# Guide Teaching Fractions through Situations: A Fundamental Experiment

Hartrup, Analaysis of arithmetic for mathematics teaching pp. Hillsdale, New Jersey: Erlbaum. Rational and fractional numbers: From quotient fields to recursive understanding. Mahwah, New Jersey: Erlbaum. Kolmogorov, A. Lebesgue, Ob izmerenii velichin [On measuring magnitudes]. Second edition. Moscow: Uchpedgiz. Lamon, S. Presenting and representing: From fractions to rational numbers. Teaching fractions and ratios for understanding. New York: Routledge. Lerman, S. Theories in mathematics education: Is plurality a problem? English, Theories of mathematics education.

Seeking new frontiers pp. New York: Springer. Lompscher, J. The sociohistorical school and the acquisition of mathematics. Biehler, R. Scholtz, R. Winkelman, Didactics of mathematics as a scientific discipline pp. Ma, L. Knowing and teaching elementary mathematics. Teachers' understanding of fundamental mathematics in China and the United States. McLellan, J.

The psychology of number. New York: Apple-Century-Crofts. Niemi, D. The Journal of Educational Research, 89 6 , Noelting, G. The development of proportional reasoning in the child and adolescent through combination of logic and arithmetic. Ohlsson, S. Mathematical meaning and applicational meaning in the semantics of fractions and related concepts. Behr, Number concepts and operations in the middle grades pp.

Pantziara, M. Educational Studies in Mathematics, 79 1 , Parker, T. Elementary mathematics for teachers. Okemos, Michigan: Sefton-Ash Publishing. Piaget, J. Ratsimba-Rajohn, H. Reys, R. Helping children learn mathematics. Canadian edition. Schmittau, J. The development of algebraic thinking. A Vygotskian perspective. ZDM, 37 1 , The development of algebra in the elementary mathematics curriculum of V.

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The Mathematics Educator, 8 1 , Sierpinska, A. Rational numbers. Revised Version. Inquiry-based learning aproaches and the development of theoretical thinking in the mathematics education of future elementary school teachers. Maj-Tatsis, M. Swoboda, Inquiry-based mathematical education pp.

Rzeszow: University of Rzeszow. Teaching absolute value inequalities to mature students. Educational Studies in Mathematics, 78 3 , Sowder, J. Reconceptualizing mathematics for elementary school teachers. Instructor's edition. New York: W. Freeman and Company. Thompson, P. The discrepancy in student performance can be explained by differences in the curricula of mathematics in grade four in these countries. Implications for the teaching of mathematics in these two countries were also discussed. The understanding of fractions is based on logical relationships that are associated with the idea of quotient or magnitude.

The inverse relationship between quantities is a fundamental aspect of the conceptual understanding of fractions, which means that when a whole is split into equal parts, the more parts will correspond to smaller parts. Authors have suggested that comprehending this inverse relation requires the reorganization of numerical knowledge Stafylidou; Vosniadou, , due to the need to grasp that the properties of integers do not define numbers in general Jordan et al.

A broad and diverse set of studies has contributed to the conceptual complexity of fractions and their impact on mathematical learning Stafylidou; Vosniadou, ; Mamede; Nunes; Bryant, ; Hecht; Vagi; Torgesen, ; Nunes; Bryant, ; Siegler; Thompson; Schneider, However, there are still different reasons why learning fractions represent a challenge for some students. In this case, Nunes and Bryant claim that representing one number using two numerical fields may induce the children to only regard one of these numbers or that they might not understand that there is an inverse relationship between the numerator and the denominator.

Another reason relates to the fact that the ordering and the equivalence relationships work differently in the two numerical fields Behr et al. A third reason concerns the significance that the numerator and denominator have in relation to the various situations in which fractions are used Behr et al. Regardless of the type of difficulty that exists in learning fractions, They often continue into higher education Stafylidou; Vosniadou, ; Vamvakoussi; Vosniadou, In spite of the many existing studies into the comprehension of logical relationships and the conceptual knowledge of fractions, they are still a matter of concern to researchers and educators, mainly in terms of what children can learn about fractions and how this learning occurs.

