BRIDGING FROM ORDINARY TRADING TO FRACTIONS CONCEPTS

F. Richard Singer III Edition date: June 2009

Website: conceptualstudy.org

Bridging Papers: This is one of several brief papers for educators illustrating the use of bridging. Bridging involves connecting concepts that a person finds remote to anything that seem manifest to that person. More about the concept of bridging is discussed near the end of this paper. To bridge to equivalent fractions we focus on what we will call the trading principle. This is a choice principle for making exchanges having equivalent value, when we have some other reason to make such trades. We begin with some familiar monetary examples. We then consider an imaginary monetary example that we use to directly bridge to trading equivalent fractions. Once equivalent fractions seems manifest to a person and is integrated with other concepts, then bridging to other fraction concepts can also be integrated into a person’s routine network of concepts. This approach in this paper is primarily a simplified version of what is done in my book Visualizing Fractions with Color Pieces.

Familiar Example: Jan gives Roy 2 quarters and 5 dimes for a dollar. This exchange involves equal monetary value. There are some reasons both parties might have for making this exchange. Perhaps Jan was glad to get rid of some of her excess change. Maybe she just she likes being helpful. It is easy to imagine many reasons that the change could have been useful to Roy. Perhaps he needed it to make a phone call or use a vending machine.

Side Remark: Personalized equivalent value is determined by the parties involved, as in trading baseball cards or other instance of bartering or even trading money for goods when bargaining occurs. Standardized equivalent value is publicly recognized, as in the case of monetary exchange or using money to make purchases of priced items. Altho application of the trading principle to equivalent fractions is of the standardized type, considering example of the other type may also be useful, especially if the trading principle is also used as a bridge to understanding the evolution of various social practices for meeting human needs. See Activity 0c below.

Note: Some possible responses to these some activities are given latter. Activities 0a-0c use concepts that will already seem highly manifest to anyone who understands the value of money. Activities 1a-1c use the same concepts for an imaginary money system. The other activities bridge to fraction concepts. Altho the activities are designed for educators rather than for students, hopeful they will provide ideas for activities that an educator could use with a variety of different students.

Activity 0a: After is wrapping coins for a bank deposit, Bob has 3 quarters and 8 dimes and 11 nickels. He exchanges the 3 quarters and 7 of the dimes with some of his friends for 29 nickels. Consider reasons for the exchange. Imagine more examples of ordinary exchanges and reasons for them. Also consider and whether or not the equivalent value is standardized or personalized. If any of them seem interesting to you then you might want to share them with me or with others.

Activity 0b: The trading is sometimes used merely to make comparisons, altho such trading is usually merely conceptual. Kay and Bob are working with some younger children who have a limited mastery of numerical concepts. Mike has 7 dimes and Sue has 3 quarters. Both think that they have the most money. Bob explains that 7 dimes are worth 70¢ and that 3 quarters are worth 75¢. This is a conceptual change into pennies. Kay has them (temporarily) trade their coins for nickels. Discuss this situation, especially as it relates to bridging.

Activity 0c: Value equivalence stays fixed with American Money. This is not the case with international currency exchange. Altho equivalent values are public, they fluctuate. According to www.x-rates.com, on 1/9/08 you could have exchanged $91.33 and for 1000 Mexican Pesos. On 5/16/08, you could have exchanged 1000 for Mexican Pesos for $95.88. Consider various reasons for making these or similar exchanges.

An Imaginary Money System: In this system all, the money consists of colored rectangular pieces of different sizes and colors. Pieces are named by their colors. The largest size is a white.

Value is determined by size. A white has the same size and value as two reds. It also has the same size and value as 3 yellows. It takes 4 blues to make a white. It takes 6 aquas to make a white. There is also a pink, a green, a violet.

Altho this is an imaginary money system, it resembles a monetary system based on gold in one significant way. In both, value is based on size, spatial in the imaginary one and weight in the gold.

Note: A template for making these pieces is available on the Fraction Section of the above website. Using these pieces should make the following activities seem more manifest. Altho this is not likely to be needed for the concepts involved, it will make the relationships between the pieces easier to remember, and this should facilitate the later bridging to fraction concepts.

Reasons for Trading: Thinking of these pieces as money, any of the reasons for trading might apply. However in the Activities 1a-1d, you may merely focus on trades and ignore such reasons.

