How To Teach Students To Divide By A Fraction

Transcription

1 Divison of Fractions: An Iceberg Inspired Learning Trajectory Since I started teaching I ve been very intrigued by the conceptual meaning of dividing fractions. I chose this topic for a learning trajectory to organize my own understandings and strategies for dividing factions, and to think more deeply about how I present this topic instructionally to students. According to Tirosh (2000) as quoted by Gregg, J. & Gregg, D., Division of fractions is often considered the most mechanical and least understood topic in elementary school. In addition, Van de Walle (2007) states Invert the divisor and multiply is probably one of the most mysterious rules in elementary mathematics. My experience as a student involved memorizing these kinds of rules and applying them to numbers. I never had difficulties multiplying by the reciprocal to divide fractions other than not having the slightest idea what it meant or where it came from. I hope to give my students a different experience dividing fractions that follows a progressive formalization learning trajectory. My learning trajectory starts with the prior knowledge understanding of how division works in whole number contexts. There are two ways to think about the division problem The quotative division model answers the question How many groups of 5 fit inside of 20? This is also sometimes called the measurement model. According to Davis and Pearn (2009), We call it measurement because we take this measurement unit and repeat it as many times as needed to get a quantity equal to what we are measuring. In this case the measurement unit would be the group size of 5. We are repeating this group size as many times as we need to get to 20. The partitive division model answers the question If you split 20 into 5 groups of equal size, how many objects will be in each group? According to Davis and Pearn (2009), This is called partitive division, because it focuses on partitioning, or spliiting, the collection of objects into so many groups of equal size. This is also referred to as the fair-share model because it connects to sharing or dealing out the same amount of objects to a fixed number of groups (like dealing a deck of cards). For the purpose of organizing my learning trajectory, I have colorcoded the strategies and big ideas for quotative division in pink and partitive division model in blue. Quotative Division Model Partitve Division Model

2 The learning trajectory then splits in two different directions, one following the measurement model (quotative) and another following the fair-share model (partitive). Both models begin informally dividing fractions using unifix cubes. To demonstrate the problem 2 I would have students connect three cubes to represent the group of size. Then I would ask them how many cubes we should connect to form the whole of (4 cubes). To make two wholes we need a tower of 8 cubes. So holding the towers side by side (one 8 cube tower to represent 2 and one 3 cube tower to represent ) we see that we can fit two full times and we have two cubes left over. It s easy to see that the 2 left-over cubes make up of another group so 2 = 2. This example demonstrated the quotative division model. We can also use cubes for the partitive division model, however I do a much better job explaining it in person with the cubes in front of me. 1 of another group of Two groups of 1 One group of In addition, we can use pattern blocks and/or fraction strips to model how many groups of one fractional size fit into another fractional size to support the quotative model. This strategy is basically a hands-on version of fraction bars that we might draw on paper. To support the development of the fraction bar models, we can move to Inside Math Book 6 Section F where we use fraction bar diagrams to model dividing a fraction by a fraction. These problems are out of context but can help provide a more structured visual model based on what students have been doing with the manipulatives. Fraction Bars Model Pattern Blocks How many 1/6 fit in 2/3? How many 1/6 fit inside of ½? 1/3 1/3 1/6 1/6 1/6 1/6

3 Continuing on with the formalization of the quotative model we move on to Connected Math Bits & Pieces II Problem 4.1. The objective is for students to begin using models informally representing a whole number divided by a fraction. The context is making pizza with bars of cheese. Each pizza requires ¾ of a bar of cheese and I have 5 bars of cheese. How many pizzas can I make? A sample student strategy for these problems is to draw rectangles representing the bars of cheese and partitioning them into sections according to what fraction of a bar of cheese is required (students could also use unifix cubes to model this). For this problem, the student could start drawing 5 rectangles (bars of cheese) and split the bars into fourths to make groups of ¾ to determine how many pizzas can be made. We have two pieces left over which can make us 2/3 of another pizza (since each pizza only requires three of these sized pieces). Here are two examples of how students might combine sections to create groups of ¾ Remainder The ratio table can also be used as a model to solve the pizza bars of cheese problem. We start by labeling ¾ bars for 1 pizza and scale up until we ve reached 5 bars of cheese. Here are two sample student strategies for how the ratio table could be used. Pizzas /3 6 and 2/3 Cheese Bars 3/4 1 ½ 3 4 ½ 1/2 5 X 2 X 2 3 Adding Columns Pizzas /6 = 6 and 2/3 Cheese Bars 3/ X 4 X 10 6 The pizza problem context involves the division problem 5 ¾. The context lends itself to the big idea of being able to restate this division number sentence in words. 5 ¾ really means How many ¾ s fit inside of 5? (thinking quotatively). Although students may not be working with the formal notation of a division number sentence yet, they are still answering this question within the context (How many ¾ bars of cheese will fit inside of 5 bars of cheese?)