Thus, the aims of this study are: to see how the inverse relation between quantities smaller than the unit, presented in quotient and part-whole situations, influences the learning of fractions; and to investigate whether there is any difference in the performance of Brazilian and Portuguese in relation to their understanding of the inverse relation between quantities represented by fraction.

Before generalizations could be established, the cultural proximity and language similarity were relevant aspects considered when comparing the performance of children from these two countries. A decision was made to investigate the logical relation that is presented in the learning of fractions, which was highlighted in the literature as an essential concept for algebra and mathematics at more advanced levels Nunes; Bryant, This article comprises three sections: the first presents a numerical knowledge literature review; the second describes the method of this study; and the final section discusses the results and educational implications of its empirical findings.

Research on numerical knowledge Behr et al. Learning fractions has an important role to play in the field of mathematics, which consists of representing quantities that cannot be described with integers and the relationships between quantities. Quantities smaller than one unit can arise in a division situation or a measurement situation Behr et al. A significant number of teaching approaches introduce fractions using part-whole situations.

In these situations, the denominator of a fraction indicates the number of equal parts on which a whole was divided and the numerator indicates the parts considered. The possibility of ordering fractions by magnitude tends to elude many children because they think that the larger the fractions values, the greater the quantity it represents Nunes; Bryant, In the quotient situation, the numerator represents the amount to be shared and the denominator represents the number of beneficiaries.

The problems of the part-whole situation involve a multiplicative relation between two quantities of the same magnitude and the use of fair division or partitioning of a whole into equal parts Vergnaud, ; Kieren; The problems regarding the quotient situation involve a multiplicative relation between two quantities of different magnitudes and the use of one-to-many correspondence, which may or may not involve division. Nunes et al.

Understanding the inverse relationship between quantities smaller than the unit. Behr et al. Ordering fractions that are smaller than the unit involve the knowledge of the inverse relationship between the numerator and the denominator, that is, when a whole is divided into other equal parts the parts will be smaller and that there is no compensation between the size and the number of parts. In its turn, the equivalence of fractions smaller than the unit involves understanding the inverse proportional relation, which implies that in order to double the number of parts, each part must be half of its original size to ensure that sizes are equivalent.

Hence, to understand the equivalence of fractions requires establishing compensatory relationships between the area and the number of equal parts in which the unit was divided. A study performed by Mamede, Nunes and Bryant investigated the understanding of quantities represented by fractions in quotient, part-whole and fractional operator situations with 80 children aged six and seven years old, before receiving formal instruction on fractions in school. The results indicated that children performed better in the quotient than in the part-whole situation, both in ordering and equivalence problems of fractions, and performed similarly in the naming tasks.

A study by Nunes and Bryant examined the comprehension of equivalence and ordering of children in their fourth and fifth graders from eight schools in England. In addition, the authors suggested that perception based on numeric symbols is not sufficient to understand equivalence and ordering in the context of rational numbers. Kieren had previously suggested that these representations play an amplifying role in natural and structuring abilities during activities, from which reasoning is driven. The possibility of exploring significant contexts from daily life provides the opportunity to use problem-solving strategies rather than memorized procedures , which can stimulate reasoning and communication, in addition to being a facilitator for learning mathematics Behr; Post; Lesh, For researchers, correspondence is a problem-solving strategy that involves multiplicative relationships Nunes et al.

Different models can be used for solving quantitative problems. Nunes and Bryant highlight three factors involved in solving problems and argue that each of these factors has an impact on the way we learn. These factors are as follows: reflective thinking, sociocultural interaction and use of learning tools. Reflective thinking consists of a mental activity on behalf of the student, which is expressed by imagining or relating ideas. Socio-cultural interaction advances the development of mathematical ideas by students, which occurs through their interactions with the environment.

