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Grade 6 The Number System & Mathematical Operations 2015 10 20 www.njctl.org 2
Table of Contents Addition, Natural Numbers & Whole Numbers Addition, Subtraction and Integers Multiplication, Division and Rational Numbers Absolute Value Comparing Integers Comparing and Ordering Rational Numbers Exponents Real Numbers Teacher Notes Glossary & Standards Click on a topic to go to that section. 3
Addition, Natural Numbers & Whole Numbers Return to Table of Contents 4
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. 5
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 6
Natural Numbers and Addition 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. 7
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. 8
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 9
Whole Numbers Adding zero to the Counting Numbers gives us the Whole Numbers. 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), was found about 1500 years ago. 0, 1, 2, 3, 4,... It was found in Cambodia and the dot is for the zero in the year 605. http://www.smithsonianmag.com/history/originnumber zero 180953392/?no ist 10
Why zero took so long to Invent Horses versus houses. 3 2 1 0 zero horses = zero houses 11
Why zero took so long 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? Math Practice Zero just isn't a natural number, but it is a whole number. 12
Addition, Subtraction and Integers Return to Table of Contents 13
Addition and Subtraction The simplest mathematical operation is addition. The inverse of addition is subtraction. Two operations are inverses if one "undoes" the other. Inverse operations are a very important concept, and apply to all mathematics. 14
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. 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 15
Adding Whole Numbers 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, it takes you to 4. 4 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 The number +5 is five steps to the right. Starting at 0, it takes you to 5. 5 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 16
Adding Whole Numbers 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 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 Therefore, 4 + 5 = 9. 17
The Commutative Property of Addition 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? 18
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. 5 5 + 4 4 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 5 + 4 = 9 19
The Commutative Property of Addition So, 4 + 5 = 5 + 4. Addition is commutative in this case. 4 + 5 4 5 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 5 5 + 4 4 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 20
The Commutative Property of Addition But there's nothing special about these numbers. This is true for any numbers: a + b = b + a a a + b b b b + a a 21
Inverse Operations Operations are "inverses" when one of them undoes what the other does. What would undo adding 5? Subtracting 5 click 22
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 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 23
Addition and Subtraction are Inverses This is true for any two numbers. Start with a, add b, then subtract b. You end up with the number you began with: a. b +b a a + b 24
Inverse Operations 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. 25
Subtracting Whole Numbers Here's what 9 5 looks like on the number line +9 5 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 And, here's 9 4 +9 4 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 26
Subtraction was invented to undo addition. But this new operation allows us to ask new questions. And the number system, up to that point, couldn't provide answers. For example: What is the result of subtracting 7 from 4? 4 7 =? We Need More Numbers 27
Subtracting Whole Numbers 4 7 =? 7 +4 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 There was no answer to questions like this in the whole number system, which was all there was until about 500 years ago. 28
Negative Numbers 4 7 =? 7 +4 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 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 have trouble with negative numbers, so did most everyone else. 29
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. 30
Integers The below number line shows only the integers. 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 31
Integers on the number line Negative Integers Zero Positive Integers 5 4 3 2 1 0 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 ` 32
Classify each number as an integer, or not. 9 3.2 3¾ 6 1 2.34437 x 10 3 5 21 ¾ 5 ½ integer 6.32 65 π not an integer 0 Answer 33
