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Transitioning to the Common Core State Standards – Mathematics

This workshop will focus on fraction concepts and understanding fractions on a number line. Participants will learn strategies for naming, locating, and comparing fractions, as well as techniques for adding, subtracting, multiplying, and dividing fractions. The workshop will also address simplified fractions and the application of Common Core State Standards in the classroom.

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Transitioning to the Common Core State Standards – Mathematics

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

  2. Please fill in these 3 lines: • First Name ________Last Name__________ • Primary Email______Alternate Email_______ • . • . • . • . • School____________District______________

  3. AGENDA • Fractions and Fractions on a Number Line • Naming/Locating Fractions • Whole Numbers, Mixed Numbers and Fractions • Comparing Fractions • Equivalent Fractions/Simplifying • Adding and Subtracting Fractions • Multiplying Fractions • Dividing Fractions • Stoplighting the CCSS

  4. 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?

  5. Fractions

  6. Fractions • 3.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.

  7. Fraction Concepts

  8. So what is the definition of a fraction?

  9. 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 .

  10. Fraction Concepts

  11. Fractions on a Number Line

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

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

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

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

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

  17. The denominator is so each piece represents • The numerator is and the fraction represented is ● 1 0 5 3

  18. 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

  19. 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

  20. 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 .

  21. Partner Activity 1

  22. 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

  23. 6 Definition of Fraction: 8 • 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 8 6

  24. Partner Activity 1, cont. Partner A 5A. 6A. Partner B 5B. 6B.

  25. | | | | | | | | | • The denominator is ……. • The numerator is ……… • Another way to name this point? 0 1 2 3 3 1 Page 83

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

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

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

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

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

  31. 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

  32. 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

  33. Rally Coach Partner B 6. 7. Partner A 6. 7.

  34. 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”?

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

  36. 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

  37. 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.

  38. Student Thinking Video Clips 2 – Background, cont. • Five weeks later, the teacher taught the content again, only this time approaching it her way, and again we assessed and videotaped children.

  39. Student Thinking Video Clips 2 – Rachel • First clip – After the procedural lesson on mixed numbers and improper fractions • Second clip – 5 weeks later after a conceptual lesson on mixed numbers and improper fractions

  40. Classroom Connections • Looking back at the 2 students we saw interviewed, what are the implications for instruction?

  41. Research • Students who learn rules before they learn concepts tend to score significantly lower than do students who learn concepts first • Initial rote learning of a concept can create interference to later meaningful learning

  42. Discuss at Your Tables • How is this different from the way your book currently teaches fractions? • How does it support all students in deepening their understanding of fractions?

  43. Compare Fractions Using Sense Making

  44. Comparing Fractions A. B.

  45. Comparing Fractions B. A. Common Numerator

  46. Comparing Fractions A. B. Common Numerator

  47. Comparing Fractions B. A. Hidden Common Numerator

  48. Comparing Fractions A. B. Hidden Common Numerator

  49. Benchmark Fractions | | | 0 ½ 1 • How can you tell if a fraction is: • Close to 0? • Close to but less than ½? • Close to but more than ½? • Close to 1?

  50. Comparing Fractions B. A. A. B.

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