Part 1: Dividing Fractions Using Visual Representations To divide fractions, remember that division can be represented by repeated subtraction, just like multiplication can be represented by repeated addition. Here s some terminology that will be used throughout this lesson. Section 1.1. Dividing Whole Numbers by Fractions Using Objects and Number Lines Model dividing a whole number by a fraction using both objects and number lines. For each problem, write a word problem that describes the situation. A. 3 # $ = B. 4 ' ( = C. 4 ' * = D. 3 +, = Created by Cathy Callow-Heusser, Ph.D. 1
Section 1.2. Dividing Fractions by Fractions When There is NO Remainder Model dividing a mixed number (a whole number plus a fraction) by a fraction using objects and number lines. E. 3 + * ' * = F. 2 ' (, ( = G. 1 + ' *, = H. 3 + * ( / = Section 1.3. Dividing Fractions by Fractions When There IS a Remainder Divide these fractions using number lines and fraction bars to model the problems. I. 2 + * + ' = J. 3 +, + ' = K. 3 + ' ' * = L. 2 + ' *, = Created by Cathy Callow-Heusser, Ph.D. 2
Part 2: Describing Division with Fractions Section 2.1. Describing Dividing Fractions In your own words, describe how visual representations (objects and number lines) can be used to model division with fractions. In your description, answer the question, How does division by fractions work? Make sure someone else could read your description and be able to model division with fractions. Section 2.2. But Isn t the Quotient Always Smaller When We re Dividing? What is the quotient when you divide 10 by 2? When you divide 5 by 10? Are these quotients larger or smaller than the dividends and divisors? Can you think of an example when you divide one whole number by another and get a quotient that is larger than the dividend or divisor? Look at the quotients for problems A through L. What do you notice? Write an explanation for your observation. Created by Cathy Callow-Heusser, Ph.D. 3
Section 2.3: Writing and Solving a Word Problem That Uses Dividing with Fractions Think about something you ve seen or done that involves dividing something into parts, where both the dividend and the divisor are fractions. A. Write a word problem to describe the situation. B. Write the division problem using fractions. C. Draw objects to model the situation. D. Write the division problem again with the quotient. Created by Cathy Callow-Heusser, Ph.D. 4
Part 3: Solving Division Problems When the Divisor is Larger Than the Dividend Section 3.1. Divisor Larger Than Dividend In all of the problems in Parts 1 and 2, the dividend has been larger than the divisor. However, the divisor can also be larger than the dividend, such as 3 10, which can be written as * +2. Word problems with fractions that can be modeled with a division problem in which the divisor is larger than the dividend are shown below. You re going to share ½ pound of chocolate with 3 friends, so each person gets an equal size portion. How much will each of you get? How wide is a rectangular strip of land with length of 1 # miles and $ an area of 3/4 square mile? (Remember, A = LW, or W = A/L) How many 3/4-cup servings are in 2/3 of a cup of yogurt? 1 2 3 3 4 11 2 2 3 3 4 Modeling the first problem is easy! Think about when you divide 12 by 3, or 12 3. You can repeatedly subtract groups of 3, resulting in 4 groups of size 3 (quotitive or repeated subtraction: you know how many are in each group and you ve got to find how many groups). Or you can divide 12 into 3 groups, each of size 4 (partitive or equal sharing: you know the number of groups but need to find how many parts are in each group). Previously, we did the first thing: repeatedly subtracting groups of the same size to find out how many groups. Let s try some problems where we divide into equal size groups. Use both fraction bars and number lines to model these problems. M. # 3 = N. + 3 = $, Created by Cathy Callow-Heusser, Ph.D. 5
O. * 2 = P. ' 3 =, * Section 3.2: Divisor Larger Than Dividend When Both are Fractions Q. + * = R. + + = ',, ' Created by Cathy Callow-Heusser, Ph.D. 6
S. ' ( = T. ' * = * / *, Section 3.3. Are the Quotients Larger or Smaller Than the Dividends and Divisors? Are the quotients in the problems in Part 3 larger or smaller than the dividends and divisors? Write an explanation for your observation. Created by Cathy Callow-Heusser, Ph.D. 7
Part 4: Dividing Fractions Using Procedures Section 4.1. Dividing Fractions Using the Standard Algorithm It is tedious to use drawings and number lines to divide fractions. Also, as fractions get more complex like the ones shown below, fraction bars or number lines won t work. We need to figure out procedures for dividing fractions. You need to understand these procedures and how they work in Algebra when you simplify expressions and perform operations with more complex fractions like the ones below. (2x 2 + 5x + 3) () = 2x2 + 5x + 3 4a 2 +12a + 9 4a 2 + 8a + 3 9a 2 25 6a 2 +13a + 5 25 12 + y +1 4 5 18 8y + 4 36 Follow instructions in the top row of the table on the next page to find the answers to the division problems using the Standard Algorithm. Check to see if your answers are the same as the answers you got for problems A through L in Part 1. Section 4.2. Discovering a Shortcut for Dividing Fractions Look at the original problems in the first column of the table and the problems as rewritten in column 4. Can you see a relationship between these two equivalent representations of the problem? Describe the relationship between the problems in column 1 and the equivalent representations in column 4. Write a short description for a shortcut you can use to divide fractions. Created by Cathy Callow-Heusser, Ph.D. 8
Column 1 Column 2 Column 3 Column 4 Column 5 Column 6 Original problems Rewrite with the problem improper fractions Rewrite using alternate division notation Multiply by a fraction that equals 1 to simplify the complex fraction Write the multiplication problem (without the denominator = 1) Write the answer Write the answer as a mixed number A. 3 # $ = 3 1 1 2 = 3 1 1 2 B. 3 +, = = = = 2 1 2 1 = 1 3 ( ) 2 1 1 = 3 1 2 1 = 6 1 = 6 C. 4 ' * = = = = D. 4 ' ( = = = = E. 3 + * ' * = = = = F. 2 ' (, ( = = = = G. 1 + ' *, = = = = H. 3 + * ( / = = = = I. 2 + * + ' = = = = = J. 3 +, + ' = = = = = K. 3 + ' ' * = = = = = L. 2 + ' *, = = = = = Created by Cathy Callow-Heusser, Ph.D. 9
Column 1 Column 2 Column 3 Column 4 Column 5 Column 6 Original problems Rewrite with the problem improper fractions Rewrite using alternate division notation Multiply by a fraction that equals 1 to simplify the complex fraction Write the multiplication problem (without the denominator = 1) Write the answer Write the answer as a mixed number M. # $ 3 = = = = = N. + 3 = = = = =, O. *, 2 = = = = = P. ' * 3 = = = = = Q. + ' *, = = = = = R. +, + ' = = = = = S. ' ( = = = = = * / T. ' * *, = = = = = Created by Cathy Callow-Heusser, Ph.D. 10
Part 5: Another Way to Think About Division with Fractions Here s another way to help you think about dividing fractions. Which, if any, of these questions have a different answer? How many 3s are there in 6? How many groups of 3 fives are there in 6 fives? How many groups of 3 tens are there in 6 tens? How many groups of 3 tenths are there in 6 tenths? How many groups of 3 @s are there in 6 @s? How many groups of 3 gronks are there in 6 gronks? How many groups of 3 anythings are there in 6 anythings (as long as both refer to the same unit)? The point of these questions is that the units of the problem do not matter. If the units are the same objects, they disappear by forming a fraction that equals 1 when you divide. This means that if 6/10 is divided by 3/10, 3/10 goes into 6/10 the same number of times as 3 goes into 6, or 3/12 goes into 6/12, or 3/20 goes into 6/20, or 3/29 goes into 6/29. Use the standard algorithm from Section 4 to divide 6/29 by 3/29. Describe what happened. This suggests a new algorithm for dividing with fractions: To divide two fractions, find a common denominator so the denominators form a fraction that equals 1, and then divide the numerators. Let s look at some of the problems we worked earlier that you solved. Rewrite equivalent fractions that have common denominators. Then, knowing the denominators form a fraction that equals 1 when dividing, divide the numerators. A. 3 + * ' * = B. 2 ' (, ( = C. 3 # $ = D. 4 ' * = E. * 2 = F. # 3 =, $ G. 3 +, + ' = H. 2 + * + ' = I. + ' *, = J. ' * ( / = K. ' * *, = L. 3 + * ' ( = In your own words, write a description to tell why this method works to divide fractions. Created by Cathy Callow-Heusser, Ph.D. 11