1 Which of the following are examples of integers? A 0 B 8 C 4.5 Answer D 7 E 1 3 34
2 Which of the following are examples of integers? A 2 5 3 B 6.1 C 287 Answer D 1000 E 0.33 35
3 Which of the following are examples of integers? A 1 4 B 6 C 4 D 0.75 Answer E 25% 36
Integers In Our World 37
Integers can represent everyday situations You might hear "And the quarterback is sacked for a loss of 7 yards." This can be represented as an integer: 7 Or, "The total snow fall this year has been 9 inches more than normal." This can be represented as in integer: +9 or 9 38
Write an integer to represent each situation 1. Spending $6 click $6 2. Gain of 11 pounds click 11 lbs. 3. Depositing $700 click $700 4. 10 degrees below zero click 10 degrees 5. 8 strokes under par (par = 0) click 8 Math Practice 6. 350 feet above sea level click 350 ft. 39
Sea Level The picture below shows three different people at three different elevations. Derived from 40
4 If you were to draw a vertical number line to model elevation, which person's elevation would be at zero? A scuba diver B sailor C hiker Answer What does zero represent in this situation? click Zero represents the top of the water (the water's surface). Derived from 41
Sea Level The diver is 30 feet below sea level. The sailor is at sea level. The hiker is 2 miles (10,560 feet) above sea level. Write an integer to represent each situation: Answer Diver Sailor Hiker Derived from 42
5 A submarine is submerged 800 feet below sea level. Which statement below is expressed correctly? A The depth of the submarine is 800 feet below sea level. B 800 feet below sea level can be represented by the integer 800. C Both A & B Answer Derived from 43
6 A coral reef is below sea level. Which statement below is expressed correctly? A The elevation of a coral reef with respect to sea level is given as 250 feet. B The depth of the coral reef is 250 feet below sea level. C Both A & B Answer Derived from 44
7 Alex's body temperature decreased by 20 F. Which statement below is expressed correctly? A Alex's body temperature dropped by 20 F. B The integer 2 represents the change in Alex's body temperature in degrees Fahrenheit. C Both A & B Answer Derived from 45
8 Which of the following integers best represents the following scenario: The effect on your wallet when you spend 10 dollars. A 10 B 10 C 0 D +/ 10 Answer 46
9 Which of the following integers best represents the following scenario: Earning $40 shoveling snow. A 40 B 40 C 0 D +/ 40 Answer 47
10 Which of the following integers best represents the following scenario: You dive 35 feet to explore a sunken ship. A 35 B 35 C 0 D +/ 35 Answer 48
11 Which of the following statements are true? (Select all that apply) A The integer for Temperature A is the opposite of the integer for Temperature B. B Temperature A is warmer than Temperature B. C Temperature C is warmer than Temperature A and B. D Temperature C is 10 degrees above zero. Answer A: 7 o F B: 7 o F C: 10 o F 49
12 Which inequality statement correctly shows the relationship between the three temperatures? A 7 > 7 < 10 B 7 > 7 < 10 C 7 < 7 < 10 D 10 < 7 < 7 Answer o A: 7 F o B: 7 F o C: 10 F 50
Number Line The numbers 4 and 4 are shown on the number line. 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 What do you notice about the two numbers? Answer click to reveal These numbers are called opposites. Can you think of why? 51
Opposites There are 2 ways to read: 9 "negative nine" "the opposite of nine" So, saying "negative" and "the opposite of" are interchangeable. Remember, opposites are the same distance from zero, just on different sides of the number line. 52
13 What is the opposite of 5? Each mark on the number line represents one unit. Plot a point on the number line that represents the opposite of 5 units. Answer From PARCC PBA sample test non calculator #2 53
14 What is the opposite of 25? Answer 54
15 What is the opposite of 0? Answer 55
16 What is the opposite of 18? Answer 56
17 What is the opposite of 18? Answer 57
18 What is the opposite of the opposite of 18? Answer 58
Opposites What conclusions can you draw about the opposite of the opposite of a number? It's the same as the original number! Let's look at the last three problems. 18 Opposite of 18 = 18 Opposite of the opposite of 18 = ( 18) = 18 This will be helpful to understand when we work with integer operations! 59
19 Simplify: ( 9) Answer 60
20 Simplify: ( 12) Answer 61
21 Simplify: [ ( 15)] Answer 62
Integers in Game Shows In the game of Jeopardy you: earn points for a correct response. lose points for an incorrect response. can have a positive or negative score. Teacher Notes & Math Practice 63
