Help required for lecture/workshop - CBSE (India) - Std IX?

IX and X stds. is the base for advanced maths. On the first day of my tuition of XII std. students, I used to emphasize on two fundamental concepts:

1 ) Division by zero is absurd. To explain this, I used to narrate a fallacy as under.

Let x = y

=> x^2 = xy

=> x^2 - y^2 = xy - y^2

=> (x - y) (x + y) = y (x - y)

=> x + y = y

=> x + x = x [because y = x]

=> 2x = x

=> 2 = 1.

First ask students to find the error in the steps. Some may know, while others will learn for the first time.

x = y => x - y = 0 and cancelling x - y from both sides amounted to dividing by zero which resulted in the incorrect answer.

The reason why this simple looking concept is very important is because even the best of text-book authors end up making the error of dividing by zero and getting incorrect result. That can happen if we are not careful in checking all division steps carefully.

2)

Square-root of any positive number is positive only. I used to ask students a question, what is √4. Some used to answer ±2 which is incorrect. It is true that there are two square-roots of 4, which are - 2 and + 2, but √4 = 2 and not - 2. If we write steps of the equation

x^2 = 4 => x = √4 = ±2, it would be incorrect. Proper steps are

x^2 = 4 => x = ± √4 = ±2.

Next question asked was what is √(x^2). Again, many students will answer √(x^2) = +x, thinking that now they will be right, but again that is incorrect. √(x^2) = l x l and not x as

√(x^2) is positive and if x is negative, then √(x^2) = +x will be incorrect. Thus, √(cos^2 x) = l cosx l.

Students should be advised to read the text-book thoroughly. My experience as a tutor is that most students don't do it and the teacher in the class also hurry up teaching theory and fgo to the problems of the exercise due to burden of having to complete the syllabus in time. I recall one explanation of sequences discussed in the text-book which was not taught in the class and is so important. I quote a para from XI std. text-book of Gujarat State as under.

"Let us note that if the first three, four (or more) terms of a sequence are given, then sometimes we can guess what a general nth term would be, but such a gues may or may not be right. This is because there can be more than one distinct sequences having the same first four or five terms, e.g., if the first four terms of a sequence are 1, 3, 5 and 7, we might guess that the fifth term must be 9 and the nth term must be 2n - 1for all n. But it is possible that the nth term may be

f(n) = (n-1)(n-2)(n-3)(n-4) + (2n-1).

Note that f(1) = 1, f(2) = 3, f(3) = 5 and f(4) = 7, but f(5) is not 9 but is 33.

So knowing only first few terms cannot really determine the sequence completely."

In order to make the lecture interesting, it should also contain some non-textural material like

1) simple calculation methods which the students may find amazing like how to square a number ending with 5 such as 35 x 35 = 1225 follows from writing the last two digits as 25 always and the initial digits as 3 x 4 = 12. Thus, 75 x 75 = 5625 (because 7x8=56) and 105 x 105 = 11025, 295 x 295 = 87025 (because 29x30 = 870). Multiplying a number by 11 such as 27 x 11 = 297 (obtained by inserting sum of 2 and 7 = 9 in-between) and 85 x 11 = 935 (8+5=13 => 3 in the middle and 1 added to 8). Also, any large number can be easily multiplied by 11 by keeping on adding subsequent terms. For example, 2345 x 11 = 25795 (unit place 5 as it is, tens place = 5+4, hundredths place = 4+3, then 3+2, then 2.).

2) Puzzles like making 3x3 or 5x5 magic square, in which the sum of digits in rows, columns and diagonals add up to a constant.

3) Magic tricks like telling the sum of five 4-digit numbers beforehand. Ask the student to tell a number. Suppose he tells 2549. Write the answer 22547 beforehand subtracting 2 and adding 2 in front in a piece of paper and keep it folded on the table. Let another student tell the next 4 digit number. If he says 1342, then you immediately write the third number as 8657 (note that you selected the number by subtracting each digit of the second number from 9). Then the fourth number is given by a new student. If he gives 5497, you add your number 4503 and ask the students to add them all up. When they come up with the answer 22547, open the folded paper and show that you had written it beforehand.

