Lesson 2 Operations with Rational and Irrational Numbers

1. Lesson 2 Operations with Rational and Irrational Numbers NCSCOS Obj.: 1.01; 1.02 Objective TLWState the coordinate of a point on a number line TLW Graph integers on a number line. TLW Add and Subtract Rational numbers. TLW Multiply and Divide Rational numbers

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

3. Addition Rule 1) When the signs are the same, ADD and keep the sign. (-2) + (-4) = -6 2) When the signs are different, SUBTRACT and use the sign of the larger number. (-2) + 4 = 2 2 + (-4) = -2

4. Karaoke Time! Addition Rule: Sung to the tune of “Row, row, row, your boat” Same signs add and keep,different signs subtract,keep the sign of the higher number,then it will be exact!

5. Answer Now -1 + 3 = ? • -4 • -2 • 2 • 4

6. Answer Now -6 + (-3) = ? • -9 • -3 • 3 • 9

7. The additive inverses(or opposites) of two numbers add to equal zero. -3 Proof: 3 + (-3) = 0 We will use the additive inverses for subtraction problems. Example: The additive inverse of 3 is

8. What’s the difference between7 - 3 and 7 + (-3) ? 7 - 3 = 4 and 7 + (-3) = 4 The only difference is that 7 - 3 is a subtraction problem and 7 + (-3) is an addition problem. “SUBTRACTING IS THE SAME AS ADDING THE OPPOSITE.” (Keep-change-change)

9. When subtracting, change the subtraction to adding the opposite (keep-change-change) and then follow your addition rule. Example #1: - 4 - (-7) - 4+ (+7) Diff. Signs --> Subtract and use larger sign. 3 Example #2: - 3 - 7 - 3+ (-7) Same Signs --> Add and keep the sign. -10

10. Okay, here’s one with a variable! Example #3: 11b - (-2b) 11b+ (+2b) Same Signs --> Add and keep the sign. 13b

11. Answer Now Which is equivalent to-12 – (-3)? • 12 + 3 • -12 + 3 • -12 - 3 • 12 - 3

12. Answer Now 7 – (-2) = ? • -9 • -5 • 5 • 9

13. 1) If the problem is addition, follow your addition rule.2) If the problem is subtraction, change subtraction to adding the opposite (keep-change-change) and then follow the addition rule. Review

14. Absolute Value of a number is the distance from zero. Distance can NEVER be negative! The symbol is |a|, where a is any number.

15. Examples 7 = 7 10 = 10 -100 = 100 5 - 8 = -3= 3

16. Answer Now |7| – |-2| = ? • -9 • -5 • 5 • 9

17. Answer Now |-4 – (-3)| = ? • -1 • 1 • 7 • Purple

18. 2) Find the sum.1) -2.304 + (-0.26) Get a common denominator and subtract. Line up the decimals and add (same signs). -2.564

19. Find the difference. 3) Change subtraction to adding the opposite. Get a common denominator. Subtract and keep sign of the larger number.

20. Find the difference. 4) Change subtraction to adding the opposite. Get a common denominator and subtract.

21. 5) Solve 6.32 – y if y = -3.42 Substitute for y: 6.32 - (-3.42) 6.32+ 3.42 9.74

22. Answer Now Find the solution6.5 – 9.3 = ? • -3.2 • -2.8 • 2.8 • 3.2

23. Answer Now Find the solution • .

24. A rational numberis a number that can be written as a fraction. How can these be written as a fraction? 3 =

25. Inequality Symbols

26. Ordering Rational Numbers 2 ways to order from least to greatest • Get a common denominator • Change the fractions to decimals (numerator  demoninator)

27. 4 3 < 1 1 8 Which rational number is bigger? 1) Get a common denominator. 2) or convert the fraction to a decimal. 0.363 < 0.375

28. 7 1 1 or 4 6 42 44 < 24 24 Which rational number is bigger? • Get a common denominator • or convert to a decimal 1.75 < 1.83

29. 5 3 __ 7 4 Answer Now Which symbol makes this true? • < • > • =

30. Answer Now Which symbol makes this true? -2 -1 __ 4 9 • < • > • =

31. Multiplying Rules 1) If the numbers have the same signs then the product is positive. (-7) • (-4) = 28 2) If the numbers have different signs then the product is negative. (-7) • 4 = -28

32. 2) = -16 Examples 1) (3x)(-8y) -24xy Write both numbers as a fraction. Cross-cancel if possible. Multiplying fractions: top # • top # Bottom # • bottom#

33. Answer Now When multiplying two negative numbers, the product is negative. • True • False

34. Answer Now When multiplying a negative number and a positive number, use the sign of the larger number. • True • False

35. 3) = =

36. Answer Now Multiply: (-3)(4)(-2)(-3) • 72 • -72 • 36 • -36

37. an easy way to determine the sign of the answer When you have an odd number of negatives, the answer is negative. When you have an even number of negatives, the answer is positive. 4) (-2)(-8)(3)(-10) Do you have an even or odd number of negative signs? 3 negative signs -> Odd -> answer is negative -480

38. Last one! 5) Positive or negative answer? Positive - even # of negative signs (4) Write all numbers as fractions and multiply. =12

39. Answer Now What is the sign of the product of (-3)(-4)(-5)(0)(-1)(-6)(-91)? • Positive • Negative • Zero • Huh?

40. Dividing Rules 1) If the numbers have the same signs then the quotient is positive. -32 ÷ (-8 )= 4 2) If the numbers have different signs then the quotient is negative. 81 ÷ (-9) = -9

41. Answer Now When dividing two negative numbers, the quotient is positive. • True • False

42. Answer Now When dividing a negative number and a positive number, use the sign of the larger number. • True • False

43. The reciprocal of is where a and b  0. The reciprocal of a number is called its multiplicative inverse. A number multiplied by its reciprocal/multiplicative inverse is ALWAYS equal to 1.

44. The reciprocal of is Example #2 The reciprocal of -3 is Example #1

45. Basically, you are flipping the fraction! We will use the multiplicative inverses for dividing fractions.

46. Answer Now Which statement is false about reciprocals? • Reciprocals are also called additive inverses • A number and its reciprocal have same signs • If you flip a number, you get the reciprocal • The product of a number and its reciprocal is 1

47. Examples 1) When dividing fractions, change division to multiplying by the reciprocal.

48. 2)

49. Answer Now What is the quotient of-21 ÷ -3? • 18 • -18 • 7 • -7

50. Answer Now • . • . • . • .