Building Numeracy: Fractions and Decimals

1. Building Numeracy:Fractions and Decimals Grades 3-5 Webinar March 17, 2016 Committed to Student Success

2. Agenda • Meanings of fraction • Fraction models • Foundational concepts of fractional parts • Fraction equivalence • Comparing fractions • Fraction computations • Decimal fractions Our vision is to ensure the success of each student.

3. Because fractions are students’ first serious excursion into abstraction, understanding fractions is the most critical step in preparing for algebra. Wu, H. (2009, p. 8) We believe, as a learning community, we must continuously improve.

4. GRADE 3 GRADE 4 GRADE 5 Fraction progression The meaning of fractions Fractions on the number line Equivalent fractions Equivalent fractions Comparing fractions Comparing fractions Adding and subtracting fractions Adding and subtracting fractions Multiplying a fraction by a whole number Multiplying and dividing fractions Decimal fractions Multiplication in scaling We believe, as leaders of learners, we must see students as volunteers in their learning.

5. Why fractions are difficult • There are many meanings of fractions. • Fractions are written in a unique way. • Students overgeneralize their whole-number knowledge. We believe we must focus on providing challenging, interesting, and satisfying work for students..

6. Meanings of fractions • Part-whole / partitioning • Measurement • Division • Operator • Ratio Behr, Lesh, Post, & Silver (1983) We believe we are responsible for the success of each student..

7. Part-whole • Most common construct • Easiest for young children to understand • Easy for children to relate to everyday because of their intuitive understanding of, for example, half of a pizza, sharing a chocolate bar among four friends, shading one-third of the rectangle • Can also be … …part of a group (e.g., 3/5 of the class) …part of a length (e.g., 3 ½ miles) We will provide high-level, engaging work for all learners and leaders to meet the needs of all stakeholders..

8. Fraction size is relative • A fraction by itself does not describe the size of the whole or the size of the parts. • A fraction tells us only about the relationship between the part and the whole. Which is greater, of a pizza or of a pizza? Coweta Committed to Student Success

9. Measurement Involves identifying a length and then using that length as a measurement piece to determine the length of an object Focuses on how much rather than how many parts 5 8 1 8 1 8 1 8 1 8 1 8 Our vision is to ensure the success of each student.

10. Division • Interpretation is understood through partitioning and equal sharing • Example: Sharing $10 equally with 4 people • Not a part-whole scenario, but each person will receive 1/4 of the money • Students should understand and feel comfortable with all these representations of this example We believe, as a learning community, we must continuously improve.

11. Operator • Fractions can indicate an operation • Examples: 4/5 of 20 square feet • 2/3 of the class We believe, as leaders of learners, we must see students as volunteers in their learning.

12. Ratio • Context in which fractions are used • Can be part-part or part-whole • Examples • ¾ can mean the ratio of students wearing jackets to students not wearing jackets (part-part) • ¾ can mean the ratio of students wearing jackets to all students (part-whole) We believe we must focus on providing challenging, interesting, and satisfying work for students..

13. Fraction models We believe we are responsible for the success of each student..

14. Area models Activity sheets: Fraction CirclesFraction Pattern Blocks Created by Blair Turner; free on her blog We will provide high-level, engaging work for all learners and leaders to meet the needs of all stakeholders..

15. Length models Activity sheet: Cuisenaire Rods Fractions Created by Blair Turner; free on her blog Coweta Committed to Student Success

16. Set models Our vision is to ensure the success of each student.

17. Models online • NRICH Math virtual Cuisenaire rods • PBS Cyberchase videos: • Thirteen Ways of Looking at a Half (fractions of geometric shapes) • Melvin’s Make a Match (equivalent fractions) • NCTM Illuminations fractions model • Math Playground fraction bars • National Library of Virtual Manipulatives – various models (scroll down) We believe, as a learning community, we must continuously improve.

18. Partitioning with area models • Fractional parts have to be the same size, but not necessarily the same shape • Number of equal-sized shapes determines the fractional amount • Color tiles can help with the “not necessarily the same shape” • Sometimes visuals do not show all the partitions 5 2 1 6 4 3 We believe, as leaders of learners, we must see students as volunteers in their learning.

19. Students do not understand that 2/3 means two equal-sized parts (although not necessarily equal-shaped objects). For example, students may think the following shape shows 1/4 green, rather than 1/2 green. Ask students to create their own representations of fractions across various manipulatives and on paper. Provide problems like this one in which all the partitions are not already shown. • How to help Misconception We believe we must focus on providing challenging, interesting, and satisfying work for students..

20. Fourths or Not Fourths? Half of a Whole task, lesson guide, teaching practices, sample solutions, video We believe we are responsible for the success of each student..

21. Finding (All the) Fair Shares Give students dot paper and instructions: • Use the dots to help you draw a 3-by-6 rectangle. • Find a way to partition the rectangle into thirds. • Redraw the same size rectangle, and partition it in a different way to show thirds. • See how many ways you can find. We will provide high-level, engaging work for all learners and leaders to meet the needs of all stakeholders..

22. Partitioning with length models • 3.NF.2b Partition part of a number line into fourths, for example, and realize that each section is one-fourth. • Folding paper strips can help students develop this understanding. • Transition to the number line. • What fraction of the bar is colored? Coweta Committed to Student Success

23. Transition to number line 0 0 0 1 1 1 Our vision is to ensure the success of each student.

24. Transition to number line 0 0 0 1 1 1 We believe, as a learning community, we must continuously improve.

25. Students think that a fraction such as 1/5 is smaller than a fraction such as 1/10 because 5 is less than 10. Or students may not be able to correctly compare 1/5 and 7/10 because they have been told that the bigger the denominator, the smaller the fraction. Use many visuals and contexts that show parts of the whole. For example, ask students if they would rather go outside for 1/2 of an hour, 1/4 of an hour, or 1/10 of an hour. Use the idea of fair shares: Is it fair if Mary gets 1/4 of the pizza and Laura gets 1/8? Ask students to explain. • How to help Misconception We believe, as leaders of learners, we must see students as volunteers in their learning.