Integers in Game Shows When a contestant gets a $200 question correct: Score = $200 Then a $100 question incorrect: Score = $100 Then a $300 question incorrect: Score = $200 How did the score become negative? 1. $0 + $200 = $200 2. $200 $100 = $100 3. $100 $300 = $200 click 64
22 After the following 3 responses what would the contestants score be? $100 incorrect $200 correct $50 incorrect Answer 65
23 After the following 3 responses what would the contestants score be? $200 correct $50 correct $300 incorrect Answer 66
24 After the following 3 responses what would the contestants score be? $150 incorrect $50 correct $100 correct Answer 67
To Review An integer is zero, any natural number, or its opposite. Number lines have negative numbers to the left of zero and positive numbers to the right. Zero is neither positive nor negative. Integers can be used to represent real life situations. 68
Multiplication, Division and Rational Numbers Return to Table of Contents 69
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 or (3)(2) 70
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 +2 +2 +2 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 71
Multiplication is Commutative Since addition is commutative... And multiplication is just repeated addition, multiplication is commutative: 3 2 = 2 + 2 + 2 = 6 +2 +2 +2 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 2 3 = 3 + 3 = 6 +3 +3 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 72
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. Adding "a" to itself "b" times yields "ab". +a +a +a ab Adding "b" to itself "a" times also yields "ab". +b +b ab 73
Inverse Operations 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! 74
Inverse Operations 7(4) = 28 There are two division questions that come from the inverse of this: 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?". 75
Inverse Operations 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. 76
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, get 30. 30 6 In turn, 30 = 6 5, so 30 = 5 6 = 5 because 5 is the number you multiply by 6 to Math Practice 77
New Numbers: Fractions 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 =? 78
New Numbers: Fractions 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. 79
Fractions 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: 2, 1, 1, 1, 4, 7, 80, etc. 3 4 8 3 5 5 4 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. 80
Fractions There are an infinite number of fractions between any integers. Looking closely between 0 and 1, we can locate a few of them. 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 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. 1 5 1 4 1 3 1 2 2 3 4 5 0.20 0 0.25 0.33 0.50 0.66 0.80 1 81
Rational Numbers Rational Numbers are numbers that can be expressed as a ratio of 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? Math Practice 82
Rational Numbers Fractions can be written in "fraction" form or decimal form. When written in decimal form, rational numbers are either: Terminating, such as 1 2 = 0.500000000000 = 0.5 Repeating, such as 1 7 Or, 1 3 = 0.142857142857142857... = 0.142857 = 0.333333333333333333... = 0.33 83
Dividing by Zero 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 or simplifying fractions. Dividing by zero is undefined since there is no way to say how many times zero can go into any number. 84
Absolute Value Return to Table of Contents 85
Absolute Value of Integers The absolute value is the distance a number is from zero on the number line, regardless of direction. Distance and absolute value are always non negative (positive or zero). 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 What is the distance from 0 to 5? 86
Absolute Value of Integers 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. 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 What is the distance from 0 to 5? 5 Click to Reveal 87
We can use absolute value to describe the relative size of numbers. If you dive 35 feet underwater, your depth is 35 feet. However, we often say 35 to describe the number of feet ( 35 ). If you owe someone $45, your debt is described as $45 rather than $45. We use 45 to describe the amount. 88
Absolute value is symbolized by two vertical bars 4 This is read, "the absolute value of 4". 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 What is the 4? 4 Click to Reveal 89
Absolute Value Use the number line to find absolute value. 9 = 9 click 9 = 9click 4 = 4click 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 90
25 Find 7 Answer 91
26 Find 28 Answer 92
27 Find 56 Answer 93
28 Find 8 Answer 94
29 Find 3 Answer 95
30 What is the absolute value of the number shown in the generator? Click on blank screen to generate a number. 96
