Transcript Chapter 6

Reconceptualizing
Mathematics
Part 1: Reasoning About Numbers and Quantities
Judith Sowder, Larry Sowder, Susan Nickerson
CHAPTER 6 – MEANINGS FOR
FRACTIONS
1
© 2010 by W. H. Freeman and Company. All rights reserved.
Discussion
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WRITING FRACTIONS
The most prominent way of representing a
fraction is to say:
whole number
non  zero whole number
or….
numerator
denominator
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ACTIVITY
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ACTIVITY
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6-7
ACTIVITY
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6-9
6.1
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ACTIVITY
Make drawings to show that:
3 3 5 15


4 4  5 20
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6-13
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
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ACTIVITY
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EXAMPLE
42
76
6
6
6  22
24


 2  2 2 
 .24
175 7  25 25 5
5  2 100
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Discussion
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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… = 4.3
and
1.7245245245… =
1.7245
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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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6.3
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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
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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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