Fractions

1. Fractions A Staff Tutorial

2. Workshop Format • This workshop is based around seven teaching scenarios. • From each of these scenarios will be drawn: • key ideas about fractions • how to communicate these to students

3. Scenario One • A group of students are investigating the books they have in their homes. • Steve notices that of the books in his house are fiction books, while Andrew finds that of the books his family owns are fiction. • Steve states that his family has more fiction books than Andrew’s.

4. Consider… • Is Steve necessarily correct? • Why / Why not? • What action, if any, do you take?

5. Steve is not necessarily correct because the amount of books that each fraction represents is dependent on the number of books each family owns. • For example…

6. Andrew’s family has more fiction books than Steve’s. Steve’s family has more fiction books than Andrew’s.

7. Key Idea:The size of the fractional amount depends on the size of the whole.

8. To communicate this key idea to students you could… • Demonstrate with clear examples, as in the previous tables. • Use materials or diagrams to represent the numbers involved (if appropriate). • Question the student about the size of one whole: • Is one half always more than one fifth? • What is the number of books we are finding one fifth of? How many books is that? • What is the number of books we are finding one half of? How many books is that?

9. Scenario Two • You observe the following equation in Emma’s work: • + = • Is Emma correct?

10. Consider… • You question Emma about her understanding and she explains: • “I ate 1 of the 2 sandwiches in my lunchbox, Kate ate 2 of the 3 sandwiches in her lunchbox, so together we ate 3 of the 5 sandwiches we had.” • What, if any, is the key understanding Emma needs to develop in order to solve this problem?

11. Emma needs to know that the relates to a different whole than the . • If it is clarified that both lunchboxes together represent one whole, then the correct recording is: • + = • Emma also needs to know that she has written an incorrect equation to show the addition of fractions.

12. Key Idea: When working with fractions, the whole needs to be clearly identified.

13. To communicate this key idea to students you could… • Use materials or diagrams to represent the situation. For example: • Question the student about their understanding. • The one out of two sandwiches refers to whose lunchbox? • Whose lunchbox does the two out of three sandwiches represent? • Whose lunchbox does the three out of five sandwiches represent?

14. Key Idea: When adding fractions, the units need to be the same because the answer can only have one denominator.

15. To communicate this idea to students you could… • Use a diagram or materials to demonstrate that fractions with different denominators cannot be added together unless the units are changed. For example:

16. Scenario Three • Two students are measuring the height of the plants their class is growing. • Plant A is 6 counters high. • Plant B is 9 counters high. • When they measure the plants using paper clips they find that Plant A is 4 paper clips high. • What is the height of Plant B in paper clips ?

17. Consider… • Scott thinks Plant B is 7 paper clips high. • Wendy thinks Plant B is 6 paper clips high. • Who is correct? • What is the possible reasoning behind each of their answers?

18. Wendy is correct, Plant B is 6 paper clips high. • Scott’s reasoning: • To find Plant B’s height you add 3 to the height of Plant A; 4 + 3 = 7. • Wendy’s reasoning: • Plant B is one and a half times taller than Plant A; 4 x 1.5 = 6. • The ratio of heights will remain constant. 6:9 is equivalent to 4:6. • 3 counters are the same height as 2 paper clips. There are 3 lots of 3 counters in plant B, therefore 3 x 2 = 6 paper clips.

19. Key Idea: The key to proportional thinking is being able to see combinations of factors within numbers.

20. To communicate this idea to students you could… • Draw a diagram to show the relationships between the numbers.

21. Use ratio tables to identify the multiplicative relationships between the numbers involved.

22. Use double-number lines to help visualise the relationships between the numbers.

23. Scenario Four • Anna says is not possible as a fraction. • Consider….. • Is possible as a fraction? • What action, if any, do you take?

24. is possible as a fraction. • It is read as “seven thirds.” • Seven thirds is equivalent to two and one third and can also be recorded as 2 . • The shaded area in the diagram represents .

