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COMMENTARY (as of 6/2003) General Remarks This commentary serves two purposes. It presents additional perspectives on fractions in order to help adults broaden their own understanding. This is presented from an adult viewpoint and does not directly relate to how children may think. However with a better understanding of fractions, adults will be able to come up with ideas for helping children understand fraction concepts. Some of the commentary also contains illustrations of fraction applications taken from the experience of adults working with children. While these are intended to give ideas for ways to apply fraction concepts with a child, applications for a particular child depend on a multitude of that child‟s characteristics. If you work with children, you will have to tailor what you do to fit their individual characteristics and needs. Understanding This book focuses a utilitarian way of understanding fractions. i.e. understanding how operations on measuring numbers must be as conceptualized if they are to be applicable to a variety of situations. However we have chosen one specific application as our central model. This model is described in the first lesson of Chapter 1. It uses a rectangle to picture the number one and portions of this rectangle to picture various fractions. The rectangle most frequently used has 24 cells, each of which is a one-inch square. This size is mathematically irrelevant. It was chosen to make the pieces fairly easy to handle. Even larger pieces might be convenient for young children. A more abstract way to understand fractions is in terms of basic algebraic laws such as the associative law, the commutative law, the distributive law, etc. For example the rule for adding fractions with like denominators can be seen as a specific case of the distributive law. While such an algebraic understanding of fractions provides further perspective, a utilitarian understanding based on a model is sufficient for many purposes. In fact, until the emergence of contemporary mathematics, it was the only significant type of understanding available. Thus we have not included any materials in this book that are designed to introduce an algebraic understanding of fractions. For anyone interested in an algebraic understanding of fractions we are working on an ordinary algebra unit entitled “Basic Algebra for Rational Numbers”. It will soon be available on our website. Constructivism In many traditional classrooms, children learn by absorbing and accumulating information, with active thinking and reasoning seldom emphasized. Constructivism goes beyond this type of learning. It realizes children must actively think and reason in order for learning to be rich. Students must be involved in the learning process. With guidance, they construct concepts that both fit with and go beyond the concepts they already understand. Learning this way they are able to interpret and apply these concepts. For children to learn to their fullest, they must be encouraged to explore many possibilities of solving problems. One limitation in much of the teaching of mathematics is that it often focuses on getting the “right” answer. It does not sufficiently concern itself with the process of getting to the solution. As constructivist we encourage students to explore, to understand various ways of solving problems, to not be concerned about being right and wrong in their initial attempts. We want them to think of mistakes as a natural part of the learning process, something to learn from rather than be worried about. We want children to feel free to say what they are thinking and then build discussion on their thoughts. Students then will be able to work their way through similar problems and be able to understand and explain how that problem works. 1 Children in a constructivist learning environment often acquire new concepts by exploring questions that they can understand but have not been taught how to answer. This allows them to pull ideas from their own understanding of the situation and think for themselves. Word problems can be viewed as a prelude to skills. In this book, we will be introducing new concepts with word problems before the arithmetic needed to do the problem is taught. You do not have to know how to do something well before you do something real. Consider a team of young children just learning to play basketball to illustrate this point. Most would probably be more enthusiastic about playing a basketball game rather than running basketball drills over and over again. Some people would argue that drill should come first, at least until certain skills are acquired. This can be useful, but if they hate this routine, little skill is achieved thru the drills. Most children would probably be much better off by simply playing the game badly for a while. This can provide the kind of experience that will help them realize what they are good at and what they need to worked on. When this is discovered, they will realize that if they work on the particular aspect that their coach says needs improvement then they will be a better team. This discovery will motivate them to practice particular skills, allowing for even greater learning to take place. Let‟s use the topic of division and remainders to illustrate the differences between traditional and constructivist learning. In most traditional classrooms, the topic of division would be presented first, then the concept would be drilled through the use of worksheets. Finally, word problems might be given. The constructivist method approaches the introduction of this concept differently. We illustrate this method with a scenario of young children. These children are younger than the age group this book focuses on. The teacher breaks a classroom of children into several groups of 4. Each group is given a bag containing 15 cookies. The teacher asks each group how they would distribute the cookies among themselves. Students have done a number of division problems, but they have never encountered remainders. One group counts the cookies this way “One for me, one for you, one for you, and one for you.” They continue this process until one of the children yells out, “Hey, there‟s not enough. It doesn‟t come out even! It won‟t be fair!” The teacher encourages the children to discuss this problem among themselves. She asks how they might go about solving the problem. One child declares, “We could cut up the 3 extra cookies, but how would that work?” Another student says, “We could just leave them. They are extras.” Another group quickly sets up the problem „15 divided by 4‟. However they are puzzled when the problem does not come out even. They too comment on there being extra cookies. One child says, “ Maybe our teacher gave us the wrong amount of cookies. It should have been 16 cookies. 16 divided by 4 equals 4. That would be right.” A third group sees the problem a little differently. “Just because we‟ve never seen a problem with things left over, that doesn‟t mean that it‟s wrong. I think it‟s okay to have things left over. It‟s just something different.” This type of learning stimulates children to put their minds at work and figure things out for themselves. Sharing cookies is something they can grasp much easier than theories and laws written in textbooks. 2 Chapter 1, focuses primarily on a wide variety of questions about situations involving fraction concepts. These questions are explored primarily thru conceptual reasoning in relation to visual models, with no focus on the mastery of algorithms for calculating with fraction. This does not mean that arithmetic calculation is considered unimportant. It merely means that such algorithms are usually better mastered when they follow, rather than precede, conceptual understanding and the ability to relate concepts to meaningful questions about various situations. In this chapter the only important algorithm is that of raising and reducing fractions. However the goal here is primarily to introduce rather than master that algorithm. Mastery for some students will automatically come later as they use the process in adding and subtracting fractions. For others it might be advisable to do more work on this process before going on to addition of fractions. Routine drill can be used. However more situations including a wide variety of questions about situations to which the student can easily relate will probably do more to enhance mastery. For visually oriented students I also recommend supplementing the arithmetic with manipulating color pieces. With students for whom fraction concepts come easily, only occasional use of the color model for fractions is advised for material suggested by Chapter 1. Rely instead primarily on the gray model. Also frequently use other visual representations initiated by the student. The color model is good for introducing concepts, but limits the fractions that can be used. The gray model overcomes this limitation. However manipulating color pieces, or even drawing the gray model, in order to answer questions can be tedious. Thus it is wise to encourage students to use any appropriate visualization. Furthermore when a student initiates a way of looking at the question, the quality of the student's involvement will often be of a higher level than when the model is given by someone else. Algorithms One critic of constructivist teaching claims that it ignores the hard work involved in becoming good at the algorithms used for arithmetic calculations. He compares the acquisition of arithmetic skill to the acquisition of athletic skill, remarking about the hours of tedious drill and practice needed to master such skills. There are a number of comments to make in response, first on the general idea of functional mastery and then on the role of drill and practice. The main constructivist principle is that unlike information, which a person can receive in a fairly direct manner from another, a person must acquire a conceptual understanding by augmenting and transforming ideas they already understand. In this sense concepts are constructed from within the conceptual net of the person learning the concept. This does not mean that practicing the concept once it is acquired is unimportant. Constructivist teaching should not focus only on initial acquisition of a concept, and thus ignore functional mastery. Functional mastery of many mathematical concepts involves both an understanding of related algorithms and skill in doing them. The difference between a constructivist approach and a non-constructivist one is in how mastery of algorithms is obtained. Again consider the analogy of drill and practice in the acquisition of an athletic skill such as basketball, distinguishing between the concept of drill and the concept of practice. While there is no sharp line between these concepts, drill is routine but practice is not. Drill is useful and can be tedious, but the real hard work comes in practice and the highest form of practice involves playing the game. Furthermore the utility of tedious drill depends highly on attitude and would be of little use to the athlete who did not want to play the game. The analogy to drill and practice in mathematics should be obvious. In fact when it comes to developing useful conceptual skill the role of attitude and real use may be even more crucial than it is for physical skills. Those most competent in conceptual skill are those that use them because they enjoy using them. 3 Cast of Characters Our characters are imaginary and are not intended as typical. One reason we did not choose to use typical persons is because they are hard to characterize. Each person is different; what is typical for one may not by typical for another. However our characters at least represent some of the major attitudes or styles that influence the learning of mathematics. Another reason these characters are not typical is that we are using them to communicate ideas to adults. In this book, we talk to adults through them. Our examples are often better related to adults than to children. When constructing examples for a child, use things that this child can relate to. For example, most children would relate better to a situation dealing with candy than one dealing with acres of land. Jan: good with whole numbers, likes counting, not sure about fractions Bob: needs things manifest, likes hands on and pictures, is unsure of abstractions Roy: fair at concepts, likes and is very good with algorithms Kay: good at concepts, makes detailed conceptual distinctions Like Jan, many people who are comfortable with whole numbers are not comfortable with fractions. However Jan knows how to convert questions about fractions to a question about whole numbers. She knows that a pound is 16 ounces, a foot is 12 inches, a gallon is 8 pints, a yard is 36 inches. Given problems involving these units of measure she can use whole numbers instead of fractions. In the book we talk as if Bob thinks in pictures. Actually he manipulates color pieces, and we abbreviate what he has done by a picture. To get a better feeling for the activity we recommend using these pieces. This will provide another alternative for understanding fractions that is more sensory than the numerical perspective Jan uses. Roy may seem better at arithmetic than Jan or Bob, but he has a limitation that they do not have. They have ways of thinking that allow them to answer questions that involve fractional concepts prior to leaning the arithmetic. Roy relies on arithmetic rules with little concern on understanding why they work. Not knowing a rule he tends to feel lost. However when he can turn his attention away from rules and rely on his own ability to think, he is just as capable as Jan or Bob. While Kay grasp ideas easily and often goes beyond the thinking of most adults. She will probably develop an appreciation and understanding of mathematical ideas regardless of the way she is taught. However with the help of an adult who understands concepts Kay will develop an even deeper appreciation. Some with Kay‟s potential are taught a way that stifles their imagination. An adult who does not have the understanding needed to help Kay should at least stay out of her way. In general, an adult who wants to help any child learn mathematics needs to be able to think about mathematics in various ways. The more understand about how fraction problems can be converted to whole number problems, the more I could help Jan. I also need to be able to formulate questions that can lead her from thinking in terms of whole numbers to thinking in terms of fractions. Similar remarks apply to helping Bob. Helping Roy is more difficult. While my versatile understanding of why the algorithms are useful, a large part of Roy‟s difficulty is based on his attitude that learning mathematics means learning algorithms. I would have the same problem with Jan and Bob, if they were more like students who thought the way they did outside of mathematics class but carried an attitude like Roy‟s into learning situations. You will learn more about our cast of characters as you read this book. In the process you will find that as the characters learn more about fractions some of their characteristics may change. We often discuss new ideas through their perspectives. 