1. CHAPTER 6 – MEANINGS FOR FRACTIONS
Reconceptualizing
Mathematics
Part 1: Reasoning About Numbers and Quantities
Judith Sowder, Larry Sowder, Susan Nickerson
CHAPTER 6 – MEANINGS FOR FRACTIONS
© 2010 by W. H. Freeman and Company. All rights reserved.
Chapter 12

2. Discussion

3. WRITING FRACTIONS
The most prominent way of representing a fraction is to say: or….

5. ACTIVITY

6. ACTIVITY

8. ACTIVITY

10. 6.1

12. ACTIVITY
Make drawings to show that:

14. REDUCING FRACTIONS
As you are probably aware, any number that is a factor of both the numerator and denominator (common factor) of a fraction can be divided out such that the fraction becomes “reduced.” We then say that the fraction is written in its simplest form or its “lowest terms” (i.e., 12/30 = 2/5; common factor of 6).

15. 6.2

16. Activity

17. ACTIVITY

19. EXAMPLE

20. Discussion

21. Non-terminating, repeating decimals, like what you get for 1/7 or 3/11, are abbreviated by putting a bar over the repeated digits. e.g., … = and … =

23. ACTIVITY

24. EXAMPLE
To work with a decimal with a repeating block of two digits or more, notice the following…

26. Non-terminating, non-repeating decimals cannot be represented as a fraction with whole numbers in both numerator and denominator. These are the irrational numbers. π is an example.

28. 6.3

30. ACTIVITY

31. When working with fractional values, it can be very helpful to compare them to the commonly recognized values of 0, 1/3, 1/2, 2/3, and 1. These types of comparisons can give us a feel for the relative size or magnitude of less familiar fractions.

32. DISCUSSION

33. ACTIVITY

34. ACTIVITY

35. 6.4

36. There are several critical ideas that children need to learn before operating on fractions. They include…

38. Similar problems exist in decimal notation
Similar problems exist in decimal notation. Students may not fully understand what the decimal point indicates. If, for example, they estimate to be 50.9, they are treating the 48 separately from the .85. Some children think that because hundredths are smaller than tenths, 2.34, which has hundredths, is smaller than 2.3, which has tenths. They do this because they see that a number like 234 is larger than 24, but they don’t fully understand the concept of decimals.

39. continued….