"2 rupees 20 paise for 100 grams" - Maths class or Ethics class?

Hello friends,

I did not know that I will be back with the new post so soon... But then, this summer has given me a wonderful opportunity to work with the children Daily. And hence, I am so lucky to be a part of (learn from) such stories daily. 

If some of you have been unable to read the earlier post, where Poonam is intentionally allowed to make a spate of mistakes in the multiplication and division algorithms, how she arrives at a (erroneous) conjecture by observing / analyzing some patterns in these mistakes, and finally how she spots her mistake to counter/ discard her own conjecture. A beautiful cycle of discovery that probably mathematicians and scientists would go through. 

This is the link to this story: 
http://rupeshgesota.blogspot.in/2015/06/we-can-divide-product-only-with-first.html

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It all started with just a casual inquiry to one of my students - "what do you feel how much does my bag weigh?"

From which the discussion gradually navigated to the fact that how some shopkeepers cheat the consumers by fixing the std. weights. Sachin claimed that he has seen the 'hole' on the other side of the standard 1kg weight so that no one can spot it. To which Sushma responded that in such cases, the vendor would earn more money.  This as seem did not at first augur well with Sachin because his opinion was stationed on the fact that consumer would get any quantity lesser than the weight on the other side of the balance.

It was interesting for me to learn that what is sometimes so obvious for one student (& the teacher) may not be so obvious for the other student. I allowed Sushma to explain her view and this quickly brought a smile on Sachin's face.

Sushma had already brought into this discussion her experience of purchasing 1-kg rice from the vendor. So I thought to just built upon it. I did not know that my next question would help me learn something so interesting from my students.

"If we assume that the vendor has removed the piece of weight worth 50 gms from the std. weight of 1 kg, then how much will he earn from the transaction of 1 kg of rice, sold at Rs.24 per kg?"

Of course, both the above values -- 50 gms and Rs.24 per kg were arrived at with the help of "students", they were not my (the teacher's) enforcement!  It was equally interesting to study their estimation sense and even help that develop using this real-life example) But I am not going deeper into those conversations this time. Why?  Well, Because I hardly get any acknowledgement from the readers for this effort :-)

Yes, coming back to the problem and how my students saw this and solved this.... Again, here too I am going very quick this time i.e I am not mentioning all the (beautiful) conversations that Really happened. 

Sachin finally could settle down with this argument that if 1 kg costs Rs.24, then half kg would cost Rs.12. So 50 grams would be worth Rs.2.20

It was interesting for me to see him arrive at the cost of 50 grams directly from half kg.  I would suggest you to pause for a while and answer this question. (of course, I will be much happier if you even reply to me with your answer) - 'What according to you must be the way that Sachin would have devised to arrive at Rs.2.20?'    I hope you will honestly pause and think before reading further :)

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As usual, I threw the ball back to my student's court. 

"Sachin, how did you arrive at this number?"

'Sir, we know that half kg costs Rs.12..... And half kg is 500 grams..... Now lets say my hand represents 500 grams rice.... (he starts pointing at each of his fingers now).... Now this 500 grams i.e. Rs. 12 have to be made from these 5 units, value of each being equal..... So then each of the unit has to be 2 rupees 20 paise"

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I was so stunned at his innovative approach, that I can easily forgive him for the error he had made. But I am also a maths teacher, you know? So how can I dare to allow an 'inaccurate' answer!  :-) :-)  :-)

But I intentionally paused for a while so that others too get some time to digest what Sachin has said. Further, I also wanted one of them to explain his approach again (so that I can be sure if they have understood. Also, if some one had missed some link while Sachin was elaborating, the repetition by some one else would help to pull in that student back into the loop. Also, this re-stating being done by another student (not the originator), will have its own flavor/ spice, thus allowing some other new student too to connect to, if he was unable to connect clearly to the first person.

So after some meditation, we begin. 

"Interesting approach. But how did you arrive at this exact figure--- 2 rupees 20 paise?"

"2 rupees taken 5 times would give Rs.10, and 20 paise taken five times would give Rs.2"

Oh. Now I could exactly figure out what must have simmered in his head. I knew that I was not the only person who was listening to Sachin. So I choose to be mum, so that some one else can spot the error. And Yes! This did happen. (You know a smart maths teacher can save his energy this way :-)

Sakshi crept into the picture... "How come 20 paise taken 5 times will give Rs.2 ?" 

I was happy but was also wondering why Sakshi did not tell the exact answer? Did she too want Sachin to think and spot his mistake on his own?  :-)

Sachin did bounce back. 'Oh...sorry.....sorry..... This will make Rs.11 and we want to make Rs.12....."

He had not yet embarked upon the process for the error correction. And meanwhile, Poonam creeps in. -- "Sir, it would be 2 rupees, 40 paise."

And I had started looking at Sachin again.  --- "What do you feel about Poonam's response?"

He looks up at the ceiling for couple of seconds....and then gives her a nod with a smile. (Wow, what a moment for the teacher!  :)

"Poonam, how did you correct this figure?"

"Sir, Sachin had already arrived at Rs.11 using 2 rupees 20 paise,.. Now we are only one rupee short of the desired answer i.e. Rs.12,..... If this One rupee has to be made from 5 units equally, then each unit has to be 20 paise... This gives 2 rupees 40 paise..."

Silence again.

"Does everyone agree? or someone else has something else to add?"

Sakshi crept in again. 'Sir, but this one finger would be 100 grams. We want to find out for 50 grams."

Oh, I was so happy to note that someone "still remembered" the original problem. 

"Hmm... So now?"

Sachin added immediately -- " It will be half of 2 rupees 40 paise...... Ummmm....  It's 1 rupee 20 paise...."

"Ok... So what's the solution to our problem?"

Poonam - "The vendor makes a profit of 1 rupee 20 paise for the sale of every 1 kg of rice."

I decide to make it spicier -- "So is the effort worth it? He does all this for just one rupee 20 paise?"

And almost instantly, I was gheraaaoed by their common-sensical argument that he does not sell just 1 kg rice to one customer.... To this, I took it even more ahead.....

What if there are 5 customers in a day?

So what's his monthly profit (cheat) ?

Annual figure?

and much more.... .. 

These discussions further not just gave them an interesting intellectual stretch, but also allowed me (the teacher) to study their other computation strategies and probably even clear their misunderstandings, if any, of other concepts that might surface up on the way....

In fact, we even penetrated into Ethics by discussing more about this Easy/Fast Money.... His little one--time effort yielding him lot of money for life-time  :-)  :-)   

But then, should the 'Maths' teacher care about talking about such abstract stuff - values/ ethics - in his/her 'Maths' period ?? :-)

Would love to hear from you on this :)

PS: Would like to inform you about two things:

1) People who wish to NOT receive such emails from me, may please notify me about so. I will remove your email-id from the group to avoid any further disturbance to you.

2) People who wish to be a part of the 2nd sitting of the Maths Teachers Study Group, scheduled on this Sunday - 31st May at Airoli 10am, should also please inform me immediately about the same.