4 COMMENTARY ON CHAPTER 1 Overview The commentary is divided into separate sections for each lesson in the text. Each section begins with general remarks. These are sometimes followed by some additional remarks, which can be considered as optional. For a perspective on the use of these materials and concepts with children I have included illustrations of adults working with children. For Chapter 1 we focus on one child Angela who is being taught by her sister Jennifer. These are written by Jennifer in first person. Jennifer and Angela are my granddaughters. To help in developing this book, Angela agreed to participate as a subject for learning fraction concepts. Jennifer worked with Angela on the ideas in chapter 1 for two hours a week for about 12 weeks. Angela is ten years old and prior to this experience had at best a minimal understanding of fraction concepts. Mathematical thinking has never been an easy matter for her. Jennifer is a twenty year old senior at Webster University majoring in business with a minor in dance. Her only teaching experience has been in the area of dance and her only college math course an independent study with me on developing materials for teaching fraction concepts. Other than this she has had no special preparation for teaching mathematical ideas to children. Lesson 1 The focus of this first lesson is on presenting an overview of the most basic concepts for thinking about fractions. At this point there is no need to compute with fractions. Instead the main purpose is to imagine situations and describe it in terms of fractions. This lesson also introduces the idea of using color pieces to picture fraction concepts. Some of the main ideas are sketched below. Whole numbers can be used for counting but we need additional numbers for measuring. A fraction is a type of name, a type that is used to name measuring numbers. Different fractions may name the same measuring numbers. I decided to use the term „measuring numbers‟ rather than the standard mathematical term „positive rational numbers‟ primarily because the terminology suggests their relationship to a larger mathematical structure which also contains negative numbers. I wanted terminology that did not suggest this relationship. However calling them measuring numbers also suggests the kinds of situations from which such numerical concepts have been abstracted. Pie pieces are often used to represent fractions. They easily show fractions such as 1/2, 1/3, 2/3. However it is hard to distinguish others such as 3/7 and 5/8. It becomes even harder to compare fractions as the denominators become larger. We chose to use rectangular pieces in this book because the rectangular shape has many possibilities. The rectangular shape allows for easy comparison. When we break the rectangle into parts, they are symmetrical. We use a 4 by 6 rectangle. Having 24 parts allows for the creation of a variety of commonly used fractions. We also color-code these pieces, which makes for even easier comparison. There are some limitations. The 4 by 6 rectangle does not allow for certain fractions such as 1/5 and 1/7. For this purpose we continue to use a rectangle with a heavily lined border for 1. We shade some of the parts gray to represent the fraction. For example, to illustrate 3/5 we separate a rectangle into 5 parts and shade 3 of them gray. 5 Additional Remarks This remark refers to the thought question about names. Since this is a thought question I leave it to you to find as many answers to it as you can. I will merely make some initial thoughts of my own. Ordinary language distinguishes between numbers and their names by using the word „numeral‟ to refer to the name of a number. At one time many elementary math books stressed this distinction but this never seemed very important for the purposes that most teachers had when teaching arithmetic. I suspect that the distinction between a name and what is being named is just too ordinary to matter to most of us most of the time. As long as we implicitly understand a distinction we need not make it explicit to avoid confusion. Consider the following example. Example The name of my son‟s dog is Bear. I hear this name when my son tells his dog to sit. I see this name when it is written. Which of these sentences below refers to the dog and which one refers to the dog‟s name? How can you tell? Bear begins with the letter B. Bear weighs about 100 pounds. This name does not look or sound like a dog. Would anyone confuse a dog with the dog‟s name? However confusion of a number and its name is fairly common. Perhaps it is because numbers are imagined objects and are not the kind of objects that can be seen. You can see five apples or five pennies. But how can you see five? You can see the word „five‟ or the numeral „5,‟ but numerals and words are names. They are not numbers. Numbers are imagined objects. But is it important to realize this? Does the confusion of a number with one of its names matter? The simple answer to this is that any type of confusion may cause subtle problems that limit our ability to effectively use our concepts. However the thought question asks for more pointed answers. I give an example so elementary that you may think no confusion could occur. Which is larger? 4 or 5 While this example may seem artificial it is not totally different from the example of a child who says that 3/7 is bigger than 1/2 because 3 is bigger than 1 and 7 is bigger than 2. We want to make it clear that the word fraction refers to a type of name, and not to a type of number. One way to stress that we are referring to a name rather than using it to talk about a number is to use quotes. The sentence on the left is about two fractional numbers. The one on the right is about particular names of these numbers. 1 /4 is smaller than 1/2. „1/2‟ and „1/4‟ have the same numerator. Confusion might result if someone said 2/3 and 4/6 are different because their numerators and their denominators are different. Do they really mean that „2/3‟ and „4/6‟ are different? Since the use of quotes is tedious we usually rely on context to tell whether we are talking about a name or a number. 6 Jennifer working with Angela When working with Angela on fractions, I tried to incorporate fractions into everyday life. To her fractions were an abstract math concept used only during math class in some textbook. I wanted her to see that fractions are often used and can be helpful. Here are some of the activities we did: We walked around the house finding fractions in everyday items. I would ask things such as “What fraction of people in that picture are smiling?” or “What fraction of the towels in the bathroom are pink?” or “What fraction of the flowers in this pot are blooming?” Angela plays piano, so she is somewhat familiar with musical notes. However, she had no concept of how fractions relate to playing music. We worked with these concepts and created measures of music. We figured out how to write a measure given a certain timing, such as 4/4 time. She then had to figure out what notes would fit into the measure given whole notes, half notes, quarter notes, and eighth notes. Instead of just having her count out the notes to make sure they were the right value, we made it more interesting and sang the counts. Both Angela and I dance. Since dancing is interesting to her, I knew that this was a topic she would relate well to. Dance combinations are often made up of counts of eight. I used this concept to work on fractions. I taught her a simple combination and then I asked her to do 1/4 of the combination or to do the last 1/8 of the combination. I would ask what fraction of the combination a certain step was. I also had her make up a combination and ask me about it. I know that these activities have helped Angela think of fractions in a new way. The other day, three of us were eating French fries and she told us “Two thirds of us have finished our fries” She said it just as a passing comment, like it was a completely normal thing to say. Angela and I have also been working on some questions that involve fraction concepts. Question: There are twelve cars. A third are red. How many cars are red? Angela writes: “1/3, 1/3, etc.” (she writes “1/3” twelve times) and says, “The answer is 12/3.” Looking at this she says, “No it is wrong, but I don‟t know how to do it.” Then after thinking some time she writes: “3+3+3+3 = 12” and says that the answer is 4. Jennifer: “How did you get that answer?” Angela: After thinking some time, “No it‟s not the right answer. The answer is 2.” Jennifer: “How did you get 2?” I‟m not sure why she so quickly disregarded her first answer as completely wrong. I think she is just afraid of being wrong and is very concerned with getting the right answer. When I ask a question like “How did you get that answer?” she immediately assumes that she is wrong. She is not used to having to explain her thinking process of how she gets to an answer. In the past the only task she has concerned herself with is getting to the right answer and them moving on to the next problem. Angela erases all her work and starts over. She writes 3+3+3+3 = 12. She decides that the answer really is 4. Jennifer: “Show me how you figured out that the correct answer is 4.” Angela makes twelve dots, four groups, with each group containing 3 dots = 12 7 Question Samantha baked 3 dozen sugar cookies. She burnt 1/4 of them. How many cookies were not burnt? Angela first added 12+12+12 to get 36 cookies total. Then she drew 36 marks in groups of four, finding that it made nine groups. I‟m not sure why she chose to make groups of four. I think maybe because the problem contained “1/4”, and her initial reaction to this number was to create groups containing four items. This is a typical reaction from her. She sees a number and decides to use it, without really thinking about why she should use that number. After looking at the picture Angela said: “My grouping won‟t work.” She reasoned that it would work if the question used the fraction 1/3 instead of 1/4. Then she could have three equal groups, each containing twelve marks. I was impressed with this observation because she usually does not make connections like this. She tends to follow a straight path of just finding the correct answer. She doesn‟t like to talk about things that interfere with this or take her on another path. She erased all her marks and started over. She figured that each group had to have more than eight marks, but less than twelve. She guessed nine and it worked. Angela: “1/4 of the cookies were burnt, so that was nine burnt cookies.” I reminded her that the question asked how many were not burnt and she went on to count all the “non-burnt” cookie marks. I stopped her and asked if there was an easier way, a faster way than counting all those marks. Angela: “369 = 27. So the answer is 27.” Jennifer: “Is there another way to figure that out?” With some prompting she concluded that she could also do 9+9+9 which was the same as 93. 8 Lesson 2 The focus of this lesson is to present all of the colored pieces as a way of picturing fractions. The numeral 1 is pictured by a 4 by 6 white rectangle with 24 cells. Each of the unit fractions 1/2, 1/3, 1/4, 1 /6, 1/8, 1/12, 1/24 is pictured by a different colored piece. Each piece contains the number of cells that is the appropriate fraction of this white piece. A fraction that is not a unit fraction is pictured by a piece consisting of a number of parts separated by heavy lines. The number of parts indicates the numerator of the fraction and their color indicates its denominator. We also introduce the gray model for fractions. This allows us to represent both fractions for which we do and do not have color pieces. Success in using color pieces to understand fraction concepts depends on the ability to automatically associate each fraction with its colored piece. One type of activity that might be helpful is to use a variety of strangely shaped pieces, and see how they can be measured using the fraction pieces. All these shapes should use the same one-inch squares as were used for the fractions pieces. I made some of them as multiple violets to make them consistent with the fraction pieces we had made. Others had no color are markings. If possible this activity should be done before the concept or arithmetic of equivalent fractions. Jennifer only used these two shapes with Angela. I recommend using many more and returning to this type of activity until the fraction pieces are extremely familiar. While the writing style in this book is for adults, the ideas are suitable for most children by the time they are twelve. The non-arithmetic ideas of Chapter 1 can be mastered by even younger children. Most of them could also learn to do some of the arithmetic given applications to which they can relate and which only use fractions with small denominators. There is nothing magic about the size and colors we chose to use for our fractions. You can make up your own fractions and colors as long as they stay consistent. We used a 4 by 6 rectangle, but other rectangles would work well too. A 6 by10 rectangle would be good if you want to work with fractions such as 1/5, and 1/10. A 15 by 16 rectangle would be more elaborate and supply many possibilities. In working with children the use of color pieces should be supplemented with other activities. Below is an example of a type of word puzzle that some children may find interesting. You can alter these examples for different aged children. Younger children would have difficulties with this, but you could choose topics other than names of states such as names of their friends, farm animals, colors, fruits, etc. I am thinking of the name of a state. Half of the letters in this name are vowels. Half of the vowels are i‟s. Half of the consonants are s‟s. I am thinking of the name of a state. Half of the letters in this name are vowels. 3/4 of the vowels are i‟s. Half of the consonants are l‟s. I am thinking of the name of a state. Half of the letters in this name are vowels. 2/5 of the vowels are i‟s and 2/5 of them are a‟s. I am thinking of the name of a state. Half of the letters is this name are vowels. 2/3 of the vowels are a‟s. 9 Jennifer working with Angela The first activity Angela and I worked on was to make her own color pieces by using 4 by 6 index cards and crayons. The experience of making her own color pieces was important. If I had simply handed her color pieces that were already made, she would have missed the learning experience that came from making them. As she colored and cut the fraction pieces, she was learning what fraction each represented. I frequently asked her questions about the fraction pieces while we were making them. I asked her to make comparisons between different fractions. For example I asked her what other piece was the same size as a 2blue. She was informally using the concept of equivalent fractions without needing the arithmetic to do the problem. I used this time as a visual prelude to the idea of equivalent fractions. Here is the first shape I gave her. Some of the fractions Angela came up with were: 1 2 6 3 /4 /8 /24 /12. To find these answers, she laid color pieces atop this piece. She also came up with combinations: 1/6 and 1/12 1 /8 and 1/12 and 1/24 /8 and 3/24. 1 She is adding fractions but does not realize it. After she had explored this shape for a while, I asked her about her last answer. Jennifer: “Tell me about what 1/8 and 3/24 are. Angela: “This funky looking piece.” Jennifer: “Can you think of a number for that piece?