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Welcome

Welcome. Welcome to content professional development sessions for Grades 3-5. The focus is Fractions . Fractions in Grades 3-5 lays critical foundation for proportional reasoning in Grades 6-8, which in turn lays critical foundation for high school algebra.

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Welcome

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  1. Welcome Welcome to content professional development sessions for Grades 3-5. The focus is Fractions. Fractions in Grades 3-5 lays critical foundation for proportional reasoning in Grades 6-8, which in turn lays critical foundation for high school algebra. The goal is to help you understand this mathematics better to support your implementation of the Mathematics Standards. Fractions: Grades 3-5: slide 1

  2. Introduction of Facilitators INSERT the names and affiliations of the facilitators Fractions: Grades 3-5: slide 2

  3. Introduction of Participants In a minute or two: 1. Introduce yourself. 2. Describe an important moment in your life that contributed to your becoming a mathematics educator. 3. Describe a moment in which you hit a “mathematical wall” and had to struggle with learning. Fractions: Grades 3-5: slide 3

  4. Overview Some of the problems may be appropriate for students to complete, but other problems are intended ONLY for you as teachers. As you work the problems, think about how you might adapt them for the students you teach. Also, think about what Performance Expectations these problems might exemplify. Fractions: Grades 3-5: slide 4

  5. Problem Set 1 The focus of Problem Set 1 is representing a single fraction. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others. Fractions: Grades 3-5: slide 5

  6. Problem Set 1 Think carefully about each situation and make a representation (e.g., picture, symbols) to represent the meaning of 3/4 conveyed in that situation. Fractions: Grades 3-5: slide 6

  7. Problem 1.1 John told his mother that he would be home in 45 minutes. Fractions: Grades 3-5: slide 7

  8. Problem 1.2 Melissa had three large circular cookies, all the same size – one chocolate chip, one coconut, one molasses. She cut each cookie into four equal parts and she ate one part of each cookie. Fractions: Grades 3-5: slide 8

  9. Problem 1.3 Mr. Albert has 3 boys to 4 girls in his history class. Fractions: Grades 3-5: slide 9

  10. Problem 1.4 Four little girls were arguing about how to share a package of cupcakes. The problem was that cupcakes come three to a package. Their kindergarten teacher took a knife and cut the entire package into four equal parts. Fractions: Grades 3-5: slide 10

  11. Problem 1.5 Baluka Bubble Gum comes four pieces to a package. Three children each chewed a piece from one package. Fractions: Grades 3-5: slide 11

  12. Problem 1.6 There were 12 men and 3/4 as many women at the meeting. Fractions: Grades 3-5: slide 12

  13. Problem 1.7 Mary asked Jack how much money he had. Jack reached into his pocket and pulled out three quarters. Fractions: Grades 3-5: slide 13

  14. Problem 1.8 Each fraction can be matched with a point on the number line. 3/4 must correspond to a point on the number line. Fractions: Grades 3-5: slide 14

  15. Problem 1.9 Jaw buster candies come four to a package and Nathan has 3 packages, each of a different color. He ate one from each package. Fractions: Grades 3-5: slide 15

  16. Problem 1.10 Martin’s Men Store had a big sale – 75% off. Fractions: Grades 3-5: slide 16

  17. Problem 1.11 Mary noticed that every time Jenny put 4 quarters into the exchange machine, three tokens came out. When Mary had her turn, she put in twelve quarters. Fractions: Grades 3-5: slide 17

  18. Problem 1.12 Tad has 12 blue socks and 4 black socks in his drawer. He wondered what were his chances of reaching in and pulling out a sock to match the blue one he had on his left foot. Fractions: Grades 3-5: slide 18

  19. Reflection Even a “simple” fraction, like 3/4, has different representations, depending on the situation. How do you decide which representation to use for a fraction? How can we help students learn how to choose a representation that fits a given situation? Fractions: Grades 3-5: slide 19

  20. Problem Set 2 The focus of Problem Set 2 is representing different fractions. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others. Fractions: Grades 3-5: slide 20

  21. Problem 2.1 Represent each of the following: a. I have 4 acres of land. 5/6 of my land is planted in corn. b. I have 4 cakes and 2/3 of them were eaten c. I have 2 cupcakes, but Jack as 7/4 as many as I do. Fractions: Grades 3-5: slide 21

  22. Problem 2.2 The large rectangle represents one whole that has been divided into pieces. Identify what fraction each piece is in relation to the whole rectangle. Be ready to explain how you know the fraction name for each piece. A ___ B ___ C ___ D ___ E ___ F ___ G ___ H ___ Fractions: Grades 3-5: slide 22

  23. Problem 2.3 What is the sum of your eight fractions? What should the sum be? Why? Fractions: Grades 3-5: slide 23

  24. Problem 2.4 Mom baked a rectangular birthday cake. Abby took 1/6. Ben took 1/5 of what was left. Charlie cut 1/4 of what remained. Julie ate 1/3 of the remaining cake. Marvin and Sam split the rest. Was this fair? How does the shape of the cake influence your answer? Fractions: Grades 3-5: slide 24

