slide 2 / 249 7th grade math number...
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7th Grade Math
Number System
www.njctl.org
2015-06-11
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Number System Unit Topics
· 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
Click on the topic to go to that section
· Converting Rational Numbers to Decimals· Glossary
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Links to PARCC sample questions
Non-Calculator #10
Non-Calculator #6
Non-Calculator #14
Calculator #5
End of Year
Performance Based Assessment
Non-Calculator #12
Non-Calculator #4
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Sometimes when you subtract the fractions, you find that you can't because the first numerator is smaller than the second! When this happens, you need to regroup from the whole number.
How many thirds are in 1 whole?
How many fifths are in 1 whole?
How many ninths are in 1 whole?
Vocabulary words are identified with a dotted underline.
The underline is linked to the glossary at the end of the Notebook. It can also be printed for a word wall.
(Click on the dotted underline.)
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Back to
Instruction
FactorA whole number that can divide into another number with no remainder.
15 3 5
3 is a factor of 153 x 5 = 15
3 and 5 are factors of 15
1635 .1R
3 is not a factor of 16
A whole number that multiplies with another number to make a third number.
The charts have 4 parts.
Vocab Word1
Its meaning 2
Examples/ Counterexamples
3Link to return to the instructional page.
4
(As it is used in the
lesson.)
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Number System, Opposites &
Absolute Value
Return toTable ofContents
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1 Do you know what an integer is?
Yes
No
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Number System
0.22
Natural1,2,3...
Whole
0
Integer
...-4, -3, -2, -1
Rational
1/5
5/2
8.3
-2.756
-3/4
1/3
-1/11
Real
Irrational
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{...-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7...}
Definition of Integer:
The set of whole numbers, their opposites and zero.
Define Integer
Examples of Integer:
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Definition of Rational:
A number that can be written as a simple fraction
(Set of integers and decimals that repeat or terminate)
Define Rational
0, -5, 8, 0.44, -0.23,
Examples of rational numbers:
9 , ½
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integer rational irrational
Classify each number as specific as possible:Integer, Rational or Irrational
5
-6
0
-21
-65
13.2
-6.329
2.34437 x 103½ ¾
3¾π5
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-1 0-2-3-4-5 1 2 3 4 5
Rational Numbers on a Number Line
NegativeNumbers
PositiveNumbers
Numbers to the left of zero are less than zero
Numbers to the right of zero are greater than zero
Zero is neitherpositive or negative
`
Zero
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-5 0
-3.212
12
4
5
-106
192
5.9
-1.12.9
16
Which of the following are examples of integers?(Touch all the integers)
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2 Which of the following are examples of rational numbers?
A
B -3
C 10D 0.25E 75%
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2 Which of the following are examples of rational numbers?
A
B -3
C 10D 0.25E 75%
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Ans
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A, B, C, D, E
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Numbers In Our World
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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
Numbers can represent everyday situations
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Can you think of anymore?
Words that represent positive and negative integers. gained
increased
up more
deposit
less
loss
underbelow
rose decreasewithdraw
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Niko's grandmother put 20 dollars into his bank account. How would we show this integer?
A shark swims 30ft below sea level. How would we represent this as an integer?
20
-30
click
click
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Niko's grandmother put 20 dollars into his bank account. How would we show this integer?
A shark swims 30ft below sea level. How would we represent this as an integer?
20
-30
click
click
[This object is a pull tab]
Hin
t
REMEMBER:
When writing a positive integer it is not necessary to put a + in front of the integer.
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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
Write a number to represent each situation:
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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
Write a number to represent each situation:
[This object is a pull tab]
Ans
wer
1. -6.752. 113. 7004. -105. -86. 350 2/3
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3 Which of the following numbers best represents the following scenario:
The effect on your wallet when you spend $10.25.
A -10.25B 10.25C 0D +/- 10.25
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3 Which of the following numbers best represents the following scenario:
The effect on your wallet when you spend $10.25.
