1 Addition and Subtraction

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Clementine Robinson
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1 Worsheet for 5.: Operations with Fractions Addition and Subtraction Adding/subtracting when there is a common denominator Draw a picture to work out each problem. Write your answer in lowest terms; if you are able to reduce, use your picture to show this too For each problem below: Adding/subtracting when you need to find a common denominator a picture of each fraction. Find the least common denominator for the fractions. Draw Draw a picture to show how to rename each fraction using the least common denominator. Add or subtract and draw a picture of the result. If it is possible to reduce the result, use your picture to show how to do this. Give your answer in lowest terms Mixed numbers and improper fractions improper fraction: numerator at least as big as denominator Fractions with the same number on top and bottom are equal to :,, 4 4, etc. Fractions where the numerator (top) is bigger than the denominator (bottom) have a value more than :, 5 4, 5, etc. mixed number: has a whole number part and a fraction part. The fraction part must be proper. A mixed number represents an addition problem: Example: MEANS PLUS whole number + proper fraction

2 . Converting between improper fractions and mixed numbers Algorithm for converting a mixed number to an improper fraction: algorithm : A b c = (A c)+b c Example : = ( )+ = 6+ = 7 Why it works (discussed using the example above): represents the addition problem + Recall that we can write any whole number as that whole number over. So our addition problem can be written as + The LCD is.to rewrite with a denominator of, we multiply both the top and bottom by. = = We know this is 6, but to understand the algorithm we will write it. Our addition problem becomes as + Since the denominators are the same, we keep the denominator and add the numerators. + = ( ) + = 6 + = 7

3 Algorithm for convertin an improper fraction to a mixed number: algorithm:. Write the fraction as a long division problem. a b becomes b a. Do the division. The quotient becomes the whole number part of the mixed number. The remainder becomes the numerator of a fraction with the original denominator. example: 4 Step : 4 Step : 4 8 remainder Answer: 4 Why it works (discussed using the example above.) A fraction is a division problem. If we had brownies to divide evenly among 4 children, each child would get 4 brownies. Step just writes the division problem a different way. The quotient tells us the number of whole brownies we can give to each child. If we were dealing out brownies (like we deal playing cards) we could go around times, giving each child one brownie each time, until we didn t have enough brownies to go around another full time. At this point, we would have leftover brownies. Since there are 4 children, we could cut each brownie into 4 pieces (fourths) and give each child one piece (one fourth) from each of the leftover brownies. Thus, each child would get whole brownies plus fourths of a brownie. This is + 4 = 4

4 Your turn!. Write 5 as an improper fraction. Justify your work different ways: (a) Using an area model. (b) Using a number line. (c) By writiing out the steps in the algorithm given above.. Write 4 as a mixed number. Justify your work different ways: (a) Using an area model. (b) Using a number line. (c) By writing out the steps of the algorithm given above.. Adding and subtracting with mixed numbers Work out each problem below ways: By keeping each number as a mixed number and regrouping as necessary. By converting each number to an improper fraction, doing the addition or subtraction, and then converting back to a mixed number. For both methods, you should give your answer as a mixed number

5 Multiplication. Multiplying by a whole number. Show that 5 = 0 by drawing a picture and using the fact that multiplication represents repeated addition.. Carol has 4 plates. On each plate, she has put half a cookie. How many cookies does she have in all? (a) Identify the common core problem type for this problem. (b) Solve the problem with a picture.. Max has a piece of wiood that is 4 feet long and of a foot wide. What is the area of Max s piece of wood? (a) Identify the common core problem type for this question. (b) Solve the problem with a picture. 4. Suri has apples. Frank has apples does Frank have? as many apples as Suri. How many (a) Identify the common core question type for this question. (b) Solve the problem by drawing a picture.. Multiplying by a fraction of the form n where n is a whole number. Fill in the blanks: Multiplying by 4 is the same as dividing by.. Draw a picture to demonstrate the above fact for 4.. Write a story problem to go with the number sentence: 5 4 =?. Multiplying by a fraction of the form b a, where a and b are whole numbers and b = 0. Fill in the blanks: Multiplying by b a is the same as multiplying by, which divides the given amount into b equal parts, and then multiplying by, which corresponds to repeated addition, giving us a copies of the resulting pieces.. Draw a picture to illustrate the above fact with the problem 4. 5

6 .4 Connecting multiplication to equivalent fractions, cancelling, and multiplying by improper fractions and mixed numbers Example 4 5 = 5 = 5 = ( 5 ) ( ) }{{} = 5 = 0 Your turn! Work the following multiplication problems by first cancelling, justifying your cancellation as shown in the example above. If necessary, write your answer as a mixed number, not an improper fraction

7 4 Division. In this problem, we consider what happens when we divide by a whole number. (a) Draw a picture/sequence of pictures to illustrate the division problem 4. (b) Fill in the blanks: dividing by 4 is equivalent to multiplying by. Dividing by a whole number n is equivalent to multiplying by.. In this problem, we consider what happens when we divide by a fraction of the form n, where n is an integer. (a) Draw a picture/sequence of pictures to illustrate 4. (b) Draw a picture/sequence of pictures to illustrate 4. (c) Draw a picture/sequence of pictures to illustrate (d) Draw a picture/sequence of pictures to illustrate 4. (Hint: even though we don t need a common denominator to divide, a common denominator will be helpful in illustrating this problem.) (e) Fill in the blanks: dividing by 4 is equivalent to multiplying by. For a whole number n, dividing by n is equivalent to multiplying by.. In this problem, we put the pieces together. Fill in the blanks. Dividing by is the same as dividing by and then dividing again by. Dividing by is equivalent to multiplying by, and dividing by is equivalent to multiplying by. Putting these pieces together, dividing by is equivalent to multiplying by. In general, dividing by a fraction b a is equivalent to multiplying by. We call this the of b a. 4. Work the following problems by using the standard algorithm for dividing by a fraction (which you just described in problem ). If necessary, give your answer as a mixed number rather than an improper fraction. (a) 4 5 (b) 7 89 (c) 4 5 7

8 5 Children s mistakes The following are part of question #, p. 84 of your text. For each problem, describe the mistake the child made and how you would demonstrate this to the child. d g. 4 = 89 l = 4 8

What is a Fraction? Fractions. One Way To Remember Numerator = North / 16. Example. What Fraction is Shaded? 9/16/16. Fraction = Part of a Whole

What is a Fraction? Fractions. One Way To Remember Numerator = North / 16. Example. What Fraction is Shaded? 9/16/16. Fraction = Part of a Whole // Fractions Pages What is a Fraction? Fraction Part of a Whole Top Number? Bottom Number? Page Numerator tells how many parts you have Denominator tells how many parts are in the whole Note: the fraction