Finally, the use of learning tools can help develop strategies and procedures for solving problems. A study by Hecht, Vagi and Torgesen suggests that children with learning disabilities use inaccurate mental models as well as memorized and incorrect procedures to solve mathematical problems. Teaching fractions during the early years of basic education in Brazil and Portugal. The poor achievement in learning fractions can be associated with the approach to fractions in school, where mainly one of the meanings of fractions is explored Kieren, Mathematics in the classroom often introduces fractions with models of geometric figures and with part-whole situations.

MEC, It is a study that starts in the second cycle of elementary school and it is consolidated in the final two cycles of school. According to the PCN Brasil. MEC, , teaching rational numbers start with their recognition in daily life. Also according to PCN Brasil.

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MEC, , learning rational numbers implies challenging with ideas built by students about natural numbers. Therefore, it takes time and demands the right approach. MEC, a. Thus, it begins to guide the initial notion of fraction and its different meanings, simultaneously exploring the fractional and decimal representation and developing number sense.

Initially, the PMEB emphasizes "[ MEC, a , p. The PMEB highlights the development of three essential skills: problem-solving, mathematical reasoning and mathematical communication. The PMEB proposes that the problems serve as an application context for knowledge acquired previously; they also serve for building new knowledge. To improve the quality of mathematics in school, the National Council of Teachers of Mathematics NCTM established principles and standards for curriculum and evaluation to the subject.

In view of the NCTM, numerical knowledge involves counting, quantities of comparison, and advancing the understanding of the structure of the numerical system of base ten. The document asserts the importance of understanding the learning process of mathematics and emphasizes that the compression of numbers becomes more complex when it involves fractions and whole number. This requires that the teaching of mathematics engage students in solving a variety of problems, not just as one of the goals of learning but as a major means of doing so.

The principles and standards indicate that problem-solving develops mathematical reasoning and promotes communication and representation that are part of the process of learning mathematics NCTM, In short, one can say that the official documents of Brazil and Portugal include in their curriculum guidelines the perspective of the principles of the US program proposed by the NCTM The Portuguese curriculum has extended the time in which children's activities involving rational numbers are exposed.

Brazilian curricular parameters emphasize the use of problem solving as a teaching strategy. Brazilian and Portuguese researchers have broadened the study on teaching and learning rational numbers. During an exploratory study with ten and years-old Portuguese students in their fifth grade, Ponte and Quaresma investigated the multiple representations of rational numbers with different uses and types of grandeur and showed that the comprehension of ordering and comparing fractions combines formal and informal reasoning.

The results showed a significant correlation between ordering and equivalence in both situations. The results indicated that the teachers adequately conceptualized fractions in some situations, in spite of the fact that they misrepresented fractions and ratio. However, comparative studies investigating fractions are scarcer. In Brazil, the current study found one comparative study with Brazilian and Portuguese children, aged six and seven, which was performed by Dorneles, Mamede and Nunes The study investigated ordering, equivalence and naming of fractions in quotient and part-whole situations.

If that really represented the whole of their knowledge, then complaints about the paucity of that knowledge would be entirely legitimate. What an individual, paper-and-pencil test cannot reveal is the depth of their knowledge and their degree of ownership. They will be able to invent and re-invent the con- cept of a rational number as a linear mapping until they internalize it, to assemble a collection of observations and partial understandings into some very solid knowledge about division and to use both rational numbers and decimals in most of the standard contexts.

To a large extent this knowledge will be institutionalized and testable, though there will, of course, be some speculations and queries left to fuel future exploration. The remaining two sections of the book will summarize and discuss the modules of the manual covering this last stage in the learning of rational and decimal numbers.

Module 8: Similarity The first situation put to students for study of fractions as linear mappings is the following. Instructions: Here are some puzzles Example: Fig. You are going to make some similar ones, larger than the ones I am giving you, according to the following rules: The segment that measures 4 cm on the model must measure 7 cm on your reproduction. When you have finished, you must be able to take any figure made up from pieces of the original puzzle and make the exact same figure with the corresponding pieces of the new puzzle.