Part 6: Solving Word Problems Using Fractions with Division We need to divide fractions by fractions in many situations, as shown in the following problems. For each problem, draw a diagram or number lines showing how you can use repeated subtraction to model dividing fractions. Remember to use a straight edge and draw number lines carefully. Then, set up the division problem to find the answer. Write answers as mixed numbers if quotient is not a whole number. For problems 1-5, use the standard algorithm to divide fractions and find the answer to the problem. 1) A lawn mower tank holds + gallon of gas. Ian s 5-gallon container has 3 + gallons in it. How many times will ' ' he be able to fill the lawn mower? 2) Tarik is cutting stringers, the boards in house walls that run horizontally between studs to add stability. Each stringer is 14 or 1 + ft. How many stringers can Tarik cut out of each 8 ft. stud? 3 3) Amal s mother is planning his birthday party and wants to serve pizza. Each pizza is cut into 8 equal size pieces. If each child eats three pieces, or * of a pizza, how many children can be fed with 4 pizzas? 4 4) Maria has 2 5 yard of fabric. She wants to make placemats. Each placemat needs + yard. How many 6 * placemats can Maria make? 5) Jaron has 5 + gallons of paint and needs to paint 8 chairs. Each chair needs ' ' * chairs could Jaron paint? Will he have enough for the 8 chairs? gallon of paint. How many For problems 6-9, use the shortcut you discovered in Part 4 to find the answer. 6) Angelo draws caricatures. Each caricature takes an average of 18 minutes, or * rents a booth at the fair for 5 + hours, how many caricatures will he be able to draw? ' +2 of an hour. If Angelo 7) Ann has 2 2/3 yard of ribbon. She needs 4 pieces, all the same length. How long should she cut each piece of ribbon? 8) Manuel is making banners for a client. Each banner is 2 * feet long. There are 21 + feet of paper left on, ' the banner roll. How many banners can he make? 9) Judy is in Great Britain and wants to help her friend make pancakes for a fundraising event. Judy s favorite recipe calls for ' cup of buttermilk for each batch of pancakes. Buttermilk in the UK comes in * 1 liter containers. A liter contains approximately 4 + cups. How many batches of pancakes can they make, if they have 2 liters of buttermilk? 10) Write your own word problem for Problem E in Part 1. 11) Write your own word problem for Problem G in Part 1. 12) Write your own word problem for Problem J in Part 1. Created by Cathy Callow-Heusser, Ph.D. 12
Part 7: Applying Repeated Subtraction (Grouping) to Rational Expressions When we multiply, 3 x 4 means 3 groups of 4 objects, or 4 objects + 4 objects + 4 objects using repeated addition. The result is 12. Similarly, if the problem was, 3 groups of how many objects is 12?, you d know this could be represented by the equation 3 x c = 12, or as a division problem, 12 3 = c or 12/3 = c. Rational expressions can also be rewritten using division, 2x 2 + 5x + 3 = (2x 2 + 5x + 3) () though we need to include grouping symbols. Otherwise, the division operation 3 x would need to be completed before addition operations. The rational expression can also be rewritten as a multiplication problem, ()i = 2x 2 + 5x + 3 Repeated subtraction, or grouping, can be used to represent division with rational expressions. In the problems below, rewrite the rational expression as a division problem. Draw objects representing each term of the dividend. Draw circles around the objects representing the divisor to group the objects (indicating repeated subtraction), write the expression for the remaining objects, apply the distributive property if possible, and write the result of the division, as shown in the first example. Rational Division Groups Remaining Distributive Result of Expression Problem Subtracted Objects Property Division 2x 2 + 5x + 3 = (2x 2 + 5x + 3) () } 2x 2 + 2x = 2x() 2x + 3 A. x 2 + 5x + 4 B. 4x 2 + 4 2 C. 3x 2 + 8x + 4 x + 2 Created by Cathy Callow-Heusser, Ph.D. 13
Part 8: Using Area Models with Rational Expressions Just as 3 x 4 can be represented with objects or as an area model, the rational expression below can be similarly represented with objects or as an area model 2x 2 + 5x + 3 = (2x 2 + 5x + 3) () to determine the simplified form of the rational expression or the result of the division, 2x + 3 in this case. Write the following rational expressions as division problems. Using algebra tiles or a drawing, find the result. Division Problem Area Model Drawing A. x 2 + 5x + 4 B. 4x 2 + 4 2 C. 3x 2 + 8x + 4 x + 2 Created by Cathy Callow-Heusser, Ph.D. 14