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Slide 2 / 206 Grade 6 The Number System & Mathematical Operations 2015-10-20 www.njctl.org
Slide 3 / 206 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 Glossary & Standards Click on a topic to go to that section.
Slide 4 / 206 Addition, Natural Numbers & Whole Numbers Return to Table of Contents
Slide 5 / 206 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 / 206 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
Slide 7 / 206 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.
Slide 8 / 206 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 / 206 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 / 206 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
Slide 11 / 206 Why zero took so long to Invent Horses versus houses. 3 2 1 0 zero horses = zero houses
Slide 12 / 206 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? Zero just isn't a natural number, but it is a whole number.
Slide 13 / 206 Addition, Subtraction and Integers Return to Table of Contents
Slide 14 / 206 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.
Slide 15 / 206 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
Slide 16 / 206 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
Slide 17 / 206 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.
Slide 18 / 206 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?
Slide 19 / 206 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
Slide 20 / 206 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
Slide 21 / 206 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
Slide 22 / 206 Inverse Operations Operations are "inverses" when one of them undoes what the other does. What would undo adding 5? Subtracting 5 click
Slide 23 / 206 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
Slide 24 / 206 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
Slide 25 / 206 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.
Slide 26 / 206 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
Slide 27 / 206 We Need More Numbers 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 =?
Slide 28 / 206 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.
Slide 29 / 206 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.
Slide 30 / 206 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.
Slide 31 / 206 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
Slide 32 / 206 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 `
Slide 33 / 206 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
Slide 34 / 206 1 Which of the following are examples of integers? A 0 B -8 C -4.5 D 7 E 1 3
Slide 35 / 206 2 Which of the following are examples of integers? A 2 5 3 B -6.1 C 287 D -1000 E 0.33
Slide 36 / 206 3 Which of the following are examples of integers? A 1 4 B 6 C -4 D 0.75 E 25%
Slide 37 / 206 Integers In Our World
Slide 38 / 206 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
Slide 39 / 206 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 6. 350 feet above sea level click 350 ft.
Slide 40 / 206 Sea Level The picture below shows three different people at three different elevations. Derived from
Slide 41 / 206 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 What does zero represent in this situation? click Zero represents the top of the water (the water's surface). Derived from
Slide 42 / 206 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: Diver Sailor Hiker Derived from
Slide 43 / 206 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 Derived from
Slide 44 / 206 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 Derived from
Slide 45 / 206 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 Derived from
Slide 46 / 206 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
Slide 47 / 206 9 Which of the following integers best represents the following scenario: Earning $40 shoveling snow. A -40 B 40 C 0 D +/- 40
Slide 48 / 206 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
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Slide 51 / 206 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? click to reveal These numbers are called opposites. Can you think of why?
Slide 52 / 206 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.
Slide 53 / 206 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. From PARCC PBA sample test non-calculator #2
14 What is the opposite of 25? Slide 54 / 206
15 What is the opposite of 0? Slide 55 / 206
16 What is the opposite of 18? Slide 56 / 206
Slide 57 / 206 17 What is the opposite of -18?
Slide 58 / 206 18 What is the opposite of the opposite of -18?
Slide 59 / 206 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!
19 Simplify: - (- 9) Slide 60 / 206
20 Simplify: - (- 12) Slide 61 / 206
21 Simplify: - [- (-15)] Slide 62 / 206
Slide 63 / 206 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.
Slide 64 / 206 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
Slide 65 / 206 22 After the following 3 responses what would the contestants score be? $100 incorrect $200 correct $50 incorrect
Slide 66 / 206 23 After the following 3 responses what would the contestants score be? $200 correct $50 correct $300 incorrect
Slide 67 / 206 24 After the following 3 responses what would the contestants score be? $150 incorrect $50 correct $100 correct
Slide 68 / 206 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.
Slide 69 / 206 Multiplication, Division and Rational Numbers Return to Table of Contents
Slide 70 / 206 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)
Slide 71 / 206 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
Slide 72 / 206 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
Slide 73 / 206 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
Slide 74 / 206 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!
Slide 75 / 206 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?".
Slide 76 / 206 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.
Slide 77 / 206 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 = 5 because 5 is the number you multiply by 6 to In turn, 30 = 6 5, so 30 = 5 6
Slide 78 / 206 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 =?
Slide 79 / 206 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.
Slide 80 / 206 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.
Slide 81 / 206 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
Slide 82 / 206 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?
Slide 83 / 206 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
Slide 84 / 206 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.