A -10.25B 10.25C 0D +/- 10.25
[This object is a pull tab]
Ans
wer
A
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4 Which of the following integers best represents the following scenario:
Earning $50 shoveling snow.
A -50B 50C 0D +/- 50
Ans
wer
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5 Which of the following numbers best represents the following scenario:
You dive feet to explore a sunken ship.
A
B
C 0
D
Ans
wer
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Fractions and the Negative Sign
When we have a negative fraction, the negative sign can be in different places.
The following all are negative one-half.
Why are they all negative?
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Fractions and the Negative Sign
These two fractions equal positive one-half.
Why are they both positive?
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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
The numbers -4 and 4 are shown on the number line.
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.
Opposites
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9 What is the opposite of -7?A
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10 What is the opposite of -3/2?
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11 What is the opposite of 18.2?
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What happens when you add two opposites?
Try it and see...
A number and its opposite have a sum of zero.
Numbers and their opposites are called additive inverses.
Click to Reveal
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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
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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...
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Let's organize our thoughts...
When a contestant gets a
$100 question correct
Then a $50 question incorrect
Then a $200 questionincorrect
Question Answered
Integer Representation
New Score
100 Correct
50Incorrect
200 Incorrect
-50
100 100
50
-150-200
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Now you try...
When a contestant gets a
$150 question incorrect
Then a $50 question incorrect
Then a $200 questioncorrect
Question Answered
Integer Representation
New Score
150 Incorrect
50Incorrect
200 Correct
-50
-150 -150
-200
0200
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Now you try...
When a contestant gets a
$50 question incorrect
Then a $150 question correct
Then a $200 questionincorrect
Question Answered
Integer Representation
New Score
Ans
wer
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12 After the following 3 responses what would the contestants score be?
$100 incorrect $200 correct $50 incorrect A
nsw
er
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13 After the following 3 responses what would the contestants score be?
$200 correct$50 correct $300 incorrect A
nsw
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14 After the following 3 responses what would the contestants score be?
$150 incorrect$50 correct $100 correct
Ans
wer
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15 After the following 3 responses what would the contestants score be?
$50 incorrect$50 incorrect $100 incorrect
Ans
wer
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16 After the following 3 responses what would the contestants score be?
$200 correct$50 correct $100 incorrect
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· 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.
To Review
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Absolute Value of NumbersThe 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 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
What is the distance from 0 to 5?
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
What is the distance from 0 to -5?
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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
Absolute value is symbolized by two vertical bars
4
What is the 4 ?
This is read, "the absolute value of 4"
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Remember
1 2 3 4 50-1-2-3-4-5
A number and its opposite have sum of zero. So ....
On a number line, a number and its opposite have the same distance from 0.
(Opposite numbers are on opposite sides from 0).
12341 2
34
-4 is 4 "jumps" from 0 4 is 4 "jumps" from zero
Both -4 and 4 are the same distance from zero
click
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-4 = 4
-9 = 9
= 9.6|9.6|
Use the number line to find absolute value.
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
Moveto
check
Move to
check
Moveto
check
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17 Find
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-818 Find
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19 What is A
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20 What is
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21 Find
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22 What is the absolute value of the number shown in the generator?
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23 Which numbers have 15 as their absolute value?
A -30B -15C 0
D 15
E 30
Ans
wer
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24 Which numbers have 100 as their absolute value?
A -100B -50C 0D 50E 100
Ans
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Comparing and Ordering Rational
Numbers
Return toTable ofContents
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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.
Use the Number Line
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
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Place the number tiles in the correct places on the number line.
4
-45
-3
-2
30
2 -5
-1
1
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4-4 5-3 -2 30 2-5 -1 1
Now, can you see:
Which integer is largest?
Which is smallest?
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4-4 5-3 -2 30 2-5 -1 1
Where do rational numbers go on the number line?