‟ Angela: “Can I use the fraction pieces?” Jennifer: “Yes, please do.” Angela: “It equals 1/4. That‟s the most reduced answer, but they all are the same.” It was clear to me that Angela had seen the connection between equivalent fractions. She had made the connection that while all these numbers were the same, 1/4 was the most reduced answer. She could have answered my question by using any of the fractions she had named. Another shape I gave her was made from 8 one-inch squares. This shape did not have any of the inside lines drawn because I did not want it associated with any colors. I am also using this activity as a prelude for working on equivalent fractions. However even though Angela had done some work on reducing fractions in school I think this is the first time she understood the concepts involved. She put 4 green pieces on top of this shape and determined that this shape was 4/12 of a whole. Then she came up with other equivalent fractions without putting them on top of this shape: 8 1 2 /24 /3 /6 . I was impressed that she had confidence and didn‟t feel she had to place each equivalent fraction piece on top of this one. 10 Lesson 3 The main purpose of this lesson is to obtain a good visual picture for the concept of equivalent fractions. Raising and reducing a fraction is pictured by trading a piece for another piece of the same size but of a different color. The use of the gray model for this purpose gives another alternative. A common stumbling block in the study of equivalent fractions is that children often see no purpose in raising fractions. In traditional classrooms, students are told that they will use this concept later in their school career. This response is taken by conscientious students as something that they have to do. So they learn it, but many of them just end up forgetting it later. Less conscientious students take it as something that they don‟t really have to do and they don‟t really care about it. When introducing a new concept, students need to be involved in the learning process. When introducing new concepts, we need to find situations that students can relate to, giving real-life examples. However at the beginning of this lesson I merely pose a thought question, giving no immediate indication of any answer. Instead the lesson proceeds with these concepts but with the primary emphasis on reducing. Only at the end do I give an example of why one might want to raise a fraction. The entire next lesson is devoted to situations in which raising a fraction is important. This strategy of briefly comparing raising to reducing, but to only focus on reducing, is intended as a way to introduce an important concept in a setting in which its acquisition does not matter. Additional Remarks Both reducing and raising fractions are often called renaming. We will use the more specific terms of raising and reducing in order to stress which we are doing. Simplifying is used in some books instead of reducing, but it is also used for a variety of other processes. In order to be more descriptive, we will not use the term simplify when we want to reduce. Many books rejected the term reducing because they were concerned that students might think that reducing made the number smaller. To avoid this we stress that the fraction name is what is getting smaller when reducing. The actual number does not become smaller. This is an important idea to understand when reducing or raising fractions. As an ordinary analogy consider a boy named Johnny. We could reduce his name to John. His name became smaller, but the boy did not get smaller. We will omit the dotted lines within the pieces from now on. It is important to focus on the actual pieces instead of counting the individual cells within each piece. It is also important to become familiar with the colors and the fraction each one represents. Kay‟s focus on mathematical exactness needs to be balanced with the idea that we often use exact mathematics for situations that are not that exact. To use mathematics to model a situation we only need an approximate match that is good enough for the purposes at hand. For the others this will never be a concern. They do not feel a need for an exact match. The most powerful motivation for studying mathematics is a curiosity about mathematical concepts and their relationships. Most people can acquire some degree of pure mathematical curiosity if they experience success in doing mathematical thinking that they find challenging. This is one of the main reasons for using applications that children already have some means to solve, but which they can solve more efficiently with extended mathematical understanding. For a child like Kay there is no need to use an application as a prelude to conceptual considerations, and such children should often be challenged by purely mathematical questions. However they can and should also be motivated by applications. 11 Jennifer working with Angela In order to reinforce the habit of thinking about fractions in ordinary situations and also work on the idea of equivalent fractions I did some more observation activities. I also had her ask me questions about what we were observing. Sometimes I would purposely give the wrong answer and she would have to guide me to the correct fraction. I would give her the most reduced fraction and she would have to figure out what fraction that came from. One question came from a picture that had 2 girls and 2 boys. She asked me “What fraction of the people in this picture are girls?” I responded with “1/2 of the people in this picture are girls.” She told me I was wrong and insisted that the correct answer was 2/4. She wouldn‟t believe me until she figured out for herself that this could be true by raising 1/2 to 2/4. She then realized that I had given the reduced fraction as my answer. At this point I wanted her to visualize the relationship between 1 /2 and 2/4, rather than to do arithmetic whose rationale she did not understand. In spite of having earlier seen that a 2blue was the same size as a red, she did not visualize this until we looked at it again. Later Angela confronted a similar situation. Angela was riding in the car with her grandfather, her grandmother, her uncle, and her 2 older brothers. My grandfather asked her what fraction of the people in the car are female. She easily responded 2/6. He then asked her to give a reduced answer. When she had difficulty he asked her to think about color pieces. It took a while, but she came up with the correct answer. Granddaddy said that he thought most of her problem was focus. When working with my sister on equivalent fractions, I found that she had trouble thinking abstractly about them. She could find equivalent fractions by matching the color pieces, but when it came time to talk about the fraction names or do it with only imagining the pieces, she would become confused. She had trouble visualizing concepts in her head. She had to see it or touch it, in order for her to understand the concept. For example, I asked Angela what was equivalent to 4/4. She knew that any fraction that had the same numerator and same denominator equaled one, but when I asked her if any of the other fraction pieces were also equivalent, she had to take each fraction set and place them atop the whole piece to see if they matched. I assumed that after she did this a few times, she would tell me that all of them (2/2, 6/6, 8/8, etc.) would be equivalent to one whole. I found that especially in this instance, using the fraction pieces was a great tool for her. Following up later, I went back to these concepts to see what she remembered and found that she had no trouble with concepts after she was able to visually understand them. Some additional comment on working with Angela Jennifer and I talk regularly about her work with Angela. However I have left a large part of the responsibility to Jennifer. I wanted to see how well these ideas would work when an adult with no special preparation for teaching fractions used them with a child. One thing I observed about Jennifer‟s work with Angela came as no surprise. What Angela needs to do is to relax and use her own understanding rather that think about what is expected of her. This is the case not only in learning about equivalent fractions, but in all of her learning about mathematics. For Angela, the habits she has already acquired in school are her greatest barriers to learning. She often takes more time to learn then some of her teachers think is appropriate. She wants to be right and she worries about being scolded for not being able to do something. As a result she focuses on doing what is expected rather than on thinking. This fits with characteristics she exhibits in other situations. I have taught her a multitude of games. She seldom learns strategy in the way I would expect. So I must take care in suggesting strategy ideas too early. Often she seems to suddenly go from almost no strategy to a high level in a single jump that surprises me. With a child like Angela, provide her enough relevant experience and no external pressure and her own expectations will produce quality results. 12 Lesson 4 The primary purpose of this lesson is to provide a number of situations for which raising fractions might be useful. The questions about these situations can all be answered by manipulating color pieces. This is an important prelude providing a visual basis for the arithmetic of raising and reducing fractions, which is the topic of Lesson 5. A closely related secondary purpose is to stimulate thinking about how to apply fraction concepts to a variety of situations and to answer questions about these situations without using arithmetic. This should provide a basis for understanding all the other arithmetic calculations that will be the topics of future chapters. The more experience that a person has in using fraction concepts prior to calculating with fractions the more likely it will be that they will obtain a good understanding of why fraction arithmetic works. As you will see Angela still needs considerably more non-arithmetic problem solving. At times she is able to handle questions that she could solve using fraction pieces. Often her success depends primarily on focus. She needs to focus on imagining the situation as if it is really happening. It is also important in teaching her to expect her to think about the situation rather than about arithmetic operations. Jennifer tends to think as if she were teaching primarily as a prelude to understanding the arithmetic. While this is of some importance, what Angela primarily needs is experience in problem solving and in visualizing equivalent fractions. Sometimes children learn things from their parents or siblings without understanding the reasoning behind it. Bob is uncomfortable if he does not understand the reasoning so he pictures it with the pieces. On the other hand, Roy likes using algorithms and may not always understand what he is doing. We need different strategies for dealing with different children. For children like Bob, encourage this thinking and when they understand the pictures encourage them to use algorithms, but do not push them to do this. For children like Roy put them in situations where they feel a need to explain what they are doing. Putting Roy in a group with Bob will require him to explain his thinking. If you are working with a child like Kay, you do not have to worry about this. Just let her explore anything she wants to. Encourage her unusual thinking. For children like Jan let them think in whole numbers. Find as many fractional situations as possible that they can use whole numbers to solve. Slowly, get them to think in fractional terms as they become comfortable with these fractional situations. Jan has made an observation that was not asked for in one of the questions. This is to be encouraged. For all children, encourage them to go beyond what a problem asks for and explore many possibilities. Jennifer working with Angela Angela has not been too eager to work with the fraction pieces. I wondered if it was because she had never worked with manipulatives like these before and was not confident using them. After talking with her about this, I found that this was only a small part of why she did not want to work with them. She told me something which I found interesting. She said that she did not like using the fraction pieces because she did not like setting them up. She said it takes too long to separate the fraction pieces. (We have been storing them in one big envelope). It only takes about thirty seconds to set them up, but if this was discouraging her from using them, I thought that we should find a better way to store them. So we separated the different fraction pieces into different envelopes, thus making for easier set-up. She seemed much happier with this system and has never complained about using the fraction pieces since. Sometimes when working with children we think that there is a large, complicated problem, but it turns out to only be a minor inconvenience. Make sure you talk with the child to see what is really bothering them before you try to solve a big problem that may not even exist. 13 Question Sam‟s mom baked a cake. Sam ate 1/6 of the cake. He fed his dog 1/12 of the cake. Sam‟s dad ate 1/2 of the cake. How much cake is left for Sam‟s mom to eat? I did not draw a cake for her, hoping she could focus Sam dad dad mom with only her imagination. Perhaps a picture of a cake cut Sam dad dad mom into 12 pieces might have helped. I could then have had her mark how much each one had. dog dad dad mom Angela: Angela pulls out the pieces for 1/6, 1/12, and 1/2. “The answer is 1/12.” Jennifer: “Where did you get 1/12?” I have no idea how this occurs to her. Angela: “Oh, I‟m silly! Did she bake one whole cake?” Jennifer: “Yes.” Angela lays out a white. Over that she puts the “eaten part” 1/6, 1/12, and 1/2. Then she lays 3 greens over the “uneaten part” She says: “3/12 is what is left.” Jennifer: “That answer uses 3 fraction pieces. Is there a way so that it has less than 3 pieces?” Angela pulls out 2 pink pieces, 2/8. Jennifer: “Where did you get 2/8? What made you decide to do that?” Angela: “1,2,3,4,5,6” She counted out the units in the three green pieces. “There are also 6 units in two pink pieces.” Jennifer: “Now you have two pieces instead of three. Is there a way you can make it into one?” Angela: “I need 6 units.” She looks around at the fraction pieces and chooses 1/4. “1/4! I did it!” Jennifer: “So what does that number represent?” Angela: “The mom ate 1/4 of the cake.” Jennifer: “Who ate the most cake?” Angela lays out the pieces 1/2, 1/4, 1/6, and 1/12 and tells me the order from greatest to least is: Dad ate 1 /2, Mom ate 1/4, Sam ate 1/6, and the dog ate 1/12. At another session I asked Angela to come up with an addition or subtraction problem for me to solve using the fraction pieces. This is what she came up with: Fred had some candy. He had ½ of a piece of candy. He split the candy with 4 of his friends and it came up even. How much did each friend get? Jennifer: To clarify the question I ask, “Did Fred eat any of the ½ piece?” Angela: “No.” Jennifer: “So what happened to the other ½ piece of candy.” Angela: “That‟s what Fred ate. He was giving the other half to his friends because he wasn‟t hungry for it anymore.” Jennifer: This seemed like a division problem to me. However I feel it is best to focus on the question rather than ask her why she thinks of this as a subtraction problem. This is the solution gave. I pulled out a 4 pink to use for 4/8. “Each friend gets 1/8 of a piece of candy.” Angela: “No. Yes.” She seems unsure. 14 Jennifer: “So am I done? Is that an okay answer?” Angela: “No. Yeah, you are done.” She doesn‟t look convinced that this is the answer she wants. Jennifer: “Should I reduce?” Angela: “Yes, reduce it.” Jennifer: Hoping she‟ll see a connection, I reduce 4/8 back to 1/2. Angela: “Yes. Now raise it.” Jennifer: “Ok, I‟ll raise it back to 4/8. Is that okay with you?” Angela: “Yes, now raise it more.” Jennifer: “I could raise it to 12/24. Then each friend would get 3/24. (I do this with the fraction pieces) But why would I want to do that?” Angela: “Because I want to see if you can do it.” She likes being able to make me give answers. Jennifer: “So what would the best answer be for the question you asked? And tell me why.” Angela: “1/8 is the easiest. It is only one fraction piece. If you chose 3/24 that would be three fraction pieces for each friend. They are both the same, but 1/8 is easier because it‟s more reduced than 3/24.” Question Kim is taking a walk around town. She walks 1/6 of a mile to the bakery. Then she walks 1 /12 of a mile to the library. Finally she walks to the pet store which is 2/24 of a mile. How far did Kim walk in total? To help focus her attention, this question is presented with a simple picture. Angela goes straight to the fraction pieces and easily pulls out an aqua, a green and a 2violet. Reverting to arithmetic, she looks lost and says: “I don‟t know how to add 1/6, 1/12, and 2/24.” She has not learned the process to add fractions with unlike denominators, and she still feels she must use arithmetic rather than rely on her own thinking. Jennifer: “That‟s why we are using the fraction pieces. I bet you could use those pieces to help you figure out how to do the problem.” She lays the green and the 2violet atop the aqua and discovers that putting the green with the 2violet is the same size as the aqua. She trades for another aqua and says her answer is 2/6. Jennifer: “Can you do something with 2/6? Do you want this as your final answer?” I hope that she will be able to explain her thinking to me. She shows me again how the pieces for 1/12 and 2/24 fit on top of the piece for 1/6. She says that it is best to trade for fewer pieces. She looks unsure so I ask her if she could do anymore trading to make it an easier fraction. She quickly trades the 2aqua for a yellow. We talked about why 1/3 is an easier answer, but that 2/6 was not wrong. She seemed more comfortable with trading after doing this problem. 