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.