[PDF] - 7th Grade The Number System & Mathematical Operations PDF Document

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7th Grade

The Number System & Mathematical Operations

2015-08-31 www.njctl.org

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Click on a topic to go to that section.

Natural Numbers and Whole Numbers Addition, Subtraction and Integers Multiplication and Division of Integers Exponents Glossary & Standards Addition and Subtraction of Integers Real Numbers Converting Rational Numbers to Decimals Addition and Subtraction of Rational Numbers Operations with Rational Numbers Multiplication and Division of Rational Numbers

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Natural Numbers and Whole Numbers

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Slide 5 / 262 Natural Numbers

The first numbers developed were the Natural Numbers, also called the Counting Numbers. 1, 2, 3, 4, 5, ... The three dots, (...), means that these numbers continue forever: there is no largest counting number.. Think of counting objects as you put them in a container those are the counting numbers.

Slide 6 / 262 Natural Numbers

Natural numbers were used before there was history. All people use them. This "counting stick" was made more than 35,000 years ago and was found in Lebombo, Swaziland. The cuts in this bone record the number "29."

http://www.taneter.org/math.html

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They were, and are, used to count objects > goats, > bales, > bottles, > etc. Drop a stone in a jar, or cut a line in a stick, every time a goat walks past. That jar or stick is a record of the number.

Slide 8 / 262 Numbers versus Numerals

Numbers exist even without a numeral, such as the number indicated by the cuts on the Lebombo Bone. A numeral is the name we give a number in our culture.

Slide 9 / 262 Numbers versus Numerals

If asked how many tires my car has, I could hand someone the above marbles. That number is represented by: · 4 in our Base 10 numeral system · IV in the Roman numeral system · 100 in the Base 2 numeral system

Slide 10 / 262 Whole Numbers

Adding zero to the Counting Numbers gives us the Whole Numbers. 0, 1, 2, 3, 4, ... Counting numbers were developed more than 35,000 years ago. It took 34,000 more years to invent zero. This the oldest known use of zero (the dot), about 1500 years ago. It was found in Cambodia and the dot is for the zero in the year 605.

http://www.smithsonianmag.com/history/origin- number-zero-180953392/?no-ist

Slide 11 / 262 Why zero took so long to Invent

zero horses = zero houses

3 2 1

Horses versus houses.

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Would I tell someone I have a herd of zero goats? Or a garage with zero cars? Or that my zero cars have zero tires? Zero just isn't a natural number, but it is a whole number.

Why zero took so long

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Addition, Subtraction and Integers

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Slide 14 / 262 Addition and Subtraction

The simplest mathematical operation is addition. The inverse of addition is subtraction. Two operations are inverses if one "undoes" the other. This is a very important concept, and applies to all mathematics.

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Addition and Subtraction

Each time a marble is dropped in a jar we are doing addition. Each time a marble is removed from a jar, we are doing subtraction. A number line allows us to think of addition in a new way.

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Let's find the sum of 4 and 5 on a number line. The number +4 is four steps to the right. Starting at 0, takes you to 4. The number +5 is five steps to the right. Starting at 0, takes you to 5. 4 5 1 2 3 4 5 6 7 8 9 10

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To find the sum "4 + 5" start at zero and take four steps to the right for the first number. Then, starting where you ended after those first steps, take five more steps to the right, to represent adding five. If we were walking, we could look down and see we are standing at 9. 4 5 4 + 5 Therefore, 4 + 5 = 9. 1 2 3 4 5 6 7 8 9 10

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A mathematical operation is commutative if the order doesn't matter. In this case, addition would be commutative if 4 + 5 = 5 + 4 Let's test that. We found that 4 + 5 = 9 How about 5 + 4?

The Commutative Property of Addition

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4 5 5 + 4 5 + 4 = 9 1 2 3 4 5 6 7 8 9 10

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The Commutative Property of Addition

First, take five steps to the right. Then, starting where you ended, take four more steps to the right. Once more, we could look down and see we are standing at 9.

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

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The Commutative Property of Addition

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So, 4 + 5 = 5 + 4 Addition is commutative in this case.

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a b a + b

The Commutative Property of Addition

a b b + a But there's nothing special about these numbers. This is true for any numbers: a + b = b + a

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Operations are inverse operations when one of them undoes what the other does. What would undo adding 5?