I will give a puzzle to each group of four or five students, and every student must either do at least one piece or else join up with a partner and do at least two. Development: After a brief planning phase in each group, the students separate to produce their pieces. The teacher puts or draws an enlarged representation of the complete puz- zle on the chalkboard. Strategy 1: Almost all the students think that the thing to do is to add 3 cm to every dimension. Even if a few doubt this plan, they rarely succeed in explaining them- selves to their partners and never succeed in convincing them at this point.

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The result, obviously, is that the pieces are not compatible. Discussions, diagnostics — the leaders accuse the others of being careless. They attempt verification — some students re-do all the pieces.

They need to submit to the evidence, which is not easy to do! The teacher intervenes only to give encouragement and to verify facts, without pushing them in any direction. Strategy 2: Some of them try a different strategy: they start with the outside square and try adding 3 cm to each of the segments in it. This produces two sides of length 17 cm and two of length 20 cm — not even a square. This gives a puzzle that is very similar to the original. So occasionally the students work their way out of the situation by a few snips of the scissors here and there.

Even if most of them are aware that they are fudging, a few are convinced that they have found the solution. The teacher, invited along with the other groups of students to confirm success, in this case suggests that the competitors use the model to form a figure with some of the original pieces such as Fig. Classes often get quite worked up — lively disputes, accusations, threats — but rarely discouraged. The whole class and the teacher take note of the success, but the procedure is examined in the following lesson.

Results All the children have tried out at least one strategy, and most have tried two. By the end of the class, they are all convinced that their plan of action was at fault, and they are all ready to change it so they can make the puzzle work. But not one group is bored or discouraged. Today you are going to try to find the right measurements that will let you make the puzzle right. Development: To make things easier, the teacher or sometimes a student who succeeded with the activity the day before puts the lengths up as a table:.

Right off the bat somebody always asks for the image of 8 which is of no use, but which they nonetheless add to the table.

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The students work in groups of 2 or 3, all of them having copies of the table in their notebooks. As before, the teacher goes from group to group, encouraging them and answering questions, but does not take part. Some of the procedures observed: 1. To find the image of 1, they write: half of 3 is 1.

Observation: One of the children, after having correctly found the image of 1, went on to make all of her calculations using 1. If your pencil is good and sharp you can get very close to halfway between two millimeters. Remark: For many children, measuring Comparison of methods and realization of the puzzles: As soon as all the groups have found the measurements, they compare and discuss their methods. The teacher then has them make the pieces and reconstitute the puzzle. The stu- dents would ask to do it themselves in any case.

Remark: This phase is essential, because for the children it is the only proof that is valid and convincing. But above all, it is source of pleasure and enthusiasm for them: their effort is repaid and they have succeeded. Results All the children know that the image of a whole number can always be found, and almost all of them know how to find it.

To do that, we had a model on which we knew all of the measurements and we had some information about one of the new mea- surements: we knew that what corresponded to 4 was 7. What did you look for? What they needed was the image of 1. They often need to review division here, which they do collectively. Remark: It is essential for the teacher to pull the class together on a regular basis to remind them where they are: recall or have them recall what problem was posed and what questions that problem gave rise to.

They absolutely must know what it is they are trying to solve. The teacher can even occasionally remind them in the course of an activity. The fact is that many children, in the process of working out the interme- diate steps of a problem, forget why it is they are carrying out their calculations. You also know that you can designate a measurement by a fraction — what did you do that for? Today you are going to try to find the image of a fraction. Development: First she asks the students to think a bit and make sure they all understand the prob- lem posed.

Spontaneously the children suggest adding 1 into the table of measurements, which the teacher does. She then asks them to find the image of 1. One of the students comes to the board and writes this image of 1, which gives one more rapid review. The new table of measurements now reads. At this point, for this particular piece of the problem, the students work in groups of two or three. Behaviors observed: 1. The teacher goes from group to group, asking questions, giving encouragement. They remain stuck. So she organizes a collective discussion.