Activity 1a: We can trade 4 pinks for 3 aquas. Using pieces, we could fit them together to see that they are of equivalent value. We could also see that both could be traded either for a red or for 6 greens. Discuss other examples of equivalent value and indicate how this could be demonstrated.

Activity 1b: Kay has 2 yellows and Bob has 3 blues. Both can easily visualize that Bob has more money. Jan agrees but wants to demonstrate this by trading for greens. She says that there are cases where seeing who has the most by visualizing may be difficult, especially when more pieces are involved. She says that she cannot tell by visualizing that 7 pinks are worth more than 5 aquas. She claims trading will always work, because we could always trade for violets. Roy says that he could tell this without trading since 4 pinks is equivalent to 3 aquas and 3 pinks are more than 2 aquas. Describe this situation, including the trades Jan might make. Discuss other such comparisons.

Activity 1c: The comparison above involved comparing several pieces of one color with several pieces of another color. In order to compare 2 yellows and a red with 3 blues and 3 pinks, Bob trades the first for 28 violets the other for 27 violets. He says this is like trading for pennies. Jan trades 2 yellows and a red for a white and an aqua. She trades 3 blues and two pinks for a white and a pink. Check his trades and explain why this also settles the comparison. Kay continues from the trade Bob made to obtain 1 white and 4 violets in one case and a white and three violets in the other case. She says that this is like trading for dollars and pennies and writes her results as 1&4 and 1&3. Indicate how you could numerically convert her numbers to violets.

Activity 1d: Suppose we use all the pieces from the above activity. Then using Kay’s notation, their value is 2&7. Suppose we take one piece of each type. Using Kay’s notation, indicate their combined value. Describe a general way to determine the value of a combination of pieces.

Fraction Concepts: All pieces can be thought of as a fraction of a white. For instance, since 2 reds are equivalent to a white a red is ½ of a white. Comparing sizes is related to comparing fractions. For instance, saying that 7 pinks are worth more than 5 aquas can be bridged to saying ⁷/₈ > ⁵/₆. Trading provides one way to see this. Trading 7 pinks for violets gives 21 of them, and so ⁷/₈ = ²¹/₂₄. Trading 5 aquas gives 20 violets, and so ⁵/₆ = ²⁰/₂₄. This gives an initial reason for raising fractions that does not involve the more remote concepts of adding or subtracting fractions. Moreover comparison can be bridged to subtraction by asking how the difference between 7 pinks and 5 aquas.

Activity 2a: Comparing 2 yellows and a red with 3 blues and 3 pinks can be bridged both to adding fractions and to changing improper fractions to mixed numerals. Explain how to think about this. Give additional examples.

Activity 2b: Trading can be purely conceptual, i.e. it need not be implement with actual pieces. A person with a plot that is ¾ of an acre and a plot that is ½ an acre does not have to make an actual trade to know that he has 1¼ acres. Adding fractions, is a form of making a conceptual trade of ¾+½ for 1¼. Discuss reasons for making this conceptual trade or for making it in the opposite direction.

Activity 2c: Sue and Joe each have 3 plots of land. Sue’s plots are {1²/₃, 1, ¹/₂} acres. Joe’s plots are {1¹/₃, ⁵/₆, ³/₄} acres. Without using either fraction pieces or numerical calculations try to decide whether Sue has more than 3 acres of land. Do the same for Joe. Also see if you can tell who has the most land. Using faction pieces determine how much land each of them has. Use this activity as a bridge to adding fractions. Add any additional comments on this situation that you want to make.

Note: A multitude of such activities involving fractions and these colored fraction pieces is given in my book entitled Understanding Fractions. This book provides materials and ideas that relate fraction concepts to a simple understanding of counting and ordinary skills in visualization. This book is free and can be downloaded from my website. I wish that all adults who can imagine some need to deepen their understanding of fraction concepts would use it. However I realize that while a desire to learn is an essential part of a person’s motivation, an incentive beyond this is also useful. The incentive I propose is that a deep understanding of fraction concepts is essential for adults who want to help children in constructing these concepts. Because of this, the book includes the types of materials, activities, and questions that might be used as a resource for helping children acquire fraction concepts.