Inverse Operations

Subtracting 5

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Slide 23 / 262 Addition and Subtraction are Inverses

Addition and subtraction are inverses. Adding a number and then subtracting that same number leaves you where you started. Starting at 4, add 5 and then subtract 5. You end up where you started. subtract 5 add 5 1 2 3 4 5 6 7 8 9 10

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This is true for any two numbers. Start with a, then add b, then subtract b. You end up with the number you began with: a. +b a a + b

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We started with the addition question: what number results when we add 5 and 4? The answer is 9. 5 + 4 = 9 That leads to two new related subtraction questions. Starting with 9, what number do we get when we subtract 5? 9 - 5 = 4 Starting with 9, what number do we get when we subtract 4. 9 - 4 = 5 Subtraction was invented to undo addition, but it now can be used to ask new questions.

Inverse Operations Slide 26 / 262

Here's what 9 - 5 looks like on the number line

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And, here's 9 - 4

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Subtracting Whole Numbers

+9 +9

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Subtraction was invented to undo addition. But this new operation allows us to ask new questions. And, the number system to that point couldn't provide answers. For example: What is the result of subtracting 7 from 4? 4 - 7 = ?

We Need More Numbers Slide 28 / 262

4 - 7 = ?

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There was no answer to questions like this in the whole number systems. Which was all there was until about 500 years ago.

Subtracting Whole Numbers

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

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This led to the invention of negative numbers. They were called negative from the Latin "negare" which means "to deny," since people denied that such numbers could exist. They weren't used much until the Renaissance, and were only fully accepted in the 1800's. If you had trouble with negative numbers, so did most everyone else.

Negative Numbers

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Slide 30 / 262 Integers

Adding the negative numbers to the whole numbers yields the Integers. ...-3, -2, -1, 0, 1, 2, 3, ... In this case, "..." at the left and right, means that the sequence continues in both directions forever. There is no largest integer...nor is there a smallest integer.

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The below number line shows only the integers. 1 2 3 4 5 6 7 8 9 10

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Integers Slide 32 / 262

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Integers on the number line

Negative Integers Positive Integers Numbers to the left

f zero are less than

zero Numbers to the right of zero are greater than zero Zero is neither positive or negative

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Zero

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1 Which of the following are examples of integers? A 0 B -8 C -4.5 D 7 E 1 3

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2 Which of the following are examples of integers? A 2 B -6.1 C 287 D -1000 E 0.33 3 5

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3 Which of the following are examples of integers? A 1 B 6 C -4 D 0.75 E 25% 4

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4 What is the opposite of 25?

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5 What is the opposite of 0?

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6 What is the opposite of 18?

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7 What is the opposite of -18?

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8 What is the opposite of the opposite of -18?

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9 Simplify: - (- 9)

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10 Simplify: - (- 12)

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11 Simplify: - [- (-15)]

Slide 44 / 262 Absolute Value of Integers

The absolute value is the distance a number is from zero

n the number line, regardless of direction.

Distance and absolute value are always non-negative (positive or zero). 4

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What is the 4 ?

This is read, "the absolute value of 4".

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13 Find -28

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14 Find 56

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15 Find -8

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16 Find 3

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17 Which numbers have 12 as their absolute value?

A -24 B -12 C 0 D 12 E 24

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18 Which numbers have 50 as their absolute value? A -50 B -25 C 0 D 25 E 50

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To compare integers, plot points on the number line. The numbers farther to the right are greater. The numbers farther to the left are smaller.

Use the Number Line

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19 The integer 8 is ______ 9. A = B < C >

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20 The integer 7 is ______ 7. A = B < C >

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21 The integer 3 is ______ 5. A = B < C >

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22 The integer -4 is ______ -3.

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A = B < C >

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23 The integer -4 is ______ -5.

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A = B < C >

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24 The integer -20 is ______ -14. A = B < C >

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25 The integer -14 is ______ -6 . A = B < C >

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26 The integer -4 is ______ 6. A = B < C >

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27 The integer -3 is ______ 0. A = B < C >

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28 The integer 5 is ______ 0. A = B < C >

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29 The integer -4 is ______ -9. A = B < C >

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30 The integer 1 is ______ -54. A = B < C >

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31 The integer -480 is ______ 0. A = B < C >

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Addition and Subtraction

f Integers

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Slide 67 / 262 Adding and Subtracting Integers

Now that we have a new operation (subtraction) and new numbers (integers), we need new rules. The new rules are based on the old ones. We have to show how to do addition and subtraction with our newly invented negative integers.

