1. 1 Math CAMPPP 2012 Plenary 1 Why students struggle with fractions

2. Let’s start thinking about the complexity of fractions Create a sentence that includes these words and numbers: improper, fraction, simple, challenging, 8 22

3. Possibilities Some ideas: It is simple to write 8 as an improper fraction but more challenging to understand it when it is in that form. 3

4. Possibilities Another idea: The improper fraction 8/4 is simple to write as a whole number, but it is hard to write 88/5 as a mixed number. 4

5. Some student samples Here are a variety of student samples. There is a simple improper fraction with a denominator of 8 and the challenging part is figuring out what the numerator could be. 5

6. Some student samples The improper fraction 8/2 can prove to be simple or challenging to a grade 5 student. 6

7. Some student samples 4/8 is an improper fraction it is challenging but somewhat simple. 7

8. Some student samples An improper fraction can make fractions both simple and challenging. Let’s say you have 1 3/8 – 7/8, you can’t do it unless you change the 1 3/8 into a proper fraction. That is 1 3/8 = 11/8-7/8=4/8. The reason they can make it more challenging is you have 2 3/8 -17/8. It is still solvable but it can get confusing.. 8

9. Some student samples There are three kinds of fractions, the simple fractions and the challenging fractions and the improper fractions. 9

10. Unfortunately… Many associate the topic of fractions with difficulty, complexity, or even worse. So why do students (and lots of grown- ups) struggle? 10

11. To know why students struggle We need to think about what we really want students to know. In some ways, the curriculum tells us more about what we want students to DO. We have to figure out what we want them to KNOW. Maybe part of the struggle is not focusing on the KNOW. 11

12. On the wiki.. You will find a document that outlines specifics for Grades 1 – 9 of what the curriculum asks students to DO and what that means they need to KNOW. We will do some examples here together. 12

13. Grade 2 DoKnow Determine the relationship between number of fraction parts and sizes (investigate) Regroup fractional parts in wholes (concrete) Compare using concrete materials (no formal notation) When you divide the same thing into more parts, the parts are smaller. If you can divide a whole into many equal parts, you could put the parts together again to make a whole. If you want to know which part is more, sometimes you can just look; other times, you can overlap the parts to tell. 13

14. More parts means smaller You might ask: There were two big cakes. They were identical. You cut the first one into a LOT of equal slices. You cut the second one into NOT TOO MANY equal slices. 14 Happy Birthday!

15. So… If you used fractions to describe the size of 3 slices of each cake, what might the fractions be? Which amount of cake is smaller? 15

16. Comparing Fractions Consider this comparison situation. 16 Situation #1

17. Comparing Fractions Contrast this comparison situation. 17 Situation #2

18. Grade 4 DoKnow Represent fractions using concrete materials, words, and standard fractional notation, meaning of numerator and denominator The numerator and denominator tell different things about the fraction. You need to know what the whole is to interpret or compare fractions. The same object(s) can represent different fractions, depending on the whole. There are many ways to represent any fraction. 18

19. Grade 4 DoKnow Compare and order fractions considering size and number of parts Demonstrate and explain equivalent fractions, using concretes There is always more than one strategy for comparing fractions, but for particular fractions, one might be easier to use than another. Many of the strategies involve renaming fractions. Every fraction has more than one name. If you are using the same whole, you can tell if they are the same by overlapping the two representations. 19

20. Grade 4 DoKnow Count forward by halves, thirds, fourths and tenths beyond one It might be helpful to think of a fraction like a/b as a sets of 1/b. You can then count by 1/b s to get a sense of its size. 20

21. Which is more? Draw pictures where ½ of something is less than ¼ of something else. Why would you still write ½ > ¼ ? 21

22. In a higher grade This is like comparing x/2 to y/4. It’s not possible to know which is greater without knowing the relationship between x and y. 22

23. Same object- different fractions Show that this triangle could be ½ or could be 1/3. 23

24. Same object- different fractions 24 1/2 1/3

25. Or it could even be 2/5 25

26. Counting by fractions The ? is more than 2. How big might your hop be? Where would 1 and 2 be? Explain. 26 0 ?

27. Counting by fractions The ? is more than 2. How big might your hop be? Where would 1 and 2 be? Explain. 27 0 1 2 ? 1/2

