Sunday, April 8, 2018

Adding Fractions through Visuals/ Understanding

I really liked the visuals drawn by students while solving this problem.

The problem at hand was 4/3 + 5/2

Students had reached a point where they had understood the need / reason for fractions to be of same size i.e to have same denominators to add easily. However we had not arrived at any particular method yet to achieve this.

One of them said that each of the unit fractions above i.e. 1/2 and 1/3 to be split into 1/4. Most probably, the reason for this could be the pictorial representations of both the quantities (that were drawn on the board) looked bigger than Quarter... Quarter is generally a common / familiar fraction to students after Half..... All of them agreed with this suggestion... So I simply went ahead without showing any hesitation. This is how the picture looked like:

So now they said that we have 14 quarters in all plus 4 smaller pieces. When I asked them how to add the smaller (green) pieces to these 14 quarters, one of them argued -

Green piece is half of Quarter. So, 2 green pieces would make up 1 Quarter.

This is not the first time I have witnessed any student giving this specific argument. I have heard it several times i.e. students misinterpreting this left over piece as half of 1/4th (why do you feel would so many students be saying/ seeing it this way?)

I chose to ask the class about this viewpoint. And to no surprise, the whole class completely agreed with this, except for one student.

She said - If 2 smaller (green) pieces sum up to 1 Quarter, then 2 pieces of one-third should sum up to 3 quarters! That isn't true. So green piece is not half of Quarter.

Isn't this a beautiful argument?

I looked at the class. Not everyone understood this. So a picture was drawn where a whole was 1st divided into three thirds, and then one 1/3rd was erased. This visual made the game easier for them as they were very familiar with the visual of 3/4th. 

So, now the problem was - Whats the size/ name of this smaller piece??

It did not take much time for one of them to shout this - 

So then, THREE green pieces would make one Quarter! 

I must confess that when I heard this claim at first, I thought it was just a random guess and hence would get eliminated through another line of argument. I did not pay attention to this or did not evaluate this new claim, probably because of the tone in which it was broadcasted and also probably because of its nature ( since TWO didn't work, so it must THREE) !

However, I am glad that couple of them took it seriously and they not just agreed with this claim but even proved it correct with the help of this diagram.

 Now, isn't this too beautiful? :-)

Finally, when I probed them, they could also find out the name of this green piece.

"Because 3 pieces make one quarter, so 12 such pieces would make one whole, hence its 1/12th "

So now, we knew that 3 thirds is same as 4 quarters and the remaining one more third also had one quarter. But then we are left with one green piece alone. 

To this, one of them proposed - So lets represent each quarter in terms of this green piece now, because we know that 3 greens make one quarter.

I looked at the class again for their approval. Some required one more round of explanation but soon everyone was on the boat.

So now finally, they transformed the original problem 4/3 + 5/2  i.e.

4 thirds + 5 halves  --->  16 twelfths + 30 twelfths  =  46 twelfths.

You might have noted that they did not multiply the Numerator and Denominator by the same number to get a common denominator.... Neither they took any LCM, nor they did any cross multiplication....

So what is your view about this approach?

Thanks and Regards
Rupesh Gesota

PS: Students study in marathi medium municipal schools and hail from disadvantaged backgrounds. To know more about and support this maths enrichment program, check the website

Tuesday, April 3, 2018

An interesting date today :)

"Sir, tomorrow is an interesting day!", my students drew my attention to an old post-it note stuck on our notice board.

"Oh, really? And what's that?"

"Its 4th day of the 4th month (April), and its also falling on the 4th day of the week i.e. Wednesday."

Flash back - we had figured out this special day many months back, but I could not recollect the specific instance that had triggered us for this exploration.So I asked them.

"What had made us landed up on this?"

It did not take much time for them to recall - 

"Some one had given you the problem last year that if 1st January 2017 falls on Sunday, then 1st of which month will be Sunday again? While working on its solution, we had extended that problem to this present one..."

"Oh yes...!"

Then I thought if the year 2018 too could give us a 4 in some way. I tried for a while, but in vain.I shared my disappointment with them.

Probably, one of my students, Kanchan, took this seriously and almost within couple of minutes, came up with this marvel:

I hope you will spare at least half a minute to appreciate the beauty of this expression. No doubt I could notice several interesting properties in this, however, the villian in me pushed the student to explain everything :)

And here she goes -
1) The digits of the year 2018 are used in order.
2) Each digit has an exponent
3) Exponents too are in order (descending as well as consecutive).

And this was her final stroke, which I had failed to notice -

4) The expression evaluates to 4.

Yes, it was a 4 again... She used the year 2018 beautifully to yield a 4, which her teacher had failed to do. Now, isn't that a proud moment for a teacher?  :-)

---  Did you figure out all these interesting properties?

---  And if this is not enough to delight you, let me also tell you that my students have also figured out the next immediate date which has similar interesting property like 4th April 2018 (falling on the 4th day of the week). Can you find out this date? And probably even next date after that?  i.e. Xth day of Xth month which falls on the Xth day of the week. 

--- How about doing this activity with your students?

Do let me know your views and experiences about this.. Will be eager to know...

Rupesh Gesota

PS: Students belong to marathi medium government school (class-7 and 8) based at Navi-Mumbai.  They hail from disadvantaged backgrounds. To know more about their Maths Enrichment program, check this link: