# New Jersey Center for Teaching and Learning

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New Jersey Center for Teaching and Learning
Progressive Mathematics Initiative This material is made freely available at and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website:

Setting the PowerPoint View
Use Normal View for the Interactive Elements To use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible: On the View menu, select Normal. Close the Slides tab on the left. In the upper right corner next to the Help button, click the ^ to 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 &

Do you know what an integer is?
1 Do you know what an integer is? Yes No

Number System Real Rational 1/3 Integer 5/2 Whole 0.22 Irrational
Natural 1,2,3... Whole Integer ...-4, -3, -2, -1 Rational 1/5 5/2 8.3 -2.756 -3/4 1/3 -1/11 Real Irrational

X Define Integer Definition of Integer:
The set of whole numbers, their opposites and zero. X 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) X Examples of rational numbers: , ½ 0, -5, 8, 0.44, -0.23, 9

X Define Irrational Definition of Irrational:
A real number that cannot be written as a simple fraction X Examples of irrational numbers:

Classify each number as specific as possible: Integer, Rational or Irrational
3.2 -6 -21 1 5 π 9 5 -65 -6.32 x 103 integer rational irrational

Rational Numbers on a Number Line
Negative Numbers Positive Numbers Zero -5 -4 -3 -2 -1 1 2 3 4 5 Numbers to the left of zero are less than zero Zero is neither positive or negative Numbers to the right of zero are greater than zero

Which of the following are examples of integers?
2 Which of the following are examples of integers? A -5 B C -3.2 D 12 Answer: A, B, D 1 2 E

Which of the following are examples of rational numbers?
3 Which of the following are examples of rational numbers? 1 3 A B -3 C 10 Answer: A, B, C, D, E D 0.25 E 75%

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:
Spending \$6.75 Gain of 11 pounds Depositing \$700 10 degrees below zero 8 strokes under par (par = 0) feet above sea level

Which of the following numbers best represents the following scenario:
4 Which of the following numbers best represents the following scenario: The effect on your wallet when you spend \$10.25. A -10.25 Answer: A B 10.25 C D +/

Earning \$50 shoveling snow.
Which of the following integers best represents the following scenario: Earning \$50 shoveling snow. A -50 B 50 Answer: B C D +/- 50

Which of the following numbers best represents the following scenario:
6 Which of the following numbers best represents the following scenario: You dive feet to explore a sunken ship. A Answer: A B C D

Opposites The numbers -4 and 4 are shown on the number line. 1 2 3 4 5
2 3 4 5 6 7 8 9 10 -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? Answer: -18.2

What happens when you add two opposites? Try it and see...
Click to Reveal A number and its opposite have a sum of zero. 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 Then a \$200 question Question Answered Integer Representation New Score 100 Correct 50 Incorrect 200 -50 -150 -200

Integer Representation
Now you try... Question Answered Integer Representation New Score 150 Incorrect 50 200 Correct -50 -150 -200 When a contestant gets a \$150 question incorrect Then a \$50 question incorrect Then a \$200 question correct

Integer Representation
Now you try... Question Answered Integer Representation New Score When a contestant gets a \$50 question incorrect Then a \$150 question correct Then a \$200 question incorrect

After the following 3 responses what would the contestants score be?
9 After the following 3 responses what would the contestants score be? \$100 incorrect \$200 correct \$50 incorrect Answer: 50

After the following 3 responses what would the contestants score be?
10 After the following 3 responses what would the contestants score be? \$200 correct \$50 correct \$300 incorrect Answer: -50

After the following 3 responses what would the contestants score be?
11 After the following 3 responses what would the contestants score be? \$150 incorrect \$50 correct \$100 correct Answer: 0

After the following 3 responses what would the contestants score be?
12 After the following 3 responses what would the contestants score be? \$50 incorrect \$50 incorrect \$100 incorrect Answer: -200

After the following 3 responses what would the contestants score be?
13 After the following 3 responses what would the contestants score be? \$200 correct \$50 correct \$100 incorrect Answer: 150

X 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.

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). 1 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? 1 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?

Absolute value is symbolized by two vertical bars
|4| This is read, "the absolute value of 4" 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 What is the | 4 | ?

