Slide 1 / 262 Slide 2 / 262 7th Grade The Number System & Mathematical Operations 2015-08-31 www.njctl.org Slide 3 / 262 Slide 4 / 262 Table of Contents Natural Numbers and Whole Numbers Addition, Subtraction and Integers Addition and Subtraction of Integers Natural Numbers and Multiplication and Division of Integers Whole Numbers Operations with Rational Numbers Addition and Subtraction of Rational Numbers Multiplication and Division of Rational Numbers Converting Rational Numbers to Decimals Exponents Return to Table Real Numbers of Contents Click on a topic to Glossary & Standards go to that section. Slide 5 / 262 Slide 6 / 262 Natural Numbers Natural Numbers Natural numbers were used before there was history. The first numbers developed were the Natural Numbers , also called the Counting Numbers . All people use them. 1, 2, 3, 4, 5, ... This "counting stick" was made more than 35,000 years ago and was found in Lebombo, Swaziland. The three dots, (...), means that these numbers continue forever: there is no largest counting number.. The cuts in this bone record the number "29." Think of counting objects as you put them in a container those are the counting numbers. http://www.taneter.org/math.html
Slide 7 / 262 Slide 8 / 262 Natural Numbers and Addition Numbers versus Numerals They were, and are, used to count objects Numbers exist even without a numeral, such as the number > goats, indicated by the cuts on the Lebombo Bone. > bales, > bottles, A numeral is the name we give a number in our culture. > 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 9 / 262 Slide 10 / 262 Numbers versus Numerals Whole Numbers Adding zero to the Counting Numbers gives us the Whole Numbers . 0, 1, 2, 3, 4, ... If asked how many tires my car has, I could hand Counting numbers were someone the above marbles. developed more than 35,000 years ago. That number is represented by: It took 34,000 more years to 4 in our Base 10 numeral system · invent zero. IV in the Roman numeral system · 100 in the Base 2 numeral system · 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 Slide 12 / 262 Why zero took so long to Invent Why zero took so long Horses versus houses. Would I tell someone I have a herd of zero goats? 3 Or a garage with zero cars? 2 Or that my zero cars have zero tires? Zero just isn't a natural number, but it is a whole number. 1 0 zero horses = zero houses
Slide 13 / 262 Slide 14 / 262 Addition and Subtraction The simplest mathematical operation is addition. The inverse of addition is subtraction. Addition, Subtraction Two operations are inverses if one "undoes" the other. and Integers This is a very important concept, and applies to all mathematics. Return to Table of Contents Slide 15 / 262 Slide 16 / 262 Adding Whole Numbers Addition and Subtraction Let's find the sum of 4 and 5 on a number line. Each time a marble is dropped in a jar we are doing addition. The number +4 is four steps to the right. Starting at 0, takes you to 4. Each time a marble is removed from a jar, we are doing subtraction. 4 A number line allows us to think of addition in a new way. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 The number +5 is five steps to the right. Starting at 0, takes you to 5. 5 -1 1 2 3 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 0 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 17 / 262 Slide 18 / 262 Adding Whole Numbers The Commutative Property of Addition To find the sum "4 + 5" start at zero and take four steps to the right for A mathematical operation is commutative if the order doesn't matter. the first number. In this case, addition would be commutative if 4 + 5 = 5 + 4 Then, starting where you ended after those first steps, take five more steps to the right, to represent adding five. Let's test that. If we were walking, we could look down and see we are standing at 9. We found that 4 + 5 = 9 How about 5 + 4? 4 + 5 4 5 -1 1 2 3 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 0 5 6 7 8 9 10 Therefore, 4 + 5 = 9.
