1. FRACTIONS PGSR 5F March 2011

2. Fractions in Daily Life Situations

3. Fraction Computations
Food for thought
Children find it difficult to link operations with whole numbers to operations with fractions
Emphasising algorithms before developing conceptual understanding leads to frustrations
The four goals of instruction
Recognising situations that involve fraction computations
Using models to find answers
Making reasonable estimates
Finding exact answers

4. The Prerequisites
Understanding fraction concepts, comparisons and equivalence
Understanding the meaning of operations on whole numbers

5. Addition of Fractions Beginning from what children already know
Adding of whole numbers
Treating fractions like whole numbers
Helping children see the links

6. Addition of fractions
Simple addition of fractions can be demonstrated in many different ways.
One easy way for pupils to understand addition is to use fraction strips.
THE REGION MODEL

7. THE LENGTH (NUMBER LINE) MODEL
Addition of fractions
Number lines can also help pupils understand addition of fractions.
THE LENGTH (NUMBER LINE) MODEL

8. Modeling addition of unlike fractions
Using fraction bars or fraction strips

9. Modeling addition of unlike fractions
Using the region model

10. Modeling addition of unlike fractions
Using fraction rods (Cuisenaire rods)

11. Modeling addition of unlike fractions
Using fraction circles

12. Modeling addition of unlike fractions
Using concept of fraction as part of a set

13. Addition Word Problems
I ate ¼ of a cake and my brother ate another ¼ of the cake. How much of the cake was eaten?
If your pupils were asked to solve this problem by building a model for it, what likely solutions would they provide you?

14. Addition Word Problems
I ate ⅓ of a cake and my brother ate another ¼ of the cake. How much of the cake was eaten?
If your pupils were asked to solve this problem by building a model for it, what likely solutions would they provide you?

15. Extending Fraction Additions
Getting children to think about fraction addition
Kim practiced playing the piano for ¾ of an hour on Monday and another ¾ of an hour on Tuesday. How long did she practice on the two days?
If your pupils were asked to solve this problem by building a model for it, what likely solutions would they provide you?

16. The Addition Algorithm
The symbolic level of adding unlike fractions involves finding a common denominator
How to find a common denominator?
Listing multiples of the denominators
Listing the equivalent fractions until a common denominator is found

17. The Addition Algorithm
Reflection
How many of your students have difficulties with this algorithm?
What are their common mistakes?
How do you normally deal with those mistakes?

18. Addition of mixed numbers
Reflection
How many of your students have difficulties with this algorithm?
What are their common mistakes?
How do you normally deal with those mistakes?

19. Meanings of subtraction
Reflection
What kinds of situations will require the use of the subtraction operation?

20. Meanings of Fraction Subtraction
The meanings of subtraction
Take-away
Comparison
Completion
Whole/part/part
My teacher bought 2⅔ meters of cloth and used 1⅓ meters to sew some curtains. How many meters of the cloth does she still have?

21. Meanings of fraction subtraction
If the height of a wall is 2⅝ meters and the height a fence is 1¼ meters, how much higher is the wall?
James is laying some tiles along his garden path which is 5 ⅘ meters long. If he has finished 3⅓ meters, how many meters of the garden path has he still got to tile?

22. Meanings of fraction subtraction
I bought 3¼ kg of mangoes and oranges at the market. If there were 1 ⅚ kg of mangoes, how many kg of oranges did I buy?
Reflection
In what ways are these problems different from one another?
What are the different ways in which these subtraction problems can be solved?

23. The Most Difficult Operation
SUBTRACTION OF FRACTIONS IS THE TOUGHEST FOR KIDS

24. Modeling subtraction problems
COMPARISON MODEL
TAKE-AWA Y MODEL

25. Children’s algorithm
Children often invent their own algorithms to reflect their unique understanding
These algorithms usually make better sense to them than their teacher’s!
See Kennedy (2004) pp for a discussion of children’s algorithm

26. The subtraction algorithm
SUBTRACTION OF FRACTIONS USING EQUIVALENT FRACTIONS

27. Formal algorithms

28. Assessing children’s understanding
The strategies
Observe performance on specialised tasks
Check / review written responses
Talk aloud – get children to explain their solutions