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 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10

16 What is ? Answer: 7.3

17 What is ? Answer: 22 3/5

19 What is the absolute value of the number shown in the generator? (Click for web site) Answer: Answers will vary.

Which numbers have 15 as their absolute value?
20 Which numbers have 15 as their absolute value? A -30 B -15 C Answer: B, D D 15 E 30

Which numbers have 100 as their absolute value?
21 Which numbers have 100 as their absolute value? A -100 B -50 C Answer: A, E D 50 E 100

Comparing and Ordering Rational Numbers

Use the Number Line To compare rational numbers, plot points on
The numbers farther to the right are larger. The numbers farther to the left are smaller. 1 2 3 4 5 6 7 8 9 10 -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 = < > 2 When using < or >, remember that the smaller side points at the smaller number.

22 10.5 is ______ 15.2. A = B < C > Answer: B

23 7.5 is ______ 7.5 A = B < C > Answer: A

24 3.2 is ______ 5.7 A = B < C > Answer: B

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 = > < -2 1 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.

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 A = B < C > Answer: A

26 -4 ______ -5 A = B < C > 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5
-4 ______ -5 A = B < C > Answer: C 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10

27 A = B < C > Answer: B

28 is ______ -6.2 A = B < C > Answer: B

29 -14.2 is ______ -14.3 A = B < C > Answer: C

Comparing All Numbers Any positive number is greater than zero or any negative number. 1 2 3 4 5 6 7 8 9 10 -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) 5) 7) 9) 2) 4) 6) 8) 10)

30 A = B < C > Answer: B

31 A = B < C > Answer: B

32 A = B < C > Answer: C

33 A = B < C > Answer: C

34 A = B < C > Answer: C

35 A = B < C > Answer: B

A thermometer can be thought of as a vertical number line
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? Answer: 7°C

37 If the temperature reading on a thermometer is 10℃, what will the new reading be if the temperature: rises 5 degrees? Answer: 15°C

38 If the temperature reading on a thermometer is 10℃, what will the new reading be if the temperature: falls 12 degrees? Answer: -2°C

If the temperature reading on a thermometer is
39 If the temperature reading on a thermometer is -3℃, what will the new reading be if the temperature: falls 3 degrees? Answer: -6°C

If the temperature reading on a thermometer is
40 If the temperature reading on a thermometer is -3℃, what will the new reading be if the temperature: rises 5 degrees? Answer: 2°C

If the temperature reading on a thermometer is
41 If the temperature reading on a thermometer is -3℃, what will the new reading be if the temperature: falls 12 degrees? Answer: -15°C

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 Or if we want to subtract -4 from -5 we will write: -5 - (-4), which is clearer than

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. 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.
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.
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. Go to the right for positive numbers

Let's do 3 + 4 on the number line.
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. Go to the right for positive numbers

Let's do -4 + (-5) on the number line.
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.
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. Go to the left for negative numbers

Let's do -4 + (-5) on the number line.
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. Go to the left for negative numbers

Let's do 5 + -7 on the number line.
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. 1 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.
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. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Go to the right for positive numbers

Let's do 5 + -7 on the number line.
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. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Go to the left for negative numbers

Let's do -4 + 9.5 on the number line.
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. 1 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.
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. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Go to the left for negative numbers

Let's do -4 + 9.5 on the number line.
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. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Go to the right for positive numbers

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.

3 + 4 = 7 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -4 + (-5) = -9 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 = 5.5 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 5 + (-7) = -2 1 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 (Same sign, find the sum)
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 1 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 = 7 -4 + (-5) = -9 1 2 3 4 5 6 7 8 9 10 -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.

(Different signs, find the difference)
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) 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 = 5.5 = 5.5; 9.5 > 4, and 9.5 is positive; so = 5.5 5 + (-7) = -2 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 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: 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.