Third phase: the search for an in-between number Assignment: The teacher first asks the students to look closely at the table of measurements from before:. Think about the calculations you would have to do. The children think silently, then propose things out loud. The proposals are immediately put to the test while the whole class watches. They are not institutionalized, and are therefore later forgotten. A few students know and recall that they can multiply the denominator by 7 to make the fraction 7 times smaller.

The children return to their groups of two or three and get back to work on the solution they were working on in the second phase. Strategies observed: All the groups make one or the other of the following tables: Either. Remark: Not all the groups get to the end of the calculations, because the children make mistakes. They have forgotten the techniques they developed during the les- sons about operations on fractions.

This is perfectly normal, and the teacher should neither worry nor blame the children. On the contrary, this is exactly the moment to re-use the processes they discovered quite a while ago, put them to work and let the children see what the processes are good for. Remark: Because this is a regular proceeding, the children are perfectly comfortable discussing their mistakes. This is beneficial to the whole class, since exploration of errors can often contribute just as much to understanding as observation of correct procedures.

After that discussion, the teacher sends some students to the board to describe the methods they used:. The teacher adds two more fractions to the table of measurements and the students calculate their image individually. They discuss their solutions rather than turning them in. Results At the end of this session, the children understand that you can find the image of any fraction at all provided you know the image of one whole number. It would be a serious error to stop and drill the students, because the up-coming activities let them re-use these notions and progressively master them each child at his own rate.

Problem situation: construction of a tessellation Materials: 3 or 4 cardboard pieces similar to the figure piece marked 1 in Fig. Assignment: We are going to make a decorative panel for our classroom. It will be made up of pieces like the one you have, put together like this3: To do that, each of you will make one piece by enlarging the model so that 1 cm on the model corresponds to 3. Another common strategy is to do the calculations by taking apart the decimal numbers as follows: For the image of 2. Next find half of 3. For the image of 1.

Many of the calculations described above are invisible — only the results appear. The groups that have found the numbers take turns at the board explaining their method. Each child makes one or more pieces out of colored paper. As happens with the puzzle activity they are again faced with measurements: 5. Results This activity gives the children a chance to re-use procedures worked out in the previous session. It enables many of them to master some difficult calculations that they have previously been unable to carry all the way out. This presents no difficulties, and very soon the teacher is able to put a collectively produced table on the board.

She then writes up the problems and results, including some intermediate problems that the students have produced: 3. The teacher then leads them to formulate the rule: When you divide by 10 or or …, you have to move the decimal point as many places to the left as there are zeros in the number. By way of solidifying the rule and pushing the students a little farther, the teacher gives them some exercises, which are done individually and immediately corrected:. Result As was the case for multiplication of a decimal number by 10, , etc.

As a result, it is absolutely necessary to keep on regularly giving them exercises corrected immediately so that they master these calculations swiftly, because they are going to need them for lots of other activities. Module 9: Linear Mappings All of the sections of Module 9 revolve around reproductions of the drawing above. It was chosen because not long before the time the lessons were given, the class had had a whole month in which after a morning in school, they spent every afternoon together at a sailing school near-by on a boat called the Optimist.

This annual event was a source of great pleasure and of great class bonding as well. The basic drawing is on card stock and has the dimensions listed below. In addition there are 11 repro- ductions also on card stock, with specified ratios of enlargement or reduction. After introducing the drawing and having the children help her label the parts of it, the teacher puts on the board the list of dimensions:. Height of mast She then puts up, beside the original, the reproduction that has a ratio of 1.

What information would you need to do it? She sets them up in groups and announces that each group must request in writ- ing the one measurement that it wants. They are then to use that to predict what all the other measurements will be if the enlargement is proportional. Once the calcula- tions are finished, they are to take their rulers up to the reproduction and check the measurements.

If all of the actual measurements correspond to their calculated ones they will have the answer to their question. The groups work together first to find the procedure that will give them the mea- surements. Then they divide up the measurements — one does the mast, another the boom, etc. As soon as they are done they check their results by measuring. In one sad case a group that had asked for the measurement of the mast proceeded to add 8. When they are done, they have a collective discussion. First the ones who have had trouble describe where the trouble arose, then the groups that succeeded come to the board and present their methods one presentation per method.

