. Fraction Concepts • What fraction of the rectangle is shaded? How might you draw the rectangle in another way but with the same fraction shaded?

18. Fractions on the Number Line

19. Fractions NF.2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. • a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

20. How many pieces are in the unit? • Are all the pieces equal? • So each piece represents ● 1 0

21. How far (how many pieces) is the point from 0? • We name that point……. ● 1 0 1 5

22. How many pieces are in the unit? • Are all the pieces equal? • So the denominator is • And each piece represents . ● 0 1 7

23. How far is the point from 0? • So the numerator is • And the name of the point is …… ● 0 1

24. Fractions NF.2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. • b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

25. How many pieces are in the unit? • Are all the pieces equal? • So each piece represents ● 0 1

26. How far is the point from 0? • How many pieces from 0? • So the name of the point is …. ● 0 1

27. Definition of Fraction: • When we name the point , we’re talking about a distance from 0 of ___ of those ___ pieces. 4

28. ● • How many pieces are in the unit? • Are all the pieces equal? • So the denominator is and each piece represents 1 0 5

29. How far is the point from 0? • So the numerator is and the fraction represented is ● 1 0 3

30. The denominator is So each piece represents • The numerator is And the fraction is ● 1 0 6 5

31. Academic Vocabulary • What is the meaning of denominator? • What about numerator? • Definitions should be more than a location – the denominator is the bottom number • They should be what the denominator is – the number of equal parts in one unit

32. Student Talk Strategy: Rally Coach • Partner A: name the point and explain • Partner B: verify and “coach” if needed • Tip, Tip, Teach Switchroles • Partner B: name the point and explain • Partner A: verify and “coach” if needed • Tip, Tip, Teach

33. Explains – Key Phrases • Here is the unit. (SHOW) • The unit is split in ___ equal pieces • Each piece represents • The distance from 0 to the point is ___ of those pieces • The name of the point is .

34. Partner Activity 1

35. 2 Definition of Fraction: 7 • Start with a unit, 1, • Split it into __ equal pieces. • Each piece represents of the unit • The point is __ of those pieces from 0 • So this point represents ● 1 0 7 2

36. 6 Definition of Fraction: 8 • Start with a unit, 1, • Split it into __ equal pieces. • Each piece represents of the unit • The pointa is __ of those pieces from 0 • So this point represents ● 1 0 8 6

37. Partner Activity 1, cont. Partner A 5. 6. 7. Partner B 5. 6. 7.

38. | | | | | | | | | • The denominator is ……. • The numerator is ……… • Another way to name this point? 0 1 2 3 3 1

39. | | | | | | | | | • The denominator is …….. • The numerator is ……… • Another way to name this point? 0 1 2 6 3 2

40. | | | | | | | | | • The denominator is …… • The numerator is ……… • Another way to name this point? 0 1 2 5 3 2 1 3

41. | | | | | | | | | • The denominator is ….. • The numerator is ……… • Another way to name this point? 0 1 2 7 3 1 2 3

42. | | | | | | | | | • Suppose the line was shaded to 5. • How many parts would be shaded? • So the numerator would be ……… 0 1 2 15 3

43. | | | | | | | | | • Suppose the line was shaded to 10. • How many parts would be shaded? • So the numerator would be ……… 0 1 2 30 3

44. Rally Coach • Partner A goes first • Name the point as a fraction and as a mixed number. Explain your thinking • Partner B: coach SWITCH • Partner B goes • Name the point as a fraction and as a mixed number. Explain your thinking • Partner A: coach Page 93-94

45. Rally Coach Part 2 • Partner B goes first • Locate the point on the number line • Rename the point in a 2nd way (fraction or mixed number) • Explain your thinking • Partner A: coach SWITCHROLES

46. Rally Coach Partner B 6. 7. 8. Partner A 6. 7. 8.

47. Connect to traditional • Change to a fraction. • How could you have students develop a procedure for doing this without telling them “multiply the whole number by the denominator, then add the numerator”?

48. Connect to traditional • Change to a mixed number. • Again, how could you do this without just telling students to divide?

49. Student Thinking Video Clips 1 – David (5th Grade) • Two clips • First clip – 3 weeks after a conceptual lesson on mixed numbers and improper fractions • Second clip – 3.5 weeks after a procedural lesson on mixed numbers and improper fractions

50. Student Thinking Video Clips 2 – Background • Exemplary teacher because of the way she normally engages her students in reasoning mathematically • Asked to teach a lesson from a state-adopted textbook in which the focus is entirely procedural. • Lesson was videotaped; then several students were interviewed and videotaped solving problems.