THIRD AND FOURTH GRADE NUMBER AND OPERATIONS: FRACTIONS

1. THIRD AND FOURTH GRADE NUMBER AND OPERATIONS: FRACTIONS

2. Third and Fourth Grade Fractions • Students express fractions as fair sharing, parts of a whole, and parts of a set. They use various context (candy bars, fruit, cakes) and a variety of models (circles, squares, rectangles, fraction bars, and number lines) to develop understanding of fractions and represent fractions. • Important Terms: partition, equal distance (intervals), fraction, unit fraction, equivalent, denominator, numerator, compare, <, >, =, benchmark fraction • Note: In the Third Grade Common Core, the expectation is limited to fractions the denominators 2, 3, 4, 6, 8. Fourth Grade uses additional denominators of 5, 10, 12, and 100.

3. Third Grade Standards Third Grade Number Sense (California) NS3.1 Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context (e.g., 1/2 of a pizza is the same amount as 2/4 of another pizza that is the same size; show that 3/8 is larger than 1/4). NS3.2 Add and subtract simple fractions (e.g., determine that 1/8 + 3/8 is the same as 1/2). Third Grade Number and Operations Fractions (Common Core) 1.Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. 2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. 3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. Recognize that equivalencies are only valid when the two fractions refer to the same whole. b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. of positive fractions, positive mixed numbers, and positive decimals to two decimal places.

4. Fourth Grade Standards Fourth Grade Number Sense (California) NS1.7 Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line. NS1.5 Explain different interpretations of fractions, for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain equivalence of fractions (see Standard 4.0). NS1.9 Identify on a number line the relative position of positive fractions, positive mixed numbers, and positive decimals to two decimal places. Fourth Grade Number and Operations Fractions (Common Core) 1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. 3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 =8/8 + 8/8 + 1/8. c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

5. Unit Fractions • A unit fraction is a fraction with a numerator of 1. • Unit fractions are formed by partitioning a whole into equal parts • If a whole is partitioned into 4 equal parts then each part is ¼ of the whole, and 4 copies of that part make up the whole

6. Unit Fractions • Other fractions are formed by unit fractions • Seeing the numerator of 3 in ¾ is saying that ¾ is the quantity you get by putting 3 “¼’s” together. • 5/3 is the quantity you get by combining 5 parts together when the whole is divided into 3 equal parts (later introduced as an improper fraction)

7. Unit Fractions • Unit fractions are the basic building blocks of fractions, in the same sense that the number 1 is the basic building block of the whole numbers. • Just as every whole number is obtained by combining a sufficient number of 1s, every fraction is obtained by combining a sufficient number of unit fractions.

8. Unit Fractions The number line reinforces the analogy between fractions and whole numbers. • Just as 5 is the point on the number line reached by marking off 5 times the length of the unit interval from 0, so 5/3 is the point obtained in the same way using a different interval as the basic unit of length, namely the interval from 0-1/3

9. 1 0 Notice how our number line goes from 0-1. We define this as our WHOLE. We have taken our whole and divided it into equal parts.

10. 1 0 Write this fraction name on the bottom of the number line.

11. Write this fraction name on the bottom of the number line. 1 0

12. Write this fraction name on the bottom of the number line. 1 0

13. 1 0 Write this fraction name on the bottom of the number line.

14. 1 Write this fraction name on the bottom of the number line. 0

15. Write this fraction name on the bottom of the number line. 1 0

16. 1 Write this fraction name on the bottom of the number line. 0

17. Write this fraction name on the bottom of the number line. 1 0

18. 1 0 Write this fraction name on the bottom of the number line.

19. 1 Write this fraction name on the bottom of the number line. 0

20. Notice how one whole is the same as . If I have colored in of the pieces green then I have colored in the whole circle. 1 0

21. 1 0 Do you notice that our whole is divided into five equal parts? That is why our denominator is 5.

22. Notice how it only took four lines to make five equal parts. To make equal parts we take the denominator and subtract one to find out how many lines we need to draw. 1 0

23. Do you notice how the line for is the same line we used for the number 1? That is why we did not have to draw five lines. 1 0 The line for is already on the number line.

24. 1 0 Another way to show .

25. . 1 0 Here is a picture of

26. . Here is a picture of 1 0

27. . Here is a picture of 1 0

28. . Here is a picture of or 1 1 0

29. Journal • Record in your journal using more than one representation.

30. Addition of Fractions • The meaning for addition is the same for both fractions and whole numbers: • Just as the sum of 4 and 7 can be seen as the length of the segment obtained by joining together two segments of lengths 4 and 7, the sum of 3/3 and 2/3 can be seen as the length of the segment obtained by joining two segments of length 3/3 and 2/3.

