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Progressive Mathematics Initiative
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7th Grade Math
Number System
2013-01-28
www.njctl.org
Setting the PowerPoint View
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minimize the ribbon at the top of the screen.
• On the View menu, confirm that Ruler is deselected.
• On the View tab, click Fit to Window.
Use Slide Show View to Administer Assessment Items
To administer the numbered assessment items in this
presentation, use the Slide Show view. (See Slide 6 for an
example.)
Number System Unit Topics
Click on the topic to
go to that section
• Number System, Opposites & Absolute Value
•
•
•
•
•
•
•
•
Comparing and Ordering Rational Numbers
Adding Rational Numbers
Turning Subtraction Into Addition
Adding and Subtracting Rational Numbers Review
Multiplying Rational Numbers
Dividing Rational Numbers
Operations with Rational Numbers
Converting Rational Numbers to Decimals
Common Core Standards: 7.NS.1, 7.NS.2, 7.NS.3
Number System, Opposites &
Absolute Value
Return to
Table of
Contents
1
Do you know what an integer is?
Yes
No
Number System
Real
Rational
1/3
Irrational
Integer
5/2
Whole
0.22
Natural
1,2,3...
1/5
-3/4
0
...-4, -3, -2, -1
-2.756
-1/11
8.3
Define Integer
Definition of Integer:
The set of whole numbers, their opposites and
zero.
Examples of Integer:
{...-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7...}
X
Define Rational
Definition of Rational:
A number that can be written as a simple fraction
(Set of integers and decimals that repeat or terminate)
Examples of rational numbers:
0, -5, 8, 0.44, -0.23,
9 ,½
X
Define Irrational
Definition of Irrational:
A real number that cannot be written as a simple fraction
Examples of irrational numbers:
X
Classify each number as specific as possible:
Integer, Rational or Irrational
3.2
-6
5
½
5
-65
integer
-6.32
π
-21
3¾
0
rational
¾
1
9
2.34437 x 10
3
irrational
Rational Numbers on a Number Line
Negative
Numbers
-5
-4
-3
Numbers to the
left of zero are
less than zero
Positive
Numbers
Zero
-2
-1
0
1
2
Zero is neither
positive or negative
3
4
5
Numbers to the
right of zero are
greater than zero
2
Which of the following are examples of integers?
A
-5
B
0
C
-3.2
D
12
1
2
E
3
Which of the following are examples of rational
numbers?
B
1
3
-3
C
10
D
0.25
E
75%
A
Numbers In Our World
Numbers can represent everyday
situations
You might hear "And the quarterback is sacked for
a loss of 5 yards."
This can be represented as an integer: -5
Or, "The total snow fall this year has been 6 inches
more than normal."
This can be represented as an integer: +6 or 6
Write a number to represent each
situation:
1. Spending $6.75
2. Gain of 11 pounds
3. Depositing $700
4. 10 degrees below zero
5. 8 strokes under par (par = 0)
6.
feet above sea level
4
Which of the following numbers best represents
the following scenario:
The effect on your wallet when you spend $10.25.
A
-10.25
B
10.25
C
0
D
+/- 10.25
5
Which of the following integers best represents
the following scenario:
Earning $50 shoveling snow.
A
-50
B
50
C
0
D
+/- 50
6
Which of the following numbers best
represents the following scenario:
You dive
A
B
C
D
0
feet to explore a sunken ship.
Opposites
The numbers -4 and 4 are shown on the number line.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Both numbers are 4 units from 0,
but 4 is to the right of 0 and -4 is to the left of zero.
The numbers -4 and 4 are opposites.
Opposites are two numbers which are the same distance from
zero.
7
What is the opposite of -7?
8
What is the opposite of 18.2?
What happens when you add two opposites?
Try it and see...
A number and its opposite have a sum of zero.
Click to Reveal
Numbers and their opposites are called additive inverses.
Jeopardy
Integers are used in game shows.
