Fractions, Percentages, Decimals and Proportions
A Learning-Teaching Trajectory for Grade 4,5 and 6
Frans van Galen, Els Feijs, Nisa Figueiredo, Koeno Gravemeijer, Els van Herpen and Ronald Keijzer
Core insights for proportion
There are many forms of mathematical descriptions with fractions, percentages, decimals and proportions, and actually they have their own rules and procedures that students must do a lot of practices to remember them. The purpose of this article is to make a simple and narrower programme without mastering procedures but understanding the principles. Researchers refer to core insight for proportion by using contexts.
Proportions are everywhere
Students will find some problems about proportions like enlarging and reducing photos, prices comparisons, recipes, comparing probability, gears on mountain bike, shadows change, and graphs and diagrams. They already have some experience with proportions in the upper grades of primary school. This book, researchers explained general concept of proportion in grades 4-6. Proportion can be placed as fractions, percentages, and decimals. In the beginning of this chapter, researchers discuss the concept of direct proportions and then ratio table. After that, they move on the more problematic issues and finally will return to proportions. The end of this book tells about sketching out a learning-teaching trajectory.
Reasoning with proportions
Proportions are based on the concept of ratios (direct proportionality). Students can use ratio table to show clearly because They can expand the number of columns to suit their needs. In mathematical terms, a directly proportional relationship between two units is called a “linear relationship” because it can be drawn as a straight line on graph and pass through the origin. In reality, there is often only an approximately direct relationship. The graph and the ratio table are not a description of a true condition, but in fact a mental model.
The ratio table
The ratio table is an important role in the curriculum on proportions in grades 4-6. It is an ideal aid for making handy calculations and gaining insight because the table invites students to write down intermediate steps. The ratio table is much clearer because every intermediate step has meaning. It can also be used for calculating with percentages and decimals. The ratio table is a mental model as well as a work sheet. The table helps students reason with proportions and is at the same time a handy calculations tool.
The ratio table gives a good overview. But is not in itself any better than a table or list in which the number are placed under each other. The advantage of the table is that all the numbers have their own place and that the unit of measurement must stay the same. There are good reasons for choosing a ratio table as the standard form of notation, but that does not mean that the ratio table should be introduced without any discussion.
Reinventing the ratio table
The reinventing comes down to two things: the choice of a systematic notation method and discovering for themselves how useful a ratio table is for doing calculations. The first, students must realize that a systematic notation method has great advantages. They often write down a list of items more automatically than they will choose a table form and discover what they need to write down for the list to be complete. The students need to explore independently which step they can make in the table and which will not work. They can learn some rules by reasoning from a context problem and by explaining to other students how they have reasoned. Teachers can help by making the mathematical steps in the table explicit.
Comparisons using proportions
Proportional comparisons are based on a mental experiment. The students have little trouble with the mental experiment underlying the conversion, because it is so logical to star from a directly proportional relationship between two problems. Many presuppositions are required to be able to make a proportional comparison. Adults are so used to thinking in proportions that they mostly do not realize that they are making an artificial mathematical transition. For children, the difference between absolute and relative comparisons is not nearly as self-evident. Teachers should focus sufficiently on the aspect of comparing absolute terms versus comparing in relative terms. They should encourage students to think a lot about the function of mathematical tools. This can be done via investigative activities and open questions without a pre-determined approach. Students should do too many activities in which the method of comparison is pre-determined.
Composite units are different types of units and combining them leads to a new unit. In the history of mathematics it took a long time for such unit to become accepted. For students composite units are troublesome, for example, velocity is a difficult concept. The ratio table simplifies calculations with composite units since in such a table students can work with the separate units rather than with the composite unit directly. Composite units are special as in everyday life people often calculate with a standardized proportion. The advantage of such standardization is a direct comparison that people can make. How things are standardization is no more than a mutual agreement and different from one country to other areas.
Directly proportional relationships are very common, but students must still consider about not directly proportional relationships. For example, The cost of a mobile telephone cell already costs a few cents before the conversation can begin, in fact the cost of it is usually made up of a starting rate and a charge related to the length of the cell. Another example of proportional relationship is that between length and area when enlarging a photo. Students are often confused by making something twice as big does not make the area twice as big. The relationship between length and area is not linear,but quadratic: the area increases by the square of the length.
The global learning-teaching trajectory for proportions
Students have stated with a broad exploration of proportions before grade 3. This exploration primarily has a qualitative nature and covers situations dealing with dimensions and measuring. Students learn about simple visualization. There are four parts of learning-teaching trajectory about proportions.
Ratio table is one of the part of the learning-teaching trajectory. After introducing it, students must develop their feelings for recognizing situation in which they can be useful. They must develop some strategies for making clever use of the ratio table. They can start by using more formal strategy into standard method. People can distinguish between situations where both numbers in the ratio refer to the same unit and situation where the units differ. The level of working with a ratio table usually does not create any extra complications to make composite unit to be easy to understand.
Percentages and decimals
Using the ratio table for percentages and decimals is a part of the learning-teaching trajectory. Percentages involve converting proportions to percentages as well as calculating with percentage. Proportions or ratios can be converted into percentages by calculating. The ratio table is well suited to global calculations and estimating the result using in iterative approach.
An important first step of proportion comparisons is to distinguish between absolute and relative comparisons. The next step is to extend into situations in which there is no strictly linear relationship, but where people act as if there is one.
There are some subjects that are not linear standing apart from others. To avoid student misunderstanding on those materials so teachers must pay attention to these relationships with a certain regularity.