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7/12/2010
Section 5.2
Multiplication, Division,
and Other Properties of Integers
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Use black counters for positive integers, red
counters for negative integers.
Start with a bag containing an _________ number
of red and black counters.
Use putting counters in a bag a certain number
of times as _________ integers.
Use taking counters out of a bag a certain
number of times as _________ integers.
The first factor tells you how many _________ you
are putting in or taking out.
The second factor tells you the _________ of the
group that you are actually putting in or taking
out.
Remove zero _________.
Remaining counters is answer to problem.
Start with a zero-charged field. (equal number
of positive and negative counters)
Take out or put in # of groups according to
_______ and absolute value of first integer.
The
h second
d integer is the
h amount/size off each
h
group and type (positive or negative) of counter
to be taken out or put in.
The resulting “charge” is your answer.
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Using a Counters Model
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Using a Charged Field Model
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Using the Number Line Model
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Examples
1) 4 x 2
Put in 4 groups of 2 positive counters
2) - 5 x 3
groups
p of 3 positive
p
counters
Take out 5 g
3) 2 x (- 1)
Put in 2 groups of 1 negative counter
4) - 2 x (- 3)
Take out 2 groups of 3 negative counters
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Examples
1) - 2 x 3
2) - 4 x – 1
3) 3 x - 2
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Start at 0.
First factor:
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◦ If positive: Face right, take that # of steps. (Walk east)
◦ If negative: Face left, take that # of steps. (Walk west)
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Examples
1) - 2 x 4
2) - 3 x - 2
Second factor:
◦ If positive, walk forward that # of units.
◦ If negative, walk backwards that # of units.
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Product:
◦ Positive integers are modeled by miles east of 0.
◦ Negative integers are modeled by miles west of 0.
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Multiplying integers with the _________ sign:
Multiply the absolute values. The product is
_________.
Multiplying integers with _________ signs:
Multiply the absolute values. The product is
_________.
(When multiplying more than two integers,
count the number of negative signs. If the
number of negative signs is odd, the product
will be negative. If the number of negative
signs is even, the product will be positive.)
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For all integers a, b and c, b ≠ 0,
a ÷ b = c iff c x b = a.
Procedures for Dividing Integers
` Dividing two integers with the same sign:
Divide
values.
Di id the
h absolute
b l
l
Quotient is positive.
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Dividing two integers with different signs:
Divide the absolute values.
Quotient is negative.
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Closure Property
For all integers a and b, ab is a unique integer.
Multiplicative Identity Property
1 is the unique integer such that for each
integer a, a x 1 = 1 x a = a.
Commutative Property
For all integers a and b, ab = ba.
Associative Property
For all integers a, b and c, (ab)c = a(bc).
Distributive Property
For all integers a, b and c, a(b +c) = ab + ac
and (b +c)a = ba + ca.
Zero Property for Multiplication
For all integers a, a(0) = 0(a) = 0.
Properties of Integer Division
For all integers a and b, a ≠ 0,
a ÷ a = 1,
a ÷ 1 = a,
0 ÷ a = 0,
0 and
d
ab ÷ a = b.
Remember: We do not divide an integer by 0,
because no unique quotient exists.
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Properties of Opposites
For all integers a and b,
-(-a) = a,
-a(-b) = ab,
(( a)b = a(( b) = -((ab
b),
) and
d
a(-1) = (-1)a = -a.
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Distributive Property for Multiplication over Subtraction:
For all integers a, b and c,
a(b – c) = ab – ac.
Distributive Property for Opposites over Addition:
For all integers a, b and c,
-(a + b) = -1(a + b) = -a + (-b) = -a + -b.
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