In the game of Jeopardy you:
• gain points for a correct response
• lose points for an incorrect response
• can have a positive or negative score
When a contestant gets a $100 question correct:
Score = $100
Then a $50 question incorrect:
Score = $50
Then a $200 question incorrect:
Score = -$150
How did the score become negative?
Let's take a look...
Let's organize our thoughts...
When a contestant gets a
$100 question correct
Then a $50 question
incorrect
Question
Answered
Integer
Representation
New
Score
100
Correct
100
100
50
Incorrect
-50
50
200
Incorrect
-200
-150
Then a $200 question
incorrect
Now you try...
When a contestant gets a
$150 question incorrect
Then a $50 question
incorrect
Then a $200 question
correct
Question
Answered
Integer
Representation
New
Score
150
Incorrect
-150
-150
50
Incorrect
-50
-200
200
Correct
200
0
Now you try...
When a contestant gets a
$50 question incorrect
Then a $150 question
correct
Then a $200 question
incorrect
Question
Answered
Integer
Representation
New
Score
9
After the following 3 responses what would the
contestants score be?
$100 incorrect
$200 correct
$50 incorrect
10
After the following 3 responses what would the
contestants score be?
$200 correct
$50 correct
$300 incorrect
11
After the following 3 responses what would the
contestants score be?
$150 incorrect
$50 correct
$100 correct
12
After the following 3 responses what would the
contestants score be?
$50 incorrect
$50 incorrect
$100 incorrect
13
After the following 3 responses what would the
contestants score be?
$200 correct
$50 correct
$100 incorrect
To Review
• An integer is a whole number, zero or its opposite.
• A rational number is a number that can be written as a
simple fraction.
• An irrational number is a number that cannot be written
as a simple fraction.
• Number lines have negative numbers to the left of zero
and then positive numbers to the right.
• Zero is neither positive nor negative.
• Numbers can represent real life situations.
X
Absolute Value of Numbers
The absolute value is the distance a number is from
zero on the number line, regardless of
direction.
Distance and absolute value are
always non-negative (positive or zero).
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
1
2 3 4
5 6
7 8 9 10
What is the distance from 0 to 5?
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2 3 4
5 6
What is the distance from 0 to -5?
7 8
9 10
Absolute value is symbolized by two
vertical bars
|4|
-10 -9 -8 -7 -6
This is read, "the absolute value of 4"
-5 -4 -3 -2 -1
0
1
2
3
What is the | 4 | ?
4
5
6
7
8
9 10
Use the number line to find absolute value.
Move
to
check
|9.6| = 9.6
Move
to check
|-9| = 9
Move
to check
|-4| = 4
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
3 4
5 6
7
8
9 10
14
Find
15
Find |-8|
16
What is
?
17
What is
?
18
Find
19
What is the absolute value of the number shown
in the generator? (Click for web site)
20
Which numbers have 15 as their absolute value?
A
-30
B
-15
C
0
D
15
E
30
21
Which numbers have 100 as their absolute value?
A
-100
B
-50
C
0
D
50
E
100
Comparing and
Ordering Rational
Numbers
Return to
Table of
Contents
Use the Number Line
To compare rational numbers, plot points on
the number line.
The numbers farther to the right are larger.
The numbers farther to the left are smaller.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
3 4
5 6
7
8
9 10
Place the number tiles in the correct places on the
number line.
Now, can you see:
Which integer is largest?
Which is smallest?
Where do rational numbers go on the number line?
Go to the board and write in the following numbers:
Put these numbers on the number line.
Which number is the largest? The smallest?
Comparing Positive Numbers
Numbers can be equal to; less than; or more than
another number.
The symbols that we use are:
Equals "="
Less than "<"
Greater than ">"
For example:
4=4
4<6
4>2
When using < or >, remember that the smaller side
points at the smaller number.
22
10.5 is ______ 15.2.
