Hello friends,
As promised.. I am back, with the part-2 :-)
I wanted to write and share the part-2, the next day itself. However some day-long assignments used to drain me out completely - making difficult for me to sit and type the (long) conversation after reaching home in the night... However, I had made up my mind to complete this task today.... You know what motivated me to do this?
It is the honest appreciation (and even confessions) from some of you who could see the 'value' in the patience demonstrated during the previous conversation... Thank you so much for these acknowledgements,.. It tells me that I am on the right track...
By the way, could any of you try out that problem (32 x 8 using 22 x 8) with your children/ students? If yes, then please share your experiences....
If you have not yet read my previous post i.e part-1, then I would suggest you to first read that before you scroll down.. Here is the link:
http://rupeshgesota.
http://rupeshgesota.
Hmmm... So where did we stop last time?
The students could (surprisingly and beautifully and confidently) think of and even communicate four different strategies for computing 32x8... But as my student Poonam had (rightly) pointed out in the end, they were still far from the approach that I was specifically looking for....
Considering this scenario, I decided to walk along the same path that was traced out by them.... .of pattern recognition.
"Yes sir....I am already working on this....", said Rajesh
And immediately, Saif shouted out the product - "Sir, it will be 336."
"And how did you do that? Can you please explain to us on board?"
He said that he had simply extended the pattern that he had seen in the former products.
12 x 8 = 96
22 x 8 = 176
32 x 8 = 256..... So,
42 x 8 = 336
" the one's place will have 6..... ten's place decreases by 2...... and hundred's place increases by 1...... You see, 12 x 8 = 96.... i.e. it also has '0' in the hundred's place...."
All fine, but his last observation about being able to see the invisible zero really delighted me! Mathematician at work !!
"hmmm.... interesting... Let's predict 52 x 8 then..", I prodded him.
And it took no time for others to join the bandwagon....
They quickly slid down the products, till there was no further space left :)
Please check the image above...
52 x 8 = 416 was an easy one for them... However, it was 62 x 8 that invited some pause and then chaos, followed by some reasoning.
When one of them said 62 x 8 would be 506, (why?), there was another who quickly argued about its impossibility.
"There is no even number in the ten's place.... it only comprises of decreasing odd numbers..."
I looked at him, for any further counter.... but his smile conveyed his conformation... and so we moved on.... only to get hit by another impediment...
"Hmmm.... so how do we decide the number after 596?"
"Sir,,, the last digit is 6.... and first digit is 6.... and the middle digit would be................."
"7", - shouted another chap.
I played neutral.
Silence.
"Yes... yes.... it will be 7.... So the answer is 676." the gang jumped up.
"And why so?"
"the middle digit has decreased from 9 to 1 and reached its termination... The cycle will restart from 9 now..."
"Oh...ok.... But are you sure about this?"
"Yes" - a confident chorus.
I wanted to hear from at least one of them -- "we think so.... but not sure...." Never mind, let's dig the hole deeper... :-)
And then.... they went 3rd gear... It took less than 30 seconds for them to run down their pattern from 676 to 1096.
(I had suggested them to avoid writing the ... x ... part of the equation) (why?)
While the class was engrossed with the further products, Rajesh sprang up with joy -- :Sir, 676 is a palindrome number!"
I congratulated him for his observation and alertness.
"Fine, so then the last product 1096 corresponds to whose product?"
The counting started down the column till they hit the bottom..
"62...72....82....92...102...
"Ok...And how can we verify now if this answer (prediction) is correct? i.e how to verify if 112 x 8 = 1096 really?"
And I saw them hurrying for the standard algorithm...(check the snap above)....
"Hold on... You are NOT allowed to write or use the algorithm to find 112 x 8.."
The command stopped them...
Sachin, the master of distributive property, worked out... " Sir, 112 means 100 + 12..... 100 x 8 = 800 and 12 x 8 = 96.... So, 112 x 8 = 800 + 96 = ..................."
Silence.......
"Why did you stop, Sachin?"
They started looking at each other and then finally at me, with that embarrassing smile.... Probably, they had realized that they had goofed-up somewhere...
