Chapter 1: Number Sense Section 3: Real Numbers
Topics In Algebra 1 © 2001 Texas Instruments Teacher Notes 1-30
Number Sense: Real Numbers Teacher Notes
Objectives
•
To illustrate the real number system in a Venn diagram.
•
To identify real numbers as rational numbers
∪
irrational numbers.
•
To review writing rational numbers as terminating or repeating decimals.
•
To review writing irrational numbers as nonterminating, nonrepeating decimals.
•
To show physical representations of the irrational numbers,
À
2
and
p
, and to review the
Pythagorean theorem and the formula for finding the circumference of a circle.
•
To state the real number system properties—commutative, associative, and distributive—as
well as the identity and inverse properties.
Math Highlights
This section starts with the building of the Venn diagram of the real number system. Definitions of
rational and irrational numbers are given. Two examples of irrational numbers,
À
2
and
p
, are
developed.
À
2
is shown as the length of the hypotenuse of a right triangle with legs of 1 unit.
p
is
shown as the circumference divided by the diameter for any circle. The statements of the properties
of the real numbers follow.
Common Student Errors
•
Many students may not have developed a solid understanding of number sets. Remind them
that using rational and irrational numbers, they can name every location on a number line.
Later in their studies of mathematics, this will be referred to as the Completeness Property of
Real Numbers, which was an important discovery in mathematics. Later, they will also
extend the real numbers to the complex numbers, a + b
À
M
1
= a + b
i
, which are numbers used,
for example, in the study of the relationship between electricity and magnetism.
•
Students probably have used 22/7 or 3.14 as an approximation of
p
. They may think that these
values are exactly
p
, but they are not equal to
p
. This provides an opportunity to talk about
approximations to several decimal places in real problems. The worksheet problems give
students an opportunity to find exact and approximate answers. There are wonderful web
sites that show
p
to millions of places. Mathematicians are still searching for more place
values. This study of
p
requires the use of computers to assist the search.
•
Some students may not be aware that the ratio of the circumference of a circle
C
divided by
the diameter
d
is
p
.
C
/
d
=
p
. This may be confusing because they have been told that
p
is
irrational and is not a ratio. Yet
p
came from a ratio of circumference to diameter. It turns
out that either
C
or
d
is also irrational. The mathematics to prove this is not given at this
level; therefore, students have to accept this without much explanation. This is a deep
discussion that will not be of interest to some students, but other students may find it
fascinating.
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