Slide 85 / 206 Absolute Value Return to Table of Contents
Slide 86 / 206 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?
Slide 87 / 206 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
Slide 88 / 206 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.
Slide 89 / 206 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
Slide 90 / 206 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
Slide 91 / 206 25 Find -7
26 Find -28 Slide 92 / 206
27 Find 56 Slide 93 / 206
28 Find -8 Slide 94 / 206
29 Find 3 Slide 95 / 206
Slide 96 / 206 30 What is the absolute value of the number shown in the generator? Click on blank screen to generate a number.
Slide 97 / 206 31 Which numbers have 12 as their absolute value? A -24 B -12 C 0 D 12 E 24
Slide 98 / 206 32 Which numbers have 50 as their absolute value? A -50 B -25 C 0 D 25 E 50
Slide 99 / 206 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
Slide 100 / 206 Comparing Integers Return to Table of Contents
Slide 101 / 206 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
Slide 102 / 206 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.
Slide 103 / 206 34 The integer 8 is 9. A = B < C >
Slide 104 / 206 35 The integer 7 is 7. A = B < C >
Slide 105 / 206 36 The integer 3 is 5. A = B < C >
Slide 106 / 206 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
Slide 107 / 206 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
Slide 108 / 206 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.
Slide 109 / 206 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 7) -24-17 8) -2-8 9) 8-8 10) -10-7
Slide 110 / 206 37 The integer -4 is -3. A = B < C > -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10
Slide 111 / 206 38 The integer -4 is -5. A = B < C > -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10
Slide 112 / 206 39 The integer -20 is -14. A = B < C >
Slide 113 / 206 40 The integer -14 is -6. A = B < C >
Slide 114 / 206 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.
Slide 115 / 206 41 The integer -4 is 6. A = B < C >
Slide 116 / 206 42 The integer -3 is 0. A = B < C >
Slide 117 / 206 43 The integer 5 is 0. A = B < C >
Slide 118 / 206 44 The integer -4 is -9. A = B < C >
Slide 119 / 206 45 The integer 1 is -54. A = B < C >
Slide 120 / 206 46 The integer -480 is 0. A = B < C >
Thermometer Slide 121 / 206 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.)
Slide 122 / 206 47 If the temperature reading on a thermometer is 10, what will the new reading be if the temperature: falls 3 degrees?
Slide 123 / 206 48 If the temperature reading on a thermometer is 10, what will the new reading be if the temperature: rises 5 degrees?
Slide 124 / 206 49 If the temperature reading on a thermometer is 10, what will the new reading be if the temperature: falls 12 degrees?
Slide 125 / 206 50 If the temperature reading on a thermometer is -3, what will the new reading be if the temperature: falls 3 degrees?
Slide 126 / 206 51 If the temperature reading on a thermometer is -3, what will the new reading be if the temperature: rises 5 degrees?
Slide 127 / 206 52 If the temperature reading on a thermometer is -3, what will the new reading be if the temperature: falls 12 degrees?
Slide 128 / 206 53 On some thermometers each line does not equal one degree. What does each line equal on this thermometer?
Slide 129 / 206 54 This picture shows part of a thermometer measuring temperature in degrees Fahrenheit. 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
Slide 130 / 206 Comparing and Ordering Rational Numbers Return to Table of Contents
Slide 131 / 206 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
Slide 132 / 206 55 What is the position of the dot on the number line below? A -1 B -1 C -2 1 2 1 4 1 4-10-9-8 -7-6 -5-4 -3-2 -1 1 0 2 3 4 5 6 7 8 9 10
Slide 133 / 206 56 What is the position of the dot on the number line below? A -5.5 B -6.5 C -5.2-10-9-8 -7-6 -5-4 -3-2 -1 1 0 2 3 4 5 6 7 8 9 10
Slide 134 / 206 57 What is the position of the dot on the number line below? A B 1 1 2 1 3 5 C 1 5 2-1 0 1 2 3
Slide 135 / 206 58 What is the position of the dot on the number line below? A -0.8 B -0.5 C -0.6-1 0 1 2 3
Slide 136 / 206 59 Select the point on the number line located at A B C D E F G H I From PARCC EOY sample test non-calculator #16
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Slide 139 / 206 Number Line Drag and drop the given rational numbers into the correct order on the number line from least to greatest. From PARCC EOY sample test non-calculator #5
Slide 140 / 206 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
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Slide 143 / 206 61 A = B < C > -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10
Slide 144 / 206 62 A = B < C > -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10
Slide 145 / 206 63 A = B < C > -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10
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Slide 148 / 206 Exponents Return to Table of Contents
Slide 149 / 206 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."
