Download Algebra 1 – 2.5

Algebra 1 – 2.5
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1. The order in which you
add numbers in a sum
doesn’t matter.
2. If you add more than 2
numbers, the order in
which you group them
doesn’t matter.
3. When you add 0 to any
number, the result is
the number itself.
4. When you add any
number to its opposite,
the result is 0. When
the sum of two
numbers is 0, each
number is the opposite
of the other.
A. For any three
numbers, a, b, and c,
a+(b+c)=(a+b)+c.
B. For any number a,
a+0=a.
C. For any two numbers a
and b, a+(-a)=0 and if
a+b=0, b=-a.
D. For any two numbers a
and b, a+b=b+a.
1.
2.
3.
4.
5.
The order in which you
multiply two numbers
doesn’t matter.
If you multiply more than 2
numbers, the order in
which you group them
doesn’t matter.
Multiplying a number by a
sum is the same as
multiplying the number by
each term in the sum and
then adding the result.
When you multiply 1 by
any number, the result is
that number.
When you multiply a
nonzero number to its
reciprocal, the result is 1.
When the product of two
numbers is 1, each number
is the reciprocal of the
other.
A. For any three
numbers, a, b, and c,
a(bc)=(ab)c.
B. For any number a,
a·1=a.
C. For any two numbers a
and b, a· (1/a)=0 and if
a·b=1, b=1/a.
D. For any three numbers
a, b, and c,
a(b+c)=ab+ac.
E. For any two numbers a
and b, a·b=b·a.
They all have names, of
course.
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Commutative Property of Addition
Associative Property of Addition
Commutative Property of Multiplication
Associative Property of Multiplication
Additive Identity
Additive Inverse
Multiplicative Identity
Multiplicative Inverse
Distributive Property
Rewrite these with letters.
1. Dividing is the same as multiplying by the
reciprocal.
2. Subtracting is the same as adding the
opposite.
3. If you have a product of two numbers, and you
find the products of the opposites of the
numbers, you get the same result.
4. If you multiply two numbers together and the
result is 1, then the numbers are reciprocals.
Zero Product Property
If ab=0, then a=0 or b=0.
Ok so prove it.
• Assume ab=0.
• If a=0, you can stop doing the proof.
– WHY?
• Assume a isn’t zero. Then it has a
reciprocal, 1/a.
– WHY?
• Multiply both sides of the equation by 1/a.
– What happens?
How does this prove the property?
Making up new arithmetic.
• The binary operation ♥ is defined by the
following rule: x ♥y = 3x + y
• Explain how to find 416 ♥18.
• Evaluate it.
• Is this property commutative?
• Is either 1 or 0 the ♥-identity?
REFLECTIONS
• To build a rectangular dog pen, Cheng uses a
wall of his house for one of the long sides. Let L
equal the length of the longer side. Let W be the
length of the shorter.
• Write an expression for the amount of fencing
Cheng needs to buy.
• How much should he buy if he wants a length of
8 ft and a width of 12 feet?
• How much should he buy if he wants a length of
5 ft and a width of 20 feet?
• Suppose the length is 9 ft more than the width.
Use only ONE variable to write an expression.
Evaluate 4(2x+3) + 2(x+1) – 7 for…
x=
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1
6
-2
½
Simplify the expression. When you
evaluate the simplified expression for x=1
and x=6, do you get the same result this
time?
Why are variables useful?
How can you invent a number
trick that always gives you the
same ending number?