By means of formulas that students have become familiar with in arithmetic, algebra can be presented as a generalized arithmetic. It is more powerful than arithmetic, as it furnishes easier and more reliable ways of solving problems. It can be shown that algebraic equations have definite solutions and beginners can be shown that they can check their work very readily and have a positive assurance that their work is correct. It can be shown that the language of algebra makes reasoning easier, ?that algebraic notation is a method of expressing numbers by letters and figures and that the number relations expressing opposite directions, kind or quality, can be indicated by positive and negative numbers, while in arithmetic size is the only relation that can be expressed. The important objective that must be made apparent is the use of algebra as a powerful tool for problem solving. The fact that algebra is a source of mental training and logical reasoning should be kept in the background. The early work in algebra should not be done hurriedly. The power to interpret the meaning of algebraic expressions and the power to express relations in algebraic language should be most carefully developed. Concrete presentations by means of effective devices make the work real. Many examples may be given to illustrate the nature of positive and negative numbers. The thermometer may be used successfully here. Indicate the reading above zero, positive and below, negative and then calculate the temperate according to changes or calculate changes from successive readings. Concrete situations may also be gotten from pupils’ activities and interests. A boy who was having some difficulty with his algebra after having several days work on positive and negative numbers was made to see the significance of negative numbers by the use of the familiar expression “so many in the hole” and, thereafter, was able to compute readily in terms of signed numbers, for now they had some real meaning to him. Errors in algebra are often due to wrong experiences or to lack of experiences from the first. The old adage that an ounce of prevention is worth a pound of cure is applicable to all learning. Much remedial work in algebra would be un-necessary if the proper precautions were taken to lay the foundation of the subject on actual experience, so that students may be able to see the reason in certain procedures rather than rely on mere memorization of certain rules and formulas. After students grasp underlying principles, they are able to formulate rules for themselves. A background of reasoning serves as a bulwark at all times against future errors. The “general lack of understanding of mathematical method” referred to by Dr. Webber in his article “Some Common Errors in Algebra” is partly to lack of understanding of fundamental principles in the first learning and, also, to lack of proper drill provisions to fix those principles during the learning period. The drills are frequently weak to the extent that they give over-practice on simple phases of the work and little practice on those phases that give greater difficulty.
The difference in emphases between Standards-based and more traditional algebra curricula suggests a need for a detailed comparison of such curricula and the performance of students who study from them. It has been argued that detailed cognitive analyses of students’ responses, including solution strategies, justifications, and modes of representation, are important in understanding similarities and differences between groups in cross-national studies. We suggest that detailed analyses of students’ solutions are also important in understanding similarities and differences in achievement in comparative studies of mathematics curricula within a country. Specifically, the following questions are addressed in this brief report:
• What strategies do students studying algebra use to solve algebra problems?
• How are students’ strategies related to their achievement in solving these problems?
• How are students’ strategies related to the curriculum?
The success of any undertaking depends upon two vitally significant facts:?first, a well-organized plan of what you expect to accomplish, with the necessary materials and tools needed; and, second, cooperation between all concerned to bring about desired results. Pupils come up from the grades knowing little about the content of high school subjects or their significance. Their background is limited and most frequently they are prejudiced in favor of or against a subject. Certain subjects would be tabooed if students had their say in the matter. Algebra is one of these subjects and has been classed as “hard” because of the number of failures among beginners. These failures in algebra can be assigned to one or more of three causes psychological, physical or pedagogical. From the psychological point of view the student may be lacking in mathematical abilities a deficiency which may or may not be corrected, according to whether they are inherent or due to lack of training. But the student may have approached algebra from a wrong angle and this is an important point in the teaching of the subject and must be considered. From the physical point of view, apart from any personal physical handicaps, the atmosphere of the classroom, including the physical situation and the teacher’s attitude, has much to do with the advancement of a student in his progress in learning. An active spirit of cooperation between teacher and pupil is absolutely necessary to a student’s progress. From the pedagogical point of view, even with the advance that has been made in the teaching of mathematics, educators are confident that still more can be done to improve the pedagogy of the subject in order that greater interest may be aroused in the students and better results attained.
An “inspirational preview” or “a bird’s eye view” of the subject is one of the most valuable approaches to the study of algebra for awakening interest and putting before the class reasons for studying algebra and some of the things that are expected during the course. The important thing is to give the students an inviting introduction to it so that they will anticipate pleasure in the study of the subject. Then, it is the privilege of the instructor to see that this interest is kept up.
Most high school students dread algebra as it is considered as one of the tough subjects. Being an advanced discipline, algebra contains difficult syntaxes and abstract concepts which often make students struggle. While you can find tomes of knowledge on algebra and how to handle the subject, it sometimes becomes difficult to solve the problems without taking outside help.
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Till date, you have learnt any number, negative or positive, when squared can’t ever be a negative integer. Very true, it’s also practically proven that when you multiply a negative with another negative, it’s got to be positive. But till date, you haven’t come across “iota”. Yes, iota denoted by “i “, is an imaginary number which squares up to –1.
i² = -1
For argument’s sake we can say, since i=√-1,
Hence, i²= (√-1)² = √(-1)² = √1 = 1 , but this is not correct and is not even necessary in this case as we already have two numbers, ±1 , which squares up to 1, so “iota” wouldn’t have been needed.
Also, if we see the exponential series of iota (i) follow a cyclic pattern.
i= √-1 i5= (i4) (i1) = 1 (i1) = (i)
i²= -1 i6= (i4) (i2) = 1(-1) = -1
i³= (-i) i7= (i4) (i3) = 1(-i) = (-i)
i4= 1 i8= (i4) (i4) = 1(1) = 1
So, it’s evident that whenever you need to calculate any higher exponents of (i) it’s always easy and advisable that you convert the figure in the exponent into multiples of 4 and the solution will be either (i) or (i2) or (i3). In few cases it can (i0) too, i.e. 1, as anything raised to the power of zero is equals to 1 which also applies to (i). For example:
i400= i4×100+0= 1100 X i0= 1 X 1 = 1
This number, imaginary or iota as you may call it, helps you solve negative figures within square roots, but it also restricts you to use the traditional rules of squares and square roots. Hence, whenever you are solving a complex number numerical, solve the i-part first.
But what are complex numbers then?
Complex numbers are any number that has a real as well as imaginary part to it. For example, (a + bi) is a basic ‘standard’ complex number where “a” is the real part and “bi” is the imaginary part. We will now see mathematical operations on complex numbers.
(3+4i) + (4–3i) = (3+4) + (4i–3i) = (7+i) [Note: you can only simplify the operation in these cases]
(6–7i) – (4–2i) = (6–4–7i+2i) = (2–5i)
(3–2i)(2+3i) = (6–4i–4i–6i²) = (6–8i–6(–1)) [since i²= -1] i.e.= (6–8i+6) = (12–8i)
A simple one would be (4/5i). Now as (i) can’t be kept in the denominator hence we will multiply both numerator and denominator by (i) i.e. (4i/5i²) = (4i/5(-1)) = (4i/-5) = -4/5(i)
Now let’s see a tough one:
6/(3+i) Even if we multiply by (i) it won’t help. Here we need “conjugates”. A conjugate is a complex number similar to another complex number with only the sign different in the middle i.e. (a+bi) is conjugate of (a–bi) & if we multiply both:
(a+bi)(a–bi) = a²-b²(i²) = a² – b²(-1) = a²+b²
6(3–i)/(3+i)(3–i) =(18–6i)/(3²+1²) = (18–6i)/10 = ( 18/10)– (6/10)(i)
This is all about complex numbers!