1. Welcome to Seminar 2 Agenda Questions about last week Discussion/MML Reminder Fraction Basics Week 2 Overview Questions

2. Definition: A fraction is an ordered pair of whole numbers, the 1 st one is usually written on top of the other, such as ½ or ¾. The denominator tells us how many congruent pieces the whole is divided into. The numerator tells us how many such pieces are being considered. numerator denominator Same measure/ size

3. Equivalent fractions a fraction can have many different appearances, these are called equivalent fractions In the following picture we have ½ of a cake because the whole cake is divided into two congruent parts and we have only one of those parts. But if we cut the cake into smaller congruent pieces, we can see that = Or we can cut the original cake into 6 congruent pieces,

4. Equivalent fractions a fraction can have many different appearances, these are called equivalent fractions Now we have 3 pieces out of 6 equal pieces, but the total amount we have is still the same. Therefore, == If you don’t like this, we can cut the original cake into 8 congruent pieces,

5. Equivalent fractions a fraction can have many different appearances, they are called equivalent fractions then we have 4 pieces out of 8 equal pieces, but the total amount we have is still the same. === Therefore,

6. The Whole 1/2 1/3 1/4 1/5 1/6 1/7 1/8

7. How do we know that two fractions are the same? we cannot tell whether two fractions are the same until we reduce them to their lowest terms. A fraction is in its lowest terms (or is reduced) if we cannot find a whole number (other than 1) that can divide into both its numerator and denominator. Examples: is not reduced because 2 can divide into both 6 and 10. How does this look? 6/2 = 3 10/2= 5 6/10=3/5 is not reduced because ??? divides into both 35 and 40.

8. How do we know that two fractions are the same?. To find out whether two fraction are equal, we need to reduce them to their lowest terms or simply… 35/5= 7 40/5= 8 35/40=7/8

9. How do we know that two fractions are the same? Are andequal?

10. How do we know that two fractions are the same? Are andequal? reduce Now we know that these two fractions are actually the same! =

11. Improper Fractions and Mixed Numbers An improper fraction can be converted to a mixed number and vice versa. An improper fraction is a fraction with the numerator larger than or equal to the denominator. A mixed number is a whole number and a fraction together Any whole number can be transformed into an improper fraction.

12. Improper Fractions and Mixed Numbers Converting improper fractions into mixed numbers: - divide the numerator by the denominator - the quotient is the leading number, - the remainder as the new numerator. -Denominator stays the same

13. Improper Fractions and Mixed Numbers Converting improper fractions into mixed numbers: - divide the numerator by the denominator - the quotient is the leading number, - the remainder as the new numerator. More examples:

14. Converting mixed numbers into improper fractions. Think about the order of operations. PEMDAS what comes first? Multiply the denominator by the whole number, then add the numerator.

15. Improper Fractions and Mixed Numbers Converting mixed numbers into improper fractions. Think about the order of operations. PEMDAS what comes first? Multiply the denominator by the whole number, then add the numerator.

16. Improper Fractions and Mixed Numbers Converting mixed numbers into improper fractions.

17. Addition of Fractions - addition means combining objects in two or more sets - the objects must be of the same type, i.e. we combine bundles with bundles and sticks with sticks. - in fractions, we can only combine pieces of the same size. In other words, the denominators must be the same.

18. Addition of Fractions with equal denominators + = ? Example:

19. Addition of Fractions with equal denominators More examples

20. Addition of Fractions with equal denominators More examples

21. Addition of Fractions with different denominators In this case, we need to first convert them into equivalent fraction with the same denominator. Example: An easy choice for a common denominator is 3×5 = 15 Now we have our same denominator + +

22. Addition of Fractions with different denominators In this case, we need to first convert them into equivalent fraction with the same denominator. Example: An easy choice for a common denominator is 3×5 = 15 Now we have our same denominator Therefore, +

23. More Exercises: What is the lowest common multiple? =

24. === = 8/4=2 2x3=6

25. More Exercises: 2 primes =

26. More Exercises: = = == 5x7 2 primes

27. More Exercises: == =

28. Adding Mixed Numbers Example:

29. Adding Mixed Numbers Another Example:

30. Adding Mixed Numbers Another Example:

31. Subtraction of Fractions - subtraction means taking objects away. - the objects must be of the same type, i.e. we can only take away apples from a group of apples. - in fractions, we can only take away pieces of the same size. In other words, the denominators must be the same.

32. Subtraction of Fractions with equal denominators Example:

33. Subtraction of Fractions More examples:

34. Subtraction of Fractions More examples:

35. Subtraction of Fractions More examples:

36. Examples of multiplying fractions 2 ½ X ¼ 51 24 X 5858

39. Examples of dividing fractions 5 10 9 12 KEEP SWITCH to multiply FLIP number following the division sign 5959 x 12 10

42. For Project Unit 3 A recipe for a drink calls for 1/5 quart water and ¾ quart apple juice. How much liquid is needed?

43. 2/5 + 1/4 = 8/20 + 5/20 = 13/20 Now if the recipe is doubled?

44. 13/20 13/20 + 13/20 = 26/20 =1 6/20= 1 3/10 Or 13/20 * 2 = 13/20 *2/1 =26/20 = 1 6/20 = 1 3/10 If the recipe is halved?

45. 13/20 13/20 / 2 = 13/20 / 2/1 = 13/20 * ½= 13/40

46. 42. 3245 4 tens + 2 ones + 3 tenths + 2 hundredths + 4 thousandths + 5 ten- thousandths We read this number as “Forty-two and three thousand two hundred forty-five ten- thousandths.” The decimal point is read as “and”.

47. Write a word name for the number in this sentence: The top women’s time for the 50 yard freestyle is 22.62 seconds.

48. Twenty-two and sixty-two hundredths

49. Slide 3- 49 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To convert from decimal to fraction notation, a) count the number of decimal places, b) move the decimal point that many places to the right, and c) write the answer over a denominator with a 1 followed by that number of zeros 4.98 2 zeros 2 places Move 2 places.

50. Slide 3- 50 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Write fraction notation for 0.924. Do not simplify. 0.924 =

51. Slide 3- 51 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Write fraction notation for 0.924. Do not simplify. Solution 3 places 3 zeros 0.924.

52. Slide 3- 52 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Write 17.77 as a fraction and as a mixed numeral.

53. Slide 3- 53 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Write 17.77 as a fraction and as a mixed numeral. Solution To write as a fraction: 17.77 2 zeros 2 places 17.77 To write as a mixed numeral, we rewrite the whole number part and express the rest in fraction form:

54. Slide 3- 54 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To convert from fraction notation to decimal notation when the denominator is 10, 100, 1000 and so on, a) count the number of zeros, and b) move the decimal point that number of places to the left. Leave off the denominator. 8.679. Move 3 places. 3 zeros

55. Add: 4.31 + 0.146 + 14.2 Solution Line up the decimal points and write extra zeros 4. 3 1 0. 1 4 6 1 4. 2 0 0 1 8. 6 5 6

56. Slide 3- 56 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example D Subtract 574 – 570.175 Solution 5 7 4. 0 0 0 57 3. 8 2 5 70. 5 9 9 10 3 1

57. Slide 3- 57 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To multiply using decimals: 0.8  0.43 a) Ignore the decimal points, and multiply as though both factors were whole numbers. b) Then place the decimal point in the result. The number of decimal places in the product is the sum of the number of places in the factors. (count places from the right). 2 (2 decimal places) (1 decimal place) (3 decimal places) Ignore the decimal points for now.

58. Unit 2 Participate in Discussion Complete Quiz Practice in MML Complete Reading Practice Problems