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Chapter 1

Chapter 1. Operations with Whole Numbers. 1-1: Mathematical Expression. Variable. A variable is a symbol used to represent one or more numbers. The number that the variable represents is called the value of the variable. Examples: b + 90 3 X n 18 – m y ÷ 24.

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Chapter 1

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  1. Chapter 1 Operations with Whole Numbers

  2. 1-1: Mathematical Expression

  3. Variable A variable is a symbol used to represent one or more numbers. The number that the variable represents is called the value of the variable. Examples: • b + 90 • 3 X n • 18 – m • y ÷ 24

  4. Variable and Numerical Expressions An expression such as b + 90 Is called a variable expression. An expression such as 3 X 2 Is called a numerical expression.

  5. Multiplication Symbols We can used a raised dot as a multiplication symbol. 9 x 7 can be written as 9·7 2 x a x b can be written as 2·a·b In a variable expression we can use the raised dot or omit the multiplication symbol. 3 x n can be written as 3·n 3·n can be written as 3n 2 x a x b can be written as 2·a·b 2·a·b can be written as 2ab

  6. Equation When a mathematical sentence uses an equal sign, it is called an equation. An equation tells us that two expressions name the same number. The expression to the left of the equal sign is the left side of the equation. The expression to the right of the equal sign is called the right side. 5 ̶ 5 = 0 4+ 5 = 9 2 · 4 = 8 18 ÷ 9 = 2

  7. An Equation is like a balance scale. Everything must be equal on both sides. = 10 5 + 5

  8. An Equation is like a balance scale. Everything must be equal on both sides. = 12 6 + 6

  9. An Equation is like a balance scale. Everything must be equal on both sides. = 7 n + 2

  10. An Equation is like a balance scale. Everything must be equal on both sides. = 7 5 n + 2

  11. Substitution When a number is substituted for a variable in a variable expression and the operation is carried out, we say that the variable has been evaluated. If n = 6, evaluate 3 · n If n = 6, 3 · 6 = 18 If x = 2, evaluate 3 + x If x = 2, 3 + 2 = 5 If y = 9, evaluate 18 ÷ y If y = 9, 18 ÷ 9= 2

  12. x = 10; y = 20. Evaluate • x̶ 5 • y ̶ x + 50 • 50 ̶ x + y • 50 + y ̶ x • y ̶ 10 • xy 10 ̶ 5 = 5 20 ̶ 10 + 50 = 60 50 ̶ 10 + 20 = 60 50 + 20 ̶ 10 = 60 20 ̶ 10 = 10 10 ·20 = 200

  13. 1-2: Properties of Addition and Multiplication

  14. The Set of Numbers Counting numbers 1, 2, 3, 4, 5, . . . Whole numbers 0, 1, 2, 3, 4, . . .

  15. Commutative Property of Addition and Multiplication The order in which two whole numbers are added or multiplied does not change their sum or their product. 3 + 4 = 7 and 4 + 3 = 7 3 x 4 = 12 and 4 x 3 = 12 a + b = b + a a x b = b x a

  16. Associative Property of Addition and Multiplication Add 6 + 5 + 7 = • (6 + 5) + 7 = • 6 + (5 + 7) = • (6 + 7) + 5 = Multiply 9 x 2 x 5 = • (9 x 2) x 5 = • 9 x (2 x 5) = • (9 x 5) x 2 = 18 18 18 90 90 90

  17. Exercise Simplify Using the Commutative and Associative Properties • 13 + 8 + 7 • 5 x 7 x 2

  18. Addition Property of Zero • 7 + 0 = • a + 0 = • 8 + 0 = • c + 0 = • 2 + 0 = 7 a 8 c 2

  19. Multiplication Property of One • 7 x 1 = • a x 1 = • 8 x 1 = • c x 1 = • 2 x 1 = 7 a 8 c 2

  20. Multiplication Property of Zero • 7 x 0 = • a x 0 = • 8 x 0 = • c x 0 = • 2 x 0 = 0 0 0 0 0

  21. 1-4: The Distributive Property

  22. The fee for each person entering the state park is $4. If the person rents a bicycle, he has to pay an additional $2. How much will a group of 12 people have to spend if each will enter the park and each will rent a bicycle? 12 · (4 + 2)Two Methods: = 12 · 6 = 72 = (12· 4) + (12· 2) = 48 + 24 = 72

  23. The fee for each person entering the state park is $4. If the person has a coupon, he gets a $2 discount. How much will a group of 12 people have to spend if each will enter the park with a coupon? 12 · (4 ̶ 2)Two Methods: = 12 · 2 = 24 = (12 · 4) ̶ (12 · 2) = 48 ̶ 24 = 24

  24. Simplify using the distributive property. • (11 · 4) + (11 · 6) =11 · (4 + 6) = 11 · 10 = 110 • 13 · 15 = (13 · 10) + (13 · 5) = 130 + 65 = 195

