"Sir, we have done this many times! ... 1/6 is Two one-twelfths....so (half)/6 is One-twelfth..."

Setting: Some 6th std government school students in an After-school Maths enrichment program

I knew that they had been taught the fraction arithmetic in their school. So when I gave them couple of problems to work out, either some of them arrived at non-sense answers (of course, not their mistake) or some of them were applying the 'bunch of rules' incorrectly and there were also some who were able to successfully recall and apply the rules to get the correct answers, but it didn't take much probing from my side to make them realize that this was mere answer-getting and not Understanding ! 

So we spent some time (few sessions - really few!) to understand fractions - through context based problems and pictures (but no manipulatives). And after that I gave few problems to them for practice. The only rule we applied was to solve without any rules! 

I would like to share with you the approach of one of these students to solve these 3 problems. I am sure, it will delight you as much as it did to me :)

This is how she had presented her work to me. I am sure, what would most probably catch your attention is her answer to the second problem (b).

Our (holy) text books don't have space for such weird-looking fractions, but my students are very comfortable understanding & playing with such creatures :)

I first spent enough time to carefully study her work, made a guess of how she might have thought and solved to get these results, and then finally called her & asked her to provide an explanation for each of these results. 

And this is what she gave me :

I would strongly suggest you to stop reading further and spare some time studying each of her 3 solutions.. Don't panic or give up if it takes more time than usual :)

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This work was a delicious dish for me. So when my heart had enough of it, I decided to call her. I wanted to give her a big hug for this feat !!  But then I contained my emotions and rather asked her a (stupid?) question -

# I saw your pictorial explanation for your first solution. Good attempt. But I am unable to understand it very clearly. Can you plz explain?

"Sir, we want to remove 4 and 3/4 from 17 and 1/2. We can remove 4 from 17 but cant remove 1/2 from 3/4. So I split 17 into 2 parts: 16 +  1 1/2.  

Now 16 - 4 = 12   and   1  1/2 - 3/4  = 3/4.  So the answer is 12 and 3/4

# Okay. Well done. What about the 2nd (b) problem now... 

"We want to do 3/4 + 5/6 ..  Both 3/4 and 5/6 are bigger than Half. So we first add these two halves to get one whole.... Now what remains to be added is 1/4 and 2/6.......... Now I know that 1/4 + 1/4 = 1/2 ..... I observed that 2/6 has 1/4 included in it ........ So I....."

# Wait..Wait... How do you know that 2/6 is more than 1/4 ?

"Because 6/6 is whole.... so 3/6 is Half..... So (1.5)/6 is 1/4....... and 2/6 is more than (1.5)/ 6

So 2/6 = 1/4 + (Half)/6 ...."

# Oh..good one ! Then?

"So now we can add the previous whole and this new half to get 1 and 1/2 . And we further need to add the left over (half)/6 .... Now we know that this (half)/6  is same as 1/12..."

# How?

"Sir, we have done this many times! ... 1/6 is Two one-twelfths....so (half)/6 is One-twelfth..."

# Oh yes! 

(How dumb of me to not remember that we had discussed this many times, isn't it ? :-)

# So then?

" Now to add 1/2 and 1/12, we see that 1/2 has 6 twelfths in it.... So their sum = 7/12..  And hence final answer is 1 whole and 7/12"

# Got it...... But your earlier answer is (9 1/2) / 6. And now you have got 1 and 7/12... How come two different sums for the same pair of numbers?

I could see that she was struggling to figure out how she had got  (9.5)/6 .... So I rather asked her --

# Okay, can we check if both these answers are equivalent or not...."

"Sir, (9.5)/6 is more than 6/6 ( whole ).... whats left is 3.5 / 6 .....   3/6  is Half......  and now (0.5) / 6  which is = 1/12  .... so yes we are getting the same answer as second one...."

#  Nice...  But I am curious to know how you got (9.5)/6 .... 

"Sir, even I am unable to find out now.... "  And she started laughing aloud :)

# Hmm.... So I hope you 'now' understand the importance of 'writing' an explanation..

"Yes sir...."

So I thought of helping her now.... (Remember I had studied and guessed?)

# I think you kept your whole as 6/6 here rather than writing it as 1 whole...

I paused here.... And as expected, she picked up from here....

"Yes,..... I got it...... 3/4 + 1/4 = 6/6 .....   So now, whats left to be added is 5/6 - (1.5)/6 = (4.5)/6 .....  6/6 + (4.5)/6 = (9.5)/6 ....."   

And she again started laughing at this point :-))

I asked her to explain the 3rd solution also......

Yes, she did beautifully and confidently explained it.... But I would allow / invite you to study her work to know what she has thought and how she has solved this one.....

I am sure you will get it (after learning so much from her :), but if you can't then let me know, I will share with you her approach....

I will be glad, if you can share your views about this post, her approach, my approach etc.

Thanks and Regards

Rupesh Gesota