Some of the methods presented by the children: First strategy — measurement requested was the length of the boom. The same strategy was used by some groups who started with the measurement of the side of the pennant. One such group began by calculating the image of all the inte- gers: 17, 3, 4, 5 and 17 and the image of 0. A third and fourth strategy were developed by groups who started with the height of the mast. One was to multiply both sides first by 10, so as to have whole numbers to deal with.

Another was to convert Both strategies then match those of the second strategy. Commentary Like any other lesson that involves making actual measurements and comparing them with the results of computations, this one brings up issues related to approximation. The teacher needs to establish very gradually over the course of all such lessons an understanding within the class of how to treat values arrived at by measuring and those arrived at by calculation.

Questions of how large a discrepancy is acceptable should be treated case by case, with student opinion always underlying the decision so that they never think the answer is handed down from on high. Eventually error intervals and the algebra thereof should work their way in, but not as a topic in themselves, always as a means of dealing with a particular situation. The next lesson is highly similar to the first. The only difference is that the new reproduction is the one with proportionality factor 1.

They settle down and swiftly work out the new lengths using one or another of the other procedures. After the solutions have been duly discussed, they discuss which one they found the most effective and institutionalize it as the one to be used in the following activities gen- erally the one that starts by turning everything into a whole number. This is followed by a very challenging lesson that uses a bunch of the reproductions and poses a new problem.

The teacher holds up five of the reproductions, some larger than the original, some smaller, and some very close in size. She fills in the column of measurements for the model, then the row of stem measurements for all of the reproductions which settles whether they got the order right. For instance, one tactic would be to calculate the ratio of Another would be to calculate the lengths of all of the masts and see which one comes out to be Or then again, one could calculate the stem length of the unknown boat and compare it with the given lengths.

This is the kind of situation in which the teacher must firmly resist temptation. If she reduces the scope of the problem by pointing out which numbers are relevant and what to calculate, she will take all the interest out of it. But which ones? They think it over a while, and after some hesitations and tentative efforts, one of them comes to the board and writes:. The children work it out in groups of two, and the teacher chooses one to write the correct process on the board. The standard mode of calculation produces a stem length that matches that of boat C.

As a follow-up, the class does a series of problems individually, so that each child can figure out whether he actually understands, and whether he knows how to find whatever measurement he needs. The rest of the lesson is a swift activity aimed chiefly at the motivation and introduction of a new notation.

The original drawing and the six reproductions labeled A through F are still posted on the board. In the interim a spare reproduction of the Optimist seems to have turned up! One at a time she gives a single measurement from each of them, and the class quickly tells her where to put it. In a short, almost playful time, all 11 are lined up on the board. We need to be able to talk about them. The teacher says she has another one that goes between A1 and A2, after which the class realizes that letters are not going to suffice.

They set to work finding an alternative method. Often one of them will suggest using the image of one, since that way they can tell whether it is enlarged by a little bit or a lot. If none of them thinks of that possibility, the teacher suggests it, and asks them to verify that it provides the information needed. It should a let them find the image of any of the measurements and b let them put any enlargement she gives them in the right place.

She then has students go to the board to show where to put something enlarged by 1. Then she has them reverse the process and find enlargements to go between ones that are already up there. This they do on their own, on scratch paper. Meanwhile the teacher writes under each reproduction the corresponding image of Enlargements, reductions, 0 or 1? They decide it ought to have a number, and the class splits between those who propose 0 and those who propose 1.

It does nothing. Enlarging by nothing should mean zero! And 1. What would you mean by an enlargement by 2? Is that a contradiction? Rather than setting up a Situation of communication, like the ones with which ratio- nal numbers as measurements were introduced in Module 1 the teacher con- tents herself with talking about communication, because here the issue is familiar to the students and nothing new would come of such a Situation. This internalization saves time later without losing any of the meaning of the knowledge being created.

