Math 6, Unit 5 Notes: Fractions Enrichment: This section will relate to students the relationship between fractions and decimals. It is recommended as a unit opener to review fraction and decimal skills and how the two are linked to each other. Relating fractions to decimals Rational Number is a number that can be written as a quotient of two integers. Decimals are special fractions whose denominators are powers of 0 Since decimals are special fractions, all the rules we have learned for fractions should work for decimals as well. The only difference is the denominators for decimals are powers of 0. i.e. 0, 0 2, 0, 0 4, etc Students normally think of powers of 0 in standard form as 0, 00, 000, 0000. In a decimal, the numerator is the number to the right of the decimal point. The denominator is not written, in is implied by the number of digits to the right of the decimal point. The number of digits to the right of the decimal point is the same as the number of zeros in 0, 00, 0000,.Therefore one place is tenths, two places is hundredths, three places is thousandths, and so on. Examples: ).56 2 places to the right of decimal 2).52 places to the right of decimal ).2 place to the right of decimal 56 00 52 000 2 0 The correct way to say a decimal numeral is to: ) Forget the decimal point 2) Say the number ) Then say its denominator and add the suffix ths Examples: ).5 5 hundredths 2).702 702 thousandths ).2 2 tenths 4) 5.6 5 and 6 hundredths Math 6 Notes Unit 5: Operation of Fractions Page of 5
Notice: When there are numbers on both sides of the decimal point, the decimal point is read as and. Say the number on the left side, the decimal point is read as and, then say the number on the right with its denominator. Example: Write 5.20 in word form: Fifteen and two hundred three thousandths Convert Fractions to Decimals There are two methods for converting fractions to decimals: I. Make equivalent fractions II. Division Method I: converting fractions to decimals by making equivalent fractions: Since decimals are simply fractions that have a power of 0 as a denominator, convert the fraction so that the denominator is a power of 0, then rewrite as a decimal. Example: Convert 2 to a decimal. First, change the fraction so that the denominator is a power of 0. Then rewrite as a decimal. Since the denominator is 2, what is the smallest power of 0 number that 2 can go into evenly? 0. Change the fraction so that it has a 0 as the denominator. 5 2 0 This is rewritten as.5 Example: Convert 4 to a decimal. Again, change the fraction so that the denominator is a power of 0. Then rewrite as a decimal. Since the denominator is 4, what is the smallest power of 0 number that 4 can go into evenly? 00. Change the fraction so that it has a 00 as the denominator. 75 4 00 This is rewritten as.75 Math 6 Notes Unit 5: Operation of Fractions Page 2 of 5
Notice: There are denominators that will never divide into any powers of 0 evenly. Since that happens, we look for an alternative way of converting fractions to decimals. To recognize that a number is not a factor of powers of ten, use the rules of divisibility. Factors of powers of ten can only have prime factors of 2 or 5. That means 2 will never divide into a power of ten because the prime factors of 2 are 2,2,. The result of that is a fraction such as 5 2will not terminate. It will be a repeating decimal. Because not all fractions can be rewritten with a power of 0 as the denominator, it is necessary to look at another way to convert a fraction to a decimal. That leads to method II. Method II: Converting fractions to decimals by dividing: Example: Convert 8 to a decimal. Since the prime factors of 8 are 2, 2, 2, 8 is not a factor of a power of 0. Therefore it is not possible to rewrite this fraction so that it has a power of 0 as a denominator. However, this can be done by division..75 8.000 CRT example: Converting Decimals to Fractions To convert a decimal to a fraction: ) Determine the denominator by counting the number of digits to the right of the decimal point. 2) The numerator is the number to the right of the decimal point. ) Simplify. Math 6 Notes Unit 5: Operation of Fractions Page of 5
Examples: ) Convert.52 to a fraction. Step Step 2 Determine the denominator. Since there are two numbers to the right of the decimal, the denominator will be 00. The number to the right of the decimal is 52 therefore 52 will be the numerator. Step Simplify: 52 26 00 50 25 2) Convert.60 to a fraction. Step Step 2 Determine the denominator. Since there are three numbers to the right of the decimal, the denominator will be 000. The number to the right of the decimal is 60 therefore 60 will be the numerator. Step Simplify: 60 000 since this does not reduce the answer is 60 000 ) Convert 8.2 to a fraction. Step Step 2 Determine the denominator. Since there are two numbers to the right of the decimal, the denominator will be 00. The number to the right of the decimal is 2 therefore 2 will be the numerator. Step Simplify: 2 6 8 00 50 25 The fraction is 8 8 25 Convert the following decimals to fractions: ).2 2).5 ).8 4).5 Math 6 Notes Unit 5: Operation of Fractions Page 4 of 5
Objectives: (-) The student will calculate with rational numbers expressed as fractions, mixed numbers, decimals, and percents. (-2) The student will translate written forms of fractions, decimals, and percents to numerical form. (-) The student will use models to translate among fractions, decimals, and percents. Methods of finding the LCM There are three methods for finding the Least Common Multiple (LCM) between two numbers: Method I: Make a list. Write the multiples of each numbers until there is a common number. Example: Find the LCM of 2 and 6. Multiples of 2: 2, 24, 6, 48, 60, 72, Multiples of 6: 6, 2, 48, 48 is the smallest multiple of both numbers, therefore 48 is the LCM. Method II: Prime factorization. Write the prime factorization of both numbers. The LCM has to contain ALL the factors of both numbers. Write the prime factors, use the highest exponent. Example: Find the LCM of 72 and 60. Prime factors of 72: 2 22 =72 Prime factors of 60: 2 2 5=60 The Prime factors of 60 and 72 are made up of 2 s, s, and a 5. How many of each prime factors are used to make up the LCM? 72 has three factors of 2 (2 ) and 60 only has two factors of 2 (2 2 ). Since the highest exponent is use 2. 72 has two factors of ( 2 ) and 60 has one factor of ( ). Since the highest exponent is use 2. 72 has no factors of 5 (5 0 ) and 60 has one factor of 5 (5 ). Since the highest exponent is use 5. The LCM = 2 2 5 =60. Math 6 Notes Unit 5: Operation of Fractions Page 5 of 5
Method III: Reduce the fraction. Write the two numbers as a fraction, reduce and cross multiply. The product is the LCM. Example: Find the LCM of 8 and 24. Make a fraction using 8 and 24. Order does not matter. 8 24 Reduce that fraction. 8 24 4 Cross multiply. Either 8 4 or 24. Both product equals 72. The LCM is 72. Note: When adding or subtracting fractions, the LCM is referred to as the Least Common Denominator or LCD. Adding and Subtracting Fractions With Unlike Denominators There is a giant laundry pile of dirty clothes in the laundry room. If 4 of the dirty clothes is Ashton s dirty clothes and of the dirty clothes belongs to Catherine, what fraction of the dirty clothes are Ashton and Catherine s? To find the fraction of the dirty clothes that belong to Ashton and Catherine, you can add, which are fractions with unlike denominators. 4 Here is a diagram representing the two fractions: 4 Math 6 Notes Unit 5: Operation of Fractions Page 6 of 5
Notice the pieces are not the same size. Making the same cuts in each fraction bar will result in equal size pieces: 4 Now the two fraction bars are divided into 2 equal pieces: can be re-written as 4 2 and 4 can be re-written as. This problem is now much 2 easier because the two fractions have the same denominator. = 4 2 4 = 2 7 2 It is inconvenient to have to draw fraction bars for every addition or subtraction problem. The following is the algorithm for adding or subtraction fractions that do not have the same denominator: Algorithm For Adding/Subtracting Fractions Step. Find a common denominator Step 2. Make equivalent fractions Step. Add or subtract the numerators, keep the common denominator, then reduce if possible. Math 6 Notes Unit 5: Operation of Fractions Page 7 of 5
Example: Add 2. 5 Step Step 2 Step Find a common denominator: The common denominator of 5 and is 5. Make equivalent fractions: 2 0 and 5 5 5 Add the numerators, keep the common denominator: 0 5 5 5 CRT example: Example: Subtract 5. This problem can also be written vertically. Using the 7 algorithm, first find the common denominator then make equal fractions. Once you complete that, you add the numerators and place that result over the common denominator and simplify. 5 5 7 2 7-2 8 2 Math 6 Notes Unit 5: Operation of Fractions Page 8 of 5
Example: Add 4. This problem can also be written vertically. In fact, it may 5 20 be easier to separate the whole numbers from the fractions if the problem is written vertically. 4 5 + 20 Step : Find a common denominator 2 4 20 + 20 Step 2: Make equivalent fractions 4 20 + 20 5 7 7 20 4 Step : Add the numerators and keep the common denominator. Then reduce. Notice that the whole numbers are just added together. 2 Example: Subtract 7 2. Another approach to evaluating fractions that involve 5 7 mixed numbers is to change all mixed numbers to improper fractions then follow the algorithm. 2 Change 7 2 to improper fractions: 5 7 subtracting fractions. 5 6 5 7. Now follow the algorithm for Step Find the common denominator: The common denominator of 5 and 7 is 5. Step 2 Make equivalent fraction using the common denominator: 57 6 5 = 5 7 7 5 245 80 5 5 Step Subtract the numerators and keep the common denominator, reduce, change back to a mixed number: 245 80 = 65 5 5 5 = 7 = 5 4. 7 Math 6 Notes Unit 5: Operation of Fractions Page 9 of 5
