Workshop presented at National Numeracy Facilitators Conference, Auckland, February 2006

1. Thinking about ... FRACTIONS Workshop presented at National Numeracy Facilitators Conference, Auckland, February 2006 Charles Darr and Jonathan Fisher

2. Introduction/Background • We develop assessment resources in mathematics for the Assessment Resource Banks. • The current approach is to select a “big idea” in mathematics and develop this area in conjunction with classroom-based research.

3. Our Aim • To develop new assessment resources that help teachers unpack their students’ understanding of fractions.

4. Agenda • The study • What are fractions? • Focuses: Partitioning Part whole Fractions as numbers • Other resources • What next?

5. The Classroom Context • Class of Year 5 students • Local Wellington school • Mid-high decile school • Toward the end of term 3 • “Wriggly” class

6. Guiding Questions • What questions can be used to unpack students’ understanding of fractions? • What are important junctures in the development of understanding of fractions? • What common misconceptions regarding fractions are observable in the classroom?

7. More guiding Questions • What are the general characteristics of tasks that produce insights into fractional understanding? • What strategies do students bring to their learning and fractional problem solving? • What learning experiences, and teaching strategies, lead to productive work with fractions?

8. The process • Pilot for the pre-test • Pre and post interviews (videoed) • Pre and post test • Twelve maths lessons (approximately one hour each) covering a range of learning experiences about fractions

9. Our Approach • Looking for students’ ideas - explanations - not just solving problems, but explaining how they solved them. • Students invited to comment on their strategy use, how difficult they found the tasks and their general approaches to fraction problems. • Encourage students to represent the fractional relationship through drawing and imaging. • Provide contexts that help students relate the maths ideas to their own experiences.

10. What is a Fraction? Fractions come in many guises • Quotients • Measures • Part whole • Ratios • Operators

11. Important ideas about fractions • Partitioning • Unitizing (Lamon) • Proportional reasoning (relative thinking) • Equivalence • Fraction size (ordering) • Discrete/ continuous • Notation

12. What is Partitioning? • Partitioning involves the ability to divide an object or objects into a given number of non-overlapping parts. • Students have usually had opportunities in their everyday lives to practice partitioning, especially through creating fair shares.

13. What is partitioning? • Partitioning can involve dividing a number of objects into even sized groups (a discrete context), or partitioning one object or shape into same-sized pieces (a continuous context). • Partitioning usually begins with a repeated halving (splitting) strategy that enables students to create fractions such as quarters or eighths. • The ability to create fractions that involve an odd number of parts, for instance thirds and fifths, develops later and requires practice.

14. Partitioning - Equal sharing Hierarchy of understanding • Simple partitions: • halves, quarters, eighths … (halving) • evenness/equal-sized parts • More complex partitions: • larger number of pieces (e.g., 6ths, 10ths, 12ths …) • odd number of pieces (e.g., partition into 3rds, 5ths …) • Partitioning complex shapes: e.g., two squares

15. What is partitioning? • Different shapes can be more or less easy to partition. For instance, for many students it is more difficult to partition a circle into thirds than a rectangle. • Students who are more mature in terms of understanding fractions are able to use larger size pieces or “chunks” when partitioning. An example of this is shown below. Example: Share 2 cakes between 6 people

16. Partitioning - Equal sharing Equal sharing … • A continuous region (e.g., an area or length) • Sets of numbers • An amount into a given number of groups An important context that builds on the experiences children have growing up and learning to work and play with other children.

17. Partitioning – simple shapes Share these shapes evenly amongst … 4 people ? 3 people

18. Partitioning – shapes Some shapes need to be partitioned in different ways. What about other shapes?

19. Partitioning – Checking How can you check they’re equal sized? • Cut out and overlay • Folding/halving • Measure slices • Find the centre and rule lines out from it

20. Partitioning – grids Divide the grid into … 2 equal parts 16 equal parts 4 equal parts 3 equal parts

21. Partitioning - Resources • P6637 • NM1216 • P6855 • P6840 • P7029 *explaining examples of incorrect partitioning

22. The great pizza cut-up

23. The great pizza cut-up Jenny and Jeff work for a pizza shop. They have been asked to cut up the pizzas for different size groups. They must make sure that each person in a group gets the same amount of pizza. For each group draw a picture to show how the pizza can be cut up. Can you use a fraction to write how much each person in the group should get? Is there more than one way to cut up the pizzas?