A
=
B
<
C
>
23
7.5 is ______ 7.5
A
=
B
<
C
>
24
3.2 is ______ 5.7
A
=
B
<
C
>
Comparing Negative Numbers
The larger the absolute value of a negative number, the smaller
the number. That's because it is farther from zero, but in the
negative direction.
For example:
-4 = -4
-4 > -6
-4 < -2
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
3 4
5 6
7
8
9 10
Remember, the number farther to the right on a number line is
larger.
Comparing Negative Numbers
One way to think of this is in terms of money. You'd
rather have $20 than $10.
But you'd rather owe someone $10 than $20.
Owing money can be thought of as having a
negative amount of money, since you need to get
that much money back just to get to zero.
So owing $10 can be thought of as -$10.
25
-4.75 ______ -4.75
A
=
B
<
C
>
26
-4 ______ -5
A
=
B
<
C
>
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
3 4
5 6
7
8
9 10
27
A
=
B
<
C
>
28
-14.75 is ______ -6.2
A
=
B
<
C
>
29
-14.2 is ______ -14.3
A
=
B
<
C
>
Comparing All Numbers
Any positive number is greater than zero or any
negative number.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
3 4
5 6
7
8
9 10
Any negative number is less than zero or any positive
number.
Drag the appropriate inequality symbol between the
following pairs of numbers:
< >
1)
-3.2
5.8
2)
-237
-259
3)
63
36
4)
-10.2
-15.4
5)
-6.7
-3.9
6)
127
172
7)
-24
-17
8)
9)
-8.75
-8.25
10)
-10
-7
30
A
=
B
<
C
>
31
A
=
B
<
C
>
32
A
=
B
<
C
>
33
A
=
B
<
C
>
34
A
=
B
<
C
>
35
A
=
B
<
C
>
A thermometer can be thought
of as a vertical number line.
Positive numbers are above
zero and negative numbers are
below zero.
36
If the temperature reading on a thermometer is
10℃, what will the new reading be if the
temperature:
falls 3 degrees?
37
If the temperature reading on a thermometer is
10℃, what will the new reading be if the
temperature:
rises 5 degrees?
38
If the temperature reading on a thermometer is
10℃, what will the new reading be if the
temperature:
falls 12 degrees?
39
If the temperature reading on a thermometer is
-3℃, what will the new reading be if the
temperature:
falls 3 degrees?
40
If the temperature reading on a thermometer is
-3℃, what will the new reading be if the
temperature:
rises 5 degrees?
41
If the temperature reading on a thermometer is
-3℃, what will the new reading be if the
temperature:
falls 12 degrees?
Adding Rational
Numbers
Return to
Table of
Contents
Symbols
We will use "+" to indicate addition and "-" for subtraction.
Parentheses will also be used to show things more clearly. For
instance, if we want to add -3 to 4 we will write:
4 + (-3), which is clearer than 4 + -3.
Or if we want to subtract -4 from -5 we will write:
-5 - (-4), which is clearer than -5 - -4.
Addition: A walk on the number line.
While this section is titled "Addition" we're going to
learn here how to both add and subtract using the
number line.
Addition and subtraction are inverse operations (they
have the opposite effect). If you add a number and then
subtract the same number you haven't changed
anything.
Addition undoes subtraction, and vice versa.
Addition: A walk on the number line.
Here's how it works.
1.
2.
3.
4.
Start at zero
Walk the number of steps indicated by the first number.
Walk the number of steps given by the second number.
Look down, you're standing on the answer.
Rules for directions
• Go to the right for positive numbers
• Go to the left for negative numbers
• Go in the opposite direction when subtracting, rather
than adding, the second number
• Subtracting a negative number means you move to the
right: since that's the opposite of moving to the left
Let's do 3 + 4 on the number line.
1.
2.
3.
4.
Start at zero
Walk the number of steps indicated by the first number.
Take the number of steps given by the second number.
Look down, you're standing on the answer.
Let's do 3 + 4 on the number line.
1. Start at zero
2. Walk the number of steps indicated by the first number.
3. Take the number of steps given by the second number.
4. Look down, you're standing on the answer.
• Go to the right for positive numbers
Let's do 3 + 4 on the number line.
