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MCTM 2016 (Final)

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MCTM 2016 (Final)

  1. 1. Turning Down the Volume Fractions Dr. Patrick Sullivan Missouri State University patricksullivan@missouristate.edu https://goo.gl/wKGEWS
  2. 2. 2 0.2 200 2 3
  3. 3. 2 ones 2 tenths 2 hundreds2 thirds
  4. 4. Goals • Establish a clear connection between operations with whole numbers and fractions • Engage in contextual problems • Reason/sense-making of operations involving fractions • Reflect on “your” thinking as you do the problems • Reflect on important language
  5. 5. Mathematics Teaching Practices • Establish mathematics goals to focus learning. • Implement tasks that promote reasoning and problem solving. • Use and connect mathematical representations. • Facilitate meaningful mathematical discourse. • Pose purposeful questions. • Build procedural fluency from conceptual understanding. • Support productive struggle in learning mathematics. • Elicit and use evidence of student thinking.
  6. 6. Volume Lowering Strategies • Strategy #1: Notation/Comparison • Strategy #2: Context • Strategy #3: Connect to Whole Number Problems
  7. 7. Tensions • “Their” models vs. “Our” models • Concrete vs. Abstract • Notation and Language • Fractions as part-whole, scalar, operator, ratio
  8. 8. 2 3
  9. 9. How do we “interpret” 2 3 ? • “2 over 3” • ”2 thirds” • 2 “copies” or “groups of” 1 3 • “multiply by 2 and divided by 3” • ”2 hits in 3 total at-bats” • ”2 parts out of 3 total parts”
  10. 10. How do we “write” it? • 2 thirds • 2 * 1 3 • 2 1 3 • 2 3
  11. 11. How do we “represent” it?
  12. 12. Strategy #1: Notation Which fraction is greater? 1 3 1 6 As loud as it gets!
  13. 13. Reasoning Strategies • Whole Number Reasoning-–”they are equal” (especially when comparing 5 6 and 7 8 ) • Gap Reasoning—”one-third is 2 from the whole and one-sixth is 5 from the whole so one-third is greater”
  14. 14. Strategy #1: Notation Which fraction is greater? 1 third 1 sixth Turning down the volume a little!
  15. 15. Strategy #2 (Context) • Drew ate 1 sixth of a whole Twizzler. Brad ate 1 third of a whole Twizzler. Who ate more? • Group A is going to equally share 1 Twizzler with 6 people. Group B is going to equally share 1 Twizzler with 3 people. The people in which group will get more get more Twizzler? How much will each a person get in each group? Turning the Volume Way Down!
  16. 16. Strategy #3 (Connect to Whole Numbers)
  17. 17. Addition • Jenny has 4 whole apples and her friend Sue has 9 whole apples. Between the two of them how many total whole apples do they have? • Jenny has 1 3 of a whole pound of cheese and her friend Sally has 3 4 of a whole pound of cheese. Between the two of them how much cheese do they have?
  18. 18. Addition with Whole Numbers 4 groups of 1 apple + 9 groups of 1 apple = 13 groups of 1 apple 4 whole apples + 9 whole apples = 13 whole apples
  19. 19. Addition with Fractions 1 third + 3 fourths = ???? 1 copy of 1 third + 3 copies of 1 fourths = ????
  20. 20. 4 twelfths + 9 twelfths = 13 twelfths 4 groups of 1 twelfth + 9 groups of 1 twelfth = 13 groups of 1 twelfth 4 1 12 + 9 1 12 = 13 1 12 = 4 12 + 9 12 = 13 12
  21. 21. Addition “Simplified” 1 2 + 3 8 1 half + 3 eighths “There are 4 eighths in 1 half” 4 eighths + 3 eighths = 7 eighths
  22. 22. 5 ÷ 1 3 What does the expression mean?
  23. 23. Thinking about “Division” • Lily has 12 Starbursts. She is going to give them away to her 3 friends. How many Starbursts will each friend receive? • Gwen has 12 Starbursts. She is going to give 3 Starbursts to as many friends as she can. How many friend will receive 3 Starbursts? What are the similarities/differences in the structure of the two problems?