Topics of algebra:

1) Ask students to do the tricky factorization like x^4 + 64 and teach them to do it.

2) Ask them to prove if a+b+c=0, then a^3+b^3+c^3=3abc and them teach them.

Topics of geometry:

1) There are a large number of proofs of Pythagorus theorem. Teach them one easy proof.

2) Explain how Pythagorean triples can be generated using

n^2 - m^2, 2mn and n^2 + m^2 taking two distinct values of m and n.

m=1 and n=2 gives (3, 4, 5)

m=2 and n=3 gives (5, 12, 13) and so on.

3) Ask them how many circles can pass through any three non-collinear points in space and then explain that a unique circle can pass and method of constructing it.

Trigo:

Ask and teach to find the value of

secx given that secx + tanx = a.

As limited time is left to answer, I sum up with the following:

I used the following method to study math which I had found very useful in getting me good results in the exam.

(1) I used to read the text-book carefully including all illustrative examples.

(2) I used to prepare my notes of useful formula to be memorized.

(3) I used to solve all problems of the exercise and then check the answers at the back. I used to tick mark those that I failed to answer.

(4) In the next revision, I used to attempt only those which were tick-marked earlier and again tick mark the problems that I failed to do in the revision. They had 2 tick marks.

(5) In the third revision, there will be problems that will get third tick mark. These are the ones that I failed to do even in the third revision.

(6) Day before the exam, I used to practise only the problems that had 3 tick marks which used to few and which I needed to revise.

(7) In the examination, while rechecking the paper, student should check that he copied the question correctly. Sometimes it so happens that one does everything correct, but there was an error in the question copied from the question paper.

Any amount of writeup is inadequate to answer this question and I could not write in organized way due to the wide coverage of the question.

I wish you prove to be such a good lecturer in whose lecture no one sleeps and you are loudly applauded at the end of the lecture. In entertainmengt programs, people shout "once more" if they like it. I wish students want you to give more lectures after hearing you.

It is usually better if one has some idea about

the kind of students one would be lecturing to.

Since it is your son's school, I am sure his class

has some idea of your forays into teaching of

math and you have some idea of the quality of

students you will be talking to.

What I would implore you to do, however, is to

motivate your impressionable audience to start

preparing for the competitive exams like AIEEE

and IIT-JEE right from grades 8, 9. The sooner the

children get an idea of the demands made by these

exams on their mental ability, the better.

I suggest you include a few simple questions in your talk

which are not a single-topic oriented but make the students

combine their knowledge of more than one topic in their

answer. For example, ask them : If a, b, c are in A.P., then

the line ax + by + c = 0 always passes through a fixed point

whose co-ordinates are ( ... , ... ). Fill in the blanks.

This question combines progressions and geometry.

The response of students to such questions will also tell

you which of them are IIT-material.

Since you have good knowledge of all three: Physics,

Chemistry and Math, I am sure your lecture is going to

be a memorable one for students.

I wish I could attend this lecture.

Good Luck.

r u sure ur not being arrogant? how do u know u know enough to study Class X. on what baiss do u say u've completed the syllabus at home? just reading the book? solving sums from RD Sharma? learning the maps? i'm sorry but a lot more needs to be done. ur either very smart or too smart. DO NOT challange the authority of the CBSE, u may not think so, but it is a VERY VERY powerful organisation that controls the future of millions of kids. according to them, u HAVE TO complete ONE YEAR in class 9 to be promoted. if u try to make a fake doc or something of the like, u r liable to be banned from ANY (of all boards) education institutions for 3 or 4 yrs. so be careful and these are not arrogant rules. they are meant just for people like u: who do not know WHAT makes a complete course. no offence, but just learning the books DOES NOT make u intelligent. so have fun in class 9, enjoy ur teens, and DO NOT challange CBSE PS: how did u complete ur practicals at home my dear? if u donno how to write a record book, u gotta learn in class 9, or ur gonna flop in class 10

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You will probably get a better response if you post this under the category

"Education & Reference" / "Teaching".

Good luck.

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