Regrouping To Subtract Mixed Numbers Review: Before teaching how to borrow with fractions, it is a good idea to revisit the following topic (from Unit 4): Borrowing when subtracting mixed numbers: The concept of borrowing when subtracting with fractions has been typically a difficult area for kids to master. For example, when subtracting 2 4 5, students usually 6 6 answer 8 4 6 if they subtract this problem incorrectly. In order to ease the borrowing concept for fraction, it would be a good idea to go back and review borrowing concepts that kids are familiar with. Example: Take away hours 47 minutes from 5 hours 6 minutes. 5 hrs 6 min hrs 47 min????????? Subtracting the hours is not a problem but students will see that 47 minutes cannot be subtracted from 6 minutes. In this case, students will see that hour must be borrowed from 5 hrs and added to 6 minutes: 4hrs 5 hrs 6 min 6minhr 6min60min 76min hrs 47 min????????? Now the subtraction problem can be rewritten as: 4 hrs 76 min 4 hrs 76 min hrs 47 min hrs 47 min??????????? hr 29 min If students can understand the borrowing concept from the previous example, the same concept can be linked to borrowing with mixed numbers. Lets go back to the first example: 2 5 6 4 6. It may be easier to link the borrowing concept if the problem is 6 7 2 6 6 6 6 6 7 rewritten vertically: 5 6 4 6 5 4??????? 6 2 7 7 6 Math 6 Notes Unit 5: Operation of Fractions Page 0 of 5
Example: Subtract 2 5 7 2 Step. Find a common denominator: The common denominator is 0. Step 2. Make Equivalent fractions using 0 as the denominator. 4 0 5 7 0 Step. It is not possible to subtract the numerators. You cannot take 5 from 4!! Use the concept of borrowing as described in the above examples to re-write this problem. Borrow from from and add ( 0 0 ) to 4 0. 2 4 0 0 0 5 7 0 4 2 0 5 7 0 9 5 0 Example: Catherine has a canister filled with flour to bake a cake. How much flour is left in the canister? Subtract 5. 2 4 5 cups of flour. She used cups of 2 4 Step. Find a common denominator: The common denominator between 2 and 4 is 4. Step 2. Make equivalent fractions using 4 as the common denominator. 2 5 4 4 Step. When subtracting the numerators, it is not possible to take from 2, therefore borrow. It may be easier to follow the borrowing if the problem is rewritten vertically. Math 6 Notes Unit 5: Operation of Fractions Page of 5
4 2 5 4 4 4 4 6 4 4 4 4 4 There are cups of flour left in the canister. Multiplying Fractions Multiplying fractions is pretty straight-forward. Here is the algorithm for multiplying fractions: Algorithm for Multiplying Fractions: Step. Make sure you have proper or improper fractions Step 2. Cancel if possible Step. Multiply numerators Step 4. Multiply denominators Step 5. Reduce Example: Multiply 4. 2 5 Step. Make sure that the entire problem is a proper or improper fraction. Change 2 to an improper fraction and rewrite as follows: 7 4 2 5 Step 2. Cancel if possible: 7 4 2 = 2 5 7 2 5 Math 6 Notes Unit 5: Operation of Fractions Page 2 of 5
Step and Step 4. Multiply numerators and denominators: 7 2 4 5 5 Step 5. Reduce: 4 4 2 5 5 Multiply the following fractions: 2 ) 2) 4 5 4 ) 2 4 2 2 4 CRT example: CRT example: Dividing Fractions Before learning how to divide fractions, it is worthwhile to revisit the concept of division using whole numbers. When asked how many 2 s are in 8, this can be written mathematically three ways: 28 8 2 8 2 Math 6 Notes Unit 5: Operation of Fractions Page of 5
To find out how many 2 s there are in 8, use the subtraction model: 8-2=6 6-2=4 4-2=2 2-2=0 Notice that 2 was subtracted 4 times therefore there are 4 two s in eight. Mathematically, 82 4. The above example defines division as repeated subtraction. This concept can be used for dividing fractions as well. Example: Let s evaluate this example by repeated subtractions. 4 8 5 2 4 8 8 8 8 8 5 4 2 8 8 8 8 8 8 4 0 8 8 8 8 8 Notice 8 is subtracted six times. This means that there are six 8 s in 4. This method of division can be tedious and time consuming. After doing several examples of division of fractions, students will see a pattern which will lead to the following algorithm: Algorithm for Division of Fractions Step. Make sure you have fractions Step 2. Change the division to Multiplication and invert the divisor. (Keep Change Flip) Step. Multiply, reduce if possible. Another way to explain the above algorithm is to show students that division is simply the multiplication of the reciprocal. Let s demonstrate this concept by using simple whole numbers: Math 6 Notes Unit 5: Operation of Fractions Page 4 of 5
Example: Divide 2 by 4: In other words, 2 4=? Students will know that 2 4=. Show them that you get the same answer if 2 you do 2. 4 4 Here are more examples: 0 0 5 0 2 5 5 24 24 8 24 8 8 6 626 2 2 2 Lead students to notice that dividing is the same as multiplying by the reciprocal of the divisor! Example: Multiply 2 9 2. 5 0 Step. Make sure you have fractions. Change both mixed numbers to improper fractions. 2 9 7 29 2 5 0 5 0 Step 2. Keep Change Flip: 7 29 7 0 5 0 5 29 Keep Change Flip Step. Multiply, reduce if possible: 2 7 0 4 5 5 29 29 29 Math 6 Notes Unit 5: Operation of Fractions Page 5 of 5