CHAPTER 6 – MEANINGS FOR FRACTIONS

Reconceptualizing MathematicsPart 1: Reasoning About Numbers and Quantities

Judith Sowder, Larry Sowder, Susan Nickerson

© 2010 by W. H. Freeman and Company. All rights reserved.

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Discussion

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WRITING FRACTIONS

The most prominent way of representing a fraction is to say:

or….

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numberwholezeronon

numberwhole

rdenominato

numerator

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ACTIVITY

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ACTIVITY

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ACTIVITY

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6.1

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ACTIVITY

Make drawings to show that:

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3 53 154 4 5 20

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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).

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6.2

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Activity

ACTIVITY

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EXAMPLE

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24.100

24

25

26

5

6

25

6

257

67

175

4222

2

2

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Discussion

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., 4.3333… = and 1.7245245245… =

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3.4

2457.1

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ACTIVITY

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EXAMPLE To work with a decimal with a repeating block of two digits or more,

notice the following…

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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.

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ACTIVITY

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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.

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DISCUSSION

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ACTIVITY

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ACTIVITY

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6.4

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

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Similar problems exist in decimal notation. Students may not fully understand what the decimal point indicates. If, for example, they estimate 48.85 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.

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continued….

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