This one requires some special preparation. The teacher needs to make a special transformation of the model that enlarges the model with a horizontal ratio of 1. The resulting reproduction will be called Z. As usual, the lesson starts with a review of the preceding lessons, including in this case listing on the board the images of 1 they found for all 11 of the reproduc- tions. You measure its dimensions. You enlarge or shrink all of its dimensions the same way You get a bigger or smaller picture. Development: The children in the first three groups decide to calculate all the dimen- sions using the proposed directions and make the corresponding design with the resulting dimensions.

They decide who is to calculate which dimension, then settle down and do it. Class discussion: When they are all done, the teacher has the first three groups put all of their numbers on the board, and the class checks them. Looking at the designs, the children are completely satisfied that 1 gives a proper reproduction. Using 1. Then they measured it and got They went back and checked all their calculations, but they were all right.

The teacher suggested that they calculate the image of 1 using the boom, so they did that and found 1. The child who had noticed that the design seemed elongated suggested that they should find the image of the hull with the new image for 1, so they did and it checked out with their measure- ments. When they presented this to the rest of the class, someone wondered whether all the vertical enlargements might be the same. So the teacher immediately encour- aged them to calculate the image of 1 first using the height of the hull and then the height of the pennant.

Now they know for sure: there is more than one image for 1. To finish up, the teacher gives them a swift introduction to linearity. She points out that the height of the boat is the height of the mast plus the height of the hull, and writes all three of those measurements on the board. Then she has them com- pute the images of all three under each of the first three mappings. The first one duly gives images that add up properly, but adding 5, or multiplying by 2 and then adding 3 both give non-matching images.

Addition just messes things up! Presentation of information: The teacher confirms their conclusion by telling them that if the sum of the images is equal to the image of the sum, then we say that the mapping is linear, or that the numbers are proportional. Some examples might be:. The whole flock of reproductions is still on the board. The teacher reminds the class of all they have learned working with the puzzle and the boat reproductions. For instance, suppose I chose C as my model rather than M. Would I be able to designate the enlargement that gives F by the same number?

Could you find a number that desig- nates this enlargement? If I take C as the model, will 1 1. They choose the boom measurement, which is easy to work with. Some re- calculate the image of 1 in F starting from M, thus con- firming that it is 1. It can be represented by as many different numbers as there are models to choose from.

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Reproduction: the action and the image. It has to be put between the model and the reproduction in order to designate the map- ping. They are the numbers of the reproduction-action, not of the reproduction-image. It turns out to be the same as the mapping that took us from the former 0. Which one was the model? What table could we set up to find it? They use the ratios in F to find the measurements in the model:.

To determine a proportional representation, how many reproductions do you have to show? Two, the model and its image. For sure, the same proportional reproduction can make each model correspond to a different image. For example, in the first two of the three questions we just worked on we saw the mapping 1 3 first taking 0. How can we find all the enlargements realized in our collection of figures? The teacher holds out for a system that represents each and every reproduction. She puts a grid on the board with all of the images designated down the side as models and across the top as images, and gets the students to fill it in.

The largest? One between this number and that? Remark This lesson can be omitted for fifth graders, but it demonstrates very nicely the need to distinguish between the mapping that produces the reproduc- tion and the image that it produces. Students can get by with thinking of enlarge- ments as operations or the result of operations without being required to make a formal distinction, but the moment the problems start getting complicated, the teacher is left without any way to explain things to the students who are the least competent at constructing their own models.

Teachers then have recourse either to teaching algorithms the traditional solution or waiting until the questions can be presented formally current solution. In either case, there is no negotiation and no teaching of the meaning. The difficulty is not resolved, it is just disguised. Now we want to know what would happen if we took E as a model and M as the copy. Every length on E corresponds to a length on M. Is it a good propor- tional reproduction?

And if so what is the enlargement factor? The teacher writes up the beginnings of a table, with E and M at the top, and the children immediately propose to put in the corresponding measurements, starting with what 1 in E maps to in M. But they still have to verify that this reduction stays the same for all the measurements.