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First, let's add a positive and negative integer: 4 + (-5) = ? Before we start, we must understand that -5 means to move 5 spaces to the left of 0, as shown below. 4

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Now, add the -5 to the end of the 4.

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Addition is commutative, so we get the same answer for (-5) + 4 = ? +4

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Adding Integers Slide 70 / 262

Now let's add two negative integers: (-4) + (-5) = ? Like the last example, we start off going to the left 4 spaces, but then we continue to the left 5 spaces.

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Therefore, (-4) + (-5) = -9. Don't memorize "rules" to add positive & negative numbers.. Instead, sketch the number line to perform each addition problem. That way you know that you are correct every time. 1 2 3 4 5 6 7 8 9 10

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32 Which number line would be used to show 9 + (-1)? A B C D

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33 Which number line would be used to show (-6) + (-3)? A B C D

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34 Which number line would be used to show (-7) + 2? A B C D

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35 Which number line would be used to show 2 + 8? A B C D

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36 ) 11 + (-4) =

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37 ) -4 + (-4) =

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38 ) 17 + (-20) =

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39 ) -15 + (-30) =

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40 ) -5 + 10 =

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41 ) 11 + (-4) =

Slide 81 / 262 Subtracting Integers

8 - 2 = 6 Since 8 is positive, we need to travel to the right 8 steps. Next, instead of moving to the right 2 spaces, as in addition, subtraction moves in the opposite direction, which means we move 2 spaces to the left. Our answer is 6. subtract 2 8 1 2 3 4 5 6 7 8 9 10

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Slide 82 / 262 Subtracting Integers

How would we show 3 - 8? As in the last example, we would start off going right 3 and then left 8. Therefore, 3 - 8 = -5 subtract 8 3 1 2 3 4 5 6 7 8 9 10

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Slide 83 / 262 Adding & Subtracting Integers

How would we show (-4) - 6? Remember that (-4) is shown by going 4 spaces to the left. Subtracting 6, take us 6 more spaces to the left. Therefore, (-4) - 6 = -10 subtract +6

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Slide 84 / 262 Adding & Subtracting Integers

The most potentially confusing case is subtracting a negative integer. How would we show 1 - (-6)? First, we would go 1 space to the right, because of the 1. Adding -6 spaces would take us to the left, but we must do the

pposite since we are subtracting, moving 6 steps to the right.

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Slide 85 / 262 Adding & Subtracting Integers

Therefore, 1 - (-6) = 7 subtract -6 +1 When we subtracted -6, it was the same as adding +6. This is because -6 is six spaces to the left, and so when you go in the

pposite direction, you go 6 spaces to the right.

The same as adding 6. Subtracting a negative number is the same as adding a positive one: "two minuses make a plus". 1 2 3 4 5 6 7 8 9 10

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Slide 86 / 262 Adding & Subtracting Integers

How would we show -7 - (-9)? First, go 7 spaces to the left, because we're adding -7. If we were adding -9 we would go 9 spaces to the left, but since we are subtracting -9, we must go in the opposite direction, 9 to the right. Therefore, -7 - (-9) = 2 Let's look at this another way. subtract -9

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Slide 87 / 262 Adding & Subtracting Integers

That yields the same answer: +2 + 9

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We could rewrite -7 - (-9) to be -7 +9, since subtracting a negative number is the same as adding a positive one.

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42 Which number line would be used to show 2 - 8? A B C D

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43 Which number line would be used to show (-7) - 2? A B C D

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44 Which number line would be used to show (-6) - (-3)? A B C D

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45 Which number line would be used to show 9 - (-1)? A B C D

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46 ) -2 - 6 =

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47 ) 4 - 10 =

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48 ) 3 - (-8) =

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49 ) -5 - (-3) =

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50 ) -7 - 8 =

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51 ) -7 - 2 =

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52 Convert the subtraction problem into an addition problem.

8 - (-4)

A 8 + 4 B 8 + (-4) C -8 + 4 D -8 + (-4)

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53 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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54 Convert the subtraction problem into an addition problem. 12 - (-5) A 12 + 5 B 12 + (-5) C -12 + 5 D -12 + (-5)

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Subtraction would be commutative if a - b = b - a Let's test that with some numbers 4 - 5 = -1 5 - 4 = 1 Since -1 ≠ 1, then 4 - 5 ≠ 5 - 4 In general, a - b ≠ b - a Subtraction is not commutative.