1. Start at zero
2. Walk the number of steps indicated by the first number.
3. Take the number of steps given by the second number.
4. Look down, you're standing on the answer.
• Go to the right for positive numbers
Let's do -4 + (-5) on the number line.
1.
2.
3.
4.
Start at zero
Walk the number of steps indicated by the first number.
Take the number of steps given by the second number.
Look down, you're standing on the answer.
Let's do -4 + (-5) on the number line.
1. Start at zero
2. Walk the number of steps indicated by the first number.
3. Take the number of steps given by the second number.
4. Look down, you're standing on the answer.
• Go to the left for negative numbers
Let's do -4 + (-5) on the number line.
1. Start at zero
2. Walk the number of steps indicated by the first number.
3. Take the number of steps given by the second number.
4. Look down, you're standing on the answer.
• Go to the left for negative numbers
Let's do 5 + -7 on the number line.
1.
2.
3.
4.
Start at zero
Walk the number of steps indicated by the first number.
Take the number of steps given by the second number.
Look down, you're standing on the answer.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
3 4
5 6
7
8
9 10
Let's do 5 + -7 on the number line.
1. Start at zero
2. Walk the number of steps indicated by the first number.
3. Take the number of steps given by the second number.
4. Look down, you're standing on the answer.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
• Go to the right for positive numbers
3 4
5 6
7
8
9 10
Let's do 5 + -7 on the number line.
1. Start at zero
2. Walk the number of steps indicated by the first number.
3. Take the number of steps given by the second number.
4. Look down, you're standing on the answer.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
• Go to the left for negative numbers
3 4
5 6
7
8
9 10
Let's do -4 + 9.5 on the number line.
1.
2.
3.
4.
Start at zero
Walk the number of steps indicated by the first number.
Take the number of steps given by the second number.
Look down, you're standing on the answer.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
3 4
5 6
7
8
9 10
Let's do -4 + 9.5 on the number line.
1. Start at zero
2. Walk the number of steps indicated by the first number.
3. Take the number of steps given by the second number.
4. Look down, you're standing on the answer.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
• Go to the left for negative numbers
3 4
5 6
7
8
9 10
Let's do -4 + 9.5 on the number line.
1. Start at zero
2. Walk the number of steps indicated by the first number.
3. Take the number of steps given by the second number.
4. Look down, you're standing on the answer.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
• Go to the right for positive numbers
3 4
5 6
7
8
9 10
Addition: Using Absolute Values
You can always add using the number line.
But if we study our results, we can see how to get the
same answers without having to draw the number line.
We'll get the same answers, but more easily.
Addition: Using Absolute Values
3+4=7
-4 + (-5) = -9
-10-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9 10
-10 -9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9 10
-4 + 9.5 = 5.5
-10 -9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9 10
5 + (-7) = -2
-10 -9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 910
We can see some patterns here that allow us to create rules to
get these answers without drawing.
Addition: Using Absolute Values
To add integers with the same sign
1. Add the absolute value of the integers.
2. The sign stays the same.
(Same sign, find the sum)
3+4=7
-10-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9 10
3 + 4 = 7; both signs are positive; so 3 + 4 = 7
-4 + (-5) = -9
-10 -9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9 10
4 + 5 = 9; both signs are negative; so -4 + (-5) = -9
Interpreting the Absolute Value Approach
The reason the absolute value approach works, if the signs of
the integers are the same, is:
The absolute value is the distance you travel in a direction,
positive or negative.
If both numbers have the same sign, the distances will add
together, since they're both asking you to travel in the same
direction.
If you walk one mile west and then two miles west, you'll be
three miles west of where you started.
Addition: Using Absolute Values
To add integers with different signs
1.
Find the difference of the absolute values of
the integers.
2.
Keep the sign of the integer with the greater
absolute value.
