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Exploring Division of Fractions: Models and Computation

This session at the College and Career-Readiness Conference focuses on interpreting and computing quotients of fractions through the use of models and equations. Participants will explore various models and learn how to compute and interpret quotients.

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Exploring Division of Fractions: Models and Computation

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  1. MIDDLE SCHOOL MATHEMATICS Some standards are more difficult to interpret and understand in mathematics College and Career-Readiness Conference Summer 2014

  2. Session Protocol Based on RIGOR: • Reduce Side Chatter • Involve Yourself in the Process • Give Your Thoughts and Ideas • Open Your Mind to How You Can Change Instruction • Remember to Silence Electronic Devices

  3. 6.NS.A.1 Cluster A. Apply and extend previous understandings of multiplication and division to divide fractions by fractions. Standard 1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fractions models and equations to represent the problem.

  4. INVERT AND MULTIPLY

  5. TODAY’S OUTCOMES Participants will: Briefly review the instructional shift, COHERENCE, and make connections between division of fractions by fractions and other content from elementary school and middle school standards. Explore a variety of models that each show what happens mathematically when values are divided by a fraction. Compute quotients of fractions divided by fractions, and interpret the quotients.

  6. OUTCOME #1 Participants will: 1. Review the instructional shift of COHERENCE, and make connections between division of fractions by fractions and other content from elementary school and middle school standards.

  7. COHERENCE A purposeful placement of standards to create logical sequences of content topics that bridge across the grades, as well as across standards within each grade.

  8. https://www.turnonccmath.net

  9. 1.OA.4: Understand subtraction as the unknown-addend problem. For example, find 10 – 8 by finding the number that makes 10 when added to 8. inverse operations 10 – 8 = x 8 + x = 10

  10. 3.OA.6: Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. inverse operations 32 ÷ 8 = a 8 xa = 32

  11. 4.NF.3b: Decompose a fraction into a sum of fractions with the same denominator… for example: 4.NF.4a: Understand a fraction a/b as a multiple of 1/b… for example:

  12. 5.NF.3b: Interpret a fraction as division of the numerator by the denominator. Students use models for “equal sharing” to explain their understanding.

  13. Grade 5 and Grade 6

  14. OUTCOME #2 Participants will: 2. Explore a variety of models that each show what happens mathematically when values are divided by a fraction.

  15. MODELS • Area Model • Number line model • Tape diagram model • Common denominator model

  16. 5.NF.B.7b: Interpret division of a whole number by a unit fraction, and compute such quotients

  17. 8 AREA MODEL – 5.NF.B.7b How many parts can be partitioned from 2 “wholes”? How many times does a part fit into 2 “wholes”? ? ? 2 eight

  18. 8 TAPE DIAGRAM – 5.NF.B.7b • How many parts can be partitioned from 2 “wholes”? 2 eight ?

  19. 8 NUMBER LINE – 5.NF.B.7b • How many parts can be partitioned from 2 “wholes”? one two three four five six seven eight ? eight

  20. COMMON DENOMINATOR – 5.NF.B.7b

  21. ExtensionAREA MODEL – 5.NF.B.7b ? remainder 1

  22. ExtensionAREA MODEL – 5.NF.B.7b remainder remainder ? ? 1

  23. ExtensionAREA MODEL – 5.NF.B.7b The remainder equals of one part. So the answer is 1 .

  24. ExtensionNUMBER LINE – 5.NF.B.7b remainder

  25. ExtensionTAPE DIAGRAM – 5.NF.B.7b 1 remainder remainder remainder

  26. COMMON DENOMINATOR – 5.NF.B.7b

  27. 5.NF.B.7a: Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.

  28. AREA MODEL – 5.NF.B.7a 1

  29. AREA MODEL – 5.NF.B.7a The quotient must be a fraction!

  30. NUMBER LINE – 5.NF.B.7a

  31. TAPE DIAGRAM – 5.NF.B.7a 1 1 1

  32. COMMON DENOMINATOR – 5.NF.B.7a

  33. ExtensionAREA MODEL – 5.NF.B.7a 1

  34. ExtensionNUMBER LINE – 5.NF.B.7a 15

  35. ExtensionTAPE DIAGRAM – 5.NF.B.7a 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

  36. ExtensionCOMMON DENOMINATOR – 5.NF.B.7a

  37. OUTCOME #3 Participants will: 3. Compute quotients of fractions divided by fractions, and interpret the quotients.

  38. 6.NS.A.1: Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fractions models and equations to represent the problem.

  39. 6.NS.A.1 - First Example When the dividend is greater than the divisor…

  40. 2 AREA MODEL – 6.NS.A.1 Two parts two one

  41. 2 NUMBER LINE – 6.NS.A.1 one two

  42. 2 TAPE DIAGRAM – 6.NS.A.1 one two

  43. COMMON DENOMINATOR – 6.NS.A.1

  44. 6.NS.A.1 - Second Example: When the dividend is greater than the divisor, with a remainder…

  45. AREA MODEL – 6.NS.A.1

  46. AREA MODEL – 6.NS.A.1

  47. AREA MODEL – 6.NS.A.1

  48. TAPE DIAGRAM – 6.NS.A.1 one two three four one two three

  49. NUMBER LINE – 6.NS.A.1 one one

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