Some time back, I had shared how a group of students added fractions visually, without using any procedure or rule. This is the link to that classroom experience:

The above post also points out how students (with weaker conceptual understanding of fractions) generally confuse or misinterpret 1/3 as sum of 1/4 and (Half of 1/4)... this guess probably gets triggered because the small extra amount (corresponding to 1/3 - 1/4) seems to be quite close to Half of 1/4, if the figures are drawn roughly or not so accurately.....and they dont have enough experience / exposure that looks can be deceptive over here and there can be two different fractions with very small difference......

I would suggest you to read that post (if you haven't yet) to know about the argument given by one of the students as to why this extra amount (i.e. 1/3 - 1/4) cant be Half of 1/4. It was music to a maths teacher :)

Little did I know that I would be facing a similar situation so soon... However, what motivated me to share this experience with you is - that there is some twist in this tale :-)

__The problem at hand was -- (about proportional reasoning)__

**If a group of workers can make 4 walls in 6 days, then how long will they take to make 1 wall?**

This is how some of them started -

6 days --> 4 walls

3 days --> 2 walls

1.5 days --> 1 wall

1 day --> 3/4 wall

I am sure you would have noticed the (common) flaw in the last step...

So I asked them how to verify this result...

**One of them suggested lets trace back...**
1 day --> 3/4 wall

2 days ---> 1.5 walls

6 days ----> 4.5 walls

So they realized that since we didnt get back to the given condition from 3/4 wall, it means its incorrect.

So then they were stuck.. How to figure out?

I had realized that they were stuck because of the arrival of fraction - 1.5 days - in the second last step... Also, I was sure they generally remember more common fractions like quarter, half, 3-quarters and tend to forget the other not-so-commonly-used-fractions. So first I asked them -

*What do I get if I remove some amount from half?*

Most of them, as expected, shouted immediately -- Quarter...

*Are you sure?*

Yes.. (instant uproar)

I paused for a while...

*No one has any doubt...??*

To this, one of them - Yash - said --- "we can have 1/3 also...."

Yes !! I was so desperately waiting for such an un-common fraction :-))

I looked at others.... Many looked puzzled... So I asked him to come forward and explain it on the board... He drew a circle and divided it into 3 equal parts, and explained how each is 1/3 and we saw that there are 3 one-thirds in a whole... Class agreed with him....

So with this background now, I headed for the second part --

*What do we mean by 1.5 days?*

1 and half days...

*Ok... So this means it has how many half days?*

3 half days...

*So now, if 3 half days correspond to 1 wall... then 1 half day corresponds to.... ??*(I paused)

Silence for @ 5-6 seconds... And again, the same voice --- "Sir, it will be 2/3 wall"

I was so glad..... but the class was still wondering.... I invited him to again come & explain with the help of pictures.....

This is what he did on the board:

He didn't draw both the steps as I showed above.... He erased one part from both the sides in the 1st step (equation) so that he can find the result corresponding to 1 day (2 halves).. which is 2/3 wall.....

Not everyone in the class understood him well.. But some did.... And so, I called up one among these -- Sania - some to come and explain....

She rephrased it very well.... and I could see a sense of satisfation on everyone's face :)

But then as we were celebrating this understanding, Sania popped up again, but with a tone of surprise this time ----

"Sir, if we remove half from these 2/3, then the remaining two small pieces will join together to make a quarter...."

Oh !! Her remark took me to Past...... Because exactly same comment / guess was made by a student in another class, few months back (this incident is described in the previous post whose link is shared above)

I didnt do the mistake of losing this golden opportunity - I just grabbed it !!

*So it means 2/3 is same as 3-quarters, isn't it ?,*I asked her with confident tone :-)

And she agreed to this...

I drew the attention of class to this unfolding interesting conversation.... To this, most of them agreed like her... (see the diagrams / scribbling above...)... but Adarsh argued --

Sir, how can be 2/3 same as 3/4 ? We just saw that 3/4 wall was an incorrect result.... It gave us 4.5 walls and does not take us back to the given condition i.e. 4 walls....

Again, there were some toggles.... rest got this point when he pointed out to the matter written on the board....

Though this was a pretty good counter to why 2/3 and 3/4 are not same, however I realized that we are missing something very important at this junction of misunderstanding....

So I again sparked off the debate --

*Luckily, we had this problem where there was reference to 3/4.... and we had worked with 3/4 to conclude its incorrect...... But what if we didn't have this problem to refer to? If we were in some other context..... How would we then find out if 2/3 and 3/4 are same or different ?*

They got my point.....and some of them agai started drawing the pictures of 2/3, 1/2, 3/4 etc.... After some time, I saw that they were unable to find a lead, I invited them for a whole class collctive disussion..... We started drawing the pictures and discussing about it...

Yash again bounced back after some time,

**"We know that double of 2/3 is 4/3 .... and double of 3/4 is one-and-half...... (pictures of 4/3 and 1.5 were drawn while this was said) ... and because 4/3 is less than one-and- half..... we can say that 2/3 is less than 3/4...."**

I hope you will pause and think about his arguement for a while.....

Isn't this beautiful ??

He had compared the doubles of quantities to find the relation between original quantities...

I looked at others for their views, and almost everyone bought this idea !!

One of them even appreciated him :-))

So now I turned to the person who had sparked this exploration - Sania...

*What do you feel now about these two smaller quantities? Do they add up to Quarter?*

No sir..

Why?

Because we saw that 2/3 is smaller than 3/4.... So if the 2 pieces add up to 1/4, then 2/3 will become equal to 3/4 .... she said this to me with a confident smile... :)

Students were about to disperse now.... but how could I let them go without a germ of thought again? :)

*So whats the relation between 2/3 and 3/4...?*

"2/3 is smaller than 3/4....."

Correct... So now my doubt is -- Its smaller than 3/4, by how much??

Few again didnt get what I asked.... Those who got explained others.....

Again, there were some quick guesses -- "Its smaller by half of quarter...."

Do you see ? They still rush to interpret such a piece (shape) as 'half of quarter' :-)

Just for my satisfaction, I probed them -- Do you mean that this missing piece is same as that extra piece (1/3 - 1/4)?

No.. No.. Sir.... we have just now proved that those two extra pieces are not halves of 1/4......

They didnt know that the answer to the question was Yes.... that the size of this missing piece is same as that of that extra piece :)

I told them to figure out this at home-work, which they happily agreed ! We will be discussing this in our next class.....

I am pretty excited, what will unfold now..... What about you? :-)

Thanks and Regards

Rupesh Gesota

__PS:__These students are from grade-7 and 8 Marathi medium government school and are part of a maths enrichment program- MENTOR. To know more, check www.supportmentor.weebly.com