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2.1 Basics of Fractions

2.1 Basics of Fractions. A fraction shows part of something. Most of us were taught to think of fractions as: part of a whole such as ½ means 1 out of two equal pieces. 2.1 Basics of Fractions.

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2.1 Basics of Fractions

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  1. 2.1 Basics of Fractions A fraction shows part of something. Most of us were taught to think of fractions as: part of a whole such as ½ means 1 out of two equal pieces

  2. 2.1 Basics of Fractions Many times we shade pictures of pies and cut-up boxes to illustrate fractions. On your homework, you will be asked to identify diagrams and their coordinating fractions.

  3. 2.1 Fraction Terms The top of a fraction is called the numerator. The bottom is called the denominator. think downstairs=denominator ½ , ¾ , ⅓ , ⅔ , ⅛ , ⅞ ½ numerator is 1; denominator is 2 ¾ numerator is 3; denominator is 4

  4. 2.1 Basics of Fractions Another way to think about fractions follows: I have found this method to be more helpful as I work with fractions. The top number tells you how many, but the bottom number tells you what they are.

  5. 2.1 Basics of Fractions ½ Read one-half; means there is one and it is a “half” ¾ Read three-fourths; means there are 3 of them and they are “fourths”

  6. 2.2 Simplifying Fractions --Factors First, let’s review the divisibility rules we learned in chapter 1 A number is divisible by: -2 if the ones digit is even -3 if the sum of the digits is divisible by 3 -5 if it ends in 5 or 0 -9 if the sum of the digits is divisible by 9 -10 if it ends in 0

  7. 2.2 Simplifying Fractions --Factors What is a factor? factors are numbers that multiply resulting in a product. We can find all the factors of a number for example, list the factors of 12 1,2,3,4,6,12 Think of them in pairs 1,12 and 2,6 and 3,4

  8. 2.2 Simplifying Fractions --Factors What are the factors of 60? (think pairs) 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 Use your divisibility rules to help you come up with the factors

  9. 2.2 Simplifying Fractions -Prime and Composite Prime numbers have only two factors-one and the number itself. examples: 3,7,11,13,19 etc Except for 2, all prime numbers are odd, BUT not all odd numbers are prime!!! Composite numbers are not prime.

  10. 2.2 Simplifying Fractions -Prime Factorizations The prime factorization is what you get when you break a number down until all the factors are prime. There are two methods for finding prime factorizations • Factor tree • Division or box method

  11. 2.2 Simplifying Fractions --Factors We’ll do some example of each on the board

  12. 2.2 Simplifying Fractions A fraction is said to be reduced, or simplified, or in lowest terms when the numerator and denominator have no factors in common except for 1. To put a fraction in lowest terms, we divide out any common factors that exist between the top and bottom. Once we have divided out all common factors, the fraction is reduced.

  13. 2.2 Simplifying Fractions -Equality of fractions A quick trick to tell if two fractions are equal is to set them equal and then take the cross product. If the cross products are equal, then the fractions are equal as well. Take 4(28) and take 16(7) . . . Are the products equal?

  14. 2.3 Fractions in all their forms-Proper and Improper Proper fractions have a numerator that is smaller than the denominator. Proper fractions are less than 1. ½ , ¾ , ⅓ , ⅔ , ⅛ , ⅞ These are all proper fractions.

  15. 2.3 Fractions in all their forms-Proper and Improper Improper fractions have a numerator that is greater than the denominator. Improper fractions are greater than or equal to 1.

  16. 2.3 Fractions in all their Forms -Mixed Numbers Mixed Numbers have a whole number part and a fraction part. 3 ½ three wholes and one half 5 ⅔ five wholes and two thirds If I bought three pizzas, but ate ½ of one on the way home, I have 2 ½ pizzas left to share.

  17. 2.3 Fractions in all their Forms -Mixed Numbers Mixed Numbers are closely related to improper fractions. We can go back and forth between the two different forms. Mixed Numbers to Improper Fractions OR Improper Fractions to Mixed Numbers

  18. Mixed to improper (circle trick) A nice trick for changing from a mixed number to an improper fraction is: -multiply the denominator and the whole number -add this to the numerator -keep the same denominator

  19. Improper to mixed The method for changing from a improper fraction to a mixed number is to divide. Remember the fraction bar is a divide sign. Can you see the 3 ½ ?

  20. 2.4 Multiplying fractions To multiply fractions: Multiply the numerators, multiply the denominators and reduce In other words, take top times top; bottom times bottom and reduce

  21. a nice trick to remember Note: you can only cross cancel across a multiplication sign-never do this across an add, subtract, or division sign. Below you can cross cancel the two’s and then multiply. (like reducing before you multiply)

  22. Multiplying by a whole number When multiplying a fraction by a whole number, remember there is a 1 in the denominator of the whole number.

  23. Multiplying by a mixed number We cannot multiply in mixed number form. Use the circle trick to change the mixed number into an improper fraction.