31 Which numbers have 12 as their absolute value? A 24 B 12 C 0 D 12 E 24 Answer 97
32 Which numbers have 50 as their absolute value? A 50 B 25 C 0 D 25 E 50 Answer 98
33 Johnny says that for temperatures below zero, as the temperature increases, the absolute value of the temperature decreases. Is his thinking correct? Use a number line to prove your answer. Yes No Answer & Math Practice 99
Comparing Integers Return to Table of Contents 100
Use the Number Line 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. 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 101
Comparing Positive Integers An integer can be equal to, less than, or greater than another integer. 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. 102
34 The integer 8 is 9. A = B < C > Answer 103
35 The integer 7 is 7. A = B < C > Answer 104
36 The integer 3 is 5. A = B < C > Answer 105
Number Line Place the number tiles in the correct places on the number line. 2 3 1 0 1 2 5 3 4 5 4 106
Comparing Negative Integers The greater the absolute value of a negative integer, the smaller the integer. That's because it is farther from zero, but in the negative direction. For example: 4 = 4 4 > 6 4 < 2 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 107
Comparing Negative Integers 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. 108
Drag the appropriate inequality symbol between the pairs of integers. 1) 3 5 Inequality Symbol < > 2) 237 259 3) 63 36 4) 10 15 5) 6 3 6) 127 172 Answer 7) 24 17 8) 2 8 9) 8 8 10) 10 7 109
37 The integer 4 is 3. A = B < C > Answer 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 110
38 The integer 4 is 5. A = B < C > Answer 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 111
39 The integer 20 is 14. A = B < C > Answer 112
40 The integer 14 is 6. A = B < C > Answer 113
Comparing All Integers Any positive number is greater than zero and any negative number. Any negative number is less than zero and any positive number. 114
41 The integer 4 is 6. A = B < C > Answer 115
42 The integer 3 is 0. A = B < C > Answer 116
43 The integer 5 is 0. A = B < C > Answer 117
44 The integer 4 is 9. A = B < C > Answer 118
45 The integer 1 is 54. A = B < C > Answer 119
46 The integer 480 is 0. A = B < C > Answer 120
Thermometer A thermometer can be thought of as a vertical number line. Positive numbers are above zero and negative numbers are below zero. (Interact with the vertical number line.) Math Practice 121
47 If the temperature reading on a thermometer is 10, what will the new reading be if the temperature: falls 3 degrees? Answer 122
48 If the temperature reading on a thermometer is 10, what will the new reading be if the temperature: rises 5 degrees? Answer 123
49 If the temperature reading on a thermometer is 10, what will the new reading be if the temperature: falls 12 degrees? Answer 124
50 If the temperature reading on a thermometer is 3, what will the new reading be if the temperature: falls 3 degrees? Answer 125
51 If the temperature reading on a thermometer is 3, what will the new reading be if the temperature: rises 5 degrees? Answer 126
52 If the temperature reading on a thermometer is 3, what will the new reading be if the temperature: falls 12 degrees? Answer 127
53 On some thermometers each line does not equal one degree. What does each line equal on this thermometer? Answer 128
54 This picture shows part of a thermometer measuring temperature in degrees Fahrenheit. Answer What is the temperature, in degrees Fahrenheit, shown on the thermometer to the nearest integer? Enter your integer answer in the box. From PARCC PBA sample test non calculator #4 129
Comparing and Ordering Rational Numbers Return to Table of Contents 130
Use the Number Line To compare rational numbers, plot points on the number line. The numbers farther to the right are larger. The numbers farther to the left are smaller. 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 131
55 What is the position of the dot on the number line below? A 1 1 2 B 1 C 2 1 4 1 4 Answer 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 132
56 What is the position of the dot on the number line below? A 5.5 B 6.5 C 5.2 Answer 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 133
57 What is the position of the dot on the number line below? A B C 1 1 2 1 3 5 1 2 5 1 0 1 2 3 Answer 134
58 What is the position of the dot on the number line below? A 0.8 B 0.5 C 0.6 Answer 1 0 1 2 3 135
59 Select the point on the number line located at A B C D E F G H I Answer From PARCC EOY sample test non calculator #16 136
Number Line Where do rational numbers go on the number line? Go to the board and write in the following numbers: 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 137
