DIVIDE AND CONQUER IN MATHS

Contents

Introduction…………………………………………………………………………………..2

Fractions………………………………………………………………………………………2

The Approach to solve the fractions ………………………………………………………….4

Conclusion…………………………………………………………………………………….6

References……………………………………………………………………………………..7

Introduction

“Divide and Conquer” that a famous saying tells us, to divide your problem and you win it. Whatever we may find is no exception to the rule. Division reduces the size of the problem as multiplication increases it. We may always want to overrun the problems with this. But be aware dividing anything into very small parts. You would be busted. The very-very small parts of the problem would be difficult to collect, they would be too many.( www.tlrp.org/pub/documents,2012) You would not be able to guess how many they could be.

So, you might have guessed till the time that we are talking about the fractions. The most difficult mathematical problems, till we have few years spent solving the arithmetic. They make their face up every time you cross a new standard of class. (Pollard, 2011)They are never away. Almost every concept is explained in terms of a fraction or a rational number which are only another form of the fraction, with certain conditions for the numerator and the denominator. We know that if we count the integers they are never going to end up. But, ever looked at the x-y plane you would know that the integers lay in the sea of fractions-proper or improper. Every minute point between the integers represents a fraction.

The rational numbers are also described as fractions. The limits also take the form of fractions in most of the cases. Most of the problems become complex, because they are fractions, like the limits where the denominator is either zero or infinity. Even the trigonometric functions are fractions. The fractions or the ratios, of the sides of the triangle. They have values mostly between 0 and 1.Ratios and the proportions is again a field which exists because of the concept of fractions. (Petit et.al 2009)We get into problems of what is the result of the ratios after we add a number to the numerator or denominator. Thus, was a brief with the fractions, we now talk how to study the fractions.

Fractions

The fractions are always a problem at high school children. There should a lot of research about how to make the topic easily comprehensible to the children. It should be taken a note that teaching a decimal number system is quite easier for the children to understand. They can easily comprehend the decimal numbers, the addition of it, the subtraction, the multiplication and the division of a number by another decimal number. Every number can be represented as a sum of the powers of ten each multiplied by a factor equivalent to the number in its place. But the roots to the decimal numbers concept lie in the understanding of the fractions. The digits to the left of the decimal place are after all the fractions with power ten as the denominators.(Lamon,2008)

With transition of the children from simple arithmetic calculation to the fractions, their arise many confusions in their little brains, out of curiosity as to know that 5 is grater than 4, but the fraction with base 4 is greater than the fraction to the base 5.Even serious confusions can come if they are given to add two fractions.

While I was teaching a kid, I found him worried to see the sum of two fraction numbers, though he could do the arithmetic of plain numbers quite with ease. So, what worried him? First the look of the fractions seem inconvenient to the children. When see such an odd looking form, they start thinking, as how are they useful in the practical scenarios. Everywhere, from the grocery shop to playing numbers game they find it unbelievable where from such a number comes from.

Then, we teach them that like dividing a chocolate or cake among friends we divide the numbers too, to get a solution of the problems we find with the division of things. That might seem okay, but then comes some special rules for adding two fractions. We did not need any such rules for adding two whole numbers, then why with the fractions? The questions seem unending, till we are bound to make them feel that the fractions should be treated differently from the other numbers. For fractions we need to teach the children, the concept of least common multiple (L.C.M) which has a quite different algorithm to find out. Even after that problem is not resolved, you need to do the cross multiplications, even more confusing and again a why.

At first the children should be brought to know the real fractions. The possible ways to do this is to make them understand the number of ways a system could be divided. He should learn to convert a big box into identical boxes of smaller sizes. He should be given opportunity to know how the time is divided into hours, how the hours are divided into minutes, minutes to seconds and hence on. Thus letting the child know how the things, problems and real life situations are sorted out by the knowledge of fractions.

The Approach to solve the fractions

The children most of the time are not aware that there can be different methods for solving the problems. Actually, it would be confusing for their age to remember the different ways to solve the same problem, unless the methods are for different problems. The best way to teach them is to illustrate them with pictures and real life things. Like, if we say them to get us the first part of a pizza it would be easier for them to correlate  with a half of pizza. Thus, the pictorial understanding is better for the case of children and it is true in case of learning fractions.

Then, there is also confusion among the children regarding the proper and improper fractions. A fraction which is represented as an integral part with fractional part denoting the fraction from 0 to 1.That is 14/3 should be denote as a number with  4 as its integral part and 2/3 as the fractional part. Thus the number could be denoted as   , the fractional part is not the denoted by the fractional part of 14 but a smaller number 1, with the integral part of the number being now equal to 4.This means that the fraction in question , is four times here. Let us suppose that we have a cake and we have to divide the cake into three parts. Then we got two pieces of that cake together to denote . Now, we should have two thirds of the same cake four times, so we would need four cakes this time, each cake divided into two thirds. This way it becomes easier for us to make the child understand like this way. So, we were talking about   , which is a proper fraction because the fractional part we see here is only  which is less than 1.In proper fraction, we have a fractional part and an integral part. Here, the integral part is 4 because we have got to multiply the fraction  , four times. So, what are the improper fractions then,   which may seem odd in appearance? We can detect an improper fraction by asking a simple question that weather the number of parts we have to get from the cake is less than the total number of divisions we have done of the cake. Here, we have got to get 14 parts of the cake from a cake with only three parts! Obviously, it’s not possible. This is the case we have got with the improper fraction, but they do exist. We need to only change it in a way to make our understanding about them clear. This is the reason we took a numerically equivalent number  before  to understand the concept of improper and proper fractions.