Is Subtraction Commutative? Slide 102 / 262

Multiplication and Division

f Integers

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Slide 103 / 262 Multiplication

Multiplication can be indicated by putting a dot between two numbers, or by putting the numbers into parentheses. (We won't generally use "x" to indicate multiplication since that letter is used a lot in algebra for variables.) So multiplying 3 times 2 will be written as: 3 ∙ 2

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Slide 104 / 262 Multiplication

Multiplication is repeated addition. So, to find the product of 3 ∙ 2 we would add the number 2 to itself three times: 3 ∙ 2 = 2 + 2 + 2 1 2 3 4 5 6 7 8 9 10

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Slide 105 / 262 Multiplication is Commutative

Since addition is commutative... And multiplication is just repeated addition, multiplication is commutative: 3 ∙ 2 = 2 + 2 + 2 = 6 2 ∙ 3 = 3 + 3 = 6 1 2 3 4 5 6 7 8 9 10

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Slide 106 / 262 Multiplication is Commutative

a ∙ b = b ∙ a Adding a number "a" to itself "b" times yields the same result as adding a number "b" to itself "a" times. +a +a +a +b +b ab ab Adding "a" to itself "b" times yields "ab". Adding "b" to itself "a" times also yields "ab".

Slide 107 / 262 Multiplying Negative Integers

It works the same way when you multiply a negative number by a positive number. So, 3 ∙ (-2) just indicates to add -2 to itself three times: 3 ∙ (-2) = (-2) + (-2) + (-2) = -6 1 2 3 4 5 6 7 8 9 10

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Since multiplication is commutative, (-3) ∙ 2 = 2 ∙ (-3) = (-3) + (-3) = -6 So, this just becomes adding -3 to itself 2 times. 1 2 3 4 5 6 7 8 9 10

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Multiplying Negative Integers

Multiplying a positive and negative integer results in a negative integer: (a)(-b) = (-a)(b) = -ab

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When multiplying two numbers, if both are positive, the answer is positive. If one is negative and the other is positive, the answer is negative. How about if both numbers being multiplied are negative?

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How do we interpret (-3)(-2)? Since, (3)(-2) means to add -2 to itself 3 times We could interpret (-3)(-2) to mean to subtract (-2) from itself three times. We already learned that addition and subtraction are inverses, so subtracting -2 is the same as adding 2. So, (-3)(-2) indicates to add 2 to itself 3 times. So, (-3)(-2) = (3)(2) = 6

Multiplying Positive & Negative Integers Slide 111 / 262 Multiplying & Dividing Integers

While showing how to multiply integers, we have come across some "shortcuts" for determining the sign of our product. What generalizations can you make about the sign of your product? (Look at the patterns below to help.) 3(3) = 9

5(3) = -15

3(2) = 6

5(2) = -10

3(1) = 3

5(1) = -5

3(0) = 0

5(0) = 0

3(-1) = -3

5(-1) = 5

3(-2) = -6

5(-2) = 10

3(-3) = -9

5(-3) = 15

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55 What will be the sign of the product (-6) ∙ 8? A Positive B Negative

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56 What will be the sign of the product (-4) (-9)? A Positive B Negative

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57 Find the product: 9 ∙ (-11)

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58 Find the product: (-12)(-11)

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59 Find the product: (-7) ∙ 9

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60 Find the product: (-3) ∙ (-9)

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7(4) = 28 This equation provides the answer "28" to the multiplication question "what is the product of 7 and 4". What are the two inverse questions that can be asked and answered based on the above multiplication fact? Which mathematical operation is the inverse of multiplication? DISCUSS!

Inverse Operations Slide 119 / 262

7(4) = 28 There are two division questions are the inverse of this are: 28÷4 = 7 This provides the answer "7" to the question "what is 28 divided by 4". 28÷7 = 4 This provides the answer "4" to the question "what is 28 divided by 7".

Inverse Operations Slide 120 / 262

Division asks: If I divide something into pieces of equal size, what will be the size of each piece? For instance, 15÷3 asks if I divide 15 into 3 pieces, what will be the size of each piece? The answer is 5, since 5 + 5 + 5 = 15 Three equal pieces of 5 will add to equal 15.