Responses to Some of the Activities: There are many appropriate responses that could be made to the activities. Below are some samples of them.

Response 0a: Since 40 nickels are needed for a roll, it seems likely that Bob wants to make such a roll and thus reduce the amount number of lose coins hat he has. His friends are probably just trying to be helpful, altho some of them may also prefer having fewer nickels. It is easy to imagine more examples of ordinary exchanges. For instance, Mac gives Jim three comic books for one that Jim gives him. Mac is interested it fills a gap in his collection. Jim just wants to increase the number of books he has to trade. The value equivalence involved is personalized rather than standardized.

Response 0b: In neither case is there a reason for an actual permanent trade. Trading is a way of bridging from the knowledge of how to trade specific coins to how value can be measured. Bob wants them to see how we normally measure small values in cents. He does it conceptually, since trading for that many pennies is tedious. Kay’s goal is less ambitious. She uses actual nickels, making the trades more manifest. Kay tells Bob that what she is doing is like using the smallest common denominator to compare fractions.

Response 0c: A possible response the dollar-peso activity could involve a tourist going into Mexico and wanting to use the local currency. It could also involve a currency trader making about a 5% profit altho such a trader would not bother with such a small amount of money.

Response 1c: Using Jan’s trade, since an aqua is larger than a pink, a white and an aqua is larger than a white and a pink. To convert to 1&4 and 1&3, use 24+4 and 24+3. In general x&y converts to x+24y.

Response 1d: Using all the pieces from the above activity and using Kay’s notation their value is 2&7. If we take one piece of each type the value is 2&12. For a general way to determine the value of a combination of pieces with w whites, r reds, …, v violets, calculate 24w+12r+8y+6b+4a+3p+2g+1. The value is x&y where x is the quotient and y is the remainder when this number is divided by 24.

Response 2a: 2 yellows and a red can be traded for 4 aquas and 3 aquas, which give 7 aquas. Bridging to adding fractions a follows ²/₃+¹/₂ = ⁴/₆+³/₆ = ⁷/₆. Trading 6 of these for a white gives a white and aqua, which bridges to changing ⁷/₆ to 1¹/₆. Likewise 3 blues can be traded for 6 pinks, and combining give 9 pinks, which can be traded for a white and a pink. Bridging to fractions give ³/₄+³/₈ = ⁶/₈+³/₈ = ⁹/₈ = 1¹/₈. A short cut can be had by trading 2 pinks for a blue and 4 blues for a white giving a white and a pink. This corresponds to ³/₄+³/₈ = ³/₄+¹/₄ +¹/₈ = 1¹/₈.

Response 2b: The simplest reason for the conceptual trade is to see how much land he has. We might trade in the other direction if we knew how much land he had and the size of one of the plots.

Response 2c: Sue clearly has 2 acres plus ²/₃ and ¹/₂, and since ²/₃ is greater that ¹/₂, she has more than 3 acres. Joe only has one plot of more than 1 acre. The other are missing ¹/₆ and ¹/₄ and the extra ¹/₃ acre is not large enough compensate for both of these smaller plots. Since Sue has more than 3 acres, she has more land. Sue’s total land is can be represented as a white and 2 yellow plus a white plus a red. Trading the yellows and reds for aquas and then trading 6 aquas for a white give 3 whites and an aqua. This bridges to fractions as 1²/₃+1+¹/₂ = 1⁴/₆+1+³/₆ = 2⁷/₆ = 3¹/₆. Joe’s total land can be given as a white and a yellow plus 5 aquas plus 3 blues. Trading all but the white for greens gives 4+10+9, which is 23 greens. These can be traded for a white and 11 greens. Bridging to fractions give 1¹/₃+⁵/₆+³/₄ = 1⁴/₁₂+¹⁰/₁₂+⁹/₁₂ = 1²³/₁₂ = 2¹¹/₁₂. We could also use color pieces to bridge to the equation 1¹/₃+⁵/₆+³/₄ =1²/₆+⁵/₆+³/₄ = 2¹/₆+³/₄ = 2²/₁₂+⁹/₁₁ = 2¹¹/₁₂.

