Happy Teachers' Day

today, some people got their special holiday, since today is the Teacher's Day! kepada semua insan-insan yang bergelar guru, tak kira guru ugama kah, melayu kah, inggeris, guru terlatih, pegawai pelajaran, mana saja tah...dengan ikhlas diucapkan Selamat Hari Guru, semoga apa yang kita lakukan, kita lakukan kerana Allah, dan demi melihat negara dan tunas bangsa maju dan berjaya, didunia dan akhirat.

small mistake that leads to BIG mistake

not really to point out the mistakes done by students, rather to correct the mistakes that can be prevented in math.

say, if the teacher asked as simple question (on formula) such as "what is the formula to find the area of triangle?", most students would reply "half times base times height". from this, the students will then write down in their paper as 1/2 x base x height. for me, I would say that this is very incorrect. why? read on, and you will be glad that you read. this is based on real situations and discussions.

Question: A triangle has a base of 10 cm. Given that its area is 20 squared cm, calculate the height of the triangle.

Common replies/mistakes done by students:
1/2 x base x height [a]
= 1/2 x 10 x 20 [b]
= 100 cm. [c]

ok. i am not making this up. the question is given to my year 11 (form 5) students, about 2 years ago when I was doing part-time tutoring. this question is so simple that I was very confident, all 10 students in that class would get it correct. it turned out that I was WRONG! Seeing this working (as shown above), I straight away spot the mistake.

Yes they got the idea right, the question is about area of triangle, which is shown in the line [a] that they have written. HOWEVER, they made mistake in writing down the formula. I always said this to my students, "WHEN YOU ARE WRITING ANY FORMULA, WHETHER IT IS SIMPLE, SHORT OR WHATEVER IT IS, MAKE SURE YOU WRITE DOWN THE COMPLETE FORMULA!"

Back to line [a]. for me, 1/2 x base x height is meaningless. I mean, what is 1/2 x base x height? Is it a formula? I would say no, since it does not have any indication or label. It is an improper way of writing down the formula. Because of this, students would mix up the information given by the question into their own version of the formula, as shown in line [b]. this student know the value for the base (10 cm) but when it comes to the value of height, this student got confused/lost. He/she might think "what is the height?? I am supposed to find the height, but where do I put this 20 cm in this formula?" Due to this confusion, the student then bet on their luck and put the 20 cm under the height space. See what I mean?

BETTER VERSION. If and only if, the student write down the complete formula, then it is almost impossible to make the mistake mentioned above.

Area of triangle = 1/2 x base x height

To make things simple, just remember this. Take formula as a puzzle, where you need to fill in as many empty spaces according to the clue given just to solve the question. In this question, we know the area of triangle (as given in the question) is 20 squared cm, so we fill 20 in the area of triangle space.

20 squared cm = 1/2 x base x height

Then, read again the question for the next clue. Do we know the value of base? Sure we do! It is 10 cm. Again, fill in the base space with 10 cm, to get

20 cm2 = 1/2 x 10 cm x height

From this, we are left with height. We don't have any value left to be picked from the question. Moreover, we are asked to find the height, so this is the complete set already. solve this!

20 = 1/2 x 10 x height
20 = 5 x height
20/5 = height
4 = height

From this discussion, what I am trying to emphasize is the importance of writing down the complete formula in solving math. Believe me, you won't regret it if you do what I suggest here. Because, formula are meant to be written in complete set so that you won't get mixed up or lost when solving them. GOOD LUCK!

a very useful math program for students

while browsing around the internet looking for teaching materials, i came across this advert. i thought it was another of that spam thing. however, since the advert was very convincing (it even dare to give a full warranty of its originality), i tried my luck to click on it, only to find out that it IS for real.

this program is very small (the portable version), only eating around 6 mb of your computer's memory. i really recommend this program to all students since it is very useful, simple and student-friendly. it will help you to understand math better since it provides all the steps in solving math, especially algebra. the best thing is, it covers math and also additional math (such as differentiation, integration). so to students, grab this file now.

http://rapidshare.com/files/141817603/P._Algebrator_4.0.1_by_yd.rar

math brainteaser

was browsing through my last year's postings in my blog, and i came across this math brainteaser. hehe...very good question. seems very hard, challenging, however when you got the answer, you will be like "laaa..."

You are on your way to visit your auntie, who lives at the end of the valley. It's her birthday, and you want to give her the cakes you've made.

Between your house and her house, you have to cross 7 bridges, and as it goes in the land of make believe, there is a troll under every bridge! Each troll, quite rightly, insists that you pay a troll toll. Before you can cross their bridge, you have to give them half of the cakes you are carrying, but as they are kind trolls, they each give you back a single cake.

How many cakes do you have to leave home with to make sure that you arrive at auntie's with exactly 2 cakes?

your answer? post them in the message board.

assignment for 4B (updated with answers)

1. The sides of a rectangle are given as x cm and y cm, where

17.5 ≤ x ≤ 18.5 and 11 ≤ y ≤ 12.