Inverse Operations

SLIDE 21

Slide 121 / 262 Multiplying & Dividing Integers

Since multiplication and division are so closely related, we can get the rules for division using the rules of multiplication. For example, = 5 because 5 is the number you multiply by 6 to get 30. In turn, 30 = 6 ∙ 5, so = 5 30 6 30 6

Slide 122 / 262 Multiplying & Dividing Integers

What is ? Why? What is ? Why? 30

6
30
6

Slide 123 / 262 Multiplying & Dividing Integers

What generalizations can you make about the sign of the quotient?

Slide 124 / 262

61 What is the sign of the quotient of -69 ÷ (-3)? A Positive B Negative

Slide 125 / 262

62 What is the sign of the quotient of 52 ÷ (-4)? A Positive B Negative

Slide 126 / 262

63 Find the quotient of -65 ÷ (-5).

SLIDE 22

Slide 127 / 262

64 Find the quotient of -126 ÷ (-3).

Slide 128 / 262

65 Find the quotient of 104 ÷ (-4).

Slide 129 / 262

66 Find the quotient of -88 ÷ (11).

Slide 130 / 262

67 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

verall 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

ver 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 EOY sample test non-calculator #10

Slide 131 / 262

Operations with Rational Numbers

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f Contents

Slide 132 / 262

Just as subtraction led to a new set of numbers: negative integers. Division leads to a new set of numbers: fractions. This results when you ask questions like: 1÷2 = ? 1÷3 = ? 2÷3 = ? 1÷1,000,000 = ?

New Numbers: Fractions

SLIDE 23

Slide 133 / 262

1÷2 = ? asks the question: If I divide 1 into 2 equal pieces, what will be the size of each? The answer to this question cannot be found in the integers. New numbers were needed.

New Numbers: Fractions Slide 134 / 262

The space between any two integers can be divided by any integer you choose...as large a number as you can imagine. There are as many fractions between any pair of integers as there are integers. Fractions can be written as the ratio of two numbers:

,- , , , , , , etc.

Or in decimal form by dividing the numerator by the denominator:

0.666, -0.25, -0.125, 0.333, 0.8,1.4, 20, etc.

The bar over "666" and "333" means that pattern repeats forever.

Fractions

2 3 1 4 1 8 1 3 4 5 7 5 80 4

Slide 135 / 262

There are an infinite number of fractions between each any integers. Looking closely between 0 and 1, we can locate a few of them. 1 2 3 4 5 6 7 8 9 10

1
2
3
4
5
6
7
8
9
10

Fractions

1 0.66 0.25 0.33 0.80 It's easier to find their location when they are in decimal form since it's clear which integers they're between...and closest to. 0.50 0.20 1 4 1 5 1 3 1 2 2 3 4 5

Slide 136 / 262

Rational Numbers are numbers that can be expressed as a ratio

f two integers.

This includes all the fractions, as well as all the integers. What are a few ways you could write 5 as a ratio of two integers?

Rational Numbers Slide 137 / 262

Fractions can be written in "fraction" form or decimal form. When written in decimal form, rational numbers are either: Terminating, such as

= -0.500000000000 = -0.5

Repeating, such as = 0.142857142857142857... = 0.142857 Or,

= 0.333333333333333333... = 0.3

1 2 1 7 1 3

Rational Numbers Slide 138 / 262 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?

SLIDE 24

Slide 139 / 262 Fractions and the Negative Sign

These two fractions equal positive one-half. Why are they both positive?

Slide 140 / 262 Slide 141 / 262 Slide 142 / 262 Slide 143 / 262

71 What is the position of the dot on the number line below?

1 2 3 4 5 6 7 8 9 10

1
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9
10

A -1 B -1 C -2

1 1 1 2 4 4

Slide 144 / 262

72 What is the position of the dot on the number line below? A -5.5 B -6.5 C -5.2

1 2 3 4 5 6 7 8 9 10

1
2
3
4
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9
10

SLIDE 25

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73 What is the position of the dot on the number line below?

1

1 2 3

1 1 2 1 3 5 1 A B C 2 5

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74 What is the position of the dot on the number line below?

1

1 2 3

A -0.8 B -0.5 C -0.6

Slide 147 / 262 Slide 148 / 262

76

1 2 3 4 5 6 7 8 9 10

1
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9
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A = B < C >

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77

1 2 3 4 5 6 7 8 9 10

1
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9
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A = B < C >

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78

1 2 3 4 5 6 7 8 9 10

1
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9
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A = B < C >

SLIDE 26

Slide 151 / 262 Slide 152 / 262 Slide 153 / 262

One number that is not defined by our numbers and mathematical operations is dividing any number by zero. The result of that division is "undefined." This will be critical later when we are working with equations

r simplifying fractions.

Dividing by zero is undefined since there is no way to say how many times zero can go into any number.

Dividing by Zero Slide 154 / 262

Addition and Subtraction of Rational Numbers

Return to Table of Contents

Slide 155 / 262 Adding Rational Numbers

How would we show -4 + 9.5? Therefore, -4 + 9.5 = 5.5 + 9.5

4

1 2 3 4 5 6 7 8 9 10

1
2
3
4
5
6
7
8
9
10

Slide 156 / 262 Adding Rational Numbers

How would we show ? Therefore, 5.5 +(-7) = -1.5

7

+5.5 1 2 3 4 5 6 7 8 9 10

1
2
3
4
5
6
7
8
9
10

SLIDE 27

Slide 157 / 262 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, especially when working with Rational Numbers.

Slide 158 / 262

1 2 3 4 5 6 7 8 9 10

1
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9
10

1 2 3 4 5 6 7 8 9 10

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1 2 3 4 5 6 7 8 9 10

1
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9
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3 + 4 = 7

4 + 9.5 = 5.5

5 + (-7) = -2

4 + (-5) = -9

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. Discuss the patterns that you see with your group.

Addition: Using Absolute Values Slide 159 / 262

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)

1 2 3 4 5 6 7 8 9 10

1
2
3
4
5
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7
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10

1 2 3 4 5 6 7 8 9 10

1
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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 Slide 160 / 262

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)

1 2 3 4 5 6 7 8 9 10

1
2
3
4
5
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7
8
9
10
4 + 9.5 = 5.5

1 2 3 4 5 6 7 8 9 10

1
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9
10

5 + (-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 Slide 161 / 262

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)

Summary Addition: Using Absolute Values Slide 162 / 262 What About Subtraction?

Remember, any subtraction can be turned into addition by: · Changing the subtraction sign to addition. · Changing the number after the subtraction sign to its opposite. · Then you can use the addition rules to solve. Examples: 5 - (-3) is the same as 5 + 3

12 - 17 is the same as -12 + (-17)

SLIDE 28

Slide 163 / 262

81 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

Slide 164 / 262

82 Convert the subtraction problem into an addition problem. A B C D

Slide 165 / 262

83 Convert the subtraction problem into an addition problem. 6.5 - (-3.2) A -6.5 + (-3.2) B -6.5 + 3.2 C 6.5 + (-3.2) D 6.5 + 3.2

Slide 166 / 262

84 Which expressions are equivalent to Select all that apply. A B C D E F

From PARCC EOY sample test non-calculator #14

Slide 167 / 262

85 ) -10.5 + 6.2 =

Slide 168 / 262

86 ) -7.3 - (-4) =

SLIDE 29

Slide 169 / 262

87 )

Slide 170 / 262

88 ) 9.27 + (-8.38) =

Slide 171 / 262

89 ) -4.2 + (-5.9) =

Slide 172 / 262

90 ) -2 - (-3.95) =

Slide 173 / 262

91 ) 5 - 6 + (-7.5) =

Hint: Remember addition and subtraction is solved left to right in the

rder of operations!

Slide 174 / 262

92 ) 19 + (-12) - 11 =

SLIDE 30

Slide 175 / 262

93 ) -2.3 + 4.1 + (-12.7) =

Slide 176 / 262

94 ) -8.3 - (-3.7) + 5.2 =

Slide 177 / 262

95 )

Slide 178 / 262

96 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 EOY sample test non-calculator #12

1

1 n p

Slide 179 / 262

Multiplication and Division of Rational Numbers

Return to Table of Contents

Slide 180 / 262

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

SLIDE 31

Slide 181 / 262

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 Slide 182 / 262

97 ) 5(-4.82) =

Slide 183 / 262

98 ) (3.2)(-6.4) =

Slide 184 / 262

99 ) (-5.12)(-9) =

Slide 185 / 262

100 )

Slide 186 / 262

101 ) (-2)(-7.5)(-4) =

SLIDE 32

Slide 187 / 262

102 )

Slide 188 / 262

103 ) (-2.5)(-4.1)(3) =

Slide 189 / 262

Teachers: Use this Mathematical Practice Pull Tab for the next 2 SMART Response Questions that follow.

