Fractions: Grades 3-5: slide 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. 2008 May 29 Fractions: Grades 3-5: slide 1

Introduction of Facilitators INSERT the names and affiliations of the facilitators 2008 May 29 Fractions: Grades 3-5: slide 2

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. 2008 May 29 Fractions: Grades 3-5: slide 3

Fractions: Grades 3-5: slide 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. 2008 May 29 Fractions: Grades 3-5: slide 4

Fractions: Grades 3-5: slide 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. 2008 May 29 Fractions: Grades 3-5: slide 5

Fractions: Grades 3-5: slide 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. 2008 May 29 Fractions: Grades 3-5: slide 6

John told his mother that he would be home in 45 minutes. Problem 1.1 John told his mother that he would be home in 45 minutes. 2008 May 29 Fractions: Grades 3-5: slide 7

Fractions: Grades 3-5: slide 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. 2008 May 29 Fractions: Grades 3-5: slide 8

Mr. Albert has 3 boys to 4 girls in his history class. Problem 1.3 Mr. Albert has 3 boys to 4 girls in his history class. 2008 May 29 Fractions: Grades 3-5: slide 9

Fractions: Grades 3-5: slide 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. 2008 May 29 Fractions: Grades 3-5: slide 10

Fractions: Grades 3-5: slide 11 Problem 1.5 Baluka Bubble Gum comes four pieces to a package. Three children each chewed a piece from one package. 2008 May 29 Fractions: Grades 3-5: slide 11

There were 12 men and 3/4 as many women at the meeting. Problem 1.6 There were 12 men and 3/4 as many women at the meeting. 2008 May 29 Fractions: Grades 3-5: slide 12

Fractions: Grades 3-5: slide 13 Problem 1.7 Mary asked Jack how much money he had. Jack reached into his pocket and pulled out three quarters. 2008 May 29 Fractions: Grades 3-5: slide 13

Fractions: Grades 3-5: slide 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. 2008 May 29 Fractions: Grades 3-5: slide 14

Fractions: Grades 3-5: slide 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. 2008 May 29 Fractions: Grades 3-5: slide 15

Martin’s Men Store had a big sale – 75% off. Problem 1.10 Martin’s Men Store had a big sale – 75% off. 2008 May 29 Fractions: Grades 3-5: slide 16

Fractions: Grades 3-5: slide 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. 2008 May 29 Fractions: Grades 3-5: slide 17

Fractions: Grades 3-5: slide 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. 2008 May 29 Fractions: Grades 3-5: slide 18

Fractions: Grades 3-5: slide 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? 2008 May 29 Fractions: Grades 3-5: slide 19

Fractions: Grades 3-5: slide 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. 2008 May 29 Fractions: Grades 3-5: slide 20

Fractions: Grades 3-5: slide 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. 2008 May 29 Fractions: Grades 3-5: slide 21

Fractions: Grades 3-5: slide 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 ___ 2008 May 29 Fractions: Grades 3-5: slide 22

What is the sum of your eight fractions? What should the sum be? Why? Problem 2.3 What is the sum of your eight fractions? What should the sum be? Why? 2008 May 29 Fractions: Grades 3-5: slide 23

Fractions: Grades 3-5: slide 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? 2008 May 29 Fractions: Grades 3-5: slide 24

Fractions: Grades 3-5: slide 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? 2008 May 29 Fractions: Grades 3-5: slide 25

Fractions: Grades 3-5: slide 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? 2008 May 29 Fractions: Grades 3-5: slide 26

Fractions: Grades 3-5: slide 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 ---------> 2008 May 29 Fractions: Grades 3-5: slide 27

Fractions: Grades 3-5: slide 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? 2008 May 29 Fractions: Grades 3-5: slide 28

Fractions: Grades 3-5: slide 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. 2008 May 29 Fractions: Grades 3-5: slide 29

Fractions: Grades 3-5: slide 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 2008 May 29 Fractions: Grades 3-5: slide 30

Fractions: Grades 3-5: slide 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 2008 May 29 Fractions: Grades 3-5: slide 31

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 2008 May 29 Fractions: Grades 3-5: slide 32

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. 2008 May 29 Fractions: Grades 3-5: slide 33

Fractions: Grades 3-5: slide 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 2008 May 29 Fractions: Grades 3-5: slide 34