Go to the board and write in the following numbers:
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Put these numbers on the number line.
-3
Which number is the largest? The smallest?
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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.
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25 10.5 is ______ 15.2.
A =B <C >
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26 7.5 is ______ 7.5
A =B <C > A
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27 3.2 is ______ 5.7
A =B <C >
Ans
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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 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
Remember, the number farther to the right on a number line is larger.
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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.
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28 -4.75 ______ -4.75
A =B <C >
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29 -4 ______ -5
A =B <C >
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
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30
A =B <C >
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31 -14.75 ______ -6.2
A =B <C >
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32 -14.2 ______ -14.3
A =B <C >
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Comparing All Numbers
Any positive number is greater than zero or any negative number.
Any negative number is less than zero or any positive number.
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
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Drag the appropriate inequality symbol between the following pairs of numbers:
1) -3.2 5.8
3) 63 36
5) -6.7 -3.9
7) -24 -17
9) -8.75 -8.25
2) -237 -259
4) -10.2 -15.4
6) 127 172
8)
10) -10 -7
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33 -17 ______ 12
A =B <C >
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34 -5.5 ______ 0
A =B <C >
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35 5.32 ______ 0
A =B <C >
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36 -4 ______ -9
A =B <C >
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37 1 ______ -54
A =B <C >
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38 -355.5 ______ 0
A =B <C >
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A thermometer can be thought of as a vertical number line. Positive numbers are above zero and negative numbers are below zero.
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0
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If the temperature reading on a thermometer
is 90, what will the new reading be if the temperature:
falls 3 degrees?
rises 2 degrees?
falls 9 degrees?
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39 If the temperature reading on a thermometer is 10º, what will the new reading be if the temperature rises 5 degrees?
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39 If the temperature reading on a thermometer is 10º, what will the new reading be if the temperature rises 5 degrees?
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15ºC
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40 If the temperature reading on a thermometer is 10º, what will the new reading be if the temperature falls 12 degrees?
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40 If the temperature reading on a thermometer is 10º, what will the new reading be if the temperature falls 12 degrees?
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-2ºC
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41 If the temperature reading on a thermometer is -3º, what will the new reading be if the temperature falls 3 degrees?
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41 If the temperature reading on a thermometer is -3º, what will the new reading be if the temperature falls 3 degrees?
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-6ºC
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42 If the temperature reading on a thermometer is -3º, what will the new reading be if the temperature rises 5 degrees?
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42 If the temperature reading on a thermometer is -3º, what will the new reading be if the temperature rises 5 degrees?
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2ºC
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43 If the temperature reading on a thermometer is -3º, what will the new reading be if the temperature falls 12 degrees?
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43 If the temperature reading on a thermometer is -3º, what will the new reading be if the temperature falls 12 degrees?
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-15ºC
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Adding Rational Numbers
Return toTable ofContents
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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.
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While this section is titled "Addition", we're going to learn 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.
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1. Start at zero 2. Walk the number of steps indicated by the first number.3. Walk the number of steps given by the second number. 4. Look down, you're standing on the answer.
Addition: A walk on the number line.
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
Here's how it works.
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Let's do 3 + 4 on the number line.
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
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.
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Let's do 3 + 4 on the number line.
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
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
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Let's do 3 + 4 on the number line.
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
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
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Let's do -4 + (-5) on the number line.
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
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.
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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
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
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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
Let's do -4 + (-5) on the number line.
· Go to the left for negative numbers
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.
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Let's do 5 + -7 on the number line.
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
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.
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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
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.
· Go to the right for positive numbers
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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
Let's do 5 + -7 on the number line.
· Go to the left for negative numbers
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.
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Let's do -4 + 9.5 on the number line.
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
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.
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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
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.
· Go to the left for negative numbers
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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
Let's do -4 + 9.5 on the number line.
· Go to the right for positive numbers
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.
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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.