(Different signs, find the difference)
-4 + 9.5 = 5.5
-10 -9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9 10
9.5 - 4 = 5.5; 9.5 > 4, and 9.5 is positive; so -4 + 9.5 = 5.5
5 + (-7) = -2
-10 -9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 910
7 - 5 = 2; 7 > 5 and 9 is negative; so 5 + (-7) = -2
Interpreting the Absolute Value Approach
If the signs of the integers are the different:
nd
For the 2 leg of styour trip you're traveling in the opposite
direction of the 1 leg, undoing some of your original travel.
The total distance you are from your starting point will be the
difference between the two distances.
The sign of the answer must be the same as that of the larger
number, since that's the direction you traveled farther.
If you walk one mile west and then two miles east, you'll be
one mile east of where you started.
Adding Rational Numbers:
To add integers with the same sign
1.
Add the absolute value of the integers.
2.
The sign stays the same.
(Same sign, find the sum)
To add integers with different signs
1. Find the difference of the absolute values of
the integers.
2. Keep the sign of the integer with the greater
absolute value.
(Different signs, find the difference)
42
11 + (-4) =
43
-4 + (-4) =
44
17 + (-20) =
45
-15 + (-30) =
46
-5 + 10 =
47
-9 + (-2.3) =
48
5.3 + (-8.4) =
49
4.8 + (12.6) =
50
-14.3 + 7.93 =
51
52
53
Turning Subtraction Into
Addition
Return to
Table of
Contents
Subtracting Numbers
Subtracting a number is the same as adding it's opposite.
(Add a line,
change the sign of the second number)
Subtracting Numbers
Subtracting a number is the same as adding it's
opposite.
We can see this from the number line, remembering our
rules for directions. Compare these two problems: 8 - 5
and 8 + (-5).
For "8 - 5" we move 8 steps to the right and then move
5 steps to the left, since the negative sign tells us to
move in the opposite direction that we would for +5.
-10-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9 10
For "8 + (-5)" we move 8 steps to the right, and then 5
steps to the left since we're adding -5.
-10-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9 10
Either way, we end up at +3.
Subtracting Negative Numbers
Compare these two problems: 8 - (-2) and 8 + 2.
For "8 - (-2)" we move 8 steps to the right and then move
2 steps to the right, since the negative sign tells us to
move in the opposite direction that we would for -2.
-10-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9 10
For "8 + 2" we move 8 steps to the right, and then 2 steps
to the right since we're adding 2.
-10-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 9 10
Either way, we end up at +10.
Subtracting Numbers
Any subtraction can be turned into addition by:
• Changing the subtraction sign to addition.
• Changing the integer after the subtraction sign to its
opposite.
EXAMPLES:
5 - (-3) is the same as 5 + 3
-12 - 17 is the same as -12 + (-17)
54
Convert the subtraction problem into an
addition problem.
8–4
A
-8 + 4
B
8 + (-4)
C
-8 + (-4)
D
8+4
55
Convert the subtraction problem into an
addition problem.
-3.7 - (-10.1)
A
-3.7 + 10.1
B
3.7 + (-10.1)
C
-3.7 + (-10.1)
D
3.7 + 10.1
56
Convert the subtraction problem into an
addition problem.
A
B
C
D
57
Convert the subtraction problem into an
addition problem.
A
B
C
D
58
Convert the subtraction problem into an
addition problem.
1-9
A
-1 + 9
B
1 + (-9)
C
-1 + (-9)
D
1+9
Adding and Subtracting
Rational Numbers Review
Return to
Table of
Contents
59
Calculate the sum or difference.
-6 – 2
60
Calculate the sum or difference.
5 + (-5)
61
Calculate the sum or difference.
-10.5 + 6.2
62
Calculate the sum or difference.
7.3 – (-4)
63
Calculate the sum or difference.
64
Calculate the sum or difference.
9.27 + (-8.38)
65
Calculate the sum or difference.
-4.2 + (-5.9)
66
Calculate the sum or difference.