15 Question: Peg painted 1/2 of her house. Lin painted 1/3 of Peg‟s house. How much of the house was painted? Angela uses the fraction pieces, pulling out a red piece for 1/2, and a yellow piece for 1/3. She lays the red piece down and then places the yellow piece on top of the red piece. Then she picks up 2 green pieces and places them next to the yellow piece, on top of the red, covering the red piece with one yellow and two greens. Angela: “2/12 is the answer.” It seems to me that she has subtracted to get 2/12. I‟m not sure what made her subtract. After talking with her I don‟t think she even knew she was subtracting. You can tell from our following conversation that Angela is very unsure of her answers and what the problem is actually asking. Jennifer: “Explain to me what the problem is asking for. What could you do to solve it?” Angela: Not answering my question she says, “Oh, I see it. The answer is not 2/12” “Good” I think. She has now realized that she shouldn‟t be subtracting. Jennifer: “Why did you lay the yellow piece on top of the red piece? What does that mean? Could you lay a red on top of a yellow?” Angela: “You can also lay a red on top of a yellow.” She lays a yellow down, then a red on top of it, then another yellow on top of that. “The answer is 1/3.” Jennifer: “How did you get 1/3?” She reads through the problem, writing it out and drawing pictures. “Peg paints 1/2.” She draws a square and colors in 1/2 of it. “Lin paints 1/3.” She draws another square and colors in 1/3 of it. Next, she goes back to the fraction pieces and begins piling pieces like crazy. She lays out 4 reds, then on top of that she puts 6 aquas, then 2 blues, and finally 2 yellows. To me this looks like a big messy pile of fraction pieces. To my surprise she looks at her pile and tells me that the answer is 5/6. Now I‟m really confused. She came up with the right answer, but I have no idea how, or what process she used to get to that answer. Hoping that she will shed some light on my questions, I ask her to give another explanation. Jennifer: “How did you get 5/6?” Angela: “You add 1/3 and 1/2.” Jennifer: “You add them?!” I am excited that she finally sees this connection. Angela: “Well, I‟m not sure if you add them or not.” Jennifer: “Well, do you? Look at the problem.” Angela: “No, you don‟t add them.” Jennifer: “Why not?” I‟m not as excited anymore, just more confused. Angela: She tries to add them by writing the equation out 1/2 + 1/3. “It doesn‟t work.” I see what is confusing her now. She can‟t add them with different denominators; that‟s why she thinks it‟s not addition. Now that I have a better grasp on her thinking I ask her to try it with the fraction pieces. She places the 1/2 piece next to the 1/3 piece. Then she picks 5 aquas and makes a trade, coming up with the answer 5/6. It works! Jennifer: “So you do add?” Angela: “Yes.” 16 I think that after completing this problem, Angela has a better grasp on how to add fractions and is more aware of adding fractions with unlike denominators. Now she can do it with fraction pieces. What still confuses me is the fact that by looking at the word problem she didn‟t know that it was an addition problem. Even after she gave an answer, she still wasn‟t convinced that it was a problem dealing with addition. I also don‟t know how she came up with the answer 5/6. Did she guess? Or in her mind did she add and just not understand how she did it? To help Angela visualize fractions in another way, we worked with measuring cups and water. The measuring cups we used were: 1/2 cup, 1/4 cup, 1/3 cup, and 1 cup. We used regular measuring cups and also liquid measuring pitchers. These held up to 4 cups, and had the measurement marks all the way from 1/4 cup to 4 cups. Question You are baking a recipe which calls only for water. The ingredients are 1/2 cup water and 1 /4 cup water. How much water is used in this recipe? Angela fills up the 1/2 cup and the 1/4 cup with water. After looking at the cups for a while, she begins to write. Before I can see what she has written, she erases her work and tells me 3/4 cup. Jennifer: “How did you get 3/4?” Angela: “I figured it out in my head. I pictured the fraction pieces. I pictured one red and one blue, which is 1/2 and 1/4.” Jennifer: “So 3/4 is the correct answer?” Angela: Once again she doubts her answer. “No.” Jennifer: “Why not?” Angela: Thinking she asks herself “How many units are in a red? Twelve. How many are in a blue? Six. So that‟s a total of 18 units. A whole has 24 units, so it‟s less than one whole, but more than one half. I think it is 3/4.” Jennifer: “How can you check your answer?” Angela: She pours water into the 1/4 and 1/2 cups. Then she pours both into the 1 cup, telling me that that‟s how you check the answer. Jennifer: “That proves that it‟s bigger that 1/2 and less than 1 cup, but how do you know that it is 3/4? It could be anything between 1/2 and 1. I suggest she try using the pitchers to measure it. She pours 1/2 and 1/4 into the pitcher and reads the measurement as 3/4. It works! Question This recipe calls for 1/2 cup cold water, 2/3 cup cool water, 1/2 cup warm water, and 1/3 cup hot water. Angela pours all the “ingredients” into the large pitcher and reads the measurement as 1 1/2 cups. (She had done it correctly, however she wasn‟t very accurate in her measurement of the water, so I asked her to try it again, just be more careful.) This time it worked, giving us 2 cups as the answer. We talked about why it should be 2 cups and not 1 1/2 cups. We grouped all the thirds together and all the halves together. Angela determined that there were three thirds, which was equivalent to 1. There were two halves which also equaled 1. When you put these together, you came up with 2 cups. 17 Lesson 5 This is the only lesson in Chapter 1 which focuses on the use of arithmetic. In particular it focuses on the arithmetic way to raise and reduce fractions. For children who can easily multiply and divide whole numbers, the calculations involved are fairly simple. However a conceptual basis for these calculations is far from apparent to most children. Thus it is not unusual for a child to seem to have mastered the arithmetic of equivalent fractions and yet not be able to apply it at appropriate times. Kay‟s thinking about a fraction whose numerator is a mixed number may seem strange, but it is not unusual for gifted children when they are exposed to such ideas. If you are working with gifted children, pay attention to what Kay has to say. To keep such children interested, encourage them to expand their numerical imaginations. They can often be more easily motivated this way than through the use of ordinary situational problems. Jennifer working with Angela Question: Who Am I? I am equivalent to 2/3. My denominator is 6. Angela wrote 63 = 22 = 4. She said she was done and the answer was 4. Jennifer: “I am a fraction.” Angela erases all her work and starts over. She writes: 2 2 2 4 = then = 3 6 3 2 6 Jennifer: “Were you wrong when you wrote 63 = 22 = 4? Angela: “No, I was mostly right. I could have just put the 6 for the denominator.” What concerns me is that when Angela saw that this problem dealt with equivalent fractions, she just plugged numbers into a formula she knew. Angela has already been exposed to the arithmetic of raising fractions. She simply followed a routine process of finding an answer without really understanding why she should do it this way. I don‟t even think she realized that the process she used (63 = 22 = 4) was a method to figure out the numerator. To her she was just figuring out some number. My Comment on the use of Mathematical Language While I do not totally agree with the saying “sloppy language is a sign of sloppy thinking”, there are certainly times when this is the case. As a case in point here is one of the things Angela wrote and Jennifer excepted. This is certainly sloppy language. In this case I doubt that it involves sloppy thinking. 63 = 22 = 4 What this actually says is that 6 divided by 3 gives the same result as 2 multiplied by 2 and that this result is 4. What is intended is that 6 divided by 3 is 2 and that 2 multiplied by 2 is 4. The grammatically correct way to say this is more cumbersome than what was actually written: 63 = 2 and 22 = 4 How serious is this poor use of mathematical language? On scratch paper or in the process of solving a problem it seems harmless enough, so I would let this go at such times. However I would take it as a sign suggesting future activities on the use of mathematical language. A student who does not know the correct grammar of mathematical language will at some point misread this language. In particular they are likely to think of an expression like 1+23 naming 9 rather than 7. Many students never learn to make a totally automatic use of the order of operation conventions that 18 are so necessary for understanding equations. 19 Back to Jennifer working with Angela I asked Angela to pick a fraction and then come up with as many equivalent fractions as she could. One fraction she chose was 1/3. She came up with many equivalent fractions, such as 2/6, 3/9, 4/12 etc. I wanted to see what reducing/raising concepts Angela really understood. I wasn‟t sure if she realized that all these numbers she had come up with were really just raised versions of 1/3. So I put this to the test by asking her to reduce 4/12. At first she seemed really confused as if I had asked her to do an entirely different problem. It was obvious to me that she did not see the immediate connection between this question and what she had just been doing with the fraction pieces. After some thinking she said that the answer was 1/9. Jennifer: “How did you get that answer?” Angela: “I subtracted 3 from the top and bottom of 4/12.” I asked her to show me this with the fraction pieces, but she could not do this. So I asked her to write it out for me. She begins to write it, but then realizes that she had the wrong sign and wrong number. Angela: “I shouldn‟t use a subtraction sign. It is supposed to be division. And it should be divided by 4, not 3” She performed the traditional algorithm to reduce 4/12 to 1/3 by dividing both the numerator and denominator by 4. I asked her to show me this with the fraction pieces, since this was the connection I wanted her to make. She picked up several of the pieces and gave me a very confusing explanation, one which I couldn‟t make any sense of. She knew that it was off the wall too because she was even confusing herself. I finally had to show her how the 4/12 pieces traded for a 1/3 piece. I was really surprised that this concept confused her so much especially since she was the one who had chosen 4/12 as an equivalent fraction to 1/3. She has raised and reduced fractions before and seemed to understand it. However I think in the past she has relied on algorithms and just followed a process without understanding how it worked. My comment on helping Angela understand why the arithmetic works Angela has been exposed in school to the arithmetic of reducing fractions without understanding why it works. This was typical of Angela‟s work with arithmetic. She tried to learn what she was supposed to do, and she had never developed confidence in her own ability to just think about a question and use whatever she understood to find an answer. Most of the summer‟s work has been devoted to the understanding of concepts. The main barrier Jennifer had to overcome in helping Angela to learn was to help her rely on understanding. Without this understanding she may learn to do certain calculations with fractions, but she will quickly forget them. More important, only an understanding of concepts will give her the functional mastery that will allow her to apply fractions in appropriate situations. At the end of the summer we finally reached a point where she had a basis for understanding the concepts that relate to the arithmetic of equivalent fractions. However this understanding was not deep enough to give her confidence in her own competence. We only had one session left to help her begin to see that the arithmetic of equivalent fractions was not just some arbitrary process, but something that she could figure out on the basis of her own understanding. The final session which Jennifer next describes is a first step in this direction. 20 Back to Jennifer working with Angela I wanted to work some more with Angela on reducing by using arithmetic. We didn‟t use the color pieces in this exercise, instead we used the gray model. I gave her an example to help her understand how we were approaching these problems: RAISE BY 2 To do this, draw a line to split gray 2 piece into 2 pieces. You will get the 4 /3 raised fraction 4/6. /6 The first question I gave Angela was to raise 3/4 by 2. First she draws a 3/4. She draws a line only partly across two columns of the picture. I‟m not sure why she did this. I don‟t think she understands why we draw the line across the picture. She is just following a procedure that she saw me do. I don‟t know how to help her understand. I ask her to check her picture by using arithmetic. She checks by multiplying both the numerator and denominator by 2. 3 32 6 = = 4 42 2 8 Angela realizes her mistake. She extends the line across the entire picture. We talk about why this works and she seems to have a better understanding. I try another picture. I ask her to raise 7/8. She looks at me confused and asks, “What do I raise it by?” I tell her to raise it by two. It encouraged me that she picked up on this. She was aware of the fact that you could raise the fraction by other numbers this way. We also worked on raising fractions by numbers other than two. We reversed roles and she made up fractions that I could raise by different numbers. After she got past the first problem and figured out what she did wrong, she was able to do all the other questions correctly and seemed to have a better understanding of why it worked. 