Adding Rational Numbers: (Different signs, find the difference)
To add integers with the same sign 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) = Answer: 7

43 -4 + (-4) = Answer: -8

44 17 + (-20) = Answer: -3

45 -15 + (-30) = Answer: -45

47 -9 + (-2.3) = Answer: -11.3

48 5.3 + (-8.4) = Answer: -3.1

49 4.8 + (12.6) = Answer: 17.4

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: 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. 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. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10

Subtracting Negative Numbers
Compare these two problems: 8 - (-2) and 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. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -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 is the same as (-17)

Convert the subtraction problem into an addition problem.
54 Convert the subtraction problem into an addition problem. 8 – 4 A -8 + 4 B 8 + (-4) Answer: B C -8 + (-4) D 8 + 4

Convert the subtraction problem into an addition problem.
55 Convert the subtraction problem into an addition problem. (-10.1) A B 3.7 + (-10.1) Answer: A C (-10.1) D

Convert the subtraction problem into an addition problem.
56 Convert the subtraction problem into an addition problem. A Answer: C B C D

Convert the subtraction problem into an addition problem.
57 Convert the subtraction problem into an addition problem. A B Answer: D C D

Convert the subtraction problem into an addition problem.
58 Convert the subtraction problem into an addition problem. 1 - 9 A -1 + 9 B 1 + (-9) Answer: B C -1 + (-9) D 1 + 9

Adding and Subtracting Rational Numbers Review

Calculate the sum or difference. -6 – 2
59 Calculate the sum or difference. -6 – 2 Answer: -8

Calculate the sum or difference. 5 + (-5)
60 Calculate the sum or difference. 5 + (-5) Answer: 0

Calculate the sum or difference. -10.5 + 6.2
61 Calculate the sum or difference. Answer: -4.3

Calculate the sum or difference. 7.3 – (-4)
62 Calculate the sum or difference. 7.3 – (-4) Answer: 11.3

Calculate the sum or difference.
63 Calculate the sum or difference. Answer: -3 1/6

Calculate the sum or difference. 9.27 + (-8.38)
64 Calculate the sum or difference. (-8.38) Answer: 0.89

Calculate the sum or difference. -4.2 + (-5.9)
65 Calculate the sum or difference. (-5.9) Answer: -10.1

Calculate the sum or difference. -2 – (-3.95)
66 Calculate the sum or difference. -2 – (-3.95) Answer: 1.95

Calculate the sum or difference. 5 - 6 + (-7)
67 Calculate the sum or difference. (-7) Answer: -8

Calculate the sum or difference. 19 + (-12) - 11
68 Calculate the sum or difference. 19 + (-12) - 11 Answer: -4

Calculate the sum or difference. -2.3 + 4.1 + (-12.7)
69 Calculate the sum or difference. (-12.7) Answer: -10.9

Calculate the sum or difference. -8.3 - (-3.7) + 5.2
70 Calculate the sum or difference. (-3.7) + 5.2 Answer: -6.8

Calculate the sum or difference.
71 Calculate the sum or difference. Answer: 4 1/10

Multiplying Rational Numbers

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 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 ÷ 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 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 or We should get the same result either way, 12.

Let's do 4 x 3 on the number line.
We'll do it as and as 4+4+4 -3 -2 -1 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 1 2 3 4 5
2 3 4 5 6 7 8 9 10 -1 -2 -3 11 13 12 14 16 17 15

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)
-13 -14 -12 -11 -10 -9 -8 -7 -6 -5 -4 -15 -16 -17 -3 -1 -2 2 3 1 So we can see that 4 x (-3) = -12

Multiplying Rational Numbers
Sign Rules for Multiplying Rational Numbers 4 ∙ 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) 4(-3) (-3) + (-3) + (-3) -12 ?

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.

Multiplying Rational Numbers
Sign Rules for Multiplying Rational Numbers 4 ∙ 3 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. 4(-3) (-4) + (-4) + (-4) -12 (-4)(-3) -((-4) + (-4) + (-4)) -(-12) 12

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) = (-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) = (3) = -15 3(2) = (2) = -10 3(1) = (1) = -5 3(0) = (0) = 0 3(-1) = (-1) = 5 3(-2) = (-2) = 10 3(-3) = (-3) = 15 2.5(3) = (3)(-2) = 18.3 2.5(2) = (2)(-2) = 12.4 2.5(1) = (1)(-2) = 6.2 2.5(0) = (0)(-2) = 0 2.5(-1) = (-1)(-2) = -6.2 2.5(-2) = (-2)(-2) = -12.4 2.5(-3) = (-3)(-2) = -18.3