4 These problems also lend to the big ideas of keeping track of the measurement unit (the size of the group you are making) and what to do with the remainder (if there is one). The number line model can be used as another strategy to help students with these two ideas. If we create a number line listing whole numbers from 0 to 5 then partition each section into fourths, students can circle groups of ¾ and then they will see two sections left over (as the remainder). At first, many students think the remainder is 2/4 because each small partition is the size of a fourth right? Wrong. We have to keep track of the unit of measurement which is our group size of 3/4. So if we have 2/4 left over and we re trying to make another group of size 3/4, then we have made 2/3 of another group. This is a difficult concept for students. 1 pizza 2 pizzas 3 pizzas 4 pizzas 5 pizzas 6 pizzas /3 of a 7 th pizza After working with whole numbers divided by fractions, we can move toward dividing a fraction by a fraction using Connected Math Bits & Pieces II Problem 4.3. The context for these problems is making bows out of ribbon. For example, if it takes 1/2 yard of ribbon to make a bow, how many bows can you make from 3/8 yard of ribbon? Restating the problem in words, we are really asking How many ½ s fit inside of 3/8? A sample student strategy involves the use of fraction bars. Start with two bars of the same size. One has 1/2 shaded in to represent how much is needed for a bow. The other has 3/8 shaded in representing the amount of ribbon we have. Comparing the two bars, students might start by dividing the shaded regions into equal size pieces. These pieces will be the size of eighths and will start to support to strategy of finding common denominators. Once we have the same sized pieces, we can see visually that ¾ of a half inside of 3/8 so. Only ¾ of a bow can be made with the material provided. The big idea here is that if we have the same size pieces (common denominators) then the size of the pieces (the denominator) is irrelevant and all that matters is how many pieces there are (the numerator). This big idea leads to the common denominator algorithm. This concludes the quotative division pathway. = How many ½ s fit inside of 3/8? 1/2 1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8 Common Denominator Algorithm We see that ¾ of a half fits inside of 3/8

5 The partitive division pathway eventually leads to the invert and multiply algorithm. I believe that the partitive model can be taught simultaneously with the quotative model but I have not experienced doing this. When I taught division of fractions in the past, I only focused on the quotative division model because it made the most sense to me. Throughout this project I was challenged to make sense of the partitive model. Now I believe it will be very valuable to present both models for division of fractions. Lets go back to the meaning of partitive division for whole numbers. Remember that 2054 represents If you split 20 into 5 groups of equal size, how many objects will be in each group? We can start to think about this in terms of division fractions with the example 6. Here were are asking, If you split 6 equally into 3/5 of a group, how many objects will be one group? The diagram below models how this could work. There would be a total of 10 hearts in one group 1/5 1/5 1/5 Another example of partitive division that might be clearer to students can be found in Connected Math Bits & Pieces II Problem 4.2. Take this problem for example, I have of a cake to split equally among 3 people. How much cake will each person get? This problem is easily modeled with a diagram of cake with shaded in. Then we split the shaded region into thirds again to see that each person would receive of a piece of cake so 3. Person 1 Person 2 Person 3 1/9 1/9 1/9 1/3 of whole cake Yet another example, I have of a cake and it fills up of my container. How much cake will fit in one whole container? Again, drawing a visual diagram helps to represent what s going

6 on. A ratio table to support the diagram can then help us solve what the question is asking. The two things we are comparing are the amount of cake and how much of a container it holds. Starting with of a cake filling up of a container, we can scale down dividing by 3 to find out how much of a container holds and then scale up multiplying by 3 to find out how much the entire container holds. Cake 3/4 3/8 9/8 = 1 and 1/8 Container 2/3 1/3 1 ¾ of the cake 2 x 3 Within this context, the big idea is to scale down to a unit fraction for how much of the container is filler and then scale up to see how much cake would fit in one container. In other words, divide by the numerator (to create a unit fraction) and multiply by the denominator (to get back to the whole). We can start to think about the invert and multiply algorithm in these two separate steps and then eventually put them together in one step. How cool is that? It s so rewarding to finally make a deep conceptual connection for a procedural rule I ve known and followed my whole life. To solve Two Steps 23 One Step This completes my learning trajectory for division of fractions based on the iceberg philosophy. I believe students will have reached the tip of the ice-berg when have constructed a conceptual understanding for what it means to divide fractions and they ve connected it to the formal algorithms. Some of the models used in this progression are limited, so eventually we want students to develop a method that will work easily for them with difficult numbers. Once students have reached this point, the next thing I would like them to be able to do is give a reallife situation that could be represent by a given division of fractions problem. For example, given the problem If of a wall can be painted in of an hour, how much can be painted in 1 hour? As mentioned in the Bay article (2003), this is a difficult task for many students (and adults) but if it can be done, a deep conceptual understanding of the operation will become apparent. The tip of the iceberg involves the ability to divide fractions procedurally (with computational fluency) and to understand the concepts behind the rule.

7 References Bay, J. (2003). Thinking Rationally about Number and Operations in the Middle School. Mathematics Teaching in the Middle School. p Davis G. & Pearn, C. (2009). Division of Fractions. A Republic of Mathematics Publication. p Tirosh, D. Enhancing Prospective Teachers Knowledge of Children s Conceptions: The Case of Division of Fractions. Journal for Research in Mathematics Education. 31 (January 2000): 5-25 Van de Walle, J. Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Allyn & Bacon, 2007.