Transcript 2 - Mr. Hood

Integer Operations
Example 1 Let yellow = positive
This piece has a value of 1.
1
53
8
Write the value of these pieces.
I will add more pieces. Write a numerical expression
and simplify.
Example 2 Let red = negative
This piece has a value of –1.
1
 6   5 
 11
Write the value of these pieces.
I will add more pieces. Write a numerical expression
and simplify.
Rules of Addition
To add two numbers with the same sign:
Step 1 ADD
Step 2 ATTACH the common sign
–8 + (–5) = – 13
8 + 5 = 13
If the common
sign is positive,
the + is not
written!
Example 3 Let yellow = positive
Let red = negative
1   1 
0
Write a numerical expression and simplify.
This expression has a special name. Do you know
what it is?
zero pair
Adding or subtracting zero pairs will not affect
the value of an expression.
Let yellow = positive
Let red = negative
4   7 
3
When you add two numbers with opposite signs,
you will need to subtract (take away) the zero pairs.
Example 4 Let yellow = positive
Let red = negative
7   3
4
Write the value of these pieces.
I will add more pieces. Write a numerical expression.
To simplify remove all the zero pairs.
The pieces that remain model the simplified answer.
Example 5 Let yellow = positive
Let red = negative
83
5
Write the value of these pieces.
I will add more pieces. Write a numerical expression.
To simplify remove all the zero pairs.
The pieces that remain model the simplified answer.
Rules of Addition
To add two numbers with the opposite signs:
Step 1 SUBTRACT
Step 2 ATTACH the sign of the number with the
larger absolute value.
4 + (–9)
–5
Example 6 Write a numerical expression and simplify.
94
5
or
4   9 
5
Can you tell at a glance what the sign is for the
answer ?
Example 7 Write a numerical expression and simplify.
 5  11
6
or
11   5
6
Can you tell at a glance what the sign is for the
answer ?
Example 8 Find the sum.
1. Write problem.
2. Signs alike – add write
the common sign.
Example 9 Find the sum.
1. Write problem.
2. Signs unlike – subtract
write sign of the largest
absolute value.
 12   30
–42
7   15 
–8
 12   30
Properties of Addition
Commutative Property: The order in which
numbers are added does not change the sum.
x +y=y +x
9+4=4+9
Associative Property: The way numbers are
grouped when added does not change the sum.
(2 + 5) + 10 = 2 + (5 + 10)
(a + b) + c = a + (b + c)
Closure Property: The sum of any two real
numbers is a unique real number.
5+3=8
Properties of Addition
Identity Property: The sum of a number and 0
is the number.
–16 + 0 = –16
x+0=x
Inverse Property of addition:
number and its opposite is 0.
45   45  0
The sum of a
a  a  0
45   45  0 a    a   0
When two
signs are
needed use
parentheses or
a superscript!
Example 10 Name the property shown by the statement.
1. Write problem.
12 + –36 = –36 + 12
2. Name the property.
commutative
Example 11 Name the property shown by the statement.
1. Write problem.
0 + –8 = –8
2. Name the property.
identity
Example 12 Name the property shown by the statement.
1. Write problem.
(4 + 5) + 6 = (5 + 4) + 6
2. Name the property.
commutative
Example 13 Name the property shown by the statement.
1. Write problem.
2. Name the property.
–7
+7=0
inverse
Example 14 Name the property shown by the statement.
1. Write problem.
2. Name the property.
(9 + 6) + 5 = 9 + (6 + 5)
associative
Example 15 Find the sum.
1. Write problem.
 37  29   2
2. Follow the rules for  8   2
the order of operations
 10
to simplify.
If you override
the left to
right rule, you
must show the
step which gives
support to your
thinking!
 37  29   2
 37   2  29
 39  29
 10
Example 16 Find the sum.
1. Write problem.  12   6  15   2  12   6  15   2