Calculate

(i) The smallest possible value of the perimeter of the rectangle,

(ii) The largest possible value of the area of the rectangle

2. A restaurant owner pays a waiter an amount of $A per week. The amount is made up of a basic wage of $ 60 plus 11 cents for each of the n customers he serves.

The formula connecting A and n in this case is

A = 60 + 11n/100

(i) Calculate the amount of money the waiter received in a week when he served 240 customers. [ans : $86.40]

(ii) At the end of another week, the waiter received $ 115. How many customers did he serve? [ans : 500]

(iii) The owner of the restaurant decides to decrease the waiter’s basic wage to $ 45 but to increase the pay per customers to 17 cents. Write down the new formula connecting A and n. [ans : A = 45 + 17n/100]

(iv) Find the number of customers the waiter would have to serve in a week for him to receive the same amount of money whichever formula is used. [ans = 250]

classwork (TIME)

2. (a) A late night radio programme began at 2245 one evening and finished at 0320 on the following morning. For how many minutes did the programme last?

(b) A small bird enters its nest on average every 40 seconds, when it is feeding its young. How many visits to the nest does the bird make, on average, each hour?

6. (a) An overnight train left Singapore at 2240 and reached Kuala Lumpur at 0515 on the following day. How long did the journey take? Give your answer in hours and minutes.

(b) A worker in an electronics factory can complete one circuit panel in 3 minutes 25 seconds. Assuming that he continues to work at the same rate, calculate how long it will take him to complete ten identical circuit panels. Give your answer in minutes and seconds.

10. (a) Calculate the number of minutes there are in a day.

(b) (i) An aircraft was due to take off from Hong Kong at 1645. Because of bad weather it did not take off until 2110. For how long, in hours and minutes, was it delayed?

(ii) The aircraft developed a fault and returned to Hong Kong after a flight lasting 5 hours 25 minutes. At what time the next morning did it land?

14. (a) Calculate the number of minutes between noon and midnight.

(b) (i) An aircraft was due to take off from Singapore at 1740. Because of the bad weather it did not take off until 2215. For how long, in hours and minutes, was it delayed?

(ii) The aircraft developed a fault and returned to Singapore after a flight lasting 4 hours 10 minutes. At what time the next morning did it land?

17. (a) (i) A television programme lasted for 1 hour 25 minutes. The programme started at 09 40. At what time did it finish?

(ii) This programme was one of a serried of 5 programmes each of which lasted for 1 hour 25 minutes. How long did the 5 programmes last together?

(b) Another television programme lasted for 2 hours 12 minutes and finished at 0145. At what time did the programme start?

23. The diagram shows a timetable, part of which has been torn away, for trains from Manchester to London.

Manchester (depart) London (arrive)

08 35 11 25

12 31 15 19

17 15 20

22 37

(a) How many minutes does the train which departs at 1231 take to reach London?

(b) The 2237 train takes 2 hours 45 minutes to reach London. At what time does it arrive?

year 10 assignment

Mr Edwards, Mr Jones and Mr Parry each decided to buy a new car which was priced in the showroom at $ 7200.

(a) Mr Edwards offered his old car in part exchange and the salesman allowed him $ 5850 towards the cost of the new car. Calculate how much more Mr Edwards had to pay for his new car. [ans : $ 1350]

(b) Mr Jones paid for his new car in cash and was given a discount. Given that he paid $ 6120 for his new car, calculate the percentage discount he received. [ans : 15%]

(c) Mr Parry agreed to pay 40% of the showroom price of the car as a deposit and the balance in equal monthly instalments over a period of 2 years. Calculate the amount of each monthly instalments. [ans : $ 180]

(d) The salesman had hoped to sell each new car for $ 7200 so that he could make a profit of 20% on the cost price. Calculate the cost price of each new car. [ans : $ 6000]

Hints

(a) Mr Edwards was virtually paying the new car with his old car. The salesman agreed, but said that his old car only worth $ 5850. So, Mr Edwards need to pay the remainder, but by how much? (very easy question).

(b) The discounted price of the new car is $ 6120. So, how much is the discount? Use the “before after” method.

(c) First, find 40% of $7200.

Secondly, deduct your answer from $7200.

Lastly, this remainder is to be divided by the no. of months in 2 years. (of course you know how many months)

(d) We are looking for the original price here.

connections between fractions and decimals

do you know that, SOMETIMES when we have a problem in fractions, it is a good idea to try to change it into decimals to make things easier? also, if we have problem doing decimals, we could make things easier by changing it into fractions? yes, they are really connected, or in other words, interchangeable.

say, 1/2 + 1/4, how do you approach this question? we need the HCF of the denominator, etc...
hey, what if we change them into decimals? 1/2 is 0.5 and 1/4 is 0.25, so the answer is 0.75. wahh...so quick, isn't it?

in order to do this properly and efficiently, you need to have a very good number sense. say, 1/2 is half, so half of a number is 0.5 (everyone knows that). a quarter, 1/4 is 0.25, three-quarter, 3/4 is 0.75 etc. yes, a lot of them to be memorized. however, it makes life easier. really... here i list down some of the decimals that could be crucial in ANY calculations.