Slide 190 / 262

104 Jane has entered a baking contest. Jane uses 3.1

unces of flour to make one cinnamon roll. How

many ounces of flour does Jane need to make 7 cinnamon rolls?

Slide 191 / 262

105 Timmy is shipping 4 boxes of shirts. Each box weighs 6.3 pounds. If it cost 5.20 per pound to ship. How much does Timmy have to spend to ship them?

Slide 192 / 262

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

SLIDE 33

Slide 193 / 262

106 Find the value of:

Slide 194 / 262

107 Find the value of:

Slide 195 / 262 Slide 196 / 262

109 Find the value of:

Slide 197 / 262

110 Find the value of:

Slide 198 / 262

111 Find the value of:

SLIDE 34

Slide 199 / 262

112 Kobe put 8 toy cars in a row. The line of cars was 16.4 meters long. How long was each car?

Slide 200 / 262

113 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?

Slide 201 / 262

Converting Rational Numbers to Decimals

Return to Table of Contents

Slide 202 / 262

Definition of Rational Number: 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? Slide 203 / 262

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? Slide 204 / 262 Converting Fractions to Repeating Decimals

Example 3: 7 9 .777... 9 7.0000

63

70

63

70

63

7 )

SLIDE 35

Slide 205 / 262 Converting Fractions to Repeating Decimals

Example 4: 5 27 .18518... 27 5.0000

27

230

216

140

135

50

27

230

216

14 )

Slide 206 / 262 Converting Fractions to Decimals

Example 5: 4 4 25 .16 25 4.000

25

150

150

) When converting a mixed number to a decimal, write down the whole number followed by a decimal point. Then, use long division to convert the fraction. Therefore, our answer is 4.16

Slide 207 / 262

114 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.)

Slide 208 / 262

115 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.)

Slide 209 / 262

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

Slide 210 / 262

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.)

SLIDE 36

Slide 211 / 262

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.)

Slide 212 / 262

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.)

Slide 213 / 262

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.)

Slide 214 / 262

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.)

Slide 215 / 262

122 Mike needed meters of fabric to fix his

couch. How can this be written as a decimal?

Slide 216 / 262

123 Hannah rode her bike miles to the neighborhood pool. What is the distance she rode her bike as a decimal?

SLIDE 37

Slide 217 / 262

124 Kevin Durant made shots in the first quarter

f the NBA finals, how is that written as a

decimal?

Slide 218 / 262

125 Coral's mother wants to know show as a

decimal. How can that fraction be written as a

decimal?

Slide 219 / 262

Exponents

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f Contents

Slide 220 / 262 Exponents

When "raising a number to a power": The number we start with is called the base, the number we raise it to is called the exponent.

24

The entire expression is called a power. You read this as "two to the fourth power."

Slide 221 / 262 Powers of Integers

Just as multiplication is repeated addition, exponents are repeated multiplication. For example, 35 reads as "3 to the fifth power" = 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 In this case "3" is the base and "5" is the exponent. The base, 3, is multiplied by itself 5 times.

Slide 222 / 262

126 What is the base in this expression? 32

SLIDE 38

Slide 223 / 262

127 What is the exponent in this expression? 32

Slide 224 / 262

128 What is the base in this expression? 73

Slide 225 / 262

129 What is the exponent in this expression? 43

Slide 226 / 262

130 What is the base in this expression? 94

Slide 227 / 262

131 What is 44 in standard form?

Slide 228 / 262

132 What is 83 in standard form?

SLIDE 39

Slide 229 / 262

133 What is 25 in standard form?

Slide 230 / 262

134 An expression is shown. 3·3·3·3 What is the expression written in exponential form? A 3·4 B 81 C 43 D 34

Slide 231 / 262 Special Terms: Squared & Cubed

A number raised to the second power can be said to be squared. That's because the area of a square of length x is x2: "x squared." A number raised to the third power can be said to be cubed. That's because the volume of a cube of length x is x3: "x cubed."