  24. 24. Which is “louder”? 5 ÷ 1 3 OR There are 5 whole Twizzlers and each person is going to receive 1 3 of a whole Twizzler? Using all of the Twizzlers, how many people can you give 1 3 of a whole Twizzler?
  25. 25. Strategy #2: Context There are 5 whole Twizzlers and each person is going to receive 1 3 of a whole Twizzler? Using all of the Twizzlers, how many people can you give 1 3 of a whole Twizzler?
  26. 26. Problem Prompts • Using only pictorial representations determine the answer to each problem. • Create a numerical expression that models your pictorial representation.
  27. 27. Problem 1 You have 8 peanut butter and jelly sandwiches and each student will eat 2 3 of a whole peanut butter and jelly sandwich. How many students can you feed?
  28. 28. Problem 2 You have 24 peanut butter and jelly sandwiches and each student will eat 2 peanut butter and jelly sandwiches. How many students can you feed?
  29. 29. Making the Connection 𝟖 ÷ 𝟐 𝟑 • Question: How many groups of 2 thirds can I make from 24 thirds? • Answer: 12 groups of 2 thirds of a whole. 𝟐𝟒 ÷ 𝟐 • Question: How many groups of 2 whole can I make from 24 whole? • Answer: 12 groups of 2 whole.
  30. 30. A Closer Look • 8 whole sandwiches became 24 thirds 8 = 24 1 3 • 24 1 3 − 2 1 3 − 2 1 3 … - 2 1 3 = 0 • 24 1 3 ÷ 2 1 3 = 12 • 24(1) – 2(1) – 2(1) …. – 2(1) = 0 • 24(1) ÷ 2(1) = 12
  31. 31. Thinking about Multiplication Lily is making gift boxes of cookies. Each gift box will hold 5 cookies. If she wants to make 4 gift boxes how many cookies will she need to make? 4 * 5 = 20 (multiplier) * (multiplicative unit) = product
  32. 32. Different Scenarios • Scenario #1: Multiplier is whole number; Multiplicative Unit is a fraction • Scenario #2: Multiplier is a fraction; Multiplicative Unit is a whole number • Scenario #3: Multiplier and Multiplicative Unit are both fractions
  33. 33. Problem 3 Each batch of cookies requires 2 3 of a whole cup of sugar. How much sugar will be needed to make 5 batches of cookies?
  34. 34. Problem 4 Each batch of cookies requires 2 whole cups of sugar. How much sugar will be needed to make 5 batches of cookies?
  35. 35. Problem 3 Problem 4
  36. 36. Making the Connection 𝟓 ∗ 𝟐 𝟑 • Question: How much is 5 groups of 2 thirds? • Answer: 10 thirds 5*2 • Question: How much is 5 groups of 2 whole? • Answer: 10 whole 5 is the multiplier, 2 3 and 2 are the multiplicative unit.
  37. 37. A Closer Look 2 1 3 +2 1 3 +2 1 3 +2 1 3 +2 1 3 = 10 1 3 5 * 2 1 3 = 10 1 3 10 1 3 is how many whole? 3 1 3 2(1) + 2(1) + 2(1) + 2(1) + 2(1) = 10 5 * 2 = 10
  38. 38. Problem 5 (The “Twist”) Joann has 15 cups of sugar. She wants to give 2 3 of what she has to her friend Ann so Ann can make some cookies. How much sugar will Ann receive?
  39. 39. 2 3 * 15 (Each box represents 1 whole cup of sugar) 2 3 is the multiplier and 15 is multiplicative unit 2 3 as the multiplier implies an action on the multiplicative unit— ”partition multiplicative unit into 3 equal groups and take 2 of those groups” 2 3 * 15 = 2 * 1 3 ∗ 15 = 2 * 5 = 10
  40. 40. Moment of Reflection Think about how the fraction 2 3 is treated in each problem. • Each batch of cookies requires 2 3 of a whole cup of sugar. How much sugar will be needed to make 5 batches of cookies? • Joann has 15 cups of sugar. She wants to give 2 3 of what she has to her friend Ann so Ann can make some cookies. How much sugar will Ann receive?