21 COMMENTARY ON CHAPTER 2 Lesson 1 In Lesson 4 of Chapter 1, we saw various situations which involved adding fractions. However no arithmetic techniques for adding fractions are given in this lesson. Instead we focus on the use of color pieces to illustrate the addition of fractions as a prelude to later lessons. This present lesson illustrates a few more such situations. The simplest way to use color pieces to add fractions is just to take the piece representing each fraction and combine them to form another piece. Unless the pieces combined are of the same color, the result will be of mixed colors. In order to make it the same color we can trade for pieces of a common color. We may need to trade only one part as we did with a red&blue which was traded for a 3blue. In many cases we must trade both parts, as we did in trading a red&yellow for a 5aqua. Trading may result in a piece that can be traded for smaller pieces to reduce the fraction. A mixed piece may represent a mixed number rather than a fractional one. Recall that a red&2yellow was bigger that a white. It could be traded for a 7aqua or for a white&aqua, depending on whether we wanted to picture the sum as an improper fraction or as a mixed numeral. A mixed numeral always has one or more white parts along with one or more parts of the some other color representing a fractional number. In elementary school students are usually taught not to use an improper fraction to name the result of calculation, but to replace any improper fraction by a mixed numeral. This suggests that a mixed numeral is a better name than an improper fraction. However there is clearly a sense in which „5/4‟ is a simpler name than „1 1/4‟, after all it is a shorter name. Whether to use an improper fraction or a mixed numeral to name a number depends on your purpose. Using a mixed numeral is useful when you want to focus your attention on a number as being between two whole numbers. Later we will encounter some cases in which naming a mixed number as a mixed numeral is more useful and others in which naming a mixed number as an improper fraction is more useful. Using color pieces is only one possible way to picture adding fractions, and the more ways a person can imagine fractions the easier it will be to understand both the arithmetic of fractions and how to apply that arithmetic to various situations. The best way to augment your perspective is to apply fractions to something in which you are often involved. For example when I am in my country home I often ride my bicycle for exercise. I have a 6 mile route shaped like the letter Y. I ride 1 mile to the branch point, take the right branch 1 mile, return 1 mile to the branch point, do the same for the left branch, then return to my starting point. Since I am riding 6 sections of 1 mile, each section is 1/6 of my ride. Riding to the end point of the right branch completes 2 of the 6 sections so it is 1/3 of my ride. Returning to the branch point is half the ride. As I approach the branch point, I often think of the 1/3 already completed plus the 1/6 I am about to complete. Thus I see 1/3+1/6 as 1/2 without using any arithmetic rule for adding fractions. While adding mixed numbers is the topic of Chapter 4 rather than of this chapter, Kay gives us a preview of this topic when she asks how far Chris goes in a week. She adds 2 1/2 and 1 2/3 to get the mixed number 4 1/6. To do this she adds the whole parts and the fractional parts. Since the sum of the fractional parts is a mixed number she represents it by the mixed numeral 1 1/6. After talking about Chris‟ exercise, Bob mentioned that Chris‟ friend Tim also likes to exercise and gave another example of adding fractions. Kay then observed that Chris got more exercise. Encouraging people to explore a situation beyond the questions asked is one of the best ways to think of mathematics as something they can apply. The more a person asks and explores their own questions, the more mathematics will seem like a way of thinking to be used and enjoyed. 22 Lesson 2 The arithmetic rule for adding fractions with the same denominator seems simple enough. Just add the numerators and keep the same denominator. Why then devote a whole lesson to this topic? I had two main reasons. (1) To understand a rule involves more than just knowing it, and even being able to use it, since the rule follows from understanding, but without understanding rules are often forgotten. (2) The more situations that a person understands that involve adding fractions with like denominators, the more easily will that person be able to extend this understanding to situations that involve adding fractions with unlike denominators. One of the main barriers to understanding the rule for adding fractions with like denominators is this rule is easy to use without understanding why it works. So what constitutes understanding? One of the main ways to understanding is to understand situations that involve adding fractions. The most common situation in which we add fractions with like denominators is when using two place decimal fractions. However since the denominator of 100 is only implicit, we may not think of this as adding fractions. In fact we usually refer to a name such as „.17‟ as a decimal rather than as a decimal fraction. The utility of 2 place decimals is so great that we even have special names and notation for using them in certain ways. Perhaps the most familiar for Americans is the cent symbol, which is used because our most basic fractional coin is 1/100 of a dollar. Another is the use of percents, a writing „17%‟ rather than „.17‟. It is easy to see that 12¢ plus 13¢ = 25¢, which in terms of dollars means 12/100+13/100 = 25/100. Of course we normally use decimal fractions when expressing sums involving money. This is especially the case when the amounts involved are mixed numbers. For example that two dollars and fifteen cents combined with a dollar and eleven cents gives three dollars and twenty six cents would be written as 2.15+1.11 = 3.26, rather than as 2 15/100 + 111/100 = 3 26/100. In general, using 2 place decimal fractions is convenient because having the same denominator allows us to ignore denominators when adding. Of course this works because of the rule for adding fractions with like denominators. On the other hand lack of the explicit denominator causes many people difficulty in understanding why our rule for multiplying 2 place decimals works. While we use decimal fractions rather common fractions when dealing with money, we also have coins whose names relate directly to some common fractions of a dollar. The coin most directly named this way is a half-dollar. We also have a quarter which means 1/4, and a dime which means 1 /10. On the other hand we have a nickel which is named for a metal, perhaps because 1/20 is a fraction concept people found less comfortable. While it is easy to think of situations for adding decimal fractions, the situation for adding fractions that come most easily to my mind usually involve different denominators. I must make a special effort to think up natural situations involving like denominators. For example, the Y shaped cycling route mentioned in the commentary for Lesson 1, contains parts I think of in terms of halves and thirds and sixths. Hence I thought of my first return to the branch point as related to the fraction sum 1/3+1/6 = 1/2. However I also think of this route as having 6 basic parts, each being a sixth of the route. From this perspective I would relate this situation directly to 2/6+1/6 = 3/6. I still probably would then think my ride as 1/2 finished. I also know midpoints in each basic part of my route, so I also think in terms of twelfths. In particular I often think of the having gone 11/12 when I reach the midpoint of the last midpoint on my way back home. Another time I think in terms of fractions with like denominators is when walking. For me 60 paces is 1/16 of a mile, 120 paces 2/16 of a mile, 360 paces 3/16 of a mile, etc. There are various segments that I walk and have paced off in these terms. 23 Lessons 3 and 4 While the rule for adding fractions with like denominators is easy to remember, many people have trouble remembering how to add fractions with unlike denominators. I recall an experience from many years ago when I was teaching seventh grade mathematics. A student told me that his teacher from the year before taught him to add fractions by adding their numerators and adding their denominators. I found this extremely implausible, and told him he was confusing this with the rule he was taught for multiplying fractions. However I could not convince him. I tried to point out that this rule would mean that 1/2+1/2 was 2/4, indicating how this could make no sense. All to know avail. He did not add numerators and denominators when they had the same denominator, but only when the denominators were different. He was sure that his memory was correct, and since he had never understood the rules he was using, the absurdity of this rule did not contradict his memory. He had a rule that was easy to remember and he did no want to give it up merely because it made no sense. Why worry about making sense when he already had an entrenched idea that mathematics did not make sense. While I always understood mathematics and tried to teach for understanding, this was my first year teaching in a classroom. Before this I had only taught individuals who could easily expand their own understanding by attending to my presentation of ways to think about the ideas involved. At that time I had not yet learned strategies for helping persons who thought of mathematics primarily as a set of algorithms. A major barrier to understanding the procedure for adding fractions is the inability to correctly read the mathematical expression for fraction sums. A person who uses algorithms without understanding why they work may write the equation below. This person sees a plus sign and just adds without regard to meaning, and is not even aware of the fact that 8/12 is smaller than 3/4. 3 /4 + 5/8 = 8/12 One way to counter this is to relate this fraction sum to situations that can be thought of in terms of adding fractions, and when possible to discuss them with others. For example, someone mixing 3/4 a cup of grape juice with 5/8 of a cup of cherry juice would easily see that 8/12 was clearly wrong. A person like Jan, who likes whole numbers, thinks of a cup as 8 ounces. Jan would say that combining 3/4 of a cup with 5/8 of a cup gives 13/8 cups, since this is the same as 6 ounces plus 5 ounces which gives a cup and 3 ounces. Jan might also think in terms of other kinds of measures. 3 /4 a foot is 9 inches, while 8/12 of a foot is only 8 inches. Likewise 3/4 of a year is 9 months while 8 /12 is only 8 months. These show that the answer of 8/12 is wrong. However 3/4 of a year plus 5/8 of a year is 9 month plus 7.5 months, which is a year and 4.5 months. So unless you think of 1/8 of a year as 1.5 months, this may not be an easy way to think of 3/4+5/8. There are a multitude of situations that persons can use to enhance their thinking about fraction addition and other fraction concepts. I recommend cultivating the habit of thinking about those situations that one routinely encounters in terms of fractions, and especially those that you can easily relate to because of your interests or for any other reason. As you might imagine from my earlier comments, the situations that I use on a routine basis are those involving my aerobic exercise. I mentioned my cycling route. I also walk even more often than I cycle, and with more varied routes. I know that 120 paces is 1/16 mile, 240 paces is 1/8 mile, 360 paces is 3/8 mile, etc. Thinking about fractions of a mile is not something I do to enhance my understanding of fractions. However my understanding of fractions is one of the tools helping me carry out my exercise program. Of course the situations people relate to will differ. Choose those that you find helpful. 24 COMMENTARY ON CHAPTER 3 Unlike the commentary for Chapters 1 and 2 this commentary is very brief and is not separated by lessons. One reason for this is that subtraction is so closely related to addition that that commentary covers much of what we need to say about subtracting fractions. Early in the first lesson we stress the fact that every subtraction equation has an addition partner. Thus it should not be surprising that we subtract fractions with like denominators by subtracting numerators and that we use common denominators when subtracting fractions with unlike denominators. The focus in this present commentary is on the concept of subtraction and its relationship to the concept of addition. As remarked at the beginning of this chapter, most of us first learned to associate subtraction with the concept of removal. That is when starting with any quantity and taking away some of it, we can use subtraction to determine the remainder. Using removal situations that involve small counting numbers helps children learn to conceptualize subtraction in relation to counting down. Since counting down is easily related to counting up, this provides a way of relating subtraction to addition. While this relationship may seem fairly simple once understood, it does not come automatically to everyone. Altho the use of subtraction for thinking about removal applies to measuring numbers as well as to counting numbers, counting down is not the most useful way to think about having 2½ gallons of paint and using ¾ a gallon to paint a room. Of course we could think of this as having 10 quarts and using 3 quarts, and count down to 7 quarts. Picturing this as a 2white&red take away a 3blue is more useful for various reasons. In particular it applies to situations in which counting would be much more tedious. Furthermore the use of colored pieces allows us to focus on the relation between addition and subtraction without being distracted by counting or any other calculating type details. We also learn to use subtraction in situations where we want to make a certain type of comparison. Suppose the evening show cost $6, but the early show only costs $4. We might say that the early show costs $2 less than the evening show or that the evening show cost $2 more than the early show. We might say that the difference between these prices is $2. The use of subtraction for thinking about comparisons also applies to measuring numbers as well as to counting numbers. However the use of subtraction in comparison situations seems less natural to many children than its use in removal situations. Asking how much more does the evening show cost a child may think of addition rather than subtraction. In fact this is the way I usually think about it, since I would probably think of adding 2 to 4 to get 6 rather than of subtracting 4 from 6 to get 2. Telling me that I had to at least unconsciously do this subtraction to know I needed to add 2 is not convincing. I am so familiar with the addition fact, that when I am asked “4 plus what makes 6?”, the number 2 just pops into mind. On the other hand asking how much less does the early show cost dose suggest subtraction to me, as does asking what the difference between them is. Of course I cannot give any absolute guarantee I am not subtracting when asked about how much more, since I learned almost all my subtraction facts from their addition partners. The main thing to remember about a how-much-more question is that adding is one way to obtain the result and use this to relate addition to subtraction. If asked how much more is 102 than 75, we can add 25 to 75 to get 100 and then add 2 more to get 102. Thus anyone who does not know how to borrow and still gets 27 may be using this or some other way of adding to go from 75 up to 102. 25 COMMENTARY ON CHAPTER 4 Lesson 1 A mixed number is a measuring number that is larger than 1 but which is not a whole number. It is important to note that the term „mixed number‟ refers to a type of number rather than to a type of name. A mixed number can be named in various ways. The focus of this lesson is on the fact that we have two main ways to use fractions in naming mixed numbers. We can use a proper mixed numeral consisting of a name with a whole part and a proper fraction part. Using a proper mixed numeral makes it easy to see which whole number is closest to that mixed number. Perhaps this is why a mixed numeral is sometimes thought of as the preferred name for a mixed number and why they are called mixed. However we can also use an improper fraction to name a mixed number. Thus fractional number, counting number, mixed numbers all have fraction names; and from a the perspective of having a uniform naming scheme improper fractions are more basic than mixed numerals. The important thing about having more than one type name for a number is being able to change from one type to another and to recognize which type name is more useful for the purpose at hand. Note that we can even have a mixed numeral whose fraction part is improper, and that there are times when this may be convenient. In order to focus what fraction a piece was being used to imagine, we have kept our white and our fraction pieces the same size. This can be inconvenient when the numbers are larger than one, so we often use smaller versions of a white. It is the relationships of fractional parts to a whole that matters, rather than the actual size of the parts and the whole. There is also nothing special about using a 4 by 6 rectangle for our whole. It was chosen merely because it had a number of convenient fractional parts. In the gray model, the size and shape of the rectangle representing one whole varies in different situations. You can choose whatever you want to represent one whole. A fraction is always understood in reference to a whole and what constitutes the whole in a situation depends on your perspective. In the incident below, I was thinking of the two mile route as the whole, but Angela first answered as if the whole was a mile. One of the routes Angela takes is two miles long. She runs one mile out and then runs one mile back. Halfway on the way out, there is a pole. When she reached the pole on the way out I asked her how much of the route she had completed. She said that she had run ½ mile. I told her that this was true, but I wanted to know what part of the route had she completed. We discussed briefly that she had run ¼ of the whole route. She said she understood, but I was not sure she had fully grasped this concept. On the way back, when she reached the pole again she volunteered, “We have ¼ of the route left.” Then I knew she understood that she had run ¼ of the route and ½ of a mile and that it could be thought of either way. Notice that the pictures on page 2 that were used for 16/4 are arranged in a different ways. The size of the fractional piece is what is important, not the positioning of the picture. We need to be able to recognize the essence of a picture without being fixated on whether it looks the same each time. Just like the size of a whole varies, our pictures can change arrangements. We use what is convenient. For example, if we need more room on the page, we can shrink the whole picture. 