-2 – (-3.95)
67
Calculate the sum or difference.
5 - 6 + (-7)
68
Calculate the sum or difference.
19 + (-12) - 11
69
Calculate the sum or difference.
-2.3 + 4.1 + (-12.7)
70
Calculate the sum or difference.
-8.3 - (-3.7) + 5.2
71
Calculate the sum or difference.
Multiplying Rational Numbers
Return to
Table of
Contents
Symbols
In the past, you may have used "x" to indicate
multiplication. For example "3 times 4" would
have been written as 3 x 4.
However, that will be a problem in the future since
the letter "x" is used in algebra as a variable.
There are two ways we will indicate multiplication:
3 times 4 will be written as either 3∙4 or 3(4).
Parentheses
The second method of showing multiplication, 3(4),
is to put the second number in parentheses.
Parentheses have also been used for other purposes.
When we want to add -3 to 4 we will write that as 4 + (-3),
which is clearer than 4 + -3.
Also, whatever operation is in parentheses is done first.
The way to write that we want to subtract 4 from 6 and
then divide by 2 would be (6 - 4) ÷ 2 = 1. Removing the
parentheses would yield 6 - 4 ÷ 2 = 4, since we work from
left to right.
Multiplying Rational Numbers
Multiplication is just a quick way of writing repeated
additions.
These are all equivalent:
3∙4
3 +3 + 3 + 3
4+4+4
they all equal 12.
Multiplying Rational Numbers
We know how to add with a number line.
Let's just do the same thing with multiplication by just
doing repeated addition.
To do that, we'll start at zero and then just keep adding:
either 3+3+3+3 or 4+4+4.
We should get the same result either way, 12.
Let's do 4 x 3 on the number line.
We'll do it as 3+3+3+3
and as 4+4+4
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Try 5 x 2 on the number line.
Try it as 5+5 and as 2+2+2+2+2
-3 -2 -1 0
1
2 3 4
5 6 7
8 9 10 11 12 13 14 15 16 17
Multiplying Negative Numbers
Let's use the same approach to determine rules for
multiplying negative numbers.
If we have 4 x (-3) we know we can think of that as (-3)
added to itself 4 times. But we don't know how to think of
adding 4 to itself -3 times, so let's just get our answer this
way:
4 x (-3) = (-3)+(-3)+(-3)+(-3)
4 x (-3) On the Number Line
4 x (-3) = (-3)(-3)(-3)(-3)(-3)
-17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
So we can see that 4 x (-3) = -12
1 2
3
Sign Rules for
Multiplying Rational Numbers
4∙3
4+4+4
12
Multiplying positive numbers has a
positive value.
4(-3)
(-3) + (-3) + (-3)
-12
Multiplying a negative number and a
positive number has a negative value.
?
What about multiplying together two
negative numbers: what is the sign of
(-4)(-3)
Multiplying Negative Numbers
We can't add something to itself a negative number
of time; we don't know what that means.
But we can think of our rule from earlier, where a (-)
sign tells us to reverse direction.
So if we think of (-4)(-3) as -(4)(-3) we can then see
that the answer will be the opposite of (-12):12
Each negative sign makes us reverse direction once,
so two multiplied together gets us back to the
positive direction.
Sign Rules for
Multiplying Rational Numbers
4∙3
4+4+4
12
Multiplying positive numbers yields a
positive result.
4(-3)
(-4) + (-4) + (-4)
-12
Multiplying a negative number and a
positive number yields a negative
result.
(-4)(-3)
-((-4) + (-4) + (-4))
-(-12)
12
Multiplying two negative numbers
together yields a positive result.
Multiplying Rational Numbers
Every time you multiply by a negative number you
change the sign.
Multiplying with one negative number makes the
answer negative.
Multiplying with a second negative change the answer
back to positive.
1(-3) = -3
-3(-4) = 12
Multiplying Rational Numbers
When multiplying two numbers with the same sign (+ or -),
the product is positive.