What is the value of (-3)(-9)?
72 What is the value of (-3)(-9)? Answer: 27

73 What is the value of 3.1 ∙ 7? Answer: 21.7

74 What is the value of 5(-4.82)? Answer:

What is the value of (-3.2)(-6.4)?
75 What is the value of (-3.2)(-6.4)? Answer: 20.48

76 What is the value of -8 ∙ 7.6? Answer: -60.8

What is the value of (-5.12)(-9)?
77 What is the value of (-5.12)(-9)? Answer: 46.08

What is the value of -2(-7)(-4)?
78 What is the value of -2(-7)(-4)? Answer: -56

79 What is the value of: Answer: -47 1/4

80 What is the value of: Answer: /2

81 What is the value of: Answer: 30.75

82 What is the value of: Answer:

Dividing Rational Numbers

Division Symbols You may have mostly used the "÷" symbol to show division. We will also represent division as a fraction. Remember: ÷ 3 = 3 3 are both ways to show division. = 3

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. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 11 13 12 14 16 17 15 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. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 11 13 12 14 16 17 15 The answer is 2: the number of green arrows of length 5 that, end to end, give a total length of 10.

Each red arrow represents a step of 3,
-12 ÷ 3 On the Number Line This can be read as how many steps of 3 would it take to get to -12. -13 -14 -12 -11 -10 -9 -8 -7 -6 -5 -4 -15 -16 -17 -3 -1 -2 2 3 1 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 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

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 Answer: 8

84 Find the value of: Answer: -4

85 Find the value of: Answer: 3

86 Find the value of: Answer: -5.1

87 Find the value of: Answer: 0

88 Find the value of: Answer: -4/5

89 Find the value of: Answer: 3.1

90 Find the value of: Answer: -21

91 Find the value of: Answer: 10 1/2

92 Find the value of: Answer: -3.1

93 Find the value of: Answer: 3

Operations with Rational Numbers

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 Exponents Multiplication Division
Addition Subtraction (ALL Grouping Symbols) Complete at the same time...whichever comes first...from left to right

-7 + (-3)[5 - (-2)] Let's simplify this step by step...
What should you do first? 5 - (-2) = = 7 What should you do next? (-3)(7) = -21 What is your last step? -7 + (-21) = -28 -7 + (-3)[5 - (-2)]

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? Click to Reveal Click to Reveal Click to Reveal Click to Reveal

Simplify the expression.
94 Simplify the expression. -12÷3(-4) Answer: 16

95 [-1 - (-5)] + [7(3 - 8)] Answer: -31

96 40 - (-5)(-9)(2) Answer: -50

Simplify the expression.
98 Simplify the expression. -3(-4.7)(5-3.2) Answer: 25.38

101 Complete the first step of simplifying. What is your answer? [3.2 + (-15.6)] - 6[4.1 - (-5.3)] Answer: [4.1 – (-5.3)]

102 Complete the next step of simplifying. What is your answer? [3.2 + (-15.6)] - 6[4.1 - (-5.3)] Answer: – 6[9.4]

103 Complete the next step of simplifying. What is your answer? [3.2 + (-15.6)] - 6[4.1 - (-5.3)] Answer: – 56.4

104 Complete the next step of simplifying. What is your answer? [3.2 + (-15.6)] - 6[4.1 - (-5.3)] Answer: -68.8

Simplify the expression.
105 Simplify the expression. Answer: 1.76

Simplify the expression.
106 Simplify the expression. Answer: 32/9 = 3 5/9

109 (-4.75)(3) - (-8.3) Answer: -5.95

Solve this one in your groups.

[(-3.2)(2) + (-5)(4)][4.5 + (-1.2)] 111 Answer: -23.1

Converting Rational Numbers to Decimals

Do you recall the definition of a Rational Number?
x 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.

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) Answer: 8.4

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) Answer:

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) Answer: …

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) Answer: ….

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) Answer: 2.333…

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) Answer: …

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) Answer: 0.444…

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) Answer: -10

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) Answer: 0.333…

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) Answer: …

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) Answer: …

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) Answer: -0.4