  25. Problem 2.5 If the number of cats is 7/8 the number of dogs in the local pound, are there more cats or dogs? What is the unit for this problem? Fractions: Grades 3-5: slide 25

  26. Problem 2.6 Ralph is out walking his dog. He walks 2/3 of the way around this circular fountain. Where does he stop? Fractions: Grades 3-5: slide 26

  27. Problem 2.7 Ralph is out walking his dog. He walks 2/3 of the way around this square fountain. Where does he stop? START ---------> Fractions: Grades 3-5: slide 27

  28. Reflection Why is it important for students to connect their understanding of fractions with the ways they represent fractions? How do you keep track of the unit (that is, the value of 1) for a fraction? How can you help students learn these things? Fractions: Grades 3-5: slide 28

  29. Problem Set 3 The focus of Problem Set 3 is unitizing. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others. Fractions: Grades 3-5: slide 29

  30. Describing Unitizing Unitizing is thinking about different numbers of objects as the unit of measure. For example, a dozen eggs can be thought of as: 12 groups of 1, 6 groups of 2, 4 groups of 3, 3 groups of 4, 2 groups of 6, 1 group of 12 Fractions: Grades 3-5: slide 30

  31. Applying Unitizing 4 eggs is 1/3 of a dozen since it is 1 of the 3 groups of 4 4 eggs = 1 (group of 4) 12 eggs = 3 (group of 4) so 4 eggs / 12 eggs = 1 (group of 4) / 3 (group of 4) = 1/3 Fractions: Grades 3-5: slide 31

  32. Thinking about the Unit 4 eggs can be thought of as a unit which measures thirds of a dozen. 2/3 of a dozen = 2 groups of 4 eggs = 8 eggs 5/3 of a dozen = 5 groups of 4 eggs = 20 eggs Fractions: Grades 3-5: slide 32

  33. Usefulness of Unitizing Skill at unitizing (that is, thinking about different units for a single set of objects) helps develop flexible thinking about “the unit” for representing fractions. Flexible thinking is a critical skill in understanding fractions deeply and in developing a base for proportional reasoning. Fractions: Grades 3-5: slide 33

  34. Problem 3.1 Can you see ninths? How many cookies will you eat if you eat 4/9 of the cookies? O O O O O O O O O O O O O O O O O O Fractions: Grades 3-5: slide 34

  35. Problem 3.2 Can you see twelfths? How many cookies will you eat if you eat 5/12 of the cookies? O O O O O O O O O O O O O O O O O O Fractions: Grades 3-5: slide 35

  36. Problem 3.3 Can you see sixths? How many cookies will you eat if you eat 5/6 of the cookies? O O O O O O O O O O O O O O O O O O Fractions: Grades 3-5: slide 36

  37. Problem 3.4 Can you see thirty-sixths? How many cookies will you eat if you eat 14/36 of the cookies? O O O O O O O O O O O O O O O O O O Fractions: Grades 3-5: slide 37

  38. Problem 3.5 Can you see fourths? How many cookies will you eat if you eat 3/4 of the cookies? O O O O O O O O O O O O O O O O O O Fractions: Grades 3-5: slide 38

  39. Reflection Was it easy for you to think about different units for “measuring” the size of a set of objects? How can we help students think about different units for a set? Fractions: Grades 3-5: slide 39

  40. Problem Set 4 The focus of Problem Set 4 is more unitizing. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others. Fractions: Grades 3-5: slide 40

  41. Problem 4.1 16 eggs are how many dozens? 26 eggs are how many dozens? Fractions: Grades 3-5: slide 41

  42. Problem 4.2 You bought 32 sodas for a class party. How many 6-packs is that? How many 12-packs? How many 24-packs? Fractions: Grades 3-5: slide 42

  43. Problem 4.3 You have 14 sticks of gum. How many 6-packs is that? How many 10-packs is that? How many 18-packs is that? Fractions: Grades 3-5: slide 43

  44. Problem 4.4 There are 4 2/3 pies left in the pie case. The manager decides to sell these with this plan: Buy 1/3 of a pie and get 1/3 at no extra charge. How many servings are there? Fractions: Grades 3-5: slide 44

  45. Problem 4.5 There are 5 pies left in the pie case. The manager decides to sell these with this plan: Buy 1/3 of a pie and get 1/3 at no extra charge. How many servings are there? Fractions: Grades 3-5: slide 45

  46. Problem 4.6 Although “unitizing” is a word for adult (and not children), how might work with unitizing help children understand fractions? Fractions: Grades 3-5: slide 46

  47. Reflection Would it be easy for students to think about different units for “measuring” the size of a set of objects? How can we help them learn that? Fractions: Grades 3-5: slide 47

  48. Problem Set 5 The focus of Problem Set 5 is keeping track of the unit. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others. Fractions: Grades 3-5: slide 48

  49. Problem 5.1 How do you know that 6/8 = 9/12? Give as many justifications as you can. Fractions: Grades 3-5: slide 49

  50. Problem 5.2 Ten children went to a birthday party. Six children sat at the blue table, and four children sat at the red table. At each table, there were several cupcakes. At each table, each child got the same amount of cake; that is they “fair shared.” At which table did the children get more cake? How much more? Fractions: Grades 3-5: slide 50

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