31. Addition of Fractions • Understanding addition as putting together allows students to see the way fractions are built up from unit fractions. Just as 5 = 1+1+1+1+1 so, 5/3 = 1/3+ 1/3 + 1/3 + 1/3 + 1/3 because 5/3 is the total length of 5 copies of 1/3.

32. Addition of Fractions • With this insight, students can compose and decompose fractions with the same denominator for addition:

33. Subtracting Fractions • Students also use this understanding for the subtraction of fractions with the same denominator:

34. Adding Fractions in Word Problems Mary and Lacey decided to share a pizza. Mary ate and Lacey ate of the pizza. How much of the pizza did the girls eat together? Mary Lacey Together = + + + + = = Reasoning: The amount of pizza Mary ate can be thought of as or and and . The amount of pizza Lacey ate can be thought of as and . The total amount of pizza they ate is + + + + or of the total pizza.

35. Adding Fractions in Word Problems Mary and Lacey decided to share a pizza. Mary ate and Lacey ate of the pizza. How much of the pizza did the girls eat together?

36. Word Problems Janet and Tracy need feet of ribbon to package gift baskets. Janet has feet of ribbon and Tracy has feet of ribbon. How much ribbon do they have altogether? Will it be enough for their project? Use more than one representation to solve this problem. How would you teach this with manipulatives? I bought gallon of paint. I only used gallons of the paint. How much paint do I have left? Use more than one representation to solve this problem. How would you teach this with manipulatives?

37. Mixed Numbers • A mixed number is a whole number plus a fraction smaller than 1 written without the + sign, e.g. 5 ¾ means 5 + ¾. • Converting a mixed number to a fraction should not be viewed as a separate technique to be learned by rote, but simply as a case of fraction addition.

38. Improper Fractions • Converting an improper fraction to a mixed number is a matter of decomposing the fraction into a sum of a whole number and a number less than 1. Knowing that 1 = 3/3, students see that:

39. Word Problems with Mixed Numbers Trevor has 4 pizza left over from his soccer party. After giving some pizza to his friend, he has 2 of a pizza left. How much pizza did he give to his friend? Solve using more than one representation. How does this relate to unit fractions? Jason has 1 cups of soda in one glass and 2 cups in another glass. How much soda does he have altogether? Solve using more than one representation.

40. Equivalent Fractions • When working with number lines, students discover that many fractions label the same point on the number line, and are therefore equal, making them equivalent fractions. This can also be seen with fraction strips and area models. • Whole numbers can also be seen as fractions. The point on the number line designated by 2 is now also designated by 2/1, 4/2, 6/3, 8/4, etc. • It is also important to note the ways of writing 1 as a fraction. 1 = 2/2 = 3/3 = 4/4 = 5/5 = …

41. Using area models to reason about fraction equivalence Using an area model to show that = . The whole is the square measured by area. On the left it is divided horizontally into 3 rectangles of equal area, and the shaded region is 2 of these and so represents . On the right it is divided into 4 x 3 small rectangles of equal area, and the shaded area comprises of 4 x 2 of these, and so it represents . How can this be put into “real-life” context?

42. Using a number line to reason about fraction equivalence

43. Equivalent Fractions • A firm understanding of fraction equivalence is necessary if students are to successfully multiply fractions. When students learn about fraction multiplication, they understand equivalence as multiplying by 1 instead of “whatever you do to the top, you do to the bottom.”

44. Reasoning about fraction equivalence • Use an area model to show that = • Use a number line to show that =

45. Equivalent Fractions • Fraction Stax • Illuminations http://illuminations.nctm.org/activitydetail.aspx?id=80

46. Comparing Fractions • When comparing fractions with the same denominator, the fraction with the larger numerator is greater because it is made of more unit fractions. ¾ is less than 5/4 because it is 3 units of ¼ as opposed to 5 units of ¼. • When comparing fractions with different denominators, the fraction with the larger denominator is smaller because in order for more identical pieces to make the same whole, the pieces must be smaller. For example, 2/5 > 2/7 because 1/7 < 1/5, so 2 units of 1/7 is less than 2 units of 2/5.

47. Comparing Fractions • As students progress, they can use their understanding of equivalent fractions to compare fractions with different denominators by rewriting both fractions using the same denominator.

48. Comparing Fractions • It is important in comparing fractions to make sure that each fraction refers to the same whole.

49. Comparing Fractions • As students move towards thinking of fractions as points on the number line, they develop an understanding of order in terms of position. Given two fractions (2 points on the number line) the one to the left is said to be smaller and the one to the right is said to be larger. This understanding of order as position will become important as students learn about negative numbers.

50. Area Model: The second cake has more left over. The second cake has left which is larger than Comparing Fractions There are two cakes on the counter that are the same size. The first cake has of it left. The second cake has of it left. Which cake has more left? Verbal explanation: I know that equals . Therefore, the second cake which has left is greater than . Number Line: