Saturday, July 30, 2016

Solving the simultaneous equations 'Their' way - Part-2

Hello friends,

I am very happy that you liked my previous post on 'Solving Simultaneous Equation 'Their ' way" Thanks to many of you for explicitly expressing your interest and motivating me to continue with this sharing work.... 

To be honest, I was a bit reluctant to write the Part-2 now.... Why?

Because something more exciting has been happening in our class since past 2 days.... 
Yes, these government school students have surpassed my expectations, and I am so very happy that they keep jumping over the bar, whenever I raise it  for them !!

I will soon share that stuff too.... but first, lets complete the previous story :-)

For those, who have not yet read the Part-1 of this post, I would encourage you to do so, before you carry on ahead.... Here is the link to that story:

If you remember, they had made some interesting observations and had even used some sort of  'reasoning' before resorting to the trial and error method to find the solution....  I now wanted to see if it was possible to replace the trial and error element by the logical approach completely...

Both - Jeetu and Vaishnavi - were now working together.. I could see them struggling for the 'second' way... After allowing this struggle for some more time, I thought to intervene..... with a question....

"You have already figured out some relation between Square and Triangle (by comparing the first two figures)... Is it possible to go a little deeper?"

They listened to me with sincere attention but couldn't build upon it... 

May be my hint was not so clear... I stretched my hand further... 

"Can you figure out Triangle is bigger than Square by how much?"

Jeetu picked up !!

"Yes Sir...  It is bigger by 10. "

"Good... and how do you know this?"

And this time, I could see even Vaishnavi join him in the argument... It was a pleasure listening to them.

" If you add a square to circle, we get 10. But if we add a triangle to circle, we get 20.  This means, that extra 10 has come because of triangle."

"Ok.... And can we represent this using an equation?"

She wrote it this way:

"Ok... But what about that 10 you mentioned now?  How can you communicate about that?"

He borrowed the chalk from her and scribbled this immediately.

"Fantastic.... You have now got a new equation.....Can we use this along with the given data to solve the problem logically?"

They again seemed to be struck.... Please note that they were dealing with 'equations' for the first time.... And I was still expecting too much from them... Yes,.... too much !!  (why?)

I don't know what struck me... And I wrote a new problem besides this old one and drew their attention to this:

A + B = 50
A  - B = 18

"Looking at these two relations between the two numbers A and B, can you find out these numbers?"

Yes, this set of equations are very familiar to us (adults). We have probably studied those in our secondary school. But I would like to remind you that these students are just promoted to Class-6 and are from a municipal school. I had no idea what will unfold in next few minutes.. But one thing was sure, I trusted their instincts and abilities !! And I was also sure, that we will definitely learn something from this... 

Jeetu kept looking at these equations for a while......and......

"Sir... I got it."

"What?  You mean, you have got the two numbers? Beware! I am not going to accept your trial and error method this time."  

I tried to act smart and strict, but he continued with a confident smile.... Listen to him.....

"The two numbers differ by 18.... So first, we will remove this difference...... We will subtract this difference from their sum.....  i.e. 50 - 18 = 32......  Now since we have removed the difference between the two numbers, they have become equal... So each of the numbers is 32/2 = 16.....But both cannot be 16... as we know that they differ by 18..... So let us now add that difference to one number...... 16 + 18 = 34....... So the two numbers are 16 and 34."

What ?  Did you really get him ?? 

I am scared if your answer is Yes.... Because, I did not get this.... It seemed logical... a bit sensible.... but my logical brain hinted that there is something fishy.... Perfectly correct answer, but could not comprehend his logic immediately...

So I requested him to explain it to me again.... But he could not rephrase it in a different way.... Rather, he continued with much more confidence....  So I turned to Vaishnavi..

"Did you get what he said?"

"Yes sir... I understood.....!"

What??? This was now driving me crazy........ It was not that I did not trust his mathematical ability... In fact, Jeetu had surprised me many times before, but this time, somehow I was not sure about his logic.... I felt there is something that's got entangled...... and I am unable to see through it or unwind it......

There were two options before me now:  
1) stick with this new problem.... make him make me understand his method..... and/ or even arrive at the other method (which would make sense to me ;-)
2) accept his method for the time being & use this strategy to solve the original problem....

I chose the second option (why?) 

"Fine.... Now that you know how to solve such equations... Can you solve our original problem?"

They did not get what I meant... So I pointed at their new equation again....

"Did you see a similar equation in the previous problem?"

They pointed it out:  A - B = 18"

"What else did you require to solve this problem?"

No answer.....

"What would be the values of A and B, if I give you only this much information ---  about their difference?"

"They can have many answers then......"

"True... So why were you able to identify exactly one solution of A and B?"

"Because we had one more condition.......  A + B = 50"

"Hmmmm........ So now, are you getting where am I taking you?"

Vaishnavi screamed......

"Yes Sir.... I got it..... We have a similar addition condition in this problem also..."

Phew.................Finally !!!   :-)   :-)

She drew Jeetu's attention to Figure (iii)  -- the one that shows Triangle + Square = 24.

Jeetu looked up to me with wonder....  He was overwhelmed with the striking similarity..... I think he had also understood the reason I made them solve the new problem (of A, B)  :-)

He worked out the solution to the original problem in no time...... and got extremely delighted to see the same answer, that he had obtained some time back, using trial n error..

(I should have captured the joy on his face... I also forgot to take the snap of his written work....  I will do that on Monday and upload in the comments section of this blog)

But Vaishnavi took some time to understand this... Jeetu explained her.... Strangely, she was struggling to relate the A & B problem with this picture problem.....   He had used the same logic as before..... He was happy with this revelation...... And I was happy because he had finally got the taste of 'logical' approach...... However, there was something that was still bothering me..... I was not yet able to understand his method completely ;-(

I was going to look at it more closely after the class...... Meanwhile, other students grabbed me for some clarifications in the assignment problems..... But little did I know, that the game was not yet over.....

Vaishnavi came running to me, saying that she has figured out one more way....

"Oh really? Plz explain...."

I now decided to draw the attention of the whole class to this discussion.... I felt it was getting more interesting.... I gave them the background - what had happened till now.....  We three ensured that they understood what are we doing and why are we doing...... (some time was given to them too to think about the solution, but most of them resorted to Trial and Error methods..... Jeetu and Vaishnavi looked at me, with smile....  Probably, this reminded them of their journey :-)

So the setting was made.... Everyone was excited and curious now - to know one more logical approach, that was about to be unfolded by Vaishnavi........  

And I am sure, even you too are curious  :)

So here it is.. This is what she scribbled on the board, with her voice-over in the background...  

And this just blew me away !! She had hit a Sixer !!

Take a careful look at her diagram... how beautifully she has merged the three figures in the problem into just two figures....

I would suggest you to analyze her work before reading further....


What she said (while drawing) was:

" If we add first two diagrams of the problem, then we get a total of 4 shapes with their value = 30 (20+10).... Now we remove Square and Triangle from this combined set.....(and while saying this, she draws a box around the two lower figures, if you can see.....the Pictorial subtraction! ) ...... We are left with only two Circles now....  Further, we know the total value of Square + Triangle.... Its 24... So removing this pair from the complete set means subtracting 24 from 30..... Hence Two Circles = 6..... So one circle = 3..... Now, we can easily figure out that Square = 7 and Triangle = 17.... So the required total of all three is 3+7+17 = 27..... " 

Now, isn't that just beautiful ??

Probably, you will appreciate this much more if you consider -
a) the way we are taught /  have solved such problems (dry way)
b) the age and background of this child.

I was still having conversation with Vaishnavi as to what made her think of this approach.... and she was trying to explain me....... and meanwhile, I heard another voice ---

"Sir, even I got another way !!"

This was from Payal....  I was so happy to hear from her after long time.... 

"Yes, Payal.... Plz explain to the class..."

"No sir.... I am not sure..... You plz check first....."

"Its ok... We all will try to understand it together...."

But she insisted that I should verify it first, before its discussed with the class... I agreed. And I was about to read her work, she intervened....

"Sir, I don't know the reason.....Why it works.... I have just guessed this method..... I don't know if its correct...."

"Its ok.... Let me see...."

Her work was a bit messy, so I requested her to explain.... This is what she had done:

" Half of 10 is 5 (pointing at the figure (i) of the problem)
  Half of 20 is 10 (from the fig (ii))
  Half of 24 is 12 (from fig (iii)
  If we add 5 + 10 + 12, we get 27... Its the correct answer."

And I was like....... Again, bowled !!

Did you understand this ??

I did not (at that instant).... So  I asked Payal for the reason.... And she argued --

"Sir, I told you... don''t ask me the reason.... I have simply guessed the method..... "

So I thought to share this with the whole class.... and while explaining her method to the class, something struck me ----  I started thinking while explaining... 

Can you ponder over this for a while, before reading ahead..... as to why this works? what could be the logic behind this method that makes it work? 


"Payal has done half of the value in the fig (i)...  10/2 = 5... But what does this 5 signify? How can we interpret this operation?"

Sahil responded: " Sir, it means Half Circle + Half Square = 5"


"Then she has halved the value in fig (ii) ... 20/2 = 10..... Means?"

Many voices join this time:   " Half Circle + Half Triangle = 10"

"And how does her last operation get interpreted?" 

" It means Half Square + Half Triangle = 12"

I had got it half the way, what's happening..... and I was so very delighted to dawn upon this beautiful approach... But I chose to stay silent and ask the students now  --

"So why does adding these three numbers give us the required solution?"

Silence for about 25-30 seconds.....and ......... Vaishanvi erupts  :-)

"Sir, I understood why it works..... Shall I explain? "

All heads turn around, looking towards her with surprise...My god, the girl is on hat-trick! :-)

I ask Payal if she has been able to figure out the reason...She gives me a shy look and a hiding smile.... I probe her....

"Sir, let Vaishnavi explain.... I told you, I don't know the reason....."