Slide 19 / 262 Slide 20 / 262 The Commutative Property of Addition The Commutative Property of Addition So, 4 + 5 = 5 + 4 First, take five steps to the right. Addition is commutative in this case. 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. 4 + 5 4 5 -1 1 2 3 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 0 5 6 7 8 9 10 5 + 4 4 5 5 + 4 5 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 5 + 4 = 9 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 21 / 262 Slide 22 / 262 The Commutative Property of Addition Inverse Operations But there's nothing special about these numbers. Operations are inverse operations when one of them undoes what This is true for any numbers: the other does. a + b = b + a What would undo adding 5? a + b a b Subtracting 5 click b + a a b Slide 23 / 262 Slide 24 / 262 Addition and Subtraction are Inverses Addition and Subtraction are Inverses Addition and subtraction are inverses. This is true for any two numbers. Adding a number and then subtracting that same number Start with a, then add b, then subtract b. leaves you where you started. You end up with the number you began with: a. Starting at 4, add 5 and then subtract 5. You end up where you started. subtract 5 -b add 5 +b -1 1 2 3 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 0 5 6 7 8 9 10 a a + b
Slide 25 / 262 Slide 26 / 262 Inverse Operations Subtracting Whole Numbers We started with the addition question: what number results when we add 5 and 4? The answer is 9. Here's what 9 - 5 looks like on the number line 5 + 4 = 9 -5 That leads to two new related subtraction questions. +9 Starting with 9, what number do we get when we subtract 5? -1 1 2 3 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 0 5 6 7 8 9 10 9 - 5 = 4 And, here's 9 - 4 Starting with 9, what number do we get when we subtract 4. -4 9 - 4 = 5 +9 Subtraction was invented to undo addition, but it now can be used -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 to ask new questions. Slide 27 / 262 Slide 28 / 262 We Need More Numbers Subtracting Whole Numbers Subtraction was invented to undo addition. 4 - 7 = ? But this new operation allows us to ask new questions. -7 And, the number system to that point couldn't provide answers. +4 For example: -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 What is the result of subtracting 7 from 4? 4 - 7 = ? There was no answer to questions like this in the whole number systems. Which was all there was until about 500 years ago. Slide 29 / 262 Slide 30 / 262 Negative Numbers Integers 4 - 7 = ? Adding the negative numbers to the whole numbers yields the Integers . -7 ...-3, -2, -1, 0, 1, 2, 3, ... +4 In this case, "..." at the left and right, means that the sequence -1 1 2 3 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 0 5 6 7 8 9 10 continues in both directions forever. There is no largest integer...nor is there a smallest integer. 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.
Slide 31 / 262 Slide 32 / 262 Integers Integers on the number line The below number line shows only the integers. Negative Positive Zero Integers Integers -5 -4 -3 -2 -1 0 1 2 3 4 5 Numbers to the left Numbers to the of zero are less than right of zero are Zero is neither zero greater than zero positive or negative -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 ` Slide 33 / 262 Slide 34 / 262 1 Which of the following are examples of integers? 2 Which of the following are examples of integers? A 2 A 0 5 3 B -8 B -6.1 C -4.5 C 287 D 7 D -1000 E 0.33 E 1 3 Slide 35 / 262 Slide 36 / 262 3 Which of the following are examples of integers? 4 What is the opposite of 25? A 1 4 B 6 C -4 D 0.75 E 25%
Slide 37 / 262 Slide 38 / 262 5 What is the opposite of 0? 6 What is the opposite of 18? Slide 39 / 262 Slide 40 / 262 7 What is the opposite of -18? 8 What is the opposite of the opposite of -18? Slide 41 / 262 Slide 42 / 262 9 Simplify: - (- 9) 10 Simplify: - (- 12)
Slide 43 / 262 Slide 44 / 262 Absolute Value of Integers 11 Simplify: - [- (-15)] 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). 4 This is read, "the absolute value of 4". -1 1 2 3 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 0 5 6 7 8 9 10 What is the 4 ? Click to 4 Reveal Slide 45 / 262 Slide 46 / 262 -7 13 Find -28 12 Find Slide 47 / 262 Slide 48 / 262 15 Find -8 14 Find 56
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