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Part I: Instructor’s Notes
41
Instructor Notes for Chapter 6: Meanings for Fractions
INSTRUCTOR NOTE 6: OVERVIEW OF CHAPTER 6
We consider here the different ways that fractions are used in the school context. This chapter
includes the following sections:
6.1 Understanding the Meanings of
a
b
The fraction ba is presented as part of a whole and as
representing division.
6.2 Equivalent (Equal) Fractions Recognizing when two fractions are equivalent is
fundamental to understanding and operating on fractions.
6.3 Relating Fractions, Decimals, and Percents It is relatively easy to write a fraction as a
decimal; simply divide the numerator by the denominator. However, writing decimals as
fractions is considerably more difficult, and ways of doing so are discussed here. Writing
fractions and decimals as percents and vice versa is reviewed.
6.4 Estimating Fractional Values Estimating fractional quantities requires fractional number
sense. This section may appear to be a review of sorts, but you may be surprised at how
much difficulty many students have with this section. Be sure students spend some time
using the Fraction applet if it is available to you.
6.5 Issues for Learning: Understanding Fractions and Decimals
6.6 Check Yourself
In the first section, we review the part-whole interpretation for fractions and then discuss (briefly)
the division interpretation. (Ratios are treated in Chapters 8 and 9.) Especially important is the
notion of equivalent or equal fractions and using this knowledge to compare nonequivalent forms.
The inter-relationships among forms of rational numbers—fractions and decimals—are then
discussed.
The jury is out on how well students understand even the part-whole interpretation of fractions; if
they do, it is often restricted to parts of a pie shape. There is an advantage to using circles rather than
rectangles because students often “add on” to a rectangle to arrive at an answer, something that
cannot be done with a circle. Both should be used at various times.
Also, there is evidence that shows that many students do not realize that ba and a ÷ b are
interchangeable. In discussing the part-whole interpretation, be sure to include fractions greater than
1.
These sections are relatively brief, but with long exercise sections. Much of what is to be learned in
this chapter is developed through the exercises.
INSTRUCTOR NOTE 6.2A
Prospective teachers can usually identify equivalent fractions, but cannot explain why two fractions
are equivalent. Drawings of both discrete and continuous situations are used to show equivalence.
42 Part I: Instructor’s Notes
Some elementary teachers like to use paper folding to show equivalence of fractions. One “trick” for
folding in thirds, invented by third grade students (Kieran, 1995), is to fold one side over until you
“see halves” at which time the fold is at the one-third mark. Then fold the other side over where the
half is seen and that fold will be at the other third mark. They went on to refine this method to fold
fifths, by folding both sheets of paper over toward the center until they could “see thirds” and then
continued folding at the places where the ends of the sheets met.
In this section, we also consider nonequivalent fractions, and students are asked, in the exercises, to
compare fractions by their size. Using the part-whole notion of fractions, they should be able to
compare unit fractions and then use that information to compare fractions that are multiples of unit
fractions. Encourage students to compare fractions using their “close to” and part-whole knowledge
because that will lead to more sense-making with fractions. “Number Neighbors” in Section 6.4
could be used in Section 6.2. Using a common denominator or decimal approach can lead to a very
rote application of rules without increasing one’s fraction sense.
Pattern Blocks are excellent for showing equivalence (for carefully chosen examples).
INSTRUCTOR NOTE 6.3A
There are three main ideas to this section:
whole number
1. The decimal equivalent of every fraction ( whole
number ° 0 ) is either a terminating or a repeating
decimal. The decimal equivalents terminate when equivalent fractions can be formed such that the
denominator is a power of 10, that is, the factors are 2s and 5s. For example, 800 is
2 × 2 × 2 × 2 × 2 × 5 × 5 . When the numerator and denominator are both multiplied by 5 × 5 × 5 , the
new denominator is (2 × 5)3 , or 100,000. Consider 123 : Although 12 has a factor of 3, it matches with
the 3 in the numerator, leaving
1
4
, or
25
100
, which is 0.25.