When multiplying two numbers with different signs, the
product is negative.
When multiplying several numbers with different signs,
count the number of negatives.
An even amount of negatives = positive product
An odd amount of negatives = negative product
Multiplying Rational Numbers
We can also see these rules when we look at
the patterns below:
3(3) = 9
3(2) = 6
3(1) = 3
3(0) = 0
3(-1) = -3
3(-2) = -6
3(-3) = -9
-5(3) = -15
-5(2) = -10
-5(1) = -5
-5(0) = 0
-5(-1) = 5
-5(-2) = 10
-5(-3) = 15
2.5(3) = 7.5
2.5(2) = 5
2.5(1) = 2.5
2.5(0) = 0
2.5(-1) = -2.5
2.5(-2) = -5
2.5(-3) = -7.5
-3.1(3)(-2) = 18.3
-3.1(2)(-2) = 12.4
-3.1(1)(-2) = 6.2
-3.1(0)(-2) = 0
-3.1(-1)(-2) = -6.2
-3.1(-2)(-2) = -12.4
-3.1(-3)(-2) = -18.3
72
What is the value of (-3)(-9)?
73
What is the value of 3.1 ∙ 7?
74
What is the value of 5(-4.82)?
75
What is the value of (-3.2)(-6.4)?
76
What is the value of -8 ∙ 7.6?
77
What is the value of (-5.12)(-9)?
78
What is the value of -2(-7)(-4)?
79
What is the value of:
80
What is the value of:
81
What is the value of:
82
What is the value of:
Dividing Rational Numbers
Return to
Table of
Contents
Division Symbols
You may have mostly used the "÷" symbol to show
division.
We will also represent division as a fraction.
Remember:
9 =3
3
9÷3=3
are both ways to show division.
Dividing Rational Numbers
Division is the opposite of multiplication, just like
subtraction is the opposite of addition.
When you divide a number, by another number, you are
finding out how many of that second number would have to
add together to get the first number.
For instance, since 5∙2 = 10, that means that I could divide
10 into 5 groups of 2's, or 2 groups of 5's.
This is just what we did on the number line for
multiplication, but backwards.
Let's try 10 ÷ 2
Try 10 ÷ 2 on the number line
This means how many lengths of 2 would
be needed to add up to 10.
-3 -2 -1 0
1
2 3 4
5 6 7
8 9 10 11 12 13 14 15 16 17
The answer is 5: the number of red arrows
of length 2 that end to end give a total
length of 10.
Try 10 ÷ 5 on the number line
This means how many lengths of 5 would
be needed to add up to 10.
-3 -2 -1 0
1
2 3 4
5 6 7
8 9 10 11 12 13 14 15 16 17
The answer is 2: the number of green
arrows of length 5 that, end to end, give a
total length of 10.
-12 ÷ 3 On the Number Line
This can be read as how many steps of 3 would
it take to get to -12.
-17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
1 2
3
Each red arrow represents a step of 3,
so we can see that -12 ÷ 3 = -4 (The answer is negative
because the steps are to the left.)
Dividing Rational Numbers
-15 ÷ 3 = -5
-15
= -5
3
We know that -5(3) = -15,
so it makes sense that -15 ÷ 3 = -5.
We also know 3(-5) = -15.
So, what is the value of -15 ÷ -5
The value must be positive 3, because 3(-5) = -15
Dividing Rational Numbers
The quotient of two positive numbers is positive.
The quotient of a positive and negative number is negative.
The quotient of two negative numbers is positive.
When dividing several numbers with different signs, count
the number of negatives.
An even amount of negatives = positive quotient
An odd amount of negatives = negative quotient
83
Find the value of 32 ÷ 4
84
Find the value of:
85
Find the value of:
86
Find the value of:
87
Find the value of:
88
Find the value of:
89
Find the value of:
90
Find the value of:
91
Find the value of:
92
Find the value of:
93
Find the value of:
Operations with Rational
Numbers
Return to
Table of
Contents
When simplifying expressions with rational
numbers,
you must follow the order of operations while
remembering your rules for
positive and negative numbers!
