Chapter One Section One - Standard Notation

Transcription

1 Chapter One Section One - Standard Notation Digit 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Periods groups of three Place Value Chart - Trillions Billions Millions Thousands Ones Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones 1 Billion, 306 Million, 313 Thousands, 812 Ones Whole Numbers Positive Whole (NO FRACTIONS) with ZERO Natural Numbers Whole Numbers without ZERO Standard Notations 4,123 Expanded Notations Four Thousands, One Hundred, Two Tens, Three Word Names Four Thousand, One Hundred, Twenty-Three What does the digit 2 mean in each number 1 6? ,555 2 ten thousands ,789 2 hundred thousands 3. 42,789,654 2 millions 4. 24,789,654 2 ten millions tens 6. 5,643,201 2 hundreds Write expanded notation 7 12: ,219 2 hundred thousands; 8 ten thousands; zero thousands; 2 hundreds; 1 ten; 9 ones 1 Chapter 1

2 thousand + 8 hundreds + 9 tens + 5 ones 9. 23,416 2 ten thousands + 3 thousands + 4 hundreds + 1 ten + 6 ones thousands + 3 tens + 1 one (DON T NEED THE ZERO HUNDREDS) thousands + 1 hundred + 8 tens ,616 1 hundred thousand + 5 ten thousands + 4 thousands + 6 hundreds + 1 ten + 6 ones Write a word name 13 19: forty-nine sixteen thirty-eight two hundred four ,155 forty-five thousand, one hundred fifty-five 18. 1,879,204 one million, eight hundred seventy-nine thousand, two hundred four 19. 6,449,000,000 six billion, four hundred forty-nine million 20. Write Standard notation: two hundred thirteen million, one hundred five thousand, three hundred twenty-nine: 213,105,329 Chapter One Section Two Addition Sum The total value of items to be added. Addend The items to be added. Additive Identity is ZERO. A+0=0+A=A. 2 Chapter 1

3 Associative Law of Addition If you are just adding, you can do it in any order. A+(B+C)=(A+B)+C Commutative Law of Addition If you add A+B you get the same value when you add B+A. A+B = B+A Perimeter the distance around an object is its perimeter = 13, = = 16, = 27,474 Find perimeter 5-7: 5. 5 in + 6 in + 9 in + 5 in + 4 in = 29 in ft 15 ft 15 ft 16 ft 16 ft + 15 ft + 16 ft + 15 ft = 62 ft 7. 3 by 5 inch index card = 16 inches 5 by 8 inch = 26 inches 3 Chapter 1

4 Chapter One Section Three Subtraction Minuend The number from which another number is subtracted. Subtrahend The number being subtracted. Difference Value after subtraction. Related Addition Sentences: 5 2 = 3 then 5 = Related Subtraction Sentences: = 7 then 4 = 7 3 or 3 = 7-4 Missing Addend: 2 + [] = = difference Subtraction of Whole Numbers Borrowing Write related addition sentence 1 2: = 2 7 = or 7 = = 9 17 = or 17 = Write two related subtraction sentences 3 4: = 13 5 = 13 8 or 8 = = = 14 3 or 3 = = = 6328 check = = 4747 check = = = = = = Chapter 1

5 Chapter One Section Four Rounding and Estimating; Order Rounding a) Locate the digit to be rounded b) Consider the next digit to the right c) If it is (0,1,2,3,4) down (5,6,7,8,9) up d) Change all the digits to the right to ZEROs Estimating = means equal to means is approximately equal to Inequality means not equal to Order (Value) which number has more or less value (ARROW ALWAYS POINTS TO THE SMALL NUMBER) Inequality Symbols Greater Than B>A B is greater than A Lesser Than A<B A is less than B Equation A math sentence where both sides of the equal sign have the same value (are equal). Round to nearest ten 1 11: Chapter 1

6 Round to nearest hundred 12 15: Round to nearest thousands 16 19: ,343 19, ,500 69,000 Round to ten, hundred, and thousand 20 21: ,968 ten: 48,970; hundred: 49,000; thousand: 49, ,582 ten: 269,580; hundred: 269,600; thousands: 270, and 23. SKIP - Answers vary Round to ten first then add = = 200 Round to hundred 25 26: = = Chapter 1

7 = = 2600 Round thousands ,278 11,698 = 23,000 12,000 = 11,000 Write < or > to make sentence true: [ ] 12 < [ ] 8 > [ ] 64 > [ ] 76 < [ ] 345 < [ ] 217 > Chapter One Section Five Multiplication Multiplication = Repeated Addition 3 X 5 = = 5 X ( ) = 5 X (3) = 15 Factors are the numbers to be multiplied and the result is the Product. Notation: 3 X 5 = 3 5 = (3)(5) = 3(5) = 15 Multiplying by ZERO: A 0 = 0 Multiplication Identity: A 1 = A Distributive Law of Multiplication: A (B + C) = (A B) + (A C) Commutative Law of Multiplication: A B = B A Associative Law of Multiplication: A (B C) = (A B) C AREA: number of square units to fill object. Rectangular objects: Area = Length X Width = L W x 2 = x 4 = x 6 = Chapter 1

8 x 5 = x 23 = x 63 = x 62 = 46, x 837 = 205, x 306 = 144, x 704 = 287, x 6005 = 14,075, x 830 = 391, x 7400 = 17,345, x 562 = 56, x 562 = 562, a) 23 x 47 = 1081 b) 47 x 23 = 1081 c) same x 2 x 4 = x 1 x 3 = 15 Page Round to ten lawn tractors 1534 x x 10 = 15, Round ten 837 x x 250 = 210,000 Round hundred 837 x x 200 = 160,000 8 Chapter 1

9 21. Area of table tennis table 9 ft by 5 ft A = L x W = 9 ft x 5 ft = 5 ft x 9 ft = 45 sq ft Chapter One Section 6 - Division Division = Repeated Subtraction 20 5 = 4 Twenty divided by 5 equals 4 Dividend number being divided (20) Divisor number doing the dividing (5) Quotient answer (4) 20 5 = [] related equation 5 [] = 20 (missing factor) A B (BUT B CANNOT EQUAL ZERO) = C (Unique) and A = B C A B = C and A C = B A 1 = A A A = 1 (A 0) 0 A = 0 (A 0) A 0 = NOT DEFINED (we just don t divide by zero) Remainder: What is left over, when the Quotient is not a Whole number = 3 + R Dividend Divisor = Quotient + Remainder /6 or Related Sentence 2 4: = 5 15 = 3 x 5 or 15 = 5 x = 9 72 = 8 x 9 or 72 = 9 x x 2 = 12 6 = 12 2 or 2 = x 6 = 42 6 = 42 7 or 7 = , 45, 36, 27, 18, 9, 1 (6) 6 x 9 = R 7 6 x 9 = = 61 9 Chapter 1

10 R 5 2 x 12 = = R 13 6 x 24 = = R R R R R R 47 Chapter One Section Seven Solving Equations Variables: are used in an equation and when its value makes the equation True, we have the solution of the equation. 9 = 3 + X (note: X here is no used for multiplication) Solution: is when the value of the variable makes the equation True. X = 12 (true when X has the value of 12) X = = 8330 X + 12 = 27 X = X + 0 = 15 X = Chapter 1

11 X + A = B X - A = B X + A A = B A X - A + A = B + A X = B A X = B + A X A = B X A = B X A A = B A X A A = B A X 1 = B A X 1 = B A X = B A X = B A 1. 8 = 1 + [ ] 7 2. [ ] + 2 = [ ] + 5 = 9 is 7 solution No, = [ ] + 5 = 9 is 4 solution Yes, = 9 5. n + 3 = 8 n = 5 6. x 2 = 8 x = = y y = t = 32 t = x 65 = y y = 22, x = x = = t t = x = x = x + 9 = 17 x = = m + 32 m = = t + 78 t = = Chapter 1

12 x = 7877 x = = = h h = x = 64 x = = 9 n n = = 8 t t = y = 1728 y = n 48 = 4512 n = 94 Chapter One Section Eight Applications and Problem Solving Five Steps for Problem Solving 1. Familiarize yourself with the situation. a. Read and reread until you understand problem. b. Draw a diagram or see if a formula applies. c. Assign a variable to the UNKNOWN (understand the Units). 2. Translate problem into an equation. 3. Solve the equation. 4. Check the answer in the original wording of the problem (Units). 5. State the answer to the problem clearly with appropriate Units. Page 66 for KEY WORDS 1. Table page = 11, same = same =19, $2003 $129 = $ $228 + x = $399 x = $ x $398 = $ Chapter 1

13 7. A = L W = 102 in 90 in = 9180 sq in = 378 R cases with 1 can left over = 79 gal 10. page 66 3 x 5 = 15 6 x 6 = 36 2 x 2 = 4 18 x 7 = = 181 seats Chapter One Section Nine Exponential Notation and Order of Operations Base: is the number, which is going to be raised to the Power. Exponent: is the Power to which the base is going to be raised = = 3 3 A 3 B = 3 A+B 3 A 3 B = 3 A-B 2 nd power is that same as Squared 3 rd power is Cubed Area of a Square = Length (side) squared = L = 10 x 10 x 10 x 10 x 10 = 100, x 5 Simplifying Expressions: What if there is adding, subtracting, multiplying and dividing all in the same problem. Rules For Order of Operations: 1. Do all calculations within parentheses (), brackets [], or braces {} before operations outside. If parentheses are within parentheses, do innermost ones first. There is no differences in (), [], or {}, they just help clarify the problem. 2. Evaluate all exponential expressions. 3. Do all multiplication and divisions in order from left to right. 4. Do all additions and subtractions in order from left to right. Average: is the sum of the items divided by the number of Items. What is the average weight of these three people: 201, 175, and 155 pounds. Solution = ( ) 3 = = 177 Removing Parentheses within Parentheses - Work from the inside out. 13 Chapter 1

14 Write exponential notation (1 4) 1. 5 x 5 x 5 x x 5 x 5 x 5 x x x 10 x 10 x x 10 x 10 x 10 = 10, x 10 = x 8 x 8 = x 2 x 2 x 2 x 2 = x = = x 26 ( ) 25 x = = (83 14) = = (32 2) = 4 (64 32) = ( ) = ( ) = x 4 (20 + 4) 8 (6 2) 9 x = = x 5 x x 71 ( x 3) 5 x 5 x x 71 ( ) 5 x 5 x x = Chapter 1

15 x x x x x x x x = x 2 x 2 x 5 (24 4) 95 2 x 2 x 2 x x 2 x x x 71 ( x 3) x 71 ( ) x x = (1 + 3) x x x = x 2 (12 9) x x = x x = ( ) 4 = x 5 + {6 [14 (5 + 3)]} 9 x 5 + {6 [14 8]} 9 x 5 + {6 6} 9 x = [18 (2 + 7) 3] (31 10 x 2) [18 9 3] (31 20) [18 3] = 4 15 Chapter 1