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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
3 + 4 = 7
-4 + 9.5 = 5.5
5 + (-7) = -2
-4 + (-5) = -9 10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
We can see some patterns here that allow us to create rules to get these answers without drawing.
Addition: Using Absolute Values
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To add rational numbers with the same sign 1. Add the absolute value of the rational numbers. 2. The sign stays the same.
(Same sign, find the sum)
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-103 + 4 = 7
-4 + (-5) = -9 10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
3 + 4 = 7; both signs are positive; so 3 + 4 = 7
4 + 5 = 9; both signs are negative; so -4 + (-5) = -9
Addition: Using Absolute Values
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Interpreting the Absolute Value Approach
The reason the absolute value approach works, if the signs of the numbers 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.
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To add rational numbers with different signs 1. Find the difference of the absolute values of the rational numbers. 2. Keep the sign of the integer with the greater absolute value.
(Different signs, find the difference)
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10-4 + 9.5 = 5.5
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-105 + (-7) = -2
9.5 - 4 = 5.5; 9.5 > 4, and 9.5 is positive; so -4 + 9.5 = 5.5
7 - 5 = 2; 7 > 5 and 7 is negative; so 5 + (-7) = -2
Addition: Using Absolute Values
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Interpreting the Absolute Value Approach
If the signs of the rational numbers are different:
For the 2nd leg of your trip you're traveling in the opposite direction of the 1st 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.
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Adding Rational Numbers:
To add rational numbers with the same sign 1. Add the absolute value of the rational numbers. 2. The sign stays the same.
(Same sign, find the sum)
To add rational numbers with different signs 1. Find the difference of the absolute values of the rational numbers. 2. Keep the sign of the number with the greater absolute value.
(Different signs, find the difference)
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44
11 + (-4) =A
nsw
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45
-4 + (-4) =
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46
17 + (-20) =
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47
-15 + (-30) =
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48
-5 + 10 =
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49
-9 + (-2.3) =
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50
5.3 + (-8.4) =
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51
4.8 + (12.6) =
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52
-14.3 + 7.93 =
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53A
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54
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55
-7.34 + (-8.21) =
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Turning Subtraction Into Addition
Return toTable ofContents
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Subtracting Numbers
Subtracting a number is the same as adding it's opposite.
(Add a line,change the sign of the second number)
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Subtracting NumbersSubtracting a number is the same as adding its 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 subtraction sign tells us to move in the opposite direction that we would for +5.
For "8 + (-5)" we move 8 steps to the right, and then 5 steps to the left since we're adding -5.
Either way, we end up at +3.
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
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Subtracting Negative NumbersCompare 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.
For "8 + 2" we move 8 steps to the right, and then 2 steps to the right since we're adding 2.
Either way, we end up at +10.
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
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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)
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56 Convert the subtraction problem into an addition problem.
8 – 4
A -8 + 4B 8 + (-4)
C -8 + (-4)D 8 + 4
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57 Convert the subtraction problem into an addition problem.
-3.7 - (-10.1)
A -3.7 + 10.1B 3.7 + (-10.1)C -3.7 + (-10.1)D 3.7 + 10.1
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58 Convert the subtraction problem into an addition problem.
A
B
C
D
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59 Convert the subtraction problem into an addition problem.
A
B
C
D
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60 Convert the subtraction problem into an addition problem.
1 - 9
A -1 + 9
B 1 + (-9)
C -1 + (-9)
D 1 + 9
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61 Which expressions are equivalent to
Select all that apply.
A
B
C
D
E
F From PARCC sample test
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61 Which expressions are equivalent to
Select all that apply.
A
B
C
D
E
F From PARCC sample test
[This object is a pull tab]
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B & D
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Adding and Subtracting Rational Numbers
Review
Return toTable ofContents
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62 Calculate the sum or difference.
-6 – 2
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63 Calculate the sum or difference.
5 + (-5)
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64 Calculate the sum or difference.