2. Follow the rules for  12  9   2
the order of
operations to simplify.  3   2
5
If you override
the left to
right rule, you
must show the
step which gives
support to your
thinking!
 12   6  15   2
 18  15   2
 18   2  15
 20  15
5
Example 17 Find the sum.
58   100    78
1. Write problem.
2. Use associative
58   100    78
property to regroup.
58   178
Then simplify.
 120
Inductive Reasoning is making conclusions on patterns you
observe. What are the missing numbers in the following:
Does the pattern suggest a general rule
for multiplying a positive number by a
negative number?
The product of a positive number and
a negative number is negative.
33  9
32  6
3 1  3
30  0
3  1   3
3  2   6
3  3   9
What are the missing numbers in the following:
Does the pattern suggest a general
 3  3  9
rule for multiplying a negative number
 3  2  6
by a negative number?
 3  1  3
The product of a negative number
30  0
and a negative number is positive.
 3  1  3
 3  2  6
 3  3  9
Rules for Multiplication
The product of a positive number and a negative number is
negative.
12(–10) = –120
The product of a negative number and a negative number is
positive.
–12(–10) = 120
Summary: If the signs are the same, the product is
positive.
If the signs are not the same, the product is
negative.
Properties of Multiplication
Commutative Property: The order in which
numbers are multiplied does not change the
product. 9(4) = 4(9) x y = y x
Associative Property: The way numbers are
grouped when multiplied does not change the
product.
(2 • 5) • 10 = 2 • (5 • 10)
(a • b) • c = a • (b • c)
Properties of Multiplication
Identity Property: The product of a number and 1 is
the number. –16 • 1 = –16
x•1=x
Property of Zero: The product of a number and
0 is zero.
–16 • 0 = 0
x•0=0
Property of Negative One: The product of a
number and –1 is the opposite of the number.
x • (– 1) = – x
–16 • (–1) = 16
Inverse Property of multiplication: The product of
a number and it’s reciprocal is 1.
2 3
· =1
3 2
Example 1 Find the product.
1. Write problem.
2. Follow rules for the
order of operations to
simplify.
– 4(–3)(–5)
12(–5)
–60
Example 2 Find the product.
1
1. Write problem.
  65 
5
2. Follow order of operations
1
to simplify.
  65 
5
─6
If you alter the
left to right
rule, you must
show the step
which gives
support to your
thinking!
1
  30 
5
6
What
property
justifies
this
step?
Example 3 Find the product.
1. Write problem.
 14
2. Write in factor form.
Optional step.
(–1 ) (–1 ) (–1 ) (–1 )
3. Simplify.
Notice the negative one
is in parentheses.
What is the answer if
it is not in
parentheses?
 14  1
1
Example 4 Find the product.
 15
1. Write problem.
2. Write in factor form. (–1) (–1) (–1) (–1) (–1)
Optional step.
–1
3. Simplify.
Can you formulate a rule that works with
the product of negatives?
A product is negative if it has an odd number
of negative factors.
A product is positive if it has an even number
of negative factors.
Example 5 Simplify the expression.
1. Write problem.
– 6(y)(–y)
2. Multiply left to right.
– 6y(–y)
3. Multiply. Write variables
in power form.
Notice the
simplified answer
does NOT have
parentheses!
6y2
Example 6 Simplify the expression.
1. Write problem.
2. Write factored form.
3. Simplify.
Writing the
factored form
helps to
prevent errors!
4(–b)3
4 b  b  b 
– 4b3
Notice the
simplified answer
does NOT have
parentheses!
Example 7 Simplify the expression.
1. Write problem.
2(–x)(–x)(–x)(–x)
2. Multiply. Write the
answer in power form.
2x4
Notice the
simplified answer
does NOT have
parentheses!
Example 8 Evaluate the expression when x = –7.
1. Write problem.
2. Substitute.
2