I could see students relishing this humorous moment :-)  :-)

The Girl takes the charge again... and scribbles up with the same style...

i) She first draws the two halves - of square and of circle, and equals it to 5.....  (equation (i))
ii) then she draws half of triangle, completes the circle, and writes 10 .... (eq. (ii))
iii) finally she completes the triangle and even square, and writes 12...... (eq. (iii))
iv) then she adds up this 12, 5 and 10 to get 27.... (she has erased 12 and has written 27 at its place)

So how was it ??    :-)   :-)

In case you feel that the story ends here, then yo are grossly mistaken..... 

We explored some more 'logical' methods to solve this problem..... 
What?  Some more?  

Yes.... But I am not going to share those in this post now.......  May be in Part-3   :-)

Till then, how about YOU thinking out some more approaches?   You can discuss these with your children/ students and let me know....  I would love to know some from you.....  

Let me give you some hint....... for some more approaches....

Yes, the hint is There in the above post ... Lets see if you can get it :)

Happy Mathing !! 

Rupesh Gesota

Wednesday, July 27, 2016

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

I had been to Bangalore for Maverick Teachers Global Summit last week ( 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 !!

Thursday, July 7, 2016

"Sir, tomorrow is a Happy Date"

Do you know that we had an interesting mathematically special date few days back. Yes, it was 28th June. To be honest, I had missed this observation, and got informed about this via facebook or whatsapp, only on the other day....  But I grabbed this opportunity to talk about this with my students. 

I asked them on the next day "Can you figure out something interesting in yesterday's date?"

( Check this link to read the entire post: )

To be honest, I was NOT expecting the correct response. Why? 

Not because I don't trust them. But because we had just touched upon this category of numbers, and that too around 5-6 months back, and had never visited that territory again. 

But you know what? Kanchan shocked me. And I am so happy that she proved me wrong!

"Sir...... both 28 and 6 are Perfect Numbers! "

Super !! Isn't it ? 

(Well, you will probably appreciate this more if I tell you that this girl is from a Class-6 municipal school, with no facilities,  privileges and parental support in her 100 sq. feet home situated deep within slums. And how about asking this question to a bunch of students from posh background?)

Perfect number is a  positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3. 

I wanted to write a separate detailed post on this particular class on perfect numbers, because we did a host of interesting things on this day... 

How do I know if that was 'interesting'?  ..... Because ALL my students founded it very very interesting.. There were many Oooooohs and Aaaaaahs in our class..... Many eyes and mouths were seen wide open.......... :-)

Here is a snapshot of 'only part of the stuff' that we did on that day.. (Thanks to the 'Perfect Date' :)   

Don't get scared if you are alien to some of the terms on the board.... Either you may google for it.... Or how about visiting my students some day and learning from them ?  :-)


Hmmm..... Now when the background is set, lets begin the real story ! 

Something More interesting happened yesterday....

Can you guess??   :-)

"Sir, tomorrow is Happy Date !! "

And this time, it was like..... me with my eyes and mouth wide open :-)

"7th July....    Both - the date as well as month - are 'Happy Numbers'......"

happy number is a number defined by the following process: Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number either equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers)...For example, 19 is happy, as the associated sequence is:
12 + 92 = 82
82 + 22 = 68
62 + 82 = 100
12 + 02 + 02 = 1.

"Wow... I am so glad you could spot this... But how did you spot it ?"

"Sir, we were waiting for this date, since the day we discussed about 'Perfect Date' "

It was a Sixer !!  I had no words for their growing mathematical dedication. 

But the Math Teacher in me soon woke up -- "Hey, but what about the year 2016?"

"No sir, that's not a happy one... We checked it.", with a little unhappy face :-(

I thought for a while...... "Ok.. So Can we then know, what's the next nearest happy date?"

"Sir, you mean, even the year?"

"yes dear!"  :-)

We had just unlocked the door of our class while we were chatting about all this, without even being seated..... But this probing question pushed them to their seats immediately,... some started doing the number crunching - mentally - on their own, while others started working on this with their peers....  And in no time, there came a loud voice....

"Sir, I got it !!"

"Hey wait... Hold on....Lets give 2 more minutes for others."

Hands started going up one after the other..... 

I really didn't care much about the Happy Date at that instant..... what made me Happy was their Happy Faces and the Happy environment :)

Can you figure out the date on your own without scrolling down for the answer?    :-))  .

Hey, no cheating :)

Got it?

If you have got 1st January 2019, then you are correct !!

Student-1: "Sir, it is too far... We have to wait for THREE years!"

Student-2: "... We will be in Tenth standard in 2019"

Student-1: " Yeah..........  But We can be Happy in Tenth std. only if we get good marks"

Student-3: " Listen..... Even 10 is a Happy number... So we can be Happy in Class-10 even if we don't get good marks!"

:-)  :-)   :-)   :-)  :-) 

Student-4: "We can be Happy this year too.... We are in Class - 7 !!"

:-)   :-)    :-)     :-)   :-)


Rupesh Gesota