Number Line Put these numbers on the number line. 2 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 Which number is the largest? The smallest? 138
Number Line Drag and drop the given rational numbers into the correct order on the number line from least to greatest. Answer From PARCC EOY sample test non calculator #5 139
Comparing Rational Numbers Sometimes you will be given fractions and decimals that you need to compare. It is usually easier to convert all fractions to decimals in order to compare them on a number line. To convert a fraction to a decimal, divide the numerator by the denominator. 0.75 4 3.00 28 020 20 0 140
Drag the appropriate inequality symbol between the numbers. 1) Inequalities < > 2) 3) 4) 5) 7) 6) 8) Answer 9) 10) 141
60 A = B < C > Answer 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 142
61 A = B < C > Answer 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 143
62 A = B < C > Answer 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 144
63 A = B < C > Answer 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 145
64 A = B < C > Answer 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 146
65 A = B < C > Answer 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 147
Exponents Return to Table of Contents 148
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. The entire expression is called a power. 2 4 You read this as "two to the fourth power." 149
Exponents This example, 3 5 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. 150
Exponents Exponents, or Powers, are a quick way to write repeated multiplication, just as multiplication was a quick way to write repeated addition. These are all equivalent: 2 4 2 2 2 2 16 In this example 2 is raised to the 4 th power. That means that 2 is multiplied by itself 4 times. 151
66 What is the base in this expression? 3 2 Answer 152
67 What is the exponent in this expression? 3 2 Answer 153
68 What is the base in this expression? 7 3 Answer 154
69 What is the exponent in this expression? 4 3 Answer 155
70 What is the base in this expression? 9 4 Answer 156
If you multiply the base and simplify the answer, the number is now written in standard form. EXAMPLE: 3 5 = 3(3)(3)(3)(3) = 243 Power Expanded Notation Standard Form TRY THESE: 1. Write 5 3 in standard form. 125 Powers of Integers When a number is written as a power, it is written in exponential form. click 2. Write 7(7)(7)(7)(7)(7)(7) in exponential form. 7 7 click 157
Number Line of 2 2 2 2 = 2 x 2 Travel a distance of 2, twice. 3 2 1 0 1 2 3 4 5 6 7 8 9 1011121314151617 158
Number Line of 2 3 2 3 = 2 x 2 x 2 2 3 = (2 x 2) x 2 First, travel a distance of 2, twice = 4 2 3 = 4 x 2 = 8 Then, travel a distance of 4, twice = 8 3 2 1 0 1 2 3 4 5 6 7 8 9 1011121314151617 159
Number Line of 2 4 2 4 = 2 x 2 x 2 x 2 2 4 = 2 x 2 x 2 x 2 First, travel a distance of 2, twice = 4 2 4 = 4 x 2 = 8 x 2 Then, travel a distance of 4, twice = 8 2 4 = 8 x 2 = 16 Then, travel a distance of 8, twice = 16 3 2 1 0 1 2 3 4 5 6 7 8 9 1011121314151617 160
71 This number line shows: (Select all that apply.) A 3 3 B 3 + 3 + 3 C (3)(3) D 3 2 E (3)(2) F 3 + 3 Answer & Math Practice 3 2 1 0 1 2 3 4 5 6 7 8 9 1011121314151617 161
72 This number line shows: (Select all that apply.) A 4 4 B (4)(4) C 2 4 D 4 + 4 + 4 E 4 2 F 4 + 4 + 4 + 4 Answer 3 2 1 0 1 2 3 4 5 6 7 8 9 1011121314151617 162
73 What is 4 4 in standard form? Answer 163
74 What is 8 3 in standard form? Answer 164
75 What is 2 5 in standard form? Answer 165
76 An expression is shown. 7 7 7 7 7 7 What is the expression written in exponential form? Answer From PARCC EOY sample test non calculator #14 166
77 An expression is shown. 3 3 3 3 What is the expression written in exponential form? Answer 167
Special Term: Squared 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 x 2 : "x squared." 168
Squared 2 2 is two squared, and 4 is the square of 2 3 2 is three squared, and 9 is the square of 3 4 2 is four squared, and 16 is the square of 4 2 2 = 4 3 2 = 9 4 2 = 16 2 2 3 4 Area 3 2 x 2 = 4 units 2 Area = 4 3 x 3 = Area = 9 units 2 4 x 4 = 16 units 2 169
Squared The area of a square whose sides have length 5 is (5)(5) or 5 2 = 25; What would the area of a square with side lengths of 6 be? click 6 2 = 36 170
Special Term: Cubed 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 x 3 : "x cubed." 171
Cubed Cubed Raising a number to the power of 3 is called "cubing" it. 2 3 is read as "two cubed," and 8 is the cube of 2 3 3 is read as "three cubed," and 27 is the cube of 3 4 3 is read as "four cubed," and 64 is the cube of 4 That comes from the fact that the volume of a cube whose sides have length 3 is (3)(3)(3) or 3 3 = 27; The volume of a cube whose sides have length 5 is (5)(5)(5) or 5 3 = 125; etc. 172