  41. 41. “Tame Way” Joann has 15 16 of a whole cup of sugar. She wants to give 2 3 of what she has to her friend Ann so Ann can make some cookies. How much sugar will Ann receive? How is this problem similar/different in structure to the problem you just solved?
  42. 42. 2 3 * 15 sixteenth (Each box represents 1 sixteenth of whole cup of sugar) 2 3 is the multiplier and 15 sixteenths, 15 1 16 , is multiplicative unit 2 3 as the multiplier implies an action on the multiplicative unit—”partition multiplicative unit into 3 equal groups and take 2 of those groups” 2 3 * 15 sixteenths = 2 * 1 3 ∗ 15 sixteenths = 2 * 5 sixteenths = 10 sixteenths OR 2 3 * 15 1 16 = 2 * 1 3 ∗ 15 1 16 = 2 * 5 1 16 = 10 1 16
  43. 43. Moment of Reflection Think about how the fractions 2 3 and 15 16 are interpreted in the problem you just completed. Joann has 15 16 of a whole cup of sugar. She wants to give 2 3 of what she has to her friend Ann so Ann can make some cookies. How much sugar will Ann receive?
  44. 44. Your Challenge 2 3 * 15 16 “Why can’t we just multiply straight across?” Would the answer be less than or greater than 1?
  45. 45. 15 Sixteenths 15 Sixteenths partitioned into three equal units 2 3 * 15 sixteenths =2 1 3 ∗ 15 sixteenths = 2 * (5 sixteenths) = 10 sixteenths Modeling 𝟐 𝟑 * 𝟏𝟓 𝟏𝟔
  46. 46. 15 Sixteenths 15 Sixteenths partitioned into thirds creating 45 Forty-eighths 2 3 ∗ 45 forty − eighths = 2 1 3 ∗ 45 forty − eighths = 2 15 forty − eighths = 30 forty − eighths = 𝟑𝟎 𝟒𝟖 Modeling 𝟐 𝟑 * 𝟏𝟓 𝟏𝟔
  47. 47. Review 3 4 * 8 19 Multiplier Multiplicative Unit Interpret fraction as an operator Partition multiplicative unit into 4 equal groups and “use” 3 of those groups. Interpret fraction as a scalar multiple. 8 nineteenths or 8 copies of 1 19 3 4 *8 nineteenths = 3 * 1 4 ∗ 8 nineteenths = 6 nineteenths = 6 19
  48. 48. Session Reflection https://goo.gl/NQTxm U
  49. 49. Going Further • Jamie has 4 5 of a whole foot of string and wants to create pieces of string that are 1 3 of a foot long. How many pieces of string that are 1 3 of a foot long can she create? (Be sure to include partial pieces.) • Jamie has 12 feet of string and wants to create pieces that are 5 feet long. How many of those size pieces can she create?
  50. 50. Problem 11 You have 12 peanut butter and jelly sandwiches and each student will eat 0.6 of a peanut butter and jelly sandwich. How many students can you feed?
  51. 51. Connecting to Problem 3 Joann has 5 cups of sugar. She wants to give 2 3 of what she has to her friend Ann so Ann can make some cookies. How much of a whole pound of sugar will Anne receive?
  52. 52. Going Further Joann has 4 5 of a pound of sugar. She wants to give 2 3 of what she has to her friend Ann so Ann can make some cookies. How much sugar will Ann receive?
  53. 53. Going Even Further • Each batch of cookies requires 0.4 of a cup of sugar. How much sugar will be needed to make 5 batches of cookies? • Joann has 0.8 of a cup of sugar. She wants to give 0.2 of what she has to her friend Ann so Ann can make some cookies. How much of a whole cup of sugar will Ann receive?
  54. 54. “Division” is “Division” • 6 ÷ 2 • 6 ÷ 1 4 • 6 ÷ 0.01 • 0.06 ÷ 0.02
  55. 55. Contact Information • Dr. Patrick Sullivan (patricksullivan@missouristate.edu)

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