26. Using length models for tasks 42 6 6 6 6 6 6 6 6 6 54 42 bandages are tan. We believe we must focus on providing challenging, interesting, and satisfying work for students..

27. Partitioning with set models • Sets of objects – such as coins, counters, cards, etc. – can be partitioned into equal shares. • Students may confuse the number of counters in a share with the name of the share. • When the equal parts are not already figured out, students may not see how to partition. whole sixths Write the fraction of animals that are cats. We believe we are responsible for the success of each student..

28. Kids and cookies • Online tool for fair sharing • Begin with results that will be whole numbers, then increase difficulty • 6 brownies, 2 children • 5 brownies, 2 children • 5 brownies, 4 children • 7 brownies, 4 children • 2 brownies, 4 children • 4 brownies, 8 children • 3 brownies, 4 children We will provide high-level, engaging work for all learners and leaders to meet the needs of all stakeholders..

29. Iterating • Whole numbers • Counting • Addition/subtraction • Fractions • Counting fractional parts • Addition/subtraction of parts Top number (numerator) counts Bottom number (denominator) tells what is being counted Coweta Committed to Student Success

30. Students think that the numerator and denominator are separate values and have difficulty seeing them as a single value. It is hard for them to see 1/4 as a single number. Find fraction values on a number line. Use this as a fun warm-up activity each day. Students place particular values on a classroom number line or in their math journals. Avoid phrases like “one out of four” or “one over four.” Instead, say “one fourth.” • How to help Misconception Our vision is to ensure the success of each student.

31. Fraction equivalence We believe, as a learning community, we must continuously improve.

32. Giving students rules to help them develop facility with fractions will not help them understand the concepts. The risk is that when students forget a rule, they’ll have no way to reason through a process. Burns, M. (2009, p. 268) We believe, as leaders of learners, we must see students as volunteers in their learning.

33. Developing equivalent fraction algorithm • Apples and Oranges • 24 counters = 1 whole apples oranges We believe we must focus on providing challenging, interesting, and satisfying work for students..

34. groups are apples groups are oranges We believe we are responsible for the success of each student.. Make groups of 2, separating by color.

35. groups are apples groups are oranges We will provide high-level, engaging work for all learners and leaders to meet the needs of all stakeholders.. Make groups of 4, separating by color.

36. groups are apples groups are oranges Coweta Committed to Student Success • Pair groups by color. • 16 is 2 groups of 8 • 8 is 1 group of 8

37. Our vision is to ensure the success of each student.

38. Arrays for equivalent fractions We believe, as a learning community, we must continuously improve.

39. Arrays for equivalent fractions We believe, as leaders of learners, we must see students as volunteers in their learning.

40. Partitioning for equal fractions We believe we must focus on providing challenging, interesting, and satisfying work for students..

41. Equivalent fractions algorithm 1 1   Equivalent Fractions: tool for creating equivalent fractions Fraction Track: 2-player game to find sum of two fractions (equivalent fractions play a significant role) We believe we are responsible for the success of each student..

42. Comparing fractions We will provide high-level, engaging work for all learners and leaders to meet the needs of all stakeholders..

43. Fractions with like denominators Strategy for comparing fractions • The size of the pieces (the denominator) is the same, so we are just comparing the count of the pieces (the numerator). • Require students to justify their answers. • Allow students to use manipulatives, like fraction tiles. • Every lesson on fractions should incorporate concrete (hands-on) or representational (pictorial) learning for students to develop conceptual understanding. Coweta Committed to Student Success

44. Fractions with like numerators Strategy for comparing fractions Our vision is to ensure the success of each student.

45. One unit fraction from the whole Strategy for comparing fractions One Fraction Unit from the Whole War game (Make 2 copies of cards for each pair of students, laminate, and cut them out.) We believe, as a learning community, we must continuously improve.

46. Using a benchmark of ½ Strategy for comparing fractions • List the fractions equivalent to ½. • What do you notice? • Were there tiles you could not use to make ½? Which ones? • How do each of these fractions compare to ½? (Name fractions for comparison.) • Working in pairs and fraction tiles, use the whole tile to mark off 0 and 1 on a number line. • Find fractions that are equivalent to ½ using one tile type (halves, thirds, quarters, etc.). We believe, as leaders of learners, we must see students as volunteers in their learning.

47. Benchmark of ½ Strategy for comparing fractions • Multiple representations are important. • Use number lines for practicing using ½ as a benchmark. • Make copies of number lines with fractions, cut them into strips, and have students use them to compare fractions to ½. We believe we must focus on providing challenging, interesting, and satisfying work for students..

48. Finding common denominators Strategy for comparing fractions • Requires understanding fraction equivalency • Students need lots of experience with manipulatives, such as fraction tiles, to develop this concept and the algorithm. We believe we are responsible for the success of each student..

49. Effective fraction computation instruction • Use contextual tasks. • Explore each operation with a variety of models. • Let estimation and informal methods play a big role in the development of strategies. • Address common misconceptions regarding computational procedures. We will provide high-level, engaging work for all learners and leaders to meet the needs of all stakeholders..

50. Estimation before computation • Students should be taught to estimate answers to problems before computing the answers so they can judge the reasonableness of their computed answers. You are going to estimate a sum or difference of two fractions. You have to decide if the exact answer is more than 1 or less than 1. You will have 10 seconds for each problem. Coweta Committed to Student Success