25. Key Idea: A fraction can represent more than one whole. The denominator tells the number of equal parts into which a whole is divided. The numerator specifies the number of these parts being counted.

26. To communicate this idea to students you could… • Use materials and diagrams to illustrate. • Question students to develop understanding: • Show me 2 thirds, 3, thirds, 4 thirds… • How many thirds in one whole? two wholes? • How many wholes can we make with 7 thirds? Let’s try

27. Scenario Five • You observe the following equation in Bill’s work: • Consider….. • Is Bill correct? • What is the possible reasoning behind his answer? • What, if any, is the key understanding he needs to develop in order to solve this problem?

28. No he is not correct. The correct equation is • Possible reasoning behind his answer: • 1/2 of 2 1/2 is 1 1/4. • He is dividing by 2. • He is multiplying by 1/2. • He reasons that “division makes smaller” therefore the answer must be smaller than 2 1/2.

29. Key Idea: To divide the number A by the number B is to find out how many lots of B are in A. • For example: • There are 4 lots of 2 in 8 • There are 5 lots of 1/2 in 2 1/2

30. To communicate this idea to students you could… • Use meaningful representations for the problem. For example: • I am making hats. If each hat takes 1/2 a metre of material, how many hats can I make from 2 1/2 metres? • Use materials or diagrams to show there are 5 lots of 1/2 in 2 1/2:

31. Key Idea: Division is the opposite of multiplication. The relationship between multiplication and division can be used to help simplify the solution to problems involving the division of fractions.

32. To communicate this idea to students you could… • Use contexts that make use of the inverse operation: • A rectangular vegetable garden is 2.5 m2. If one side of the garden is 1/2 a metre long, what is the length of the other side? • Half of a skipping rope is 2.5 metres long. How long is the skipping rope?

33. Scenario Six • Which shape has of its area shaded? • Sarah insists that none of the shapes have of their area shaded. • Consider: • Do any of the shapes have of their area shaded? • What action, if any, do you take?

34. The shape on the right has of it’s area shaded. • is equivalent to , that is it represents the same quantity. The same amount of each of the circles is shaded:

35. Key Idea: Equivalent fractions have the same value.

36. To communicate this idea to students you could… • Use diagrams or materials to show equivalence. • Paper folding • Cut up pieces of fruit to show, for example, that one half is equivalent to two quarters. • Fraction tiles

37. Question students about their understanding. For example, using the fraction tiles you could ask: • How many twelfths take up the same amount of space as two sixths? • How many sixths take up the same amount of space as one third? • Can you see any other equivalent fractions in this wall? • Record the equivalent fractions as they are identified.

38. Scenario Seven • You observe the following equation in Bruce’s work: • Consider: • Is he correct? • After checking that Bruce understands what the “>” symbol means, what action, if any, do you take?

39. No he is not correct. • The correct equation is because • one sixth is less than one quarter.

40. Key Idea: The more pieces a whole is divided into, the smaller each piece will be.

41. To communicate this idea to students you could… • Demonstrate the relative size of fractions with materials or diagrams. • Question students about the relative size of each fractional piece: If we had 2 pizzas and we cut one pizza into six pieces and the other into 4 pieces, which pieces would be bigger?

42. The use of reference points 0, 1/2 and 1 can be useful for ordering fractions larger than unit fractions. For example: Which is larger is larger than one half and is less than one half, so is greater than .

43. Key Ideas about Fractions • The size of the fractional amount depends on the size of the whole. • When working with fractions, the whole needs to be clearly identified. • When adding fractions, the units need to be the same because the answer can only have one denominator. • The key to proportional thinking is being able to see combinations of factors within numbers. • A fraction can represent more than one whole.

44. The denominator tells the number of equal parts into which a whole is divided. The numerator specifies the number of these parts being counted. • Division is the opposite of multiplication. • The relationship between multiplication and division can be used to help simplify the solution to problems involving the division of fractions. • Equivalent fractions have the same value. • The more pieces a whole is divided into, the smaller each piece will be.