Fractions: Grades 3-5: slide 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 2008 May 29 Fractions: Grades 3-5: slide 35

Fractions: Grades 3-5: slide 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 2008 May 29 Fractions: Grades 3-5: slide 36

Fractions: Grades 3-5: slide 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 2008 May 29 Fractions: Grades 3-5: slide 37

Fractions: Grades 3-5: slide 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 2008 May 29 Fractions: Grades 3-5: slide 38

Fractions: Grades 3-5: slide 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? 2008 May 29 Fractions: Grades 3-5: slide 39

Fractions: Grades 3-5: slide 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. 2008 May 29 Fractions: Grades 3-5: slide 40

16 eggs are how many dozens? 26 eggs are how many dozens? Problem 4.1 16 eggs are how many dozens? 26 eggs are how many dozens? 2008 May 29 Fractions: Grades 3-5: slide 41

Fractions: Grades 3-5: slide 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? 2008 May 29 Fractions: Grades 3-5: slide 42

Fractions: Grades 3-5: slide 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? 2008 May 29 Fractions: Grades 3-5: slide 43

Fractions: Grades 3-5: slide 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? 2008 May 29 Fractions: Grades 3-5: slide 44

Fractions: Grades 3-5: slide 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? 2008 May 29 Fractions: Grades 3-5: slide 45

Fractions: Grades 3-5: slide 46 Problem 4.6 Although “unitizing” is a word for adult (and not children), how might work with unitizing help children understand fractions? 2008 May 29 Fractions: Grades 3-5: slide 46

Fractions: Grades 3-5: slide 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? 2008 May 29 Fractions: Grades 3-5: slide 47

Fractions: Grades 3-5: slide 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. 2008 May 29 Fractions: Grades 3-5: slide 48

Fractions: Grades 3-5: slide 49 Problem 5.1 How do you know that 6/8 = 9/12? Give as many justifications as you can. 2008 May 29 Fractions: Grades 3-5: slide 49

Fractions: Grades 3-5: slide 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? 2008 May 29 Fractions: Grades 3-5: slide 50

Fractions: Grades 3-5: slide 51 Problem 5.2 Blue table: 6 children Red table: 4 children (a) blue table: 12 cupcakes red table: 12 cupcakes (b) blue table: 12 cupcakes red table: 8 cupcakes 2008 May 29 Fractions: Grades 3-5: slide 51

Fractions: Grades 3-5: slide 52 Problem 5.2 Blue table: 6 children Red table: 4 children (c) blue table: 8 cupcakes red table: 6 cupcakes (d) blue table: 5 cupcakes red table: 3 cupcakes (e) blue table: 2 cupcakes red table: 1 cupcake 2008 May 29 Fractions: Grades 3-5: slide 52

Would you purchase the following poster? Why or why not? Problem 5.3 Would you purchase the following poster? Why or why not? 2008 May 29 Fractions: Grades 3-5: slide 53

Why is it so important to keep track of the unit for fractions? Reflection Why is it so important to keep track of the unit for fractions? 2008 May 29 Fractions: Grades 3-5: slide 54

Fractions: Grades 3-5: slide 55 Problem Set 6 The focus of Problem Set 6 is in between. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others. 2008 May 29 Fractions: Grades 3-5: slide 55

Fractions: Grades 3-5: slide 56 Problem 6.1 Find three fractions equally spaced between 3/5 and 4/5. Justify your solutions. 2008 May 29 Fractions: Grades 3-5: slide 56

Fractions: Grades 3-5: slide 57 Problem 6.2 We know that 3.5 is halfway between 3 and 4, but is 3.5/5 halfway between 3/5 and 4/5? Explain. 2008 May 29 Fractions: Grades 3-5: slide 57

Fractions: Grades 3-5: slide 58 Problem 6.3 Find three fractions equally spaced between 1/4 and 1/3. Justify your solutions. 2008 May 29 Fractions: Grades 3-5: slide 58

Fractions: Grades 3-5: slide 59 Problem 6.4 We know that 3.5 is halfway between 3 and 4, but is 1/3.5 halfway between 1/4 and 1/3? Explain. 2008 May 29 Fractions: Grades 3-5: slide 59

Fractions: Grades 3-5: slide 60 Reflection How do you know when fractions are equally spaced? Is it important for students in Grades 3-5 to be able to do determine this? Where would this idea appear in the K-8 Mathematics Standards? 2008 May 29 Fractions: Grades 3-5: slide 60