The teacher refrains from making objections. Do you think that every proportional reproduction that we have seen has a reciprocal? If so, would you know how to calculate it? They will also be proportional reproductions. Some of the students have to re-do the tables and calculations.

It develops rapidly as a game question and answer. The process starts with a review of everything the class knows about fractions, bringing back into focus the original construction of fractions as a measurement. We did some calculations that we could have written as one fraction times another fraction. We are going to see if we can find which calculations they were.

Here the meaning of the product of two fractions is quite different from that of the product of two natural numbers. We think that such an exhaustive examination is inappropriate with children of this age, but that it is indispensable to have them inventory a certain number of properties Either that are conserved for example distributivity over addition , Or that change for example the fact that the product of two whole numbers is equal to or greater than each of the two And, of course, to construct a new meaning for multiplication.

Definition of the product of two fractions. As you might suspect, we need to look at enlargements and reductions and not at additions to construct this new multiplica- tion. We will proceed in three steps. The teacher gives the students a few minutes to think about it and possibly write something on a small piece of paper and put it on the corner of her desk.

They will find out whether they were right in the course of the class research. See if you can understand why. Model Reproduction 1 4. Model Reproduction 1 4 5 3. Is it an enlargement or a reduction? Fourth step: Calculating the product of two fractions.

Do you know how to calculate the result? The students work in pairs. They try to reactivate the techniques they discovered in the activity from Module 8, Lesson 3. The teacher, who goes from group to group during the working phase, expresses astonishment and asks how they got it. They tell her they multiplied the two numer- ators and the two denominators. The children then solve it with one or the other of the strategies above.

Collective correction: The children who found it come to the board to demonstrate their strategies. Many observe and affirm that you can do it by multiplying numerators and then denominators. Module Multiplication by a Rational Number The teacher agrees to check out this method, which she baptizes with the name of the student who proposed it. The children set out to calculate it both ways, and the teacher helps the ones who are having a little trouble.

So she proposes a new form of verification in the form of a game. Verification of the rule First game: 1. Each child writes on scratch paper. The result is written on the board:. Remark: The children love this activity. The teacher has a student come to the board while the others look on and com- ment. The students sets up the usual format. The calculation is carried out initially by expressing the decimal numbers as decimal fractions, then directly, after a rediscovery of the way moving the decimal point represents the denominator.

The algorithm is recognized and practiced and given the status of something to be memorized, as it would be in the classical methods. Results: All the children understand the algorithm and the meaning of this multipli- cation. But it is interesting to note that it gives them great satisfaction to be able at last to multiply two decimal numbers.

In that case, the teacher has no choice but to have a collective clearing-up ses- sion. But what does 1. We will also indicate as many as possible of the conclusions and remarks that the children may make as they master different methods of solution, different types of questions, different uses of linear functions, etc. In order to avoid presenting a multi- tude of problems we will concentrate all these conclusions slightly artificially on a few examples.

In any case, from the moment that the students start solving the problems the teacher ceases to exercise control over the details of the means of coming to the conclusions, and focuses on keeping the class engaged and with its eye on the goal. The teacher demonstrates as an example what questions it is worthwhile asking oneself in the course of solving the following problem, while explaining step by step the solution of the problem: The children collected the cream from 2 l of whole milk and got 32 cl of cream.

They also collected the cream from 5 l of low-fat milk and got 40 cl of cream. Can you answer the following questions: How much cream would you get from 50 l of milk? How much milk would you need to get 4 l of cream? The students discuss the problem and end up asking the teacher to remove the ambiguity of the questions.

This helps prepare them to construct problems themselves. The teacher makes comments and indicates how to present the givens, how to express the results in the solution and how to check the use of numbers and func- tions. Then he asks the children to recall the different ways they have encountered to solve linear mappings. On a regular basis, students are to come up with problems that involve solving a linear mapping. The problems can be invented or taken from a book, but in any case the student who presents a problem must be able to give a solution if asked. The tournament will be open until the end of the year.

Only the teacher knows which student proposed which problem. The primary goal of this activity is obviously technical: it develops in the stu- dents a knowledge and culture of problems.