26 Lesson 2 In a situation in which we want to add mixed numbers we can either name them using mixed numerals or improper fractions. For example suppose one of the numbers can be named either as 7/3 or as 2 1/3, and suppose the other can be named either as 9/2 or as 4 1/2. 7 We can add improper fractions just like we add proper ones: /3 + 9/2 = 14/6 + 27/6 = 41/6 To add using mixed numeral, add the whole parts and add the fraction parts. 2 1/3 + 4 1/2 = 2 2/6 +4 3/6 = 6 5/6 In most situations in which we might want to add mixed numbers these numbers have been named using mixed numerals rather than improper fractions. Furthermore we often want to think of the sum using a mixed name. This is why we haven‟t given any examples of adding improper fractions in the text. Kay sometimes uses improper fractions for the sake of variety and because she likes to explore ideas. This is unlikely to impress any of our other characters because they don‟t see any reason to do it the way Kay does. Their way is much faster. Usually Kay will also use mixed numerals, but at other times she will do it a more complicated way just for the fun of it. While adding mixed numerals merely involves adding the whole parts and adding the sum of the fraction parts might be larger than 1. Situations of this type are considered in Lesson 4. You might want to think about this prior to that lesson For example adding 5½ and 3¾ gives 9¼. However the sum of the whole parts is 8 and the sum of the fraction parts is 5/4. As we noted in the chapter on subtraction, we normally view difference problems as subtraction problems, but children often see them as addition problems. Suppose Ellen has 2 1/6 pounds of candy and Jane has 3 1/2 pounds of candy. One way to see how much more candy Ellen has is to start adding to 2 1/6 until we reach 3 3/6. First add 1 to get 3 1/6. Then add another 2/6. By adding we can observe that Jane would need 11/3 more pounds of candy to have as much as Ellen. If asked how much less candy Jane has the answer could also be obtained by addition. However the phrase „how much less‟ is more likely to suggest subtraction. This is perfectly okay; it is part of the addition- subtraction relationship, and such examples can be taken as an opportunity to examine this relationship. This can be reinforced with examples involving numbers for which the arithmetic is easy. Suppose Peter has earned 4 gold metals and wants to win a total of 7. How many more does he need to win to reach his goal? While most adults would look at this situation using 74 = 3, many children would think about it by using addition. They would begin at 4 and count their way up to 7 by adding on numbers, figuring out that it would take 3 more medals. As with adding mixed numerals, we gave situations in which we could subtract mixed numerals by subtracting the whole parts and subtracting the fraction parts. We did not indicate what might be done in a case like 5½ 3¾ where the fraction part of the number to be subtracted is larger than the fraction part of the number which it is subtracted from. However since adding 1¾ to 3¾ gives 5½, the answer should be apparent. This topic is also part of Lesson 4. 27 Lesson 3 While most situations involving division by fractions will not be explored until chapter 6, we do focus on division of mixed numbers by whole numbers in this chapter. Our main reason for doing so at this time is to illustrate why it might be useful to change a mixed numeral to an improper fraction. Sometimes it is easy to use a mixed numeral when dividing a mixed number by a whole number. Other times it can be messy. For example, if you wanted to divide 4½ by 2, it would be easy. Just divide 4 by 2 and ½ by 2. Thus 4½ divided by 2 equals 2¼. However if you had something like 3½ divided by 2, you would get 1½ and ¼. You could then change this to 1¾. Unless the whole part of the mixed numeral is divisible by the whole number, using improper fractions will be easier. As remarked earlier the more situations you can think of in terms of fractional numbers the stronger your net for thinking about these numbers will become. The same comment applies to mixed numbers. My wife was baking cookies. Her recipe called for ¾ cup of shortening. She wanted to triple the recipe. She had a package containing four of one cup sticks. Her old habit would have been to use 3 sticks and take 3/4 of each. However since we had been talking about the fractions book, she tripled 3/4 to get 2¼. It is not that she did not already know how do this, it was more that she never developed the habit of thinking that way. The advantage was minor. However instead of having 3 small ¼ partial stick around we only had a single ¾ stick to store. Had she wanted to quadruple the recipe her old way would have left 4 small partial sticks instead of one unopened stick. Situations involving ratios provide another example in which we often think in terms of improper fractions. One of the main reasons is that using improper fractions make opposite comparisons easy. by merely using reciprocals. A company which has 11/7 ratio of factory to office workers has a 7/11 ratio of office to factory workers. Had we said there are 14/7 times as many factories as office workers, we would probably have changed 14/7 to 11/7 to determine the ration of office to factor workers. The reason that we get different fractional numbers for ratios in the same situation is that we are thinking about different quantities as our whole. In comparing ratios of adults to children, we are using the children to represent the whole. In comparing ratios of children to adults, we are using the adults to represent the whole. This is why we get two different numbers. Sometimes we also want to compare adults and children to the total number of people. Which fraction we use depends on our focus. Unlike raising or reducing fractions, we can not change fractions to reciprocals. The reciprocal of a fraction names a different number than the fraction. You can raise fractions to get a different name for the same number. A reciprocal gives a whole new fraction; this is an entirely different idea. It finds a name for a different number that can be used to think about as ratios. However ratios are only one situation where this is a useful concept. Jan‟s answer to the probability gives one possible way to have those ratios. Doubling all the marbles would still give the same ratios. For example, if you doubled all of them you would have a total of 30 marbles and 10 white marbles. This would still give the same answer of 1/3. 28 Lesson 4 An improper mixed name is one in which the fraction part is an improper fraction. We saw two types of situations in which improper mixed names might arise. When adding mixed names by adding the whole parts and adding the fraction parts, the sum of the fraction parts may be greater than 1. Sometimes the fractional part of a number we want to subtract is larger than the fractional part of the number we are subtracting it from. When this happens it may be convenient to change the number we are subtracting from to an improper mixed name. In the next chapter we will see another type of situation in which improper mixed names may arise, namely in multiplying a mix number by a whole number. For example using color pieces it is easy to see that we can find 7 times 2¾ by multiplying 7 times 2 and 7 times ¾ to obtain 14 21/4. This can then be changed to the proper mixed name 19 ¼. The use of improper mixed numbers in adding is like carrying when we add whole numbers. All of the improper mixed names we encounter by adding a pair of proper mixed names have a fractional part that is less than 2. Thus when adding, we added the fraction parts and wrote the excess over 1 and then increased the sum of the whole part. The use of improper mixed names in adding merely helps stress why this works. In a case where we are taking the sum of more than two mixed names, the sum of the fractional parts may be larger than 2. In such cases we need to increase the sum accordingly. The use of an improper mixed name merely breaks this into two parts. To find the sum below we could add the fraction parts write 1/8 and carry 3 to add with the whole parts, omitting the improper mixed name 13 25/8. 2 7/8 + 4 5/8 + 2 7/8 +5 6/8 = 13 25/8 = 16 1/8 Of course there are other ways to find this sum. We could add the first pair to obtain 7 4/8, then add 2 7/8 to this to obtain 10 3/8, then add 5 6/8 to obtain 16 1/8. Note that we would leave the first partial sum as 7 4/8 rather than reduce it to 71/2. The use of improper mixed numbers in subtracting is like borrowing when we subtract whole numbers. For example thinking of 6 3/10 2 7/10 as 52 plus 13/10 7/10 uses the same idea as we use when we think of 6327 as 5020 plus 137. Using decimal notation, the similarity is even more striking. We think of 6.32.7 as 52 plus 1.3.7. Kay‟s use of negative fractions is one way to avoid changing the number we are subtracting from to an improper mixed name. However the resulting difference will be a different type of improper mixed name, one whose fraction part is negative. 6 3/10 2 7/10 = 4 4/10 = 3 6/10 Historically, negative numbers were seldom taught prior to teaching fractions. Thus this is not a method that you are likely to encounter in most books on fractions. However this is analogous to what a few young children who know about negative numbers when subtracting whole numbers. 6327 = (6020)+(37) = 404 = 36. 29 COMMENTARY ON CHAPTER 5 Lesson 1 Some problem that students encounter is due to their limited concept of multiplication as repeated addition. While we do not add when we multiply by 1, the result is so simple that few students think about the fact that multiplication by 1 is not conceptualized in terms of addition. A similar remark applies to multiplication by 0. The need for a concept of multiplication that goes beyond the idea of repeated addition is essential if the student is to understand multiplication by fractions. The focus of this first lesson is on presenting such an expanded concept. The essence of this concept is the idea of an action on some quantity. Whenever you think about multiplication think of the first factor as acting on the second to produce some product. When the working factor is a fractional number the action is to take some of it. For instance to multiply by 1/3 we take 1 of 3 equal parts. To multiply by 2/3 we take 2 of 3 equal parts. Reconsider multiplication by 1 or 0. While such products do not involve repeated addition, they can be thought about in relation to the concept of taking. To multiply some quantity by 0 we take none of it, rather than part of it. To multiply it by 1 we take all of it, rather than part of it. This taking concept can be applied to multiplication by other counting numbers. If there are several 5 pound sacks of flower, how many pounds of flour will you have if you take 3 of them? From this expanded perspective the fact that multiplication can sometimes be conceptualized in terms of addition is a special case of taking some of a quantity, i.e. where we only take a whole number of this quantity. We have focused on multiplication as a product with a working factor and a second factor being worked on. We do so because this is a basic way of thinking about multiplication. It also is a way of thinking that applies nicely to a variety of situations. It especially applies to those in which the first factor is thought of as a pure number and the second factor is thought of as a quantity of some type, and thus the product is also of this type. For example taking ½ of $8 produces $4, and while the second factor and the product are thought of as dollars the working factor of ½ is thought of as a pure number rather than as half a dollar. There are cases in which we multiply quantities of different type to obtain a product which differs from both. A common example is when the first factor is a rate. For example rate time produces a distance. If a turtle‟s rate is ½ mile per hour for 8 hours we use „½8 = 4‟ to show it goes 4 miles. Here ½ is in miles per hour. This is a rate rather than a time or a distance. Thus the product is a different type of quantity than either the first or second factor. This would not be the case if we said a turtle went 8 miles and wanted to know how far he had gone at the halfway point. This would be a case when the working factor of a half was a pure number. Other rate situations include a yearly rate of interest times the number of years to produce amount of interest, hourly wage rate times hours worked to produce the amount of wages paid, etc. Early in this lesson Kay say‟s that the product of 3 and 2/3 is 6/3. In a traditional setting she would be expected to change 6/3 to 2. However this is not relevant to the point she is making, and in a constructivist setting her comment would not be challenged by the teacher. Her answer does not satisfy Bob, so he changes it to 2. Like many students, Bob seems most comfortable when he feels that there is a standard form for giving answers. In order to encourage flexible thinking, I might ask Kay why she was content with 6/3 and ask Bob why he preferred 2. In ordinary language we talk about dividing something in half when we want think about taking half of it, perhaps because the idea of taking half of is so closely related to dividing by 2. However this way of using language might be part of the reason that students sometime think that taking half of some number is to divide it by 1/2 rather than multiply it by 1/2. 30 Lesson 2 The rule for multiplying a fraction by a whole number is to multiple its numerator by this number. This rule so easy to use that you may wonder why we spend a whole lesson on multiplying a fraction by a whole number. One reason is to stress the idea of multiplication as an action with the first factor working on the second factor to produce a product and to focus on how this can be imagined with fraction pieces. A good understanding of this will provide a basis for products when the working factor is not a whole number. Another reason to focus on multiplication by a whole number is to illustrate in detail how a rule might be discovered. Note that our students were instructed not to reduce fractions or change improper fractions to mixed numerals. Making such changes will make the rule harder to observe. This instruction would be unnecessary for students that were more flexible in their thinking. For example, Kay would have felt no compulsion to reduce or make such changes. She makes them only when serves her purposes. Roy is rule and answer oriented, and since he looks for answers given in standard form, he is less likely to make discoveries. Jan and Bob are more flexible in their thinking. However they are as not sure of their abilities to think independently as is Kay. The product rule for multiplying a fraction by whole number can be applied to improper fractions as well as to proper ones. The result will also be an improper fraction. Roy noted that we can multiply a whole number times a mixed numeral by multiplying it by the whole part and the mixed part. Kay then noted while it is easier to multiply many mixed numbers when named as mixed numerals rather than as improper fractions, we may obtain an improper mixed numeral. Such a name can then be changed to a proper one. For example, 37/5 and 7 2/5 name the same number. To multiply 9 times this number using its mixed numeral, we take 9 times 3 and 9 times 2. 2 7 2/5 = 63 18/5 = 66 3/5 To multiply 9 times this number using its improper fraction name, we must multiple 9 times 17. 9 37/5 = 333/5 Roy‟s brother Jim has learned about multiplication of fractions in a traditional classroom. Like Roy, Jim is good at learning algorithms, but he feels no need to understand why they work. So he teaches them to Roy who also has no idea why they work. As Kay says, to understand what Roy is doing these students need to first understand more about multiplying fractions. We will discuss what Roy calls cancellation in Lesson 4. 31 Lesson 3 Although the arithmetic for multiplication by 1/2 is fairly simple, we focused on using 1/2 as our working factor because it illustrates some important basic concepts. In lesson 1 Bob‟s Method of finding 1/2 times 1/6 was to raise 1/6 to 2/12 and then take half of the numerator. Using Roy‟s Rule we would multiply the denominator of 1/6 by 2. Both results give 1/12 as the answer to 1/2 times 1/6. Since a fractional number always has a fraction name with an even number we can always use either Bob‟ Method or Roy‟s Rule when the working factor is 1/2. Using Roy‟s Rule we can find half of 3/4 by multiplying the denominator by 2: /2 /4 = /8. 1 3 3 Using Bob‟s, we can find half of 3 /4 by raising raise 3/4 to 6/8, and taking half of the numerator. The chained equation for this is: /2 /4 = /2 /8 = /8. 1 3 1 6 3 For fractions with odd numerators Roy‟ Rule is clearly faster, and it follows the more general rule taught in traditional classrooms. However Bob‟s Method is more directly related to pictures that can be used to imagine these types of products. Furthermore Bob‟s Method is closely related to the shortcut Kay mentioned for finding half of a fraction whose numerator is already even. For such fractions we could also use Roy‟s Rule and then reduce. Using Roy‟ Rule to take half of 6/7 we obtain a fraction that is not reduced: /2 /7 = /14 = /7. 1 6 6 3 Using Kay‟s shortcut we could just take half of the numerator, omitting the 6/14. After discussing further examples using ½ as the working factor we extend the ideas to products when the working factor is some other unit fraction. Since such products are often fractions for which we have no color pieces, we use the gray model. Our students examined some products of ½ times mixed numbers. Naming the second factor as an improper fraction you can multiply it by ½ in the same way you can multiply proper fractions by ½. However they discover a shortcut. You can take ½ of a mixed numeral with an even whole part by taking half of the whole part and half of the fraction part. Kay then observes that you can change a mixed numeral with an odd whole number part to an improper mixed numeral with an even whole number part, and so the same idea can be used. Their ideas can be extended to finding products of any unit fraction times a mixed number. Can you see why the use of improper fractions might seem simpler when the working factor is not ½? Roy has seen that to multiply a fraction by ½ we can multiply its denominator by 2. He has also observed that to multiply a whole number by ½ we can divide it by 2. He does not appear to make any connection between these two ideas. 32 Lesson 4 The main focus of this lesson is not only to discover a rule for multiplying fractions, but to understand why it works. However the product rule for multiplying fractions is easy to learn and remember without understanding. Merely see a multiplication sign between two fractions and multiply the numerators and denominators. Why then does a person need to understand any rationale for this rule? The main reason is that the use of any mathematical procedure rule without understanding has at least three undesirable consequences. It reinforces an attitude that learning mathematics is a matter of being able to carry out procedures rather than a matter of understanding concepts. Closely related to this is that learning procedures without understanding concepts usually results in an inability to recognize when and how to apply the procedures to relevant situations. Furthermore while some procedures are easy to remember without understanding why they work others are not, and those that are not easy to remember are then easily confused with those that are. To indicate the potential for confusion when memory is used without understanding we consider the ten procedures below, half of which are correct and half of which are not. (0) To add two fractions, add the numerators and add the denominators. (1) To add two fractions, obtain a common denominator and then only add the numerators. (2) To add two fractions, obtain common denominators and add the numerators and add the denominators. (3) To add two mixed numerals, add the whole parts and add the fractional parts. (4) To add two mixed numerals, change them to improper fractions and add the numerators and add the denominators. (5) To multiply two fractions, multiply the numerators and multiply the denominators. (6) To multiply two fractions, obtain a common denominator and then only multiply the numerators. (7) To multiply two fractions, obtain common denominators and multiply numerators and multiply the denominators. (8) To multiply two mixed numerals, multiply the whole parts and multiply the fractional parts. (9) To multiply two mixed numerals, change them to improper fractions and multiply the numerators and multiply the denominators. To understand the use of common denominators for adding may seem essential because it is not uncommon for students to learn to add fractions using (1) and later slip into using (0). For students who rely more on memory than on understanding, this often happens after they learn procedure (5) which seems much easier to remember. Occasionally someone may even remember they are supposed to get common denominators, but not understanding why, they will use (2). I do not recall having observed anyone forgetting (5) and instead using (6) or (7). Of course (7) is correct but, unless the fractions already have the same denominator, it is more tedious. It seems that even persons who rely on primarily on memory are more likely to think they remember a procedure that is simpler than the one they forgot rather than one which is more complicated. While more people have a tendency to use incorrect procedures for adding fractions than for multiplying them, the reverse is the case for working with mixed numerals. For mixed numerals adding correctly by using (3) is simpler than adding incorrectly by using (4). On the other hand, multiplying incorrectly by using (8) is simpler than multiplying correctly by using (9). 33 When finding products of mixed numbers, Roy‟s tendency to want easy rules leads him to just multiply the whole parts and fraction parts when these numbers are named using mixed numerals. Once he sees that this does not work he decides that he will instead always use improper fractions and the product rule. Roy is not interested in thinking about alternative procedures. He sees no problem with his “one way of doing things” attitude, and since Roy can conceptualize fairly well within his own way of thinking, he does well with standard procedures most of the time. He also seems to develop an intuitive understanding of why many of them work, altho he has no inclination to articulate this. Thus it is hard to motivate him to examine alternatives. A student who relies on algorithms but has almost no understanding of why they work is much harder to motivate. We have not included such persons in our cast of characters. Instead our other characters are fairly flexible. Jan and Bob want any shortcut that helps them find answers with less work. This is the main way that they can be motivated to look for alternatives and expand their conceptual perspective. Kay wants to find alternatives primarily because she finds them interesting. The best way to motivate her is by responding to her ideas and suggesting others she has not considered. Persons like Kay sometimes dislike math because they find it easy but boring. Doing a multitude of routine calculations feels like a mindless and pointless activity. This is not to say that they do not need to do a fair amount of calculating to reinforce their understanding. However practice with calculations needs to be spread out, and is best done in connection with questions that they find interesting. Most of the situations in which we used multiplication had a working factor which was a pure number. In the commentary for Lesson 1 we also mentioned those in which the working factor was a rate. An example of this is given near the end of Lesson 4. We also use multiplication of quantities of the same type to produce a quantity of a different but related type. For example our 4 inch by 6 inch pieces for a white is composed of 24 squares. The sides are measured as distances in inches, but the product is measured as an area in square inches. The use of a mixed number product for computing an area gives a visual way of understanding the rationale for using 4 partial products to find the product of two mixed numerals. Consider a rectangle with sides of 2½ and 1½. 2 ½ The area of each part is its length times its width. The area of the top left part is 21 = 2. 1 2 ½ The area of the top right part is ½1 = ½. The area of the bottom left part is 2½ = 1. The area of the bottom right part is ½½ = ¼. ½ 1 ¼ We have used the term „factorization‟ instead of „cancellation‟, primarily because some people use cancellation without understanding why it works. The term factorization‟ is intended as a reminder of why it works. While factorization can save some time in calculation, the amount saved when working with fractions having small numerators and denominators is minimal. Given the tendency to use this rule without understanding and the resulting errors that occur, this method could be skipped if the only reason for it was to save time in calculating. However there is a significant conceptual reason for understanding why factorization works. It shows a relation between the product rule and reducing fractions as this relates to changing the order of operation, and the idea of correctly changing the order of operation is a major concept in mathematics. 34 COMMENTARY ON CHAPTER 6 Lesson 1 The main focus of this lesson is on the types of situations to which we apply the concept of division. We referred to them as dividing to share, dividing for size, and dividing to compare. Before looking at how these ideas relate to dividing with fractions, we first examined these ideas as they related to dividing with whole numbers. Here is some further discussion of dividing for size and dividing to share with whole number examples. Even before people had names for numbers they could divide 150 sheep equally to share with 3 shepherds. They would merely separate them by giving one to each shepherd, then another to each shepherd, etc. However to divide 150 sheep to see how many groups of size 50 can be obtained, they would need some way to represent 50, perhaps the ability to count to fifty either symbolically or with objects such as stones. First remove 50 sheep, then 50 more, etc. This is still the way children first solve problems of dividing to share and dividing for size. To physically divide 8 padlocks to share with 2 people we start with 8 padlocks or tokens for them. First, send one to each person. Send one to each person. Send one to each person. Send one to each person. To obtain parts with size 4 keep sending groups of 4 padlocks until none are left. Send 4 into a pile which gives 1 pile and some padlocks left. Send 4 more into a pile giving 2 piles and none left When dividing with whole numbers, we can imagine many situations that involve dividing to share and many situations that involve dividing for size. However they are not always stated this way. The example below is of dividing to share because we are sharing 500 envelopes equally in each the 5 one minute intervals. In dividing to share we think of dividing envelopes by 5 to obtain envelopes, but our divisor is minutes rather than envelops. In general when we divide to share, the dividends and the quotient are of the same type but the divisor is of some other type. To share: 500 envelopes sealed in 5 minutes, do 100 envelopes in each minute: 5005 = 100. 35 We can also divide envelopes for sizes. Below we have specified the size job for each minute and are looking for how many jobs of this size. Here we divide envelopes by envelopes and the result is how many minutes. In general when we divide for size the divisor and dividend are of a similar type but the quotient is not. For size: 500 envelopes sealed at 100 envelopes each minute, taking 5 minutes: 500100 = 5. Using ordinary language, we might describe dividing 500 objects for size 100 as dividing 500 objects into groups of 100. We then talk about the arithmetic of how many 100s go into 500. Using the phrase „dividing into‟ in these situations can cause confusion for some children. This is why we recommend initial use of the phrase, „dividing for size‟ instead of „dividing into‟. Once concepts are well mastered the duality in the use of „into‟ should be clear from context. As remarked in an earlier lesson, since dividing to share with whole numbers does not always result in a whole number, this is one of the main reasons for conceptualizing measuring numbers. While we can physically divide 2 pounds of food almost equally among 3 dogs, we cannot describe this in numerical terms unless we use fractional numbers. Dividing 1½ pounds of food among 3 dogs is physically no more difficult than dividing 2 pounds among them. Just separate the food into 3 parts. Numerically, dividing a measuring number to share uses the same idea as dividing a whole number to share. To divide 1½ to obtain 3 equal part, merely think of it as 3/2. However even when the dividend and quotient are not whole numbers, most ordinary situations of dividing to share involve a divisor that is a whole number, and all the examples of dividing given in the text are of this type. Kay‟s sister has some examples in which we divide to share with a divisor that is not a whole number. However these examples may seem somewhat artificial. Dividing 1/4 gallon of paint among 1/8 gallon containers we need 2 such containers. Dividing 20 minutes between 10 minute periods gives 2 periods. This is the same as dividing 1/4 of an hour between periods of 1/8 of an hour. When the type of quantity for our divisor can be measured using measuring numbers, we can divide to share among any amount of this quantity. We can even imagine a situations in which we divide a mixed number between a mixed number and get a mixed number quotient. Bo has several 2½ gallon gasoline cans. He buys 11¼ gallons of gasoline. He can share these between 4½ how many cans Why? Two cans will hold 5 gallons, so we need 4 cans to hold 10 gallons. This leaves 1¼ gallons. He will only half fill the fifth can. Jo has 2 large dogs and one small dog. She gives a full portion to the large dogs and a half portion to the small dog. This amounts to 3¾ pounds of dog food. To find the size of a portion, divide 3¾ by 2½. We see that a portion is 1½ without knowing the standard arithmetic methods of dividing mixed numbers, which we introduce in Lesson 4. Think of 3¾ as a 15blue. We want to separate this into 2 parts of the same size and one part of half that size. To do this, use a 6blue and a 6blue and a 3blue. Since a 6blue represents one portion the size of a portion is 1½. 36 In many ordinary situations that involve dividing for size with whole numbers we want the quotient to also be a whole number. For example with 24 people and