 5  x   
 7
2

 5   7   
3. Follow rules for order
 7
of operations to simplify.
5
2

35  
 7
–10
Example 9 Evaluate the expression when x = –2.
1. Write problem.
2. Substitute.
3  x 3
3  23
3. Follow order of operations: 323
simplify within parentheses.
4. Evaluate the power.
5. Simplify.
3(8)
24
Example 10 A leaf floats down from a tree at a velocity of
–12 cm/sec. Find the displacement, which is the change in
position, of the leaf after 4.2 seconds. Note: An object’s
change in position when it drops can be found by multiplying its
velocity by the time it drops.
Let d = displacement
1. Write let statement.
2. Write verbal model. Displacement = Velocity • Time
3. Write algebraic model.
d = –12(4.2)
4. Solve.
5. Sentence.
= –50.4
The leaf’s displacement is –50.4 cm.
Subtracting Real Numbers
Rules of Subtraction
Adding the opposite of a number is equivalent
to subtracting the number.
Addition Problem
6 + (– 4) = 2
 
6  4 2
Equivalent Subtraction Problem
6– 4 = 2
 
6   4   2
Example 1 Find the difference.
1. Write problem.
– 9 – (–4)
2. Rewrite as addition.
9  4
3. Use rules of addition.
–5
For tonight’s
homework, you must
rewrite subtraction
as addition!
Example 2 Find the difference.
1. Write problem.
–9–4
2. Rewrite as addition.
 9   4 
3. Use rules of addition.
–13
Example 3 Find the difference.
5–7
5 + –7
–2
Example 4 Simplify the expression.
1. Write problem.
10 – 5 + 14 – (–18)
10 + (−5) + 14 + 18
5 + 14 + 18
3. Follow rules for order
Leave
of operations to simplify.
19 + 18
addition
37
alone!
2. Rewrite as addition.
Example 5 Simplify the expression.
1. Write problem.
2 5
7 
3 3
2  5

7      
 3  3
2. Rewrite as addition.
3. Follow rules for
order of operations
21  2   5 
to simplify.
 
 
3  3   3 
What property
19  5 
allows you to
  
3  3
move the 7 to
the far right?
14
3
2 5
7 
3 3
2  5

7      
 3  3
 2   5 
7        
 3   3 
7

7   
 3
21  7 
 
3  3 
14
3
Terms of an Expression
When an expression is written as a sum, the parts
that are added are the terms of the expression.
Find the terms of the expression.
The problem.
List the terms.
4x + y + 6
4x, y and 6
The problem.
Rewrite as addition.
List the terms.
–5 – x
–5 + – x
–5 and – x
Example 6 Find the terms of the expression.
1. Write problem.
3x – 14y – –36
2. Rewrite as addition.
3x + (–14y) + 36
3. List the terms.
3x, –14y and 36
Example 7 In February 1956 the temperature in Bismarck,
North Dakota fell 95 degrees overnight. The initial
temperature was 42 degrees Fahrenheit. What was the
final temperature?
1. Write an expression. 42 – 95
2. Rewrite as addition. 42 + (–95)
3. Use rules of addition. –53
4. Write a sentence.
The final temperature was –53 Fahrenheit.
Combining Like Terms
An algebraic expression is easier to evaluate when it is
simplified. The distributive property allows you to combine
like terms by adding their coefficients. What is a term?
A term is a number or the product of a number and
variable/s.
one term
7
x
one term
Terms are
separated by
7x
one term
addition.
7
one term 1  7
x
x
two terms
7x
two terms
–– x
7+
An algebraic expression is easier to evaluate when it is
simplified. The distributive property allows you to combine
like terms by adding their coefficients. What is a
coefficient?
In a term that is the product of a number and a variable,
the number is called the coefficient of the variable.
– 1 is the coefficient of x

–1x + 3x2

3 is the coefficient of x2
Like terms are terms in an expression that have
the same variable raised to the same power.
8x and 3x
4x2 and 4x
7m and –2m
25 and 10
Like Terms
Not Like Terms
Like Terms
Like Terms
Constants are
considered like
terms.
Example 1 Identify the like terms in the expression.
1. Write problem.
x2+–– 3x + 2x2 +–– 5 + 4x
2. Change subtraction
to addition.
3. Identify the like terms.
x2, 2x2
and
–3x, 4x
Example 2 Identify the like terms in the expression.
1. Write problem.
3m2 +– – 6m + 4m2 +–– 7 + m + 15
2. Change subtraction
to addition.
3. Identify the like terms.
3m2, 4m2
and
– 6m, m
and
–7, 15
An expression is simplified if it has: no grouping
symbols, no like terms, and no double signs.
The problem.
4c – c
Use the Identity Property 4c– – 1c–

to name the coefficient.
(4 – 1 )c
Distribute.
3c
Simplify.
If you have 4 cookies
and you eat one
cookie, how many
cookies are left?
This is the
mathematical proof
for combining like
terms!
Example 3 Simplify the expression.
1. Write problem.
5x2 +–– 7 + 3x2
2. Change subtraction to
addition.
8x2 + – 7
3. Combine like terms.
8x2 – 7
4. Undo the double sign.
It may be
helpful to
circle like
terms!
Write the
variable term
before the
constant term!
Example 4 Simplify the expression.
1. Write problem.
2. Change subtraction to
addition.
3. Combine like terms.
4. Undo the double sign.
I bet she wants
me to write that
in my notes so
I‘ll remember it!
3x2 +–– 5x + 4x2 –+– 7x
7x2 + – 12x
7x2 – 12x
Good form is
alphabetical
descending
order!