Slide 232 / 262

135 Evaluate 32.

Slide 233 / 262

136 Evaluate 52.

Slide 234 / 262

137 Evaluate 82.

SLIDE 40

Slide 235 / 262

138 Evaluate 43.

Slide 236 / 262

139 Evaluate 73.

Slide 237 / 262

When evaluating exponents of negative numbers, keep in mind the meaning of the exponent and the rules of multiplication. For example, (-3)2 = (-3)(-3) = 9, is the same as (3)2 = (3)(3) = 9. However, (-3)2 = (-3)(-3) = 9 is NOT the same as

32 = -(3)(3) = -9,

Similarly, (3)3 = (3)(3)(3) = 27 is NOT the same as (-3)3 = (-3)(-3)(-3) = -27,

Powers of Integers Slide 238 / 262

140 What is (-7)2? A 49 B -49

Slide 239 / 262

141 What is -82? A 64 B -64

Slide 240 / 262

142 What is -24? A 16 B -16

SLIDE 41

Slide 241 / 262

143 What is (-2)6? A 64 B -64

Slide 242 / 262

144 Evaluate: 43

Slide 243 / 262

145 Evaluate: (-2)7

Slide 244 / 262

146 Evaluate: (-3)4

Slide 245 / 262

Real Numbers

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f Contents

Slide 246 / 262 Real Numbers

Real Numbers are numbers that exist on a number line. Rational Numbers, Integers, Whole Numbers and Natural Numbers are all types of Real Numbers we have learned about. We will learn about Irrational Numbers in 8th Grade.

SLIDE 42

Slide 247 / 262

147 What type of number is -30? Select all that apply. A Real Number B Rational C Integer D Whole Number E Natural Number

Slide 248 / 262

148 What type of number is - ? Select all that apply. A Real Number B Rational C Integer D Whole Number E Natural Number 5 3

Slide 249 / 262

149 What type of number is 0? Select all that apply. A Real Number B Rational C Integer D Whole Number E Natural Number

Slide 250 / 262

150 What type of number is - 4 ? Select all that apply. A Real Number B Rational C Integer D Whole Number E Natural Number 1 2

Slide 251 / 262

151 What type of number is 25? Select all that apply. A Real Number B Rational C Integer D Whole Number E Natural Number

Slide 252 / 262

152 What type of number is ? Select all that apply. A Real Number B Rational C Integer D Whole Number E Natural Number 2 8

SLIDE 43

Slide 253 / 262

Glossary & Standards

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f Contents

Slide 254 / 262

Back to Instruction

Absolute Value

How far a number is from zero on the number line.

2 = 2

2 = 2

2 -1
2 -1
2 -1

0 = 0

2 1 0 1 2 1 2

Slide 255 / 262

Back to Instruction

Fractions

Numbers created through division written as the ratio of two numbers.

Dividing by zero is undefined. 2 3 1 4 1 8 1 3 4 5 7 5 80 4

Slide 256 / 262

Back to Instruction

Integers

Positive numbers, negative numbers and zero ..., -2, -1, 0, 1, 2, ...

symbol for integers

Slide 257 / 262

Back to Instruction

Inverse Operation

The operation that reverses the effect of another operation.

Addition

Subtraction Multiplication Division + _ x ÷ 11 = 3y + 2

2
2

9 = 3y ÷ 3 ÷ 3 3 = y

5 + x = 5

x = 10 + 5 + 5

Slide 258 / 262

Back to Instruction

Natural Numbers

Counting numbers 1, 2, 3, 4, ...

symbol for natural numbers

SLIDE 44

Slide 259 / 262

Back to Instruction

Rational Numbers

A number that can be expressed as a fraction. 1 4 5.3 2

symbol for rational numbers

Slide 260 / 262

Back to Instruction

Real Numbers

All the numbers that can be found on a number line.

1 2 3 4 5 6 7 8 9 10

1
2
3
4
5
6
7
8
9
10

Slide 261 / 262

Back to Instruction

Whole Numbers

Counting numbers including 0 0, 1, 2, 3, ...

symbol for whole numbers

Slide 262 / 262

Standards for Mathematical Practices

Click on each standard to bring you to an example of how to meet this standard within the unit.

MP8 Look for and express regularity in repeated reasoning. MP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision. MP7 Look for and make use of structure.