Solving the simultaneous equations 'Their' way - Part-1

I had been to Bangalore for Maverick Teachers Global Summit last week (https://www.youtube.com/watch?v=0x_I70vm9ec) and hence was unable to work with my students in this span.

You know, they have now become so fond of maths that they insist and even ensure that I give them an Assignment of challenging problems whenever I go out of city for some work... So today, we were finally meeting after @ 10 days. Most of them were done with most of the problems, sharing with me their triumphs as well as struggles... They asked for some time so that they can discuss among themselves before 'submission of their assignment to me for correction'... (Yes, these are the english jargons that these Class-6 Marathi-medium Municipal school students love to speak :)

Meanwhile, two of them -- Jeetu and Vaishnavi -- were already done with their work and demanded from me a problem to amuse themselves. I just scrolled through my WhatsApp images folder, to find this interesting piece.

So I draw this problem on the board, but without any explanation. 

After watching them glued to the problem for few seconds, I probe them "Hope you know what has to be done!"

"Yes Sir, we have to find the total value of three shapes."

Shouldn't he use the word 'weight' rather than 'value' ?  I chose to not bother him for this trivial stuff and waited patiently for them to solve the problem.

My anticipation: They will solve the problem but will probably use the 'Trial and Error' method to arrive at the solution. I thought I will then use this opportunity to introduce the 'logical' approach to them.

After about 5-10 minutes, Jeetu approached me with confidence, 

"Sir, I got the answer."

"Ok... Can you plz explain?"

"Figure (i) says Square and Circle add up to 10... So first, I tested if Square and Circle are of same value... So I gave each of them value = 5..... This gives Triangle = 15 from fig (ii)..... Now putting this value of Triangle in fig (iii) gives Square = 9, which is in conflict with the original value of 5 assigned by us to Square.... So Square and Circle have different values..

So now I wanted to know which is bigger of the two..... So I compared the 2nd and 3rd figures.... Fig (ii) says that Triangle and Circle sum up to 20, while Fig (iii) says that Triangle and Square sum up to 24........ So it means Square is heavier than the Circle."

Wowwww  !!  I just wanted to hug him for this beautiful observation as well as reasoning !! But I chose to contain my emotions for now, "Okay..... go ahead...."

"Now I saw the Fig (i) again.....It says that Square and Circle sum up to 10.... I know that Square is bigger than Circle...... So I put Square = 6 and Circle =4, but this set of values led to the same problem as before (conflicting values)...... 

So then I tried Square = 7 and Circle = 3, and this worked.........."

"How do you know it worked?"

"From fig (ii), if Circle = 3, then Triangle = 17.... Now, If we use this value of Triangle in fig (iii), then we get the same value of Square (= 7) which we had assumed....."

"Okay.... So you mean there is no conflict this time?"

"No sir..... But I thought there might be more solutions.... So I tried even other possibilities.... like (8,2) (9,1)  but these options did not work.... The only one which worked was (7,3)....."

I appreciated him for his observation and reasoning in the first part and asked him if the 'trial and error' method in the latter part can be replaced by any other way..... a 'logical' way..... 

He instantly started pondering about the 'second' approach...

Meanwhile, Vaishnavi was ready with her solution....

"Sir, even I got the answer..."

"Okay... But did you verify it....?"

"Yes Sir, I have put the values in all the shapes..... It is correct..."

"Fig (i) says Square and Circle add up to 10... So it means Square is less than 10 and Circle is also less than 10........"

I thought to interrupt and counter her by asking that what if one of them is exactly 10? However I chose to wait and just listen to her till she completes.

" Then I saw the figure (iii). Square and Triangle add up to 24..... Square is less than 10..... So Triangle is more than 14..... "

I loved this explanation !!  ( Did you get it ?  :-)

" Now I took Square = 3 and Circle = 7..... but this was not working well........  So I interchanged their values.....   I took Square = 7 and Circle = 3 and this worked well........ "

"Hmmm.... So why did you take 3 and 7 as the initial values for Square and Circle?"

"Because they add up to 10..."

"Ok... but I mean, why specifically 7 and 3...?   Why didn't you choose 8 and 2 ? Or 6 and 4 ?"

She started smiling....She had understood that I was seeking a reason for this move...

"Sir, I just took it randomly... there is no reason...."

"Hmm.... So, what would you do, if suppose this set of 7 and 3 too had not worked?"

"Then I will try other set --- 8 and 2"

"What if this set does not work too?"

"I will try another set...."  and by now she had realized that she was being pulled up by me for something else.......  (labor work)

"So do you realize how much will you need to work if these two shapes would have added to 100 instead of 10 ?"

"Yes.... I will have to try 50 options! "

"True.... So though your present approach is correct, but can you think of any other 'better'  or more 'logical' approach?"

Jeetu was enjoying our conversation from behind and I could see him welcoming Vaishnavi with a teasing smile :-)

Something struck me and I told them to share their respective methods with each other... After seeing them agreeing to each other's method, I thought of hinting them a bit towards a 'logical' approach...

So I asked them a question.......... a question that helped them arrive at a beautiful approach...  a question that led to another question......and which in turn led to another beautiful approach..... something -- that I can safely bet -- that a conventionally taught and learned teacher would mostly be Not aware about.... 

So What was that question?  And what is that 'awakening' I am talking about? 

This revelation (discussion) is quite lengthy and hence calls for a separate post... So we will look at this tomorrow in the Part-2 of this post  :)

Till then, keep Mathing .... and Stay connected !!