  25. 144 304 Simplify using the distributive property. • 6 ( 20 + 4) • 4 ( 80 – 6) • (4 x 12) + (4 x 8) • (33 x 90) + (33 x 10) • (23 x 104) – (23 x 4) • (56 x 11) – (6 x 11) • 35 (10 + 2) • 33 ( 100 – 3) • 12 x 22 • 32 x 8 • 9 x 120 • 7 x 896 80 3300 2300 550 420 3201 264 256 1080 6272

  26. Test: 1-1, 1-2, 1-3, 1-4

  27. 1-5: Order of Operations

  28. Grouping Symbols: Show which operations need to be performed first. [ ] ( ) When one pair of grouping symbols is enclosed in another, we ALWAYS perform the operation enclosed in the INNER pair of symbols FIRST. (14 + 77) ÷ 7 [3 + (4 · 5)] · 10

  29. Simplify (14 + 77) ÷ 7 ÷ 7 91 13

  30. Simplify [3 + (4 · 5)] · 10 · 10 20 ] [3 + · 10 23 230

  31. For expressions that are written without grouping symbols like, 8 + 3 – 9 x 2 ÷ 3 Rule • Do all multiplication and divisions in order from left to right. • Then do all additions and subtractions in order from left to right.

  32. Simplify 72 – 24 ÷ 3 8 72 – 64 Rule Do all multiplication and divisions in order from left to right. Then do all additions and subtractions in order from left to right.

  33. Simplify 8 + 3 – 9 x 2 ÷ 3 18 ÷ 3 8 + 3 – 11 6 – 5 Rule Do all multiplication and divisions in order from left to right. Then do all additions and subtractions in order from left to right.

  34. Solve • 12 ( 18 + 36) • (100 – 16) ÷ 7 • 4[(6 + 17)2]

  35. 1-6: A Problem Solving Model

  36. Plan for Solving Word Problems • Read the problem carefully. Make sure you understand what it says. You may need to read it more than once. • Use questions like these in planning the solution: • What is asked for? • What facts are given? • Are enough facts given? If not, what else is needed? • Are unnecessary facts given? If so, what are they? • Will a sketch or diagram help? • Determine which operations or operations can be used to solve the problem. • Carry out the operations carefully. • Check your results with the facts given in the problem. Give the answer.

  37. The Golden Gate Bridge has a span of 4200 feet. The Brooklyn Bridge has a span of 1595 feet. How much longer is the span of the Golden Gate Bridge? • What number or numbers does the problem ask for? • Are enough facts given? If not, what else is needed? • Are unnecessary facts given? If so, what are they? • What operation or operations would you use to find the answer?

  38. How many 30-second ads can a politician buy with $528,000? • What number or numbers does the problem ask for? • Are enough facts given? If not, what else is needed? • Are unnecessary facts given? If so, what are they? • What operation or operations would you use to find the answer?

  39. Paula washed 5 cars and Jim washed 4. Paula charged $3 for each car. Jim charged $4. How much money did Paula earn? • What number or numbers does the problem ask for? • Are enough facts given? If not, what else is needed? • Are unnecessary facts given? If so, what are they? • What operation or operations would you use to find the answer?

  40. Mike can type a page in 7 min. How many pages can he type in 45 min? • What number or numbers does the problem ask for? • Are enough facts given? If not, what else is needed? • Are unnecessary facts given? If so, what are they? • What operation or operations would you use to find the answer?

  41. 1-7: Problem Solving Applications

  42. During the four quarters of a basketball game, the Hoopsters scored 16 points, 21 points, 19 points, and 17 points. How many points did the Hoopsters score during the game? • What number or numbers does the problem ask for? • Are enough facts given? If not, what else is needed? • Are unnecessary facts given? If so, what are they? • What operation or operations would you use to find the answer?

  43. Simon had $165 in his checking account. He wrote checks for $32, $19, and $47. How much did Simon have left in his account? • What number or numbers does the problem ask for? • Are enough facts given? If not, what else is needed? • Are unnecessary facts given? If so, what are they? • What operation or operations would you use to find the answer?

  44. The temperature was 15°C at 8am. By noon, the temperature had increased by 13°. What was the temperature at noon? • What number or numbers does the problem ask for? • Are enough facts given? If not, what else is needed? • Are unnecessary facts given? If so, what are they? • What operation or operations would you use to find the answer?

  45. Test: 1-5, 1-6, 1-7Next: Chapter 11

  46. Chapter 11 Operations with Integers

  47. 11-1: Negative Numbers Objective: To represent negative numbers on the number line. HW: P. 368: 1-39 ODD; 40-43 ALL; Read 11-2

  48. The Number Line Natural Numbers = {1, 2, 3, …} Whole Numbers = {0, 1, 2, …} Integers = {…, -2, -1, 0, 1, 2, …} -5 0 5

  49. Definition Positive number – a number greater than zero. 0 1 2 3 4 5 6

  50. Definition Negative number – a number less than zero. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

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