28. Counting by fractions The ? is more than 2. How big might your hop be? Where would 1 and 2 be? Explain. 28 0 1 2 ? 1/3

29. At a higher level Students could be asked to solve for x so that 7/x > 2 and determine more answers. 29

30. Grade 8 DoKnow Represent, compare and order rational numbers Translate between equivalent forms of a number Use estimation with solving operations with fractions There is always more than one strategy for comparing fractions, but…. A non-unit fraction can be interpreted as a quotient. Knowing this might help determine the decimal form of that fraction. Each operation with fractions holds the same meaning as it did before; knowing this can inform estimating and calculating with fractions. 30

31. Grade 8 DoKnow Represent the multiplication and division of fractions, using a variety of tools Solve problems involving +, -, x and ÷ with simple fractions Multiplying a fraction by a fraction involves temporarily changing the whole. Dividing by a fraction involves finding how many of that fraction makes the other number or involves determining a “unit rate”. There are always many strategies for calculating with fractions. 31

32. Grade 8 DoKnow Solve problems involving proportions using concretes, pictorials and variables A proportion is a statement that two fractions are equivalent. Every proportion can be solved using multiplication or division or some combination thereof. 32

33. Fraction as a quotient What picture would you draw to show that 12/5 = 12 ÷ 5? What picture would you draw to show that 5/12 = 5 ÷ 12? 33

34. Possible pictures 12 ÷ 5 34 0 1 2 1/5

35. Possible pictures 5 ÷ 12 35

36. Possible pictures 5 ÷ 12 36

37. Multiplying- changing the whole What does 1/3 x 2/5 mean? The 2/5 temporarily becomes the whole of which we take 1/3. Then we go back to the original whole to represent the result. 37

38. Picture 1/3 x 2/5 38

39. Proportion When I solve this problem: If you drive 75 km/h, how long does it take to go 35 km? you might set up a proportion 75/60 = 35/x or the proportion 75/1 = 35/x which means you want an equivalent fraction for either 75/60 or 75/1 with a numerator of 35. 39

40. Notice You could actually solve the problem using operations, e.g. 35 ÷ 75 or 35 ÷ 75 x 60 but that is a different issue. 40

41. So let’s go back to specifics about struggling I have a task for you. Create a model or draw a picture that shows both 1/6 and 5/6 at the same time. 41

42. Misconception Some students ONLY see 1/6 below. Would it help to use two colours instead of only one? 42

43. Use your pattern blocks You can use RED, YELLOW, GREEN, BLUE blocks. Use your blocks to show each of these fractions (one at a time is fine). 1/2, 1/3, 1/6, 1/4 43

44. The struggle Students are focused on fractions as parts of wholes and do not accept that you can use 1/3 to describe this. 44 1/3 OF THE BLOCKS are red.

45. Another example 45 Less than half of the area is red but more than ½ of the blocks are red.

46. Using whole number reasoning Many students reasonably assume that 1/3 > 1/2 since they see 3 and 2 and 3 > 2. This is based on using whole number reasoning when it does not apply. This is a reason to avoid standard notation too early. 46

47. Another example of whole number reasoning getting in the way 47

48. 48

49. Comparing fraction strategies Which two of these four fractions would you find easiest to compare? Why? 5 5 4 9 912 4343 49

50. Comparison strategy struggles Some students think a picture is more reliable than reasoning to compare two fractions. But to compare 2/5 and 3/8 (which are close), most children-drawn pictures are misleading. 50

51. 2/5 and 3/8 51 √√ √√√

52. 2/5 and 3/8 But comparing fractions with the same denominator (16 anythings is more than 15 of those things) is more reliable. (16/40 vs 15/40) 52

53. 2/5 and 3/8 Or comparing fractions with the same numerator (6 big pieces is more than 6 small pieces) is more reliable. (6/15 is more than 6/16) 53