78 Evaluate 3 2. Answer 173
79 Evaluate 5 2. Answer 174
80 Evaluate 8 2. Answer 175
81 Evaluate 4 3. Answer 176
82 Evaluate 7 3. Answer 177
83 On Tuesday, you invited 2 friends to your party. On Wednesday, each of those friends invited 2 friends. This pattern continued Thursday and Friday. How many people were invited on Friday? Write your answer as a power. Answer 178
84 Sandra opened a savings account and deposited $2. Each month, she deposits twice the amount that she did the onto before. How much money will be in her savings account after 6 months? Answer 179
Real Numbers Return to Table of Contents 180
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. 181
85 What type of number is 30? Select all that apply. A Real Number B Rational C Integer D Whole Number Answer E Natural Number 182
86 What type of number is 5? Select all that apply. 3 A Real Number B Rational C Integer D Whole Number Answer E Natural Number 183
87 What type of number is 0? Select all that apply. A Real Number B Rational C Integer D Whole Number Answer E Natural Number 184
88 What type of number is 4 1? Select all that apply. 2 A Real Number B Rational C Integer D Whole Number Answer E Natural Number 185
89 What type of number is 25? Select all that apply. A Real Number B Rational C Integer D Whole Number Answer E Natural Number 186
90 What type of number is 2? Select all that apply. 8 A Real Number B Rational C Integer D Whole Number Answer E Natural Number 187
Glossary & Standards Teacher Notes Return to Table of Contents 188
Standards for Mathematical Practice MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & 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 & make use of structure. MP8: Look for & express regularity in repeated reasoning. Math Practice Additional questions are included on the slides using the "Math Practice" Pull tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull tab. 189
Absolute Value How far a number is from zero on the number line. 2 = 2 2 = 2 0 = 0 2 1 0 1 2 2 1 0 1 2 2 1 0 1 2 Back to Instruction 190
Base The number that is going to be raised to a power. Exponent 3 2 Base "3 to the second power" 2 = x 3 3 3 3 3 = 3x 3 x 3 3 2 x 2 3 3 3 x 3 3 Back to Instruction 191
Cubed A number multiplied by itself twice. 3 3 =3x3x3 = 27 Back to Instruction 192
Elevation The height of a person, place or thing above a certain reference point. This reference point is sea level. Death Valley, CA is 282 feet below sea level. Denver, CO is one mile above sea level. Back to Instruction 193
Exponent A small, raised number that shows how many times the base is used as a factor. Exponent 2 2 3 = x Base 3 "3 to the second power" 3 3 3 3 = x 3 x 3 3 3 2 x 2 3 3 3 x 3 3 Back to Instruction 194
Exponential Form Writing a number using exponents, instead of repeating factors. (3)(3) = 3 2 (2)(2)(3) = 2 2 x 3 (3)(3)(5)(5)(5) = 3 2 x 5 3 Back to Instruction 195
Fractions Numbers created through division written as the ratio of two numbers. 2 3 1 4 1 8 1 3 4 5 7 5 80 4 Dividing by zero is not allowed. Back to Instruction 196
Inequality Symbol A symbol used to compare the value of two numbers that are not equal. larger smaller smaller larger Greater than Less than Not equal Back to Instruction 197
Integers Positive numbers, negative numbers and zero..., 2, 1, 0, 1, 2,... symbol for integers Back to Instruction 198
Inverse Operation The operation that reverses the effect of another operation. Addition_ + Subtraction Multiplication x Division 5 + x = 5 + 5 + 5 x = 10 11 = 3y + 2 2 2 9 = 3y 3 3 3 = y Back to Instruction 199
Natural Numbers Counting numbers 1, 2, 3, 4,... symbol for natural numbers Back to Instruction 200
Power A power is another name for an exponent. It is a small, raised number that shows how many times to multiply the base by itself. Exponent 2 3 Base 3 "3 to the second power" 3 2 = x 3 3 = x 3 x 3 3 3 3 2 x 2 3 3 3 x 3 3 Back to Instruction 201
Real Numbers All the numbers that can be found on a number line. 10 9 8 7 6 5 4 3 2 1 1 0 2 3 4 5 6 7 8 9 10 Back to Instruction 202
Squared A number multiplied by itself once. 3 2 = = 3x3 9 Back to Instruction 203
Standard Form A general term meaning "the way most commonly written". 4 2 2 3 9 =(2)(2)(2)(2)(3) = 48 3 = standard form standard form y y 3 = 9y 2 2 standard form Back to Instruction 204
Rational Numbers A number that can be expressed as a fraction. 1 4 2 5.3 symbol for rational numbers Back to Instruction 205
Whole Numbers Counting numbers including 0 0, 1, 2, 3,... symbol for whole numbers Back to Instruction 206