tables that can seat 6 people, I need 4 tables. Of course, in such a situation the number of people may not be a multiple of 6. With 29 people, we cannot divide them into a whole number of groups of 6. I could say that I need 4 5/6 groups of people, but I am more likely to say we have 4 groups of 6 and 5 people left over. In situations involving whole numbers where we want to divide for size, we often think in terms a whole number quotient and whole number remainder. This is partially because of the fact that in this situation I do not think of fractional tables. This is even the case when dividing measuring numbers for size. If Jo had 4½ pounds of dog food and needed 1½ pound portions, she has enough for 3 dogs. With 5 pounds she has enough for 3 dog with a ½ pound extra. Since ½ pound is 1/3 of a portion we could say she has enough for 3 1/3 dogs, but this is not the ordinary way of talking about this situation. However with 5¼ pounds we obtain a quotient of 3½, and this could be interpreted as enough for 3 large dogs and a small one. When dividing for size our divisor may be any measuring number, but often it is smaller than the dividend. If not our quotient will be a fractional number. Since we are thinking of it as indicating how many parts, this may seem either artificial or even impossible for many children. The responses to the question below are all correct, given the perspective taken by the child. To insist that a child apply measuring number division methods without the perspective that makes this appropriate, is to teach that mathematics makes no sense. For children with this perspective it is fine, but until they have or seem ready to move on to this perspective, there are plenty of situations in which division for size gives whole number answers. Question May has some 2-gallon containers. Her cow gives 1/4 of a gallon of milk. How many of these containers can she fill? Child 1 You can‟t fill a 2-gallon container with 1/4 a gallon. (perspective not a division problem) Child 2 None, she needs a quart jar. (perspective: division with remainders) Child 3 She can fill 1/8 of a container. (perspective how many can be interpreted as how much) As we said, with ordinary situations that involve dividing to share we usually expect our divisor to be a whole number, and when dividing for size we tend to think of our quotient as being a whole number. However there are many ordinary situations that involve dividing to compare in which none of the numbers involved are whole numbers. Lot A is a ½ acre. Lot B is ¾ an acre. Lot A is 2/3 the size of Lot B. Lot B is 3/2 the size of lot A. More examples of this type are given in Lesson 4. 37 Lesson 2 The main focus of this lesson is to find ways to divide by unit fractions. One way is to get a common denominator. It is easy to picture why this works. It is also an easy method to use. However it is not the way that is usually taught in school. Instead schools focus on using reciprocals This rationale for this method is harder for most students to understand, probably because the explanation is more remote. Children are not the only ones who have trouble with this method. In general most people who have used this method, including those who can use it correctly, do not understand why using reciprocals works. Furthermore as illustrated by Roy‟s response to the boat drifting question, it is a rule that is often used incorrectly when its rationale is not understood. You may wonder why this is the method most commonly taught. One reason why the use of common denominators is not taught in school is that most of the teachers are unaware of this way of dividing fractions. However even if they were aware of this method they might still prefer using reciprocals. In fact even if students prefer to use the common denominator method, there are some reasons to encourage them to think about why using the reciprocal method is reasonable. Our general discussion of these reasons will be postponed until Lesson 4. For now we only comment on why using reciprocals works when dividing by unit fractions. Bob has indicated one way to understand why dividing a whole number for sizes of ½ is the same a multiplying this number by 2. Simple stated, each white is a 2red. Jan notes that the fact that each white is a 4blue helps her to understand why dividing a whole number by 1/4 is the same as multiplying it by 4. Similar remarks apply to dividing any whole number for unit fraction sizes. It is also fairly easy to see why dividing by ½ is the same as multiplying by 2 when the dividend is a mixed number whose fractional part is ½, especially when the mixed number is represented by an improper fraction. For example consider 7/2 divided for sizes of ½. Using the common denominator method we have 7 divided by 1, which gives 7. Using the reciprocal with cancellation immediately gives this same result. If we did not use cancellation we would first obtain the intermediate result of 14 /2 which reduces to 7. In general if we multiply a fraction by 2 then we double the numerator, and when we reduce a fraction with a denominator of 2 then we take half of the numerator. Thus multiplying a fraction whose denominator is 2 merely produces the numerator. Dividing the same fraction by ½ gives the same result. Likewise dividing a number whose denominator is 3 by 1/3 is easily seen to give the same result as multiplying this number by 3, dividing a number whose denominator is 4 by ¼ is easily seen to give the same result as multiplying this number by 4, etc. When the dividend is a fraction whose denominator is not 2, it is more difficult to see why dividing by ½ is the same as multiplying by 2. The result of dividing 4/3 by ½ should be a third of the result of dividing 4 by ½. To divide 4 by ½ we can multiply 4 by 2. Thus to divide 4/3 by 2 we can multiply 4 by 2 and take 1/3 of the result. This gives the same as multiplying 4/3 by 2. Likewise to divide ¾ by 1/3, we should get one fourth the result of dividing 3 by 1/3. Thus we can multiply 3 by 3 and divide by 4. In general to divide a number by a unit fraction we can multiply it by the denominator of this unit fraction. 38 Lesson 3 Lessons 3 and 4 both focus on dividing fractions when the divisor is not a unit fraction They differ primarily because Lesson 3 is applied to situations in which the quotient is a whole number. Since ordinary situations of the same type may or may not result in whole number quotients, this is an artificial distinction. For example dividing 2½ acres to obtain ½ acre plots yields 5 plots, while dividing 2¾ acres to obtain ½ acre plots yields 5½ plots. The reason for making this artificial distinction is that it is easier to picture dividing measuring numbers when the quotient is a whole number. For example to see that 2¾ divided by ½ is 5½ we can trade a 2white&3blue for a 5red&blue. We then picture this as 5½ reds. Since a blue is easily seen to be half a red this is fairly easy to picture. Imagine seeing that 6½ divided by 2/3 is 9¾. This may be harder to picture. Most of the situations in this lesson involved dividing for size and most of our divisors were fractional numbers. As a prelude to thinking about such situations consider dividing for size with whole number divisor and dividends. When we ask for how-many we are usually thinking in terms of a whole number quotient, and when dividing does not give one, we often give our result as a whole number quotient and a remainder. A gallon of milk in on sale for $2. How many can you buy for various amounts of money? Clearly $8 will buy 4 gallons. However $9 will not buy 4½ gallons at this store. Milk doesn‟t come that way. $9 will buy 4 gallons with a $1 left. The same way of thinking applies to how-many questions for fractional sizes. With the daily cost of 3/8 pounds of gold, Sam could stay 2 days with his 3/4 pounds of gold. With 7/8 pounds he could stay 2 days and have 1/8 pound left. While we could divide 7/8 by 3/8 to obtain 2 1/3, this would be interpreted as 2 days with 1/3 of the price for another day. On the other hand if an acre of land cost 3/8 pounds of gold we could ask how much land could be purchased for 7/8 pounds of gold and it might be appropriate to answer 2 1/3 acres. In general we use „how much‟ rather than „how many‟ when we do not intend to limit our quotient to being a whole number. We leave such situations to Lesson 4. In order to make the pictures easier to draw we did not give any examples in the text of how-many questions in which we had remainders. However pictures for situations in which the quotient is a whole number and the remainder is a fraction are easy to describe. Dividing 4½ acres to obtain ¾ acre plots yields 6 plots, while dividing 5½ acres to obtain ¾ acre plots yields 7 plots with ¼ an acre left over (rather than saying 71/3 plots). To picture this represent 4½ as an 18blue, separating this into six 3blues. For 5½ use a 22blue which separates into seven 3blues with a blue left over. Lesson 3 only included one example in which our divisor was a mixed number, and in order to make the picture easy to draw this mixed number was only a little larger than 1. We can easily divide 17¼ for sizes of 5¾. Trade a 17white&blue for a 69blue, and trade a 5white&3blue for a 23blue. Thus we can divide 69 by 23 to obtain a quotient of 3. Lesson 3 also provides a way of thinking about why using reciprocals works when dividing by fractions other than unit fractions. Recall Kay‟s explanation. To divide by 1/3, multiply by 3. 2 /3 is twice as large 1/3. So dividing by 2/3 should give half as much as dividing by 1/3. So to divide by 2/3 we can multiply by 3 and take half the result. This is the same as multiplying by 3/2. We discuss using reciprocals again in the commentary on Lesson 4 39 Lesson 4 We introduce dividing in which we obtain a mixed number quotient with a pure division question. Jan asks “What happens if we try to divide something like 2 by 3/8?” She sees how to get a quotient of 5 and a remainder of 1/8. However she wants an answer that is a measuring number? She does not even wonder if there is some situation in which this division problem might arise. At least for the moment she is taking a purely mathematical perspective. Division is an operation that we perform on measuring numbers. Bob relates this purely mathematical question to the application he understands best, namely separating color pieces. Jan then finds an application in which a result of 51/3 makes sense when asking a how-many question. She is able to interpret 51/3 salt shakers as filling them rather than as counting them. While Kay likes applications she does not need them, and she returns to a purely mathematical perspective and answers the question by using common denominators. Kay easily adopts a purely mathematical perspective, and this is one reason she has a tremendous advantage over most other people when it comes to learning mathematics. This is not to say that Kay also appreciates the application of mathematics to a variety of situations. In fact it is her ability to think like a pure mathematician that helps her also excel at applications. For the rest of the lesson we return to thinking about dividing measuring numbers in relation to applications. The how-much question about gold dust and cider gives a situation in which they would not think the quotient would have to be a whole number. Thus it is not surprising that if a gallon costs 2/3 of an ounce then 1½ ounces would buy 2¼ gallons. Since the price was given for a gallon it seemed natural to divide in order to see that 1½ ounces is 2¼ times as much as 2/3, and thus should buy 2¼ times as much. An alternative perspective would be to think that you could get 3/2 gallons of cider per ounce of gold, and thus 1½ times that much for 1½ ounces. Note that gallons per ounce is the reciprocal of ounces per gallon. Thus to answer this question we can either in divide 1½ by 2/3 or multiply 1½ by 3/2. Suppose that for 3/4 ounces of gold dust Sam got 2 gallons of cider. In this situation we are not given either cost per unit of quantity or quantity per unit of cost. However we can figure them from the information given. Dividing 3/4 by 2 we obtain a cost of 3/8 gallons per ounce, while dividing 2 by 3 /4 we see that we could get 8/3 gallons per ounce. Thus if we want to know how much cider Sam could have gotten for 1½ ounces we could either multiply 3/2 by 8/3 or divide 3/2 by 3/8. Thus in this case understanding of the situation shows that we should get the same answer by using reciprocals as by dividing. 3 /2 3/8 = 12/8 3/8 = 4 or 3/2 8/3 = 4 Since situations in which the divisor is smaller than the dividend seem easier to picture we used such situations to illustrate division of measuring numbers in which the quotient is not a whole number. However we could also ask a how-much where the dividend and the divisor and the quotient are all fractional numbers. We could ask how much 1/12 of an ounce of gold dust would buy. The answer is 1/8, since 1/12 divided by 2/3 is 1/8. Bob might picture this by trading a 2yellow for an 8green and observing that seeing that a green only gives 1/8 of a 2yellow. Kay might compare this to dividing 1/12 by 8/12. Jan might think in terms of pints. If 2/3 buys 8 pints, 1/3 will buy 4 pints, 1 /6 will buy 2 pints, 1/12 will buy 1 pint. 40 Returning to the question of why using reciprocals is the method taught in school, I can think of two main reasons. (1) Using reciprocals is considered to be more efficient. (2) The rationale for using reciprocals is rooted in the fundamental algebraic laws for the system of measuring numbers. The greater efficiency of using reciprocals may not be apparent unless we are dealing with fractions whose common denominators are not immediately apparent. Even then it greater efficiency depends on using cancellation. Consider 14/45 21/55. Using common denominators: 14/45 21/55 = 154/195 189/495 = 22/29 = 4 Using reciprocals and cancellation: 14/45 21/55 = 14 /45 55/21 = 2/9 11/3 = 22/29 However as long as the fractions involved have rather small denominators, this advantage is not significant, and a person like Bob will prefer using a method whose rationale is easier to understand. Later Bob may encounter situations which will allow him to appreciate the greater efficiency of using reciprocals, but for his current purposes I see no reason that he should not use a method he can more easily understand. At this stage I would merely encourage him to think about why using reciprocals is another method that also makes sense. Since most people have little need to divide fractions like the one above the main place that they might encounter the greater efficiency of using reciprocals is in dividing algebraic fractions. An understanding of why reciprocals work in this context probably needs to be related to an understanding of algebraic laws. These laws can be illustrated by numerical examples prior to the use of any algebraic notation. The main idea is similar to that used to raise fractions. Stating this idea in relation to division, multiplying the divisor and the dividend b the same number does not change the quotient. For example 21 divided by 3 is 7, while 42 divided by 14 is also 7. Using the same idea with fractions we could double the divisor and dividend to show that dividing a number by ½ is the same as dividing 2 times that number by 1. This give a perspective using the reciprocal of ½ that is different from the pictorial perspective used in the text. This idea applies to dividing by any fraction. For example 2/7 4/5 is the same as (2/75/4) (4/55/4). Since 4/55/4 = 1, this is the same as 2/75/4. Relating division directly fractions we can think of 2/74/5 as the complex fraction below. We then multiply by 5/4 to see that this gives the same result as multiplying the numerator by 5/4. 2 /7 2 /7 5 /4 2 /7 5 /4 4 = = = 2 /7 5 /4 /5 4 /5 5 /4 1 A more fundamental algebraic analysis depends on the associative and the identity and the inverse laws. For such an analysis, ask about our ordinary algebra unit entitled “More Algebra for Rational Numbers”, which we hope will soon be available on our website. 41

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