1. Chapter 12 To accompany Helping Children Learn Math Cdn Ed, Reys et al. ©2010 John Wiley & Sons Canada Ltd.

2. Guiding Questions What are three meanings of fractions, and what are models of the part-whole meaning? How can you help children make sense of fractions, and how can you use concrete and pictorial models to develop children’s understanding of ordering fractions and equivalent fractions? Describe how children can use estimation strategies for adding and subtracting by rounding to whole numbers and benchmark numbers to determine reasonableness of answers to fraction and decimal problems. How can models assist the development of children’s conceptual understanding of adding, subtracting, multiplying, and dividing fractions or decimals?

3. Conceptual Development of Fractions Three Meanings of Fractions Models of the Part-Whole Meaning Making Sense of Fractions Ordering Fractions and Equivalent Fractions Benchmarks Mixed Numbers and Improper Fractions

4. Three Meanings of Fractions Part-Whole Quotient Ratio 3 boys to 5 girls

5. Models of Part-Whole Meaning Region Set Length Area: special case of region model where parts are equal in area but not necessarily congruent

6. Making Sense of Fractions Partitioning Words Counting Symbols Drawing a Model Extending the Model Benchmarks Going From a Part to a Whole Understanding Equivalence

7. Ordering Fractions and Equivalent Fractions Concrete Models Pictorial Models Symbolic Representation 1/2 = 3/6 1/3 = 2/6

8. Benchmarks Students also need to be able to tell if a fraction is near 0 or near 1. This will let them use the benchmarks 0, 1/2 and 1 to put in order a set of fractions such as 13/25, 2/31, 5/6, 4/11, and 21/20, which would be a time-consuming task if they tried to find a common denominator. Using benchmarks as shown on the following number line makes the task rather easy:

9. Mixed Numbers and Improper Fractions You can add partitions in the model to show all the fourths, so children can see that the initial counting is 9 fourths, or the improper fraction 9/4. A mixed number is a natural symbolic representation of the adjacent model.

10. Operations with Fractions The key to helping children understand operations with fractions is to make sure they understand fractions, especially the idea of equivalent fractions. They should be able to extend what they know about operations with whole numbers to operations with fractions.

11. Adding Fractions with Like Denominators 2/6 + 1/6 = 3/6

12. Addition of Fractions

14. Addition of Fractions with Unlike Denominators

15. Subtraction of Fractions 3/4 -1/4 = 2/4 or 1/2

16. Multiplication Whole Number Times a Fraction: – You have 3 pans, each with 4/5 of a pizza. How much pizza do you have?

17. Multiplication Fraction times a whole number You have ¾ of a case of 24 bottles. How many bottles do you have?

18. Multiplication of Fractions Fraction Times a Fraction You own 3/4 of an acre of land, and 5/6 of this is planted in trees. What part of the acre is planted in trees?

19. Division of Fractions How many 2-metre lengths of rope can be made from a 10- metre length of rope? Students can then experiment with 1/2 metre and 1/4 metre sections of rope. Now ask students how they would draw a picture to find out how many 3/4 metre pieces there are in a 6-metre length of rope.

20. Conceptual Development of Decimals Relationship to Common Fractions Relationship to Place Value Ordering and Rounding Decimals

21. Relationship to Common Fractions

22. Relationship to Place Value Often, you can use a place-value grid to help students who are having difficulty with decimals. Consider, for example, this grid for the number 32.43. Point out that the decimal can be seen both as 32 and 43 hundredths and as 32 and 4 tenths and 3 hundredths. It can also be read as 3243 hundredths. What other ways could you express this number in words?

23. Ordering and Rounding Decimals Children should be able to understand the ordering and rounding of decimals based directly on their understanding of decimals and their ability to order and round whole numbers. For decimals, this understanding must include being able to interpret the decimals in terms of place value and being able to think of, for example, 0.2 as 0.20 or 0.200.

24. Operations with Decimals One advantage of decimals over fractions is that computation is much easier, since it basically follows the same rules as for whole numbers.

25. Addition and Subtraction of Decimals To deal with this difficulty, have the children first estimate by looking at the wholes (about 480). Some children may need help in lining up the like units and so may benefit from using a grid. Difficulty with adding or subtracting decimals arises mainly when the values are given in horizontal format or in terms of a story problem and the decimals are expressed to a different number of places (e.g., 51.23 + 434.7).

26. Multiplication of Decimals To help students make sense of multiplying a decimal by a decimal, rather than only remembering a rule about counting decimal places, consider this decimal grid.

27. Division of Decimals Have children talk through a decimal division problem so that they can evaluate the reasonableness of their answer.

28. For Class Discussion The following slides will show you samples of student thinking around fractions. As you review each slide please take a few minutes to share your observations with others.

29. Student Interviews Interviewer: “Which fraction is more, 1/3 or 1/4? 2/5 or 2/7?” Charles: Fifth Month of Grade Four

30. Student Interviews Interviewer: “Add these fractions: 3/8 + 2/8 2/3 + 1/4.” Amanda: Fifth Month of Grade Four

31. Student Interviews Interviewer: “Add these fractions: 3/8 + 2/8 2/3 + 1/4.” Amy: Fifth Month of Grade Four

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