Fractions: Grades 3-5: slide 61 Problem Set 7 The focus of Problem Set 7 is variations on fraction tasks. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others. 2008 May 29 Fractions: Grades 3-5: slide 61

Fractions: Grades 3-5: slide 62 Problem 7.1 What number, when added to 1/2, yields 5/4? Write at least 5 different answers. 2008 May 29 Fractions: Grades 3-5: slide 62

Fractions: Grades 3-5: slide 63 Problem 7.2 Write two fractions whose sum is 5/4. Write at least 5 different answers. 2008 May 29 Fractions: Grades 3-5: slide 63

Fractions: Grades 3-5: slide 64 Problem 7.3 Write two fractions, each with double-digit denominators, whose sum is 5/4. Write at least 5 different answers. 2008 May 29 Fractions: Grades 3-5: slide 64

Which of problems 7.1, 7.2, and 7.3 is the most “unusual”? Why? 2008 May 29 Fractions: Grades 3-5: slide 65

Fractions: Grades 3-5: slide 66 Reflection Do your curriculum materials include “unusual” problems? Why is it important for students to have experience with “unusual” problems? 2008 May 29 Fractions: Grades 3-5: slide 66

Fractions: Grades 3-5: slide 67 Problem Set 8 The focus of Problem Set 8 is modifying fractions. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others. 2008 May 29 Fractions: Grades 3-5: slide 67

Fractions: Grades 3-5: slide 68 Problem 8.1 What happens to a fraction if (a) the numerator doubles (b) the denominator doubles (c) both numerator and denominator double (d) both numerator and denominator are halved (e) numerator doubles, denominator is halved (f) numerator is halved, denominator doubles 2008 May 29 Fractions: Grades 3-5: slide 68

Fractions: Grades 3-5: slide 69 Problem 8.2 What happens to a fraction if (a) the numerator increases (b) the denominator increases (c) both numerator and denominator increase (d) both numerator and denominator decrease (e) numerator increases, denominator decreases (f) numerator decreases, denominator increases 2008 May 29 Fractions: Grades 3-5: slide 69

Fractions: Grades 3-5: slide 70 Problem 8.3 The letters a, b, c, and d each stand for a different number selected from {3, 4, 5, 6}. Solve these problems and justify each answer. (a) Write the greatest sum: a/b + c/d (b) Write the least sum: a/b + c/d (c) Write the greatest difference: a/b - c/d (d) Write the least difference: a/b - c/d 2008 May 29 Fractions: Grades 3-5: slide 70

Fractions: Grades 3-5: slide 71 Reflection Which of these problems could be presented to students as “mental math” problems? Which of these problems would students need to explore over a long period of time? 2008 May 29 Fractions: Grades 3-5: slide 71

Fractions: Grades 3-5: slide 72 Problem Set 9 The focus of Problem Set 9 is reflection on thinking. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others. 2008 May 29 Fractions: Grades 3-5: slide 72

Fractions: Grades 3-5: slide 73 Problem 9.1 Write a division story problem appropriately solved by division so that the quotient has a label different from the labels on the divisor and the dividend. What does “divisor” mean? What does “dividend” mean? 2008 May 29 Fractions: Grades 3-5: slide 73

Fractions: Grades 3-5: slide 74 Problem 9.2 Write a story problem appropriately solved by division that demonstrates that division does not always make smaller. 2008 May 29 Fractions: Grades 3-5: slide 74

Is a fraction a number? Explain. Problem 9.3 Is a fraction a number? Explain. 2008 May 29 Fractions: Grades 3-5: slide 75

Why are fractions called equivalent rather than equal? Problem 9.4 Why are fractions called equivalent rather than equal? 2008 May 29 Fractions: Grades 3-5: slide 76

Fractions: Grades 3-5: slide 77 Reflection What knowledge for teachers do these problems address? Why is this important knowledge for teachers? 2008 May 29 Fractions: Grades 3-5: slide 77

Fractions: Grades 3-5: slide 78 Closing Comments Implementing the K-8 Mathematics Standards will require a deeper focus of mathematics ideas at each grade. Personal understanding of these ideas will make the implementation process easier. 2008 May 29 Fractions: Grades 3-5: slide 78