Order of Operations
Parentheses
(ALL Grouping Symbols)
Exponents
Multiplication
Division
Addition
Subtraction
Complete at the same
time...whichever comes
first...from left to right
Let's simplify this step by step...
-7 + (-3)[5 - (-2)]
What should you do first?
5 - (-2) = 5 + 2 = 7
What should you do next?
(-3)(7) = -21
What is your last step?
-7 + (-21) = -28
Let's simplify this step by step...
What should you do first?
Click
to
Reveal
What should you do third?
Click
to
Reveal
What should you do second?
Click
to
Reveal
What should you do last?
Click
to
Reveal
94
Simplify the expression.
-12÷3(-4)
95
[-1 - (-5)] + [7(3 - 8)]
96
40 - (-5)(-9)(2)
97
5.8 - 6.3 + 2.5
98
Simplify the expression.
-3(-4.7)(5-3.2)
99
100
101 Complete the first step of simplifying. What is
your answer?
[3.2 + (-15.6)] - 6[4.1 - (-5.3)]
102 Complete the next step of simplifying. What is
your answer?
[3.2 + (-15.6)] - 6[4.1 - (-5.3)]
103 Complete the next step of simplifying. What is
your answer?
[3.2 + (-15.6)] - 6[4.1 - (-5.3)]
104 Complete the next step of simplifying. What is
your answer?
[3.2 + (-15.6)] - 6[4.1 - (-5.3)]
105 Simplify the expression.
106 Simplify the expression.
107
108
109 (-4.75)(3) - (-8.3)
Solve this one in your groups.
How about this one?
110
111
[(-3.2)(2) + (-5)(4)][4.5 + (-1.2)]
112
113
114
115
Converting Rational Numbers
to Decimals
Return to
Table of
Contents
Do you recall the definition of a
Rational Number?
Definition of Rational:
x
A number that can be written as a simple fraction
(Set of integers and decimals that repeat or terminate)
In order for a number to be rational, you should be able to divide
the fraction and have the decimal either terminate or repeat.
How can you convert Rational
Numbers into Decimals?
Use long division!
Divide the numerator by the denominator.
If the decimal terminates or repeats, then you have a
rational number.
If the decimal continues forever, then you have an
irrational number.
Long
Division
Review
116
Convert to a decimal
(if the number is repeating, use bar notation in your
notebook but enter the repeating number(s) 3 times on
your responder)
117 Convert to a decimal
(if the number is repeating, use bar notation in your
notebook but enter the repeating number(s) 3 times on
your responder)
118 Convert to a decimal
(if the number is repeating, use bar notation in your
notebook but enter the repeating number(s) 3 times on
your responder)
119
Convert to a decimal
(if the number is repeating, use bar notation in your
notebook but enter the repeating number(s) 3 times on
your responder)
120
Convert to a decimal
(if the number is repeating, use bar notation in your
notebook but enter the repeating number(s) 3 times on
your responder)
121 Convert to a decimal
(if the number is repeating, use bar notation in your
notebook but enter the repeating number(s) 3 times on
your responder)
122
Convert to a decimal
(if the number is repeating, use bar notation in your
notebook but enter the repeating number(s) 3 times on
your responder)
123 Convert to a decimal
(if the number is repeating, use bar notation in your
notebook but enter the repeating number(s) 3 times on
your responder)
124 Convert to a decimal
(if the number is repeating, use bar notation in your
notebook but enter the repeating number(s) 3 times on
your responder)
125 Convert to a decimal
(if the number is repeating, use bar notation in your
notebook but enter the repeating number(s) 3 times on
your responder)
126 Convert to a decimal
(if the number is repeating, use bar notation in your
notebook but enter the repeating number(s) 3 times on
your responder)
127
Convert to a decimal
(if the number is repeating, use bar notation in your
notebook but enter the repeating number(s) 3 times on
your responder)