2. When a whole number is divided by 7, the only possible remainders are 0, 1, 2, 3, 4, 5, and 6. But
0 cannot be a remainder, because if it were, the decimal would terminate, contrary to Point 1.
Therefore, for 17 , or 1 ÷ 7, after six (or fewer) places the digits in the quotient must start repeating.
For 113 , there will be 10 or fewer places before repeating must begin. Such is the case for any
fraction in simplest form for which the denominator has factors other than 2 and 5.
3. Every repeating or terminating decimal can be expressed as a fraction, but non-repeating, nonterminating decimals cannot. For a terminating decimal, simply write the fraction as the decimal is
read: 0.3 is three-tenths is 103 . A procedure is provided for changing repeating decimals to fractions.
Rational numbers (positive and 0) can then be defined to include either repeating or terminating
decimal numbers or fractions in which the numerator and denominator are whole numbers and the
denominator is not zero. (Negative rationals are introduced in Chapter 10.) Irrational numbers are
defined as non-repeating, non-terminating decimal numbers.
INSTRUCTOR NOTE 6.3B Finding the fraction for a repeating decimal
The method given in Section 6.3 for finding the fraction for a repeating decimal builds on
students’ likely knowledge that 0.3 = 13 . The following is another commonly used method and
Part I: Instructor’s Notes
43
can handle decimals like 7.3456 a bit more easily if you wish to spend that much time on this
topic. (The method in the text would require some facility with adding the fraction from 7.34 and
0.0056 .) Both methods can lead to shortcuts. You can decide whether to require your students to
simplify the resulting fractions.
Think of 0.5 = 0.55555... and multiply by 10:
10 × 0.5
=
10 × 0.5
1 × 0.5
=
=
5.5555…
0.5555… Subtracting gives
9 × 0.5
0.5 = 59
=
5 and, on dividing both sides by 9,
5.5555… Then
This technique works with repeating decimals that have one digit repeating endlessly because
multiplying by 10 affects the place value of each digit by 10. The result in effect “moves” the
digits one place-value to the left and keeps the endless part set up so that the subtraction gives a
difference with only a finite number of nonzero digits.
10 × 18.2 = 182.222222222222... (forever)
1 × 18.2 = 18.222222222222... (forever)
9 × 18.2 = 164 and on dividing both sides by 9
164
18.2 =
9
Suppose that the repeating block of digits has more than one digit, as in 7.3456 :
7.3456 = 7.345656565656... (forever)
10 × 7.3456 = 73.456565656565... (forever)
Multiplying by 10 does not adjust the digits so that the endless parts disappear on subtraction.
We need to move the digits two place-values:
100 × 7.3456 = 734.565656565656... (forever)
1 × 7.3456 =
7.345656565656... (forever)
Subtracting gives 99 × 7.3456 = 727.22 , or 7.3456 =
whole number
nonzero whole number
form, 7.3456 =
727.22 ×100
99 ×100
=
72722
9900
727.22
99
. To get the result in the usual
.
In general, if the repeating block is n digits long, multiplying the number by 10 n and then
subtracting the number will get rid of the infinite part and leave an easily solved equation. Some
whole number
adjustment, as in the last example, may be necessary to get the result in nonzer
o whole number form.
44 Part I: Instructor’s Notes
INSTRUCTOR NOTE 6.4A Formalizing “close to”
Many students have never been exposed to the type of thinking needed for this activity and do not
know how to tell when a fraction is close to 0, 12 , or 1. Student-invented “rules” need to be discussed
as a class before the exercises are assigned. Examples of responses you should expect from your
students: Close to 0—when the numerator is a small number; close to 1—when the numerator and
denominator are close to each other. Such responses should be praised, but you also want to make
sure that students are not reasoning additively in their responses. For example, most people would
agree that 87 is close to 1 but 23 is not close to 1, even though the numerator and denominator differ
only by 1 in both cases. (In the video clip of Ally, this type of error is made.) Make sure that students
are precise in their answers. For example: Close to 0—when the numerator is very small relative to
the denominator; close to 12 —when the denominator is about twice as large as the numerator or the
numerator is about one half the denominator, etc.