After, being done with the improper and proper fractions we would go to have a understanding of reducing a fraction to a smaller looking fraction, but they should be equivalent numerically. Children love to do, multiplication and divisions. If we tell them they have to divide a number above and a number down the fractions, that is numerator and the denominator, they would do this work happily. So, we take a fraction for the sake of making them understand better, like    Here, we have got two numbers 12 and 18, above and below in the fraction. Now, we ask a child what should be the common factor between 12 and 18, they might not get that. So, we should tell them to divide 12 and 18, with a number common to them like 2, 3 or 6 till we are not left with the numbers which do not have anything common to them. So, following this pattern we get many fractions, like   .If we see these fractions they would seem to be different, but they are indeed equivalent to each other. This is actually a difficult task to make them understand with pen and paper. So, we would have to take a practical case to divide a cake into the desired fractions as found in the example above. With the example we would be able to understand this task easily, but still cumbersome.(CILF,2011)

Even a greater task is to make them understand the addition of two fractions. This has become a greater task because we got two fractions instead of one. So, let us get two numbers for the same purpose. Let them be so we got many different cases with varying difficulties. But for the sake of making them feel comfortable, we have tried to take two simpler fractions. The fraction   , means dividing two cakes into equal five parts. Similarly,  means dividing two cakes into three equal parts. So, the child gets it easily how to add the numbers. This is all for understanding the theoretical part of adding two numbers. But, that one is indeed a longer and lengthier process, but a simpler and important stepping stone to get the addition of the fractions. If we go through the theoretical process, we would have to ask the child to first understand the concept of Least common multiple (L.C.M).Actually, LCM is nothing but a number which helps us bring the denominators of the two fractions to a common number. This common number would help us divide the numbers in the numerator into equal number of parts. Thus the addition becomes easier, two add two numbers instead of adding two fractions.

After the addition, the subtraction comes which is a little difficult if the child has come to learn the addition of two numbers. To an elder person the subtraction would seem same as addition with one of the fractions bearing a negative sign, but for a child both are completely different tasks and hence there should be a different way to make them learn the subtraction. For the subtraction of two numbers, we need to know which one of the numbers is smaller and hence place the greater number before the smaller one. Similar is the case with the fractions, we need to identify the smaller among the two of them. This can be done easily if the child has understood the calculation of LCM. It is through LCM we make the denominators of the fractions become equal. Then relevant numbers has to be multiplied to the numerators of the fractions, so that the resulting fractions become equivalent to the earlier one. See, we have got to make the child friendly with the fractions, so that they can do their operations easily. They should get a natural feeling while playing with the fractions. So, we have made the fractions with common denominator and they are equivalent to the earlier fractions, but not necessarily the same. Now, it becomes easy task to find a greater among the fractions. After realizing the greater among the fractions, we can subtract the smaller from the bigger one. The result would be a new fraction.(Nunes et.al ,2006)

The multiplication and the division of the fraction are easier as compared to the addition and the subtraction of the fractions. This could be easily performed without the knowledge of the LCM and other things. While in multiplication we need to multiply the numerator with the numerator, denominator with the denominator and the interesting thing is that the denominators and the numerators of the two fractions has a relationship of division among them. The division is similar to multiplication but different in the starting, that in order to divide a fraction with another fraction, we need to exchange the numbers in the numerator and the denominator positions in either of the fractions. Then, the division of fractions becomes similar to the multiplication process.

Conclusion

Thus, we described how we can make the process of learning fractions and their operations, easier to a child in the middle schools. The relationship with the fractions is quite long in the academic career of a person and indeed unending if he or she has to get into the contact with the numbers regularly. The solution of many important equations, trigonometric calculations, the integral and the differential calculus and the most importantly the probability all need the concept of fractions in one way or the other.

References

1. Fractions- Teaching and Learning programme ,viewed on 2^nd may 2012, <www.tlrp.org/pub/documents/no13_nunes.pdf>

2. Fractions research , viewed on 2^nd may 2012, <http://gse.berkeley.edu/faculty/gsaxe/Research.html>

3. CILF 2011 , center of improving learning of fractions.

4. Petit, Laird and Marsden 2009, A Focus on fractions

5. Issues for Research in teaching and learning of fractions ,viewed on 2^nd may 2012, <www.hbcse.tifr.res.in/data/pdf/fractions-issues-for-research-mar-08>

6. Andrew Pollard  2011, Why do pupils find understanding fractions difficult?, viewed on 1^st may 2012,

www.rtweb.info/content/view/178/43

7. Nunes et al (2006) Fractions: difficult but crucial in mathematics learning Teaching and Learning Research Programme Research Briefing no.13.  Available at:
http://www.tlrp.org/pub/documents/no13_nunes.pdf 

8. Susan J.Lamon 2008,Teaching fractions and ratios for understanding-

8.Learner’s understanding of addition of fractions,viewed on 3^rd may,

< academic.sun.ac.za/mathed/malati/Files/Fractions992.pdf>

9. Student’s conceptual understanding of fractions

<www.merga.net.au/publications/counter.php?pub=pub_conf&id…>

10. Revisiting A theoretical Models on fractions, viewed on 2^nd may 2012,

<www.emis.de/…/PME29RRPapers/PME29Vol2CharalambousEtAl.pd…>

11. Fractions are foundational, viewed on 2^nd may 2012,

<www.nctm.org › About NCTM › President’s Corner>

12.Teaching Fractions for understanding Ratios And Proportions

<http://www.hbcse.tifr.res.in/research-development/research-areas-in-stme/mathematics-education>

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