54. A possible discussion question Rebecca says that 1/3 is less than 1/2 since 3 is more than 2. Aaron says that 1/3 is more than 1/2 since 3 is more than 2. Tara says that 1/3 could be more or could be less than 1/2. It depends. With whom do you agree? Why? 54

55. Comparing fractions A fraction is REALLY CLOSE to 1/2. What might it be? Why did you use a bigger numerator and denominator than 1 and 2? 55

56. The struggle for some… Some students may have difficulty comparing fractions with big numerators and denominators because of lack of experience. 56

57. For example… You could ask whether 342/1000 or 2001/ 10 000 is greater, without using calculators. 57

58. For example… I know that 342/1000 is less than 2001/ 10 000 by thinking that the first denominator is about 3 times the first numerator and the second is about 5 times its numerator. So I’m comparing about 1/3 to about 1/5. 58

59. Bigger?? One fraction has both a greater numerator and a greater denominator than another, but it’s still smaller. What could the two fractions be? 59

60. What might students think Many students assume that 6/10 > 3/4 since 6 is more than 3 and 10 is more than 4. Maybe the difficulty is that we use such a limited set of numerators and denominators and/or we don’t confront the issue directly. 60

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62. 62

63. What fraction do you see? 63

64. Using improper fractions If the whole is not identified, it is not reasonable for students to know what it is. We need to identify and/or have students identify the whole. 64

65. Struggles with equating fractions with division We addressed this earlier, but many students accept, but don’t understand, why you divide the numerator by denominator to name a fraction as a decimal. Many will write 2/3 as 1.5 since they divide 3 by 2. They figure that makes more sense than dividing 2 by 3. 65

66. At a higher grade Ask why 5x/2 might mean 5x shared into 2 parts (division) or 5/2 multiplied by x, i.e drawing 2 ½ copies of x. 66

67. Look at the fraction tower You divide two fractions and the answer is just a little more than 2. What could the fractions have been? 67

68. 68 The answer is a little more than 2 when you divide two fractions.

69. With the tower You are looking for any fraction that is just a bit less than half of another on the tower. It could be, e.g. 1/5 and 1/2 or 3/12 and 2/3 or….. 69

70. 70 The answer is a little more than 2 when you divide two fractions.

71. 71 The answer is a little more than 2 when you divide two fractions.

72. To avoid the problem.. Maybe when students see 2 ÷ 1/3, we read it as How many 1/3s are in 2. If they read 1/2 ÷ 1/5, they think: “How many 1/5s are in ½?” 72

73. Adding fractions Imagine it rains 2 days in one week and 3 in another. You say it rained 5/14 of the days, essentially adding the numerators and denominators. But 2/7 + 3/7 = 5/7 and not 5/14. What’s going on? 73

74. Struggle The relationship between fractions and ratios is a struggle. In the ratio situation, you are combining parts of different wholes and comparing to the combined whole. In the fraction situation, you are combining parts of the same whole and comparing to that one whole. 74

75. Using common denominators Some students assume that since you use common denominators for addition and subtraction, you should for multiplication and division. That is not wrong. It is inefficient for multiplication (although not really for division). 75

76. Using common denominators This is an indication that kids are learning rules but not understanding why things work. 76

77. Why it’s not inefficient for division If I were calculating 2/3 ÷ 1/4 (which asks how many 1/4s are in 2/3), it is actually convenient to think of 8/12 ÷ 3/12 and realize you just need 8 ÷ 3 (or 8/3). 77

78. Which are correct and why? 2 x 3 = 2 2 x 5 = 5 3 5 5 3 12 3 2 x 5 = 2 2 x 5 = 5 3 5 3 9 4 18 2 78

79. Simplifying Teaching students to cross-out without meaning is not good– they need to think about why it works. The top left one is easy to show: 2/3 of 3 of anything is 2 of that thing. 79

80. Simplifying The bottom right one is explained by thinking of 2/9 x 5/4 = 2/4 x 5/9 and then renaming 2/4 as 1/2. Not writing the 1 for 1/2 does cause difficulties for some students. 80

81. This was just a start…. There is a lot of literature and lots of information about fraction struggles. I am sure you will meet many more of these issues this week. Although it can seem like a downer, knowledge is power. 81