Transcript Document

Chapter
5
Integers
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NCTM Standard: Integers
Principles and Standards recommends that
students in grades 3−5 be able to
 explore numbers less than 0 by extending the
number line and through familiar applications.
(p. 148)
 develop meaning for integers and represent
and compare quantities with them. (p. 214)
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NCTM Standard:
Integers and Number Theory
Middle-grades students should also work with
integers. In lower grades, students may have
connected negative integers in appropriate ways
to informal knowledge derived from everyday
experiences, such as below-zero winter
temperatures or lost yards on football plays.
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NCTM Standard:
Integers and Number Theory
In the middle grades, students should extend
these initial understandings of integers. Positive
and negative integers should be seen as useful
for noting relative changes or values. Students
can also appreciate the utility of negative
integers when they work with equations whose
solution requires them, such as 2x + 1 = 7
(pp. 217–218)
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5-1 Integers and the Operations of
Addition and Subtraction
Representations of Integers
Integer Addition
Number-Line Model
Absolute Value
Properties of Integer Addition
Integer Subtraction
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Representations of Integers
The set of integers is denoted by I:
The negative integers are opposites of
the positive integers.
–4
is the opposite of positive 4
3 is the opposite of –3
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Example 5-1
For each of the following, find the opposite of x.
a. x = 3
−x
= −3
b. x = −5
−x
=5
c. x = 0
−x
=0
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Integer Addition
Chip Model
Black chips represent positive integers and red
chips represent negative integers. Each pair of
black/red chips neutralize each other.
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Integer Addition
Charged-Field Model
Similar to the chip model. Positive integers are
represented by +’s and negative integers by –’s.
Positive charges neutralize negative charges.
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Integer Addition
Number-Line Model
Positive integers are represented by moving
forward (right) on the number line; negative
integers are represented by moving backward
(left).
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Example 5-2
The temperature was −4°C. In an
hour, it rose 10°C. What is the new
temperature?
−4
+ 10 = 6
The new temperature is 6°C.
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Integer Addition
Pattern Model
Beginning with whole number facts, a table of
computations is created by following a pattern.
Basic
facts
4+3=7
4+2=6
4+1=5
4+0=4
4 + −1 = 3
4 + −2 = 2
4 + −3 = 1
4 + −4 = 0
4 + −5 = −1
4 + −6 = −2
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Absolute Value
The absolute value of a number a, written |a|, is
the distance on the number line from 0 to a.
|4| = 4 and |−4| = 4
Absolute value is always positive or zero.
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Definition
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Example 5-3
Evaluate each of the following.
a. |20|
|20| = 20
b. |−5|
|−5| = 5
c. |0|
|0| = 0
d.
−|−3
|
e. |2 + −5|
−|−3|
= −3
|2 + −5| = |−3| = 3
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Properties of Integer Addition
Integer addition has all the properties of wholenumber addition.
Given integers a, b, and c.
Closure property of addition of integers
a + b is a unique integer.
Commutative property of addition of integers
a + b = b + a.
Associative property of addition of integers
(a + b) + c = a + (b + c).
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Properties of Integer Addition
Identity element of addition of integers
0 is the unique integer such that, for all integers
a, 0 + a = a = a + 0.
Uniqueness of the additive inverse
For every integer a, there exists a unique
integer −a, the additive inverse of a, such that
a + −a = 0 = −a + a.
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Properties of the Additive Inverse
By definition, the additive inverse, −a, is the
solution of the equation x + a = 0.
For any integers a and b, the equation x + a = b
has a unique solution, b + −a.
For any integers a and b
−(−a) = a and −a + −b = −(a + b).
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Example 5-4
Find the additive inverse of each of the following.
a. −(3 + x)
3+x
b. a + −4
−(a
−3
−(−3
c.
+ −x
+ −4) = −a + 4
+ −x) = 3 + x
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Integer Subtraction
Chip Model for Subtraction
To find 3 − −2, add 0 in the form 2 + −2 (two
black chips and two red chips) to the three
black chips, then take away the two red chips.
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Integer Subtraction
Charged-Field Model for Subtraction
To find −3 − −5, represent −3 so that at least five
negative charges are present. Then take away
the five negative charges.
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Integer Subtraction
Number-Line Model
While integer addition is modeled by maintaining
the same direction and moving forward or
backward depending on whether a positive or
negative integer is added, subtraction is modeled
by turning around.
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Integer Subtraction
Pattern Model for Subtraction
Using inductive reasoning and starting with
known subtraction facts, find the difference of
two integers by following a pattern.
3−2=1
3−3=0
3 − 4 = −1
3 − 5 = −2
3−2=1
3−1=2
3−0=3
3 − −1 = 4
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Integer Subtraction
Subtraction Using the Missing Addend Approach
Subtraction of integers, like subtraction of whole
numbers, can be defined in terms of addition.
We compute 3 – 7 as follows:
3 – 7 = n if and only if 3 = 7 + n.
Because 7 + –4 = 3, then n = –4.
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Definition
Subtraction
For integers a and b, a − b is the unique integer
n such that a = b + n.
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Example 5-5
Use the definition of subtraction to compute the
following:
a. 3 − 10
Let 3 − 10 = n. Then 10 + n = 3, so n = −7.
b. −2 − 10
Let −2 − 10 = n. Then 10 + n = −2, so n = −12.
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Integer Subtraction
Subtraction Using Adding the Opposite
Approach
Subtracting an integer is the same as adding its
opposite.
For all integers a and b
a − b = a + −b.
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Example 5-6
Using the fact that a − b = a + −b, compute each of
the following:
a. 2 − 8
2 − 8 = 2 + −8 = −6
b. 2 − −8
2 − −8 = 2 + −(−8) = 2 + 8 = 10
c.
−12
− −5
−12
− −5 = −12 + −(−5) = −12 + 5 = −7
d.
−12
−5
−12
− 5 = −12 + −5 = −17
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Example 5-7
Write expressions equal to each of the following
without parentheses.
a.
−(b
− c)
−(b
− c) = −(b + −c) = −b + −(−c) = −b + c
b. a − (b + c)
a − (b + c) = a + −(b + c) = a + (−b + −c)
= (a + −b) + −c = a + −b + −c
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Example 5-8
Simplify each of the following.
a. 2 − (5 − x)
2 − (5 − x) = 2 + −(5 + −x) = 2 + −5 + −(−x)
= 2 + −5 + x = −3 + x or x − 3
b. 5 − (x − 3)
5 − (x − 3) = 5 + −(x + −3) = 5 + −x + −(−3)
= 5 + −x + 3 = 8 + −x = 8 − x
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Example 5-8
(continued)
Simplify each of the following.
c.
−(x
− y) − y
−(x
− y) − y = −(x + −y) + −y = [−x + −(−y)] + −y
= (−x + y) + −y = −x + (y + −y) = −x
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Order of Operations
Recall that subtraction is neither commutative nor
associative.
An expression such as 3 − 15 − 8 is ambiguous
unless we know in which order to perform the
subtractions.
Mathematicians agree that 3 − 15 − 8 means
(3 − 15) − 8.
Subtractions are performed in order from left
to right.
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Example 5-9
Compute each of the following.
a. 2 − 5 − 5
2 − 5 − 5 = −3 − 5 = −8
b. 3 − 7 + 3
3 − 7 + 3 = −4 + 3 = −1
c. 3 − (7 − 3)
3 − (7 − 3) = 3 − 4 = −1
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