-10.5 + 6.2
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65 Calculate the sum or difference.
7.3 – (-4)
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66 Calculate the sum or difference.
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67 Calculate the sum or difference.
9.27 + (-8.38)A
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68 Calculate the sum or difference.
-4.2 + (-5.9)
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69 Calculate the sum or difference.
-2 – (-3.95)
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70 Calculate the sum or difference.
5 - 6 + (-7)
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71 Calculate the sum or difference.
19 + (-12) - 11
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72 Calculate the sum or difference.
-2.3 + 4.1 + (-12.7)
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73 Calculate the sum or difference.
-8.3 - (-3.7) + 5.2
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74 Calculate the sum or difference.
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75 Two numbers, n and p are plotted on the number line shown.
The numbers n - p, n + p, and p - n will be plotted on the number line. Select an expression from each group to make this statement true.
The number with the least value is _____,
and the number with the greatest value is _____.
A n - p B n + p C p - n
D n - p E n + p F p - n
From PARCC sample test
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75 Two numbers, n and p are plotted on the number line shown.
The numbers n - p, n + p, and p - n will be plotted on the number line. Select an expression from each group to make this statement true.
The number with the least value is _____,
and the number with the greatest value is _____.
A n - p B n + p C p - n
D n - p E n + p F p - n
From PARCC sample test
[This object is a pull tab]
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F p - n is the greatest value
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Multiplying Rational Numbers
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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).
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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.
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Multiplying Rational Numbers
Multiplication is just a quick way of writing repeated additions.
These are all equivalent:
3·43 +3 + 3 + 3 4 + 4 + 4
they all equal 12.
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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.
Multiplying Rational Numbers
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10 2 3 4 5 6 7 8 9 10-1-2-3
Let's do 4 x 3 on the number line.
11 1312 14 16 1715
We'll do it as 3+3+3+3 and as 4+4+4
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10 2 3 4 5 6 7 8 9 10-1-2-3 11 1312 14 16 1715
Try 5 x 2 on the number line.
Try it as 5+5 and as 2+2+2+2+2
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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)
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10 2 3-17 -1-2-3-4-5-6-7-8-9-10-11-13-15-16 -14 -12
4 x (-3) On the Number Line
4 x (-3) = (-3)(-3)(-3)(-3)
So we can see that 4 x (-3) = -12
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4∙34 + 4 + 412
4(-3)(-3) + (-3) + (-3)-12
Multiplying positive numbers has a positive value.
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)
Sign Rules for Multiplying Rational Numbers
?
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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.
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4∙34 + 4 + 412
4(-3)(-4) + (-4) + (-4)-12
Multiplying positive numbers yields a positive result.
Multiplying a negative number and a positive number yields a negative result.
Multiplying two negative numbers together yields a positive result.
Sign Rules forMultiplying Rational Numbers
(-4)(-3)-((-4) + (-4) + (-4))-(-12)12
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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
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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
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We can also see these rules when we look at the patterns below:
3(3) = 9 -5(3) = -153(2) = 6 -5(2) = -103(1) = 3 -5(1) = -53(0) = 0 -5(0) = 03(-1) = -3 -5(-1) = 53(-2) = -6 -5(-2) = 103(-3) = -9 -5(-3) = 15
2.5(3) = 7.5 -3.1(3)(-2) = 18.62.5(2) = 5 -3.1(2)(-2) = 12.42.5(1) = 2.5 -3.1(1)(-2) = 6.22.5(0) = 0 -3.1(0)(-2) = 02.5(-1) = -2.5 -3.1(-1)(-2) = -6.22.5(-2) = -5 -3.1(-2)(-2) = -12.42.5(-3) = -7.5 -3.1(-3)(-2) = -18.6
Multiplying Rational Numbers
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76 What is the value of (-3)(-9)?
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77 What is the value of 5(-4.82)?
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78 What is the value of (-3.2)(-6.4)? A
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79 What is the value of (-5.12)(-9)?
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80 What is the value of -2(-7)(-4)?
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81 What is the value of: A
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82 What is the value of:
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83 Jane has entered a baking contest. Jane uses 3.1 ounces of flour to make one cinnamon roll. How many ounces of flour does Jane need to make 7 cinnamon rolls? A
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84 Timmy is shipping 4 boxes of shirts. Each box weighs 6.3 pounds. If cost 5.20 per pound to ship. How much does Timmy have to spend?
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Dividing Rational Numbers
Return toTable ofContents
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Division Symbols
You may have mostly used the "÷ " symbol to show division.
We will also represent division as a fraction. Remember: 9 9÷ 3 = 33
are both ways to show division.
= 3
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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
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10 2 3 4 5 6 7 8 9 10-1-2-3 11 1312 14 16 1715
Try 10 ÷ 2 on the number lineThis means how many lengths of 2 would be needed to add up to 10.
The answer is 5: the number of red arrows of length 2 that end to end give a total length of 10.
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10 2 3 4 5 6 7 8 9 10-1-2-3 11 1312 14 16 1715
Try 10 ÷ 5 on the number lineThis means how many lengths of 5 would be needed to add up to 10.
The answer is 2: the number of green arrows of length 5 that, end to end, give a total length of 10.
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10 2 3-17 -1-2-3-4-5-6-7-8-9-10-11-13-15-16 -14 -12
-12 ÷ 3 On the Number LineThis can be read as how many steps of 3 would it take to get to -12.
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.)
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-15 ÷ 3 = -5
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
-153 = -5
Dividing Rational Numbers
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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
Dividing Rational Numbers
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85 Find the value of 32 ÷ 4A
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86 Find the value of:
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87 Find the value of:
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88 Find the value of:A
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89 Find the value of:
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90 Find the value of:
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91 Find the value of:A
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92 Find the value of:
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93 Kobe put 8 toy cars in a row. The line of cars was 16.4 meters long. How long was each car?
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94 Olivia squeezed 3/4 of a gallon of orange juice. She split the orange juice equally into 6 cups. How many gallons was in each cup?
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95 In which situation could the quotient of -24 3 be used to answer the question?A The temperature of a substance decreased by 24
degrees per minute for 3 minutes. What was the overall change of the temperature of the substance?
B A football team loses 24 yards on one play, then gains 3 yards on the next play. How many total yards did the team gain on the two plays?
C Julia withdrew a total of $24 from her bank account over 3 days. She withdrew the same amount each day. By how much did the amount in her bank account change each day?
D A cookie jar contains 24 cookies. Each child receives 3 cookies. How many children are there?
From PARCC sample test
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95 In which situation could the quotient of -24 3 be used to answer the question?A The temperature of a substance decreased by 24
degrees per minute for 3 minutes. What was the overall change of the temperature of the substance?
B A football team loses 24 yards on one play, then gains 3 yards on the next play. How many total yards did the team gain on the two plays?
C Julia withdrew a total of $24 from her bank account over 3 days. She withdrew the same amount each day. By how much did the amount in her bank account change each day?
D A cookie jar contains 24 cookies. Each child receives 3 cookies. How many children are there?
From PARCC sample test
[This object is a pull tab]
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C
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Operations with Rational Numbers
Return toTable ofContents
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When simplifying expressions with rational numbers,
you must follow the order of operations whileremembering your rules for
positive and negative numbers!
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Order of Operations
ParenthesesExponentsMultiplicationDivisionAdditionSubtraction
Complete at the same time...whichever comes first...from left to right
(ALL Grouping Symbols)
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Let's simplify this step by step...
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
-7 + (-3)[5 - (-2)]
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Let's simplify this step by step...
What should you do first? What should you do second?
What should you do third? What should you do last?