1/2 = 0.5
1/4 = 0.25
3/4 = 0.75
1/5 = 0.2

for example, take a look at this very famous question.
0.1 x 0.1
the most common mistake that students did was when they gave 0.1 as the answer. why this answer? because (most probably), they did the multiplication "directly", i.e. 0 x 0 = 1 and 1 x 1 = 1, so the answer is also 0.1.

better way?
0.1 x 0.1 can also be expressed as fraction as
1/10 x 1/10
now multiply numerators and denominators, we will get
1/100
which if we convert into decimals, we will get
0.01.

the same goes with the question 0.2 x 0.3, in which the answer is 0.06, not 0.6.

converting these numbers from one form to another is very useful, especially when we are dealing with multiplication (as shown above) and division. another example?

0.8 divide by 0.2.

some teacher taught the moving the decimal trick, but i really don't emphasize that trick to my students. we will discuss that later. now, lets convert these decimals into fraction, then we get

8/10 divide 2/10, and we can change divide to multiplication, and at the same time taking the reciprocal of the fraction on the right, to get

8/10 x 10/2

now, we cancel the 10s, and divide 8 with 2 to get 4 as our answer.

sounds tedious huh? well, first impression is not always true. again, i emphasize, this technique works very well for majority of the questions, NOT all questions. say we cannot do
1/3 + 1/2
by converting them into decimals, since 1/3 is a recurring decimal of 0.3333333...
I always emphasize this; use our own logics.

by the way, before i end this topic, i remember giving out these questions to my tutorial students:
(i) 0.2 x 0.8
(ii) 1.2 x 0.4
this "average" student tried to point out that my technique is useless, since he can calculate the first question directly, i.e. 0x0 =0 and 2x8 = 16, so the answer is 0.16. i just smiled to him, and asked him to do the second question on the whiteboard, and show his technique to the rest of his friends. proudly, he said this.
" this is easy...1x0 = 0 and 2x4 = 8, so the answer is 0.8. "

for readers, what is wrong here?
what should be the answer?

believe me, since that night, this student never argued with my technique, anymore... :-)

basics first

okay. let's first review what we were being taught in school in doing the famous four operations; + - x /. during those time, i thought that BODMAS is the proper way of doing the four operations. the fact is...it is misleading IF we are not familiar with number concepts. we were taught - do the brackets first, then division, then multiply, then addition and lastly we must do the subtraction. this is very incorrect.

the proper way of viewing order of these operations is, yes we must do the brackets first, then either division OR multiplication, and either subtraction OR addition comes last, doing the ones that comes first from the left of the expression.

1. BRACKETS
2. DIVISION OR MULTIPLICATION
3. ADDITION OR SUBTRACTION

take a look at this example. 10 - 3 + 5. looks simple? personally, i don't think so. one time (last year if i'm not mistaken) i gave this question to my form 1 (year 7) students, and out of 24 students, only 2 got it right. what was their response?

10 - 3 + 5
= 10 - 8
= 2

can you spot what is wrong?

some people thought this is funny, but believe it or not, i did try giving this same question to my form 3 students (year 9) and form 5 students (year 11) from my tutorial school, and yes some of them still got this wrong. the response? as shown above.

so, how did they end up having this as their answer? BODMAS!!! According to this system, we must do addition first, so they added 3 to 5 to get 8, then they do the subtraction to obtain 2 as their answer. I am not saying they are wrong, this is what they called misconception.

if you are very familiar with directed numbers, even if you follow the BODMAS system, you will get the correct answer. why?

10 - 3 + 5

[by directed numbers, we MUST take note of the sign in front of the number. in this case, 3 is not a positive number. instead, it is a NEGATIVE number. so we have - 3 + 5 to obtain 2.]

= 10 + 2
= 12 ans.

we will obtain the same answer if we follow the order of operations properly, i.e. addition OR subtraction, which comes first from the left should be done first. in this case, subtraction comes first, so we do them first.

10 - 3 + 5
= 7 + 5
= 12 ans.

And this morning, i purposely gave the question below to my form 4 students. believe me, out of a total of about 60 students from 3 different classes, only about 20 of them got it right. the question?

7 - 2 x 3 - 2

what is your answer? drop them in the message board and see if you got it right. of course, no calculator. hehe...cheers

My introduction

Assalamualaikum and hello to reader(s).

first of all, this tutorial blog is part of my initiative to broaden my teaching, thus reaching as many students out there as possible. Bear in mind though, that my tutorial focuses mainly on Mathematics, since i am a Math educator myself.

What I am trying to do is, Insya-Allah, post as many math questions as possible and at the same time, discuss the solution of that answer. I will try as much as possible to make the workings simple and easy to digest. At the same time, you can leave some message to me, ask me how to work out the answer and have some discussion.

One last thing though. The math that i will discuss here in this blog mainly for Year 11 below. Year 12 and above? hmmm...maybe some other time, although I do know how to do that. Again, this is part of trial, so I am trying to focus on Year 11 and below for the meantime.