24. The great pizza cut-up Pizzas Group size 7 people 3 people 5 people

25. The great pizza cut-up Name the pieces Cut it other ways How much altogether?

26. Common misconceptions • Confusion with whole number knowledge. For example, the larger the numbers in the fraction the larger the fraction • Geometric knowledge is tightly integrated with the ability to partition shapes into equal sized parts. For instance: many students use different approaches to partition a loaf of bread and a pizza.

27. Part whole fractions • Most students’ first introduction to fractions is as a part-whole comparison. A part-whole fraction compares one or more equal parts of a whole with the total number of equal parts that make up the whole. • The bottom number (the denominator) tells you how many equal parts make up the whole. The top number (the numerator) tells you how many of these parts are of interest.

28. Important ideas • The greater the number of parts needed to make a whole, the smaller each individual part is. This means for instance that 1/20 is smaller than 1/19. • The size of the unit parts must be equal. • Different fractions can be used to represent the same amount of a whole. • A fraction indicates a relative, rather than an absolute amount, that is, it tells you how much of something is present.

29. Part whole – name the part

30. The size of the part • How much of the square is shaded?

31. Part whole - resources

32. Part whole: Cuisenaire rods Cuisenaire rods can be used to model part whole relationships.

33. Part whole: Find the whole Given the part find the whole or another part

34. Fractions as Numbers • As students learn about fractions they come to understand them as a system of numbers. In particular, they realise that fractions can be ordered, rounded and represented as a point on the number line. • Students’ experiences with judging proportions in real life situations can be used to help them develop a “feel” for fraction size. E.g.: The milk bottle lessons

35. The Milk Bottle • Milk bottles filled up and placed around the room • Students guess how full each bottle is? They name the amount full with a fraction. • Links to students’ own experiences

36. The Milk Bottle

37. Strategies for ordering fractions • Students employ a range of strategies when ordering and comparing fractions. Often these strategies demonstrate their developing understanding of fractions.

38. Strategies for comparing fractions • Attempting to use whole number knowledge • Drawing pictures • Identifying fractions with the same denominator or numerator • Benchmarking fractions to well known fractions • Using equivalent fractions Less sophisticated More sophisticated

39. Fractions as numbers - ordering • Ordering unit and non unit fractions • Using imaging

40. Fraction size

41. Some lessons from the project • Use “messy fractions” – not just small numbers • Introduce “top heavy” fractions for discussion (comparison) • Opportunities to cut out, overlay, fold, and explore shapes • Encourage the drawing of diagrams • Encourage explanations – written or verbal • Sometimes students’ learned-methods almost get there – but understanding is more important (and transferable)

42. more lessons from the project • Use a range of fraction questions: involving sets, amounts/numbers, and number lines and 2-d shapes, mix the delivery of the question – find the part, find the whole, find another part. • Language is an important part of understanding, and vital to show understanding • Use a range of experiences/tools - grids, counters, number lines, shapes, milk bottles, cuisenaires … fractions are about relationships. • Make sure that fraction problems are not being solved without understanding.

43. Animations - fractions of a set • Immediate response • Computer medium • High motivation to “do again” • Independent • Formative information for the teacher • Sets of objects: NM0129 (unit fractions) & NM0130 (non unit fractions)

44. Animations - Future animations • Number line – placing on a number line with instant formative feedback • Partitioning shapes

45. What next? • Equivalence through re-unitizing • Fractions as operators • Comparing difficulty with sets, shapes (1-D and 2-D) • Locating fractions on a number line (assessment resource and animation resource) • Relative thinking