Clickto
Reveal
Clickto
Reveal
Clickto
Reveal
Clickto
Reveal
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96 Simplify the expression.
-12÷ 3(-4)
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97 Simplify the expression.[-1 - (-5)] + [7(3 - 8)]
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98 Simplify the expression.40 - (-5)(-9)(2)
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99 Simplify the expression.5.8 - 6.3 + 2.5
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100 Simplify the expression.-3(-4.7)(5-3.2)
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101 Simplify the expression.
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102 Simplify the expression.
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103 Complete the first step of simplifying. What is your answer?
[3.2 + (-15.6)] - 6[4.1 - (-5.3)]
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104 Complete the next step of simplifying. What is your answer?
[3.2 + (-15.6)] - 6[4.1 - (-5.3)]
-12.4 - 6[4.1 - (-5.3)]click to reveal step from previous slide
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105 Complete the next step of simplifying. What is your answer?
[3.2 + (-15.6)] - 6[4.1 - (-5.3)]
-12.4 - 6[9.4]
-12.4 - 6[4.1 - (-5.3)]click to reveal steps from previous slides
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106 Complete the next step of simplifying. What is your answer?
[3.2 + (-15.6)] - 6[4.1 - (-5.3)]
-12.4 - 56.4-12.4 - 6[9.4]
-12.4 - 6[4.1 - (-5.3)]click to reveal steps from previous slides
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108 Simplify the expression.
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109A
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110
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111(-4.75)(3) - (-8.3)
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Solve this one in your groups.A
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How about this one?
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112
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113[(-3.2)(2) + (-5)(4)][4.5 + (-1.2)]
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114
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115
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116A
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117 Select the correct number from each group of numbers to complete the equation.
A 2
B -2
C 3/4
D -4/3
E 2
F -2
G 4/3
H -3/4
_____ _____
From PARCC sample test
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117 Select the correct number from each group of numbers to complete the equation.
A 2
B -2
C 3/4
D -4/3
E 2
F -2
G 4/3
H -3/4
_____ _____
From PARCC sample test
[This object is a pull tab]
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H -3/4
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Converting Rational Numbers to
Decimals
Return toTable ofContents
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Definition of Rational:
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.
Do you recall the definition of a Rational Number?
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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.
How can you convert Rational Numbers into Decimals?
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Long Division Review
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119 Convert to a decimal.
(If the number is repeating, use bar notation in your notebook by entering the repeating number(s) 3 times on your responder.)
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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.) A
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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.)
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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.)
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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.) A
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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.) A
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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.)
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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.)
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127 Mike needed meters of fabric to fix his couch. How can this be written as a decimal?
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128 Hannah rode her bike miles to the neighborhood pool. What is the distance she rode her bike as a decimal?
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129 Kevin Durant made shots in the first quarter of the NBA finals, how is that written as a decimal?
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130 Coral's mother wants to know show as a decimal. How can that fraction be written as a decimal?
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Glossary
Return toTable ofContents
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Back to
Instruction
Absolute Value
How far a number is from zero on the number line.
-2 = 22 = 2
0 1 2-2 -10 1 2-2 -1
0 = 0
0 1 2-2 -1
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Back to
Instruction
IntegersZero, all whole numbers
and their opposites.
... -1, 0, 1... 35
2.3
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Back to
Instruction
Irrational Numbers
Any number that cannot be made by dividing one
integer by another.
The decimal form of a number that goes on forever
without repeating.
= 3.14159...??
(no ratio).75 = 34
= 2.718281...??
(no ratio)
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Back to
Instruction
Opposites
-3 and 3 -5 and 5
Two numbers which are the same distance from zero.
-(3) = -3 -(-3) = 3
-3 and 3
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Back to
Instruction
Rational Numbers
Any number that can be made by dividing one integer by another.
ab .75 = 34
= 3.14159...
= ??
(no ratio)
ratio
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Back to
Instruction
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Back to
Instruction
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