Comment: Calculating with fractions is clearly more efficient than manipulation color pieces, at least if a person knows how to calculate with fractions. However merely knowing how to calculate with fractions does not automatically mean fractions concepts have been adequately mastered. Nor does it mean that a person will know when to use factions or that fraction concepts are well integrated into a person’s world. Unless this happens the ability to calculate with fractions is likely to fade and the utility of fraction concepts will be limited. The advantage of using colored pieces is not that they merely allow a way to obtain answers but they do so in a way that forms a bridge to making fraction concepts seem manifest and thus become a part of a person’s world. Of course many people master fraction concepts without using this particular bridge. I once worked with a young woman who was beginning a business that involved interior decorating. She was able to bridge to fraction concepts by relating them to practical problems of estimating cost of various projects. Many persons with a good grasp of whole numbers can use this understanding to bridge to fraction concepts. The use of color pieces is especially useful for persons who like a visual model.

Bridging: There is a gap between thinking that uses concepts that a person finds manifest and thinking that uses concepts that this person finds remote. We repeatedly and routinely bridge these gaps in the course of ordinary living, primarily by seeing how concepts that seem remote relate to those that we find manifest. With experience, concepts that once seemed remote become manifest. They become integrated with other concepts. They take on utility. They can be explained to others. They can be brought into better focus. In short, they become sufficiently mastered to play a noteworthy role in our world. For many concepts, ordinary living provides the experience thru which this occurs. For more see A Constructivist Bridging Example

Example: As a toddler, Joe finds the concept ‘mother’ manifest, but it applies only to one person. The broader mother concept is too remote for Joe to grasp. Yet in a few years and with no deliberate effort the broader concept emerges as manifest. How does this happen? Joe learns that his mother is also his sister’s mother. He learns that his friends have mothers. He learns that his mother has a mother. Gradually his concept of a mother is not restricted to those he knows. The bridge from a highly manifest concept to one that once was too remote to grasp has been bridged. His mother concept is now integrated into a broader network of family relations concepts. The larger bridge he makes are built from a variety of smaller bridges, each of which has a small span between what is highly manifest to something slightly more remote. Of course, how many small bridges are needed to make an initially remote concept manifest and integrate it with other concepts and the width of a gap they can span will vary from person to person.

Deliberate Bridging: For many concepts that a person is less likes use in the ordinary course of living, deliberately designed bridging activities can be used to help make these concepts a part of a person’s world. Deliberate bridging activities can also be used to speed up the acquisition of concepts that might have eventually been obtained thru ordinary experience.

Note: Descriptive Psychology is a network of theory neutral conceptual tools. One of these tools is called parametric analysis. A parameter merely enables a person to focus on some aspect that seems useful for some purpose.

Concept Parameters: My paper Concept Parameters discusses the role that a concept may have in a person’s world, with special emphasis on ways to think about a person’s mastery of a concept. The role a concept has in a person’s world is conceptualized as complex personal state of affairs, using five parameters for thinking about its various features. Bridging relates most closely to what I call the proximity parameter. This parameter indicates and describes how close a concept is to a person’s common experiences. Proximity can vary from highly manifest to extremely remote. A concept is manifest to a person P to the extent that it is close to experiences that P finds it easily accessible and easy to identify and understand. It is remote to P to the extent that P finds it removed from such experiences. The chair a person sits at for dinner will be a highly manifest concept. A person’s concept of a chair may be somewhat less manifest, but since the function of a chair is easy to understand in terms of ordinary experience, it will also be highly manifest to people who use chairs. Altho initially not as manifest as the concept of a chair, the concept of furniture also easily becomes highly manifest to most people. In fact, all concepts in a person’s network of concepts for ordinary matters will seem manifest that person. The concept of a galaxy is likely to be at least somewhat remote to many people, altho some people find it less remote than others do. Bridging also involves what I call the integration parameter, since a person’s mastery of a concept normally depended on how it is integrated with that person’s other concept. A concept is adequately integrated if it is appropriately connected to other concepts and if these connections are those that the person would most commonly need in order to understand and communicate with others. The utility parameter indicates the uses that a person my have for a concept. Bridging is normally more effective when it enable a person to recognize the utility that the concepts will have for that person. Activities like 2b and 2c are intended to illustrate the utility that fraction concepts might have for some persons. However, what utility a person’s world, so bridging activities that allow some people to recognize utility may not serve that purpose for others.