"What?? Say something..." I ask them, while controlling my urge to smile (hint)
So Rajesh continued, but with hesitation -- "Sir, answer should be 896.... But we have got 1096"
"Hmmm.... I am glad you realized this....So now can you figure out where and why you were led to this error, if you really feel you have erred up somewhere."
All eyes hooked on the board.... Minds-ON !!
After some time, I intervened.
"So now let's write the corresponding multiplication expressions in front of their products"
He completes the equations...
Sachin started thinking aloud -- "102 means 100 + 2... So 100 x 8 = 800 and 2 x 8 = 16... So 102 x 8 = 816..."
And to this, everyone was again shocked....
Poonam almost cried -- "Sir, even this is wrong??"
"How do I know, dear? You please figure out..."
Rajesh entered into the picture -- "Wait guys, let's go back further then.... It seems we have erred up somewhere in the beginning itself..."
Students again started looking at me..... hoping that their teacher would help them in this crisis.... But they soon realized, from my teasing smile, that their teacher was cruel - unwilling to nudge !!
"Sir, plz tell naa."
"Do you really think that I will tell you?"
Eyes back to the board.
After about a minute..........
"Sir, our very first answer i.e. 32 x 8 = 256 is correct."
"How do you know?"
He proved this using distributive property.... and similarly did the same for even 42 x 8 and 52 x 8
And while he was still doing and explaining this to me.... Saif shouted -- "Sir, 62 x 8 is not equal to 596.... It is equal to 496."
All heads turned towards Saif now...
He had figured out this using distributive property... (480 + 16 = 496)
"Hmmm.... Good find Saif... So can you please replace the incorrect figure with the correct one on the board?"
While doing this, he saw something.....
"Sir, I saw the pattern.... It is not what we had seen.... It is something else...."
While, others were perhaps still in the state of wonder as to how come their pattern had collapsed. :-)
"Yes Saif... Plz continue..."
"Sir, our calculation of 72 x 8 is also incorrect.... It is not 676..... It should be 560 + 16 = 576....." And while saying this, he did the correction for this figure as well...
12 x 8 = 96
22 x 8 = 176
32 x 8 = 256
42 x 8 = 336
52 x 8 = 416
62 x 8 = 496
72 x 8 = 576
" So now the next product would be 656.....and then 736....and then....... "
The car got geared up, again... :-)
82 x 8 = 656
92 x 8 = 736
102 x 8 = 816
112 x 8 = 996
Silence... some were still hooked on the board while others were now waiting for me to say something...
"What? So are you done? "
This time they played cautious. "Sir wait.... we are checking..."
Sachin intervened -- "No, 996 is wrong again... Some time back we had calculated 112 x 8 = 896"
Slowly without saying a word, Saif started replacing 996 by 916.... The silence perhaps meant that everyone was again shocked !! :-)
So now when the board looked something like this....
12 x 8 = 096
22 x 8 = 176
32 x 8 = 256
42 x 8 = 336
52 x 8 = 416
62 x 8 = 496
72 x 8 = 576
82 x 8 = 656
92 x 8 = 736
102 x 8 = 816
112 x 8 = 896
there were some interesting interpretations for this (new) pattern...
As per Saif:- "4 gets repeated twice... 8 gets repeated twice.... then we will have 12 also repeated twice down the column.... It is in steps of 4...."
As per Poonam:- "Our earlier (incorrect) prediction after the number 416 was 596.... It is too far from 416.... How can there be such a big difference? So 496 fits the bill.... Similarly, after 816 we cannot have 996.... (too big jump from 816 for 996).... So that's why we have 896...."
This explanation really amused me.... And I grabbed this opportunity to draw their attention to something that I was desperate for....
"Ok Poonam, you mentioned that the difference cant be too big.... So, do you mean the difference should be less? If less, then how much should it be?"
They started calculating the difference between the consecutive products down the column... And in no time, they started shouting and jumping...
"Sir, the difference is constant = 80 everywhere."
"Hmmm.... Nice observation..."
I turn to Poonam for her response now... and she was already enjoying this revelation...
"Sir, I understood now.... The difference has to be 80.... Hence 816 + 80 = 896 (and not 996)... similarly 416 + 80 = 496 (and not 596)"
And now... I stood up and took the charge of the board..... As I knew that we were very close to the climax..... The volcano an erupt anytime now... :-))
"Good... Now, can you all tell me the reason for this 80... I mean - why do we get the constant difference of '80' ?"
Tube-lights got lit.... and yes, as expected the volcano too erupted !! :-))
Almost all the students started shouting together -- "80 is because of 8 x 10"
"I am not so satisfied with this reason... Can you please be more clear?"
Poonam explained beautifully --
"As we move down the column, the numbers are increasing in steps of 10... So, 22 is 12 + 10.... So if 12 x 8 = 96, then 22 x 8 = 96 + 80 = 176."
We had got so engrossed in fault finding and pattern recognition and reasoning that they did not realize that they had "finally found out" the approach I was looking for..... (32 x 8 using 22 x 8)
So I reminded them --
"Did you all realize what did you just say? Do you remember what was our problem?"
Again some more flashes of light.... "Yes sir..... This is exactly what we were looking for...... We got the method finally......"
The class was filled with laughter and joy and a sense of achievement and fulfillment... (Wish I had captured this moment for you).
After allowing them to relish for a few minute and to ensure that the learning gets cemented and does not gets evaporated in the celebration, I threw another challenge to them -
"So tell me if 13 x 5 = 65 , then what would be 33 x 5?"
Saif answered immediately -- " 13 x 5 + 20 x 5"
"Good... Did everyone get what Saif said?"
I could see some of them still thinking....
"Saif, can you plz explain again?"
So this time as he explained, I represented his thinking pictorially (check the snap -- bottom part of the board)
"Good.... now what if you wish to find out 55 x 6 = ?"
After couple of seconds Rajesh answered - "Sir, we know 15 x 6 = 90.... so we can now find out 55 x 6 from this.."
He had almost answered... I was seeking some more elaboration...
Poonam continued -- Sir, 15 x 6 = 90 .... There is a difference of 40 between 55 and 15.... So we add more 40 x 6 = 240 to the former 90 to get the answer 55 x 6 = 330."
"Fantastic! So now tell me, you know the multiplication tables till what number?"
"Any number! ............. Sir, It's so easy...."
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Could you spare few minutes to go through my (another) social venture - which keeps me excited and engaged and energized these days? Yes, I am talking about the program MENTOR, about which I had shared last time.
Besides its literal meaning, MENTOR, also stands for it's mission - Mathematics ENrichment program TOwards the Reform.
MENTOR is a 'regular' long-term program to mentor and enrich the mathematical thinking of those children -
a) who come from challenged socio-economic background AND
b) who have an aptitude or potential for Mathematics (or who are interested in Mathematics).
MENTOR is a 'regular' long-term program to mentor and enrich the mathematical thinking of those children -
a) who come from challenged socio-economic background AND
b) who have an aptitude or potential for Mathematics (or who are interested in Mathematics).
Through this focused intervention, MENTOR also aspires to spur a Reform –
a) in the society by trying to preserve & nurture the abilities of these children to think critically & creatively and reason logically AND
b) in the mindsets & practices of systems (teachers, schools & parents) by sharing with them the outcomes of this program.
a) in the society by trying to preserve & nurture the abilities of these children to think critically & creatively and reason logically AND
b) in the mindsets & practices of systems (teachers, schools & parents) by sharing with them the outcomes of this program.
To know more about the objectives, activities and dreams of MENTOR and how exactly & why it got started, you may visit this website www.supportmentor.
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An insightful article: A Teacher Explains How Today’s Education Is Destroying The Youth’s Ability To Think.
Here is the link:
Here is the link:
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Thanks & Regards
Rupesh Gesota
"And somewhere there are engineers helping others fly faster than sound. But, where are the engineers helping those who must live on the ground?"
"The true value of a teacher is determined not by what he knows, not by his ability to impart what he knows, but by his ability to stimulate in others a desire to know."
It's very delightful that our childrens started new way of thinking....regards !!!
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