Slide 150 / 206 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.
Slide 151 / 206 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.
Slide 152 / 206 66 What is the base in this expression? 3 2
Slide 153 / 206 67 What is the exponent in this expression? 3 2
Slide 154 / 206 68 What is the base in this expression? 7 3
Slide 155 / 206 69 What is the exponent in this expression? 4 3
Slide 156 / 206 70 What is the base in this expression? 9 4
Slide 157 / 206 Powers of Integers When a number is written as a power, it is written in exponential form. 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 click 2. Write 7(7)(7)(7)(7)(7)(7) in exponential form. 7click 7
Slide 158 / 206 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 10111213 14151617
Slide 159 / 206 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 10111213 14151617
Slide 160 / 206 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 10111213 14151617
Slide 161 / 206 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 + 3-3 -2-1 0 1 2 3 4 5 6 7 8 9 10111213 14151617
Slide 162 / 206 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 + 4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10111213 14151617
73 What is 4 4 in standard form? Slide 163 / 206
74 What is 8 3 in standard form? Slide 164 / 206
75 What is 2 5 in standard form? Slide 165 / 206
Slide 166 / 206 76 An expression is shown. 7 7 7 7 7 7 What is the expression written in exponential form? From PARCC EOY sample test non-calculator #14
Slide 167 / 206 77 An expression is shown. 3 3 3 3 What is the expression written in exponential form?
Slide 168 / 206 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."
Slide 169 / 206 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
Slide 170 / 206 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
Slide 171 / 206 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."
Slide 172 / 206 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.
78 Evaluate 3 2. Slide 173 / 206
79 Evaluate 5 2. Slide 174 / 206
80 Evaluate 8 2. Slide 175 / 206
81 Evaluate 4 3. Slide 176 / 206
82 Evaluate 7 3. Slide 177 / 206
Slide 178 / 206 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.
Slide 179 / 206 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?
Slide 180 / 206 Real Numbers Return to Table of Contents
Slide 181 / 206 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 182 / 206 85 What type of number is -30? Select all that apply. A Real Number B Rational C Integer D Whole Number E Natural Number
Slide 183 / 206 86 What type of number is - 5? Select all that apply. 3 A Real Number B Rational C Integer D Whole Number E Natural Number
Slide 184 / 206 87 What type of number is 0? Select all that apply. A Real Number B Rational C Integer D Whole Number E Natural Number
Slide 185 / 206 88 What type of number is - 4 1? Select all that apply. 2 A Real Number B Rational C Integer D Whole Number E Natural Number
Slide 186 / 206 89 What type of number is 25? Select all that apply. A Real Number B Rational C Integer D Whole Number E Natural Number
Slide 187 / 206 90 What type of number is 2? Select all that apply. 8 A Real Number B Rational C Integer D Whole Number E Natural Number
Slide 188 / 206 Glossary & Standards Return to Table of Contents
Slide 189 / 206 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. 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.
Slide 190 / 206 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
Slide 191 / 206 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 = 3 x 3 x 3 3 2 x 2 3 3 3 x 3 3 Back to Instruction
Slide 192 / 206 Cubed A number multiplied by itself twice. 3 3 =3x3x3 = 27 Back to Instruction
Slide 193 / 206 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
Slide 194 / 206 Exponent A small, raised number that shows how many times the base is used as a factor. Exponent 2 2 = x 3 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
Slide 195 / 206 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
Slide 196 / 206 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
Slide 197 / 206 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
Slide 198 / 206 Integers Positive numbers, negative numbers and zero..., -2, -1, 0, 1, 2,... symbol for integers Back to Instruction
Slide 199 / 206 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
Slide 200 / 206 Natural Numbers Counting numbers 1, 2, 3, 4,... symbol for natural numbers Back to Instruction
Slide 201 / 206 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
Slide 202 / 206 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
Slide 203 / 206 Squared A number multiplied by itself once. 3 2 = = 3x3 9 Back to Instruction
Slide 204 / 206 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
Slide 205 / 206 Rational Numbers A number that can be expressed as a fraction. 1 4 2 5.3 symbol for rational numbers Back to Instruction
Slide 206 / 206 Whole Numbers Counting numbers including 0 0, 1, 2, 3,... symbol for whole numbers Back to Instruction