  24. 2.5 Dividing fractions By definition, division is multiplying by the reciprocal What is a reciprocal? Two numbers are reciprocals if their product equals 1. To find the reciprocal of a number, interchange the numerator and denominator. In other words, flip it! ½ becomes 2 2/3 becomes 3/2

  25. 2.5 Dividing fractions To divide fractions: remember division is multipying by the reciprocal. So . . . -leave the first fraction as is -change the division to a multiplication -flip the second fraction -multiply (take top times top; bottom times bottom) -reduce

  26. 2.5 Dividing Mixed Numbers We cannot divide mixed numbers. We must change them to improper fraction form first. Then divide as normal. Leave first fraction alone. Flip second fraction. Then multiply: top times top; bottom times bottom. Reduce.

  27. 2.6 Least Common Multiples We tend to always learn about GCF and LCM at about the same time. They tend to get confused in our brain. In GCF ignore the G(reatest) and focus on what a factor is. Factors are things that multiply together to give you a product. The factors of 12 are: 1 and 12, 2 and 6, 3 and 4 1, 2, 3, 4, 6, 12

  28. 2.6 Least Common Multiples Somewhat like the GCF discussion we had in Chapter 2, this section about LCM will see like it has nothing to do with fractions. GCF was important because it helps us reduce fractions. LCM is important now because an LCM is the same as LCD (least common denominator) which we will need to add and subtract fractions.

  29. 2.6 Least Common Multiples Ignore the L(east) in LCM and focus on what a multiple is. The multiples of 12 are: 12, 24, 36, 48, 60, . . . The multiples of 10 are 10, 20, 30, 40, 50, . . .

  30. 2.6 LCM by the list method Finding the LCM of 10 and 12 by list You can list some of the multiples for the numbers involved and try to find one in common. 10, 20, 30, 40, 50, . . . 12, 24, 36, 48, 60, . . . Do you see a LCM yet? How about 60?

  31. 2.6 LCM by the list method The list method is okay, but it can be tedious to list all the multiples and sometimes you will make a list and not go far enough, as you saw in the last slide. It does work though and it is one option.

  32. 2.6 LCM by multiplying Finding the LCM of 10 and 12 by multiplying. This will always give you a common multiple, but it will not always give you the LEAST common multiple. It may require extra reducing at the end of the problem. 10 x 12 = 120 (remember our LCM = 60)

  33. 2.6 LCM using the larger denom Finding the LCM of 10 and 12 by counting by 12’s until you come across a number that 10 will also go into. This can be difficult if you can’t count by 12’s.  12, 24, 36, 48, 60 there it is!

  34. 2.6 LCM by Magic Finding the LCM of 10 and 12 by magic. This means you just look at the two numbers given and you just know what the LCM is. Some people have better magic than others  but a lot of times you will look and just know.

  35. 2.6 LCM by prime factorization Finding the LCM of 10 and 12 by Prime Factorization. This is the last resort. It is the most tedious method, but it will always work. Only do this method when you have tried the other options first.

  36. 2.6 LCM by prime factorization Find the prime factorizations for 10 and 12 10 = 2 x 5 12 = 2 x 2 x 3 2 x 5 22 x 3 Whatever factors appear in either of our prime factorizations, they must appear in our LCM. And to the highest power that they appear. LCM = 22 x 3 x 5 = 12 x 5 = 60

  37. 2.6 Write Equivalent Fractions The fraction ½ can take many different forms ½ is the same as 4/8 ½ is the same as 6/12 ½ is the same as 10/20 ½ is the same as 50/100 ½ is the same as 23/46 Is ½ the same as 31/63?

  38. 2.6 Write Equivalent Fractions Rewrite the given fraction with the new denominator: Ask yourself, what would I multiply the 3 times to get a 6? Multiply the top by that same number

  39. 2.7 Adding and SubtractingLike Fractions Like Fractions are fractions that have the same denominator. Example: ⅔ , ⅓ OR ⅛ ,⅜ ,⅝ Unlike Fractions are fractions that have different denominators. Example: ⅔ ,⅜

  40. 2.7 Adding Like Fractions When adding like fractions: -add the numerators -keep the same denominator -reduce if needed ⅔ + ⅓ = 3/3 = 1 ⅛ + ⅜ = 4/8 = ½

  41. 2.7 Subtracting Like Fractions When subtracting like fractions: -subtract the numerators -keep the same denominator -reduce if needed ⅝ - ⅜ = 2/8 = ¼ ⅔ - ⅓ = ⅓

  42. 2.7 Adding and SubtractingLike Fractions We can only add and subtract fractions that have the same denominators? Why? Remember when we talked about the top number tells us “how many” and the bottom number tells us “what they are”? We cannot add 2 apples and 3 tomatoes and say we have 5 grapes.

  43. 2.7 Write Equivalent Fractions Ask yourself, what would I multiply the 3 times to get a 15? Multiply the top by that same number

  44. 2.7 Adding and subtracting unlike fractions -find a common denominator (LCD) -rewrite each fraction with new denominator -add or subtract numerators as indicated -keep new denominator -reduce See appendix b for more info on LCD

  45. 2.7 Adding -horizontal

  46. 2.7 Adding - vertical

  47. 3.3 Subtracting - horizontal

  48. 3.3 Subtracting - vertical

  49. 2.8 ADD Mixed Numbers Regular Horizontal Method

  50. 2.8 ADD Mixed Numbers Regular Vertical Method

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