Masti with Multiplication -- Part - 1

They were eagerly waiting for me. I had not visited them or worked with them since long...Their teacher had told me the day before that students were complaining about my absence. "Sir, does not teach us now. He just goes to other school." This shook me & so I decided to meet them today....& make our Friday, really a Good Friday :)

As I was about to enter, I could hear their non-stop recitation of multiplication tables. The moment I entered, the tape-recorder stopped with a delightful surprise on their eyes. 

"Please continue." I suggest this to him while greeting their teacher who was managing this show.

And he started telling the table of 12, but with little hesitation this time. He completed it well and as was about to sit, 

"Hey, wait. How much did you say 12x8?"

Students generally (and unfortunately) get scared when their maths teacher responds to their solution with a question. 

"Don't worry... You were correct. I am just telling you to say again."

"12 x 8 = 96"

"True. Can you tell me how much would be 22 x 8 then? "    (I had no idea or plan to do this to them today.... but somehow, this happened :)

I could see some of them quickly resorting to pen and paper. I instruct them not to do so. Within 10-15 seconds, the same boy replied -

"It's 176"

"Okay.... and how did you do this?"

"Sir, I knew the table of 22."

Gosh!! My plan flopped. However, I didn't give up.

"Good, What if I ask you 32 x 8 now?"

"I don't know the table of 32."

"Yes, that's why I asked you so. You don't even need to know the table of 32... I was surprised when you said that you knew the table of 22. Though this pleased me, but I think, you don't need to memorize even till that."

His blank expression worried me. So I thought to reiterate the question.

"If 22x8 = 176, then what would be 32x8 ?  We need to find the product without directly multiplying 32 and 8."

To ensure that everyone understood the question, I asked one of them to share what they understood from the question.

Realizing that the entire class was still not with me, I asked the other student to write the question on the board.

And then, almost instantly, a hand went up.

"Sir, 32 x 10 = 320. Hence 32 x 8 = 320 - 64 = 256."

Though this was not the answer to my question, but his approach really delighted me. Sadly, very few students are able to think and work out this way (why?).

"Does everyone get what he is saying?" One of them reiterates this strategy and the class agrees with his approach.

"Hey, this was a good strategy. But can we now move on to our original question. Finding 32x8 using 22x8"

Silence for about 10 seconds. And again, a hand went up.

"Sir, 30x8 = 240 and 2x8 = 16. So 32x8 = 240 + 16 = 256."

Oh! Whats happening? Though this was again not what I was expecting, I was very happy with his strategy.

"Does everyone get what he is saying now? The same process again -- one of them is made to reiterate his understanding till the class agrees with this strategy.

"Hey, again a good strategy. But is this the solution to our original question?"

I could see some of them nodding their head, and I was glad that they were still with me.

Silence for about 10 seconds. And again, a hand went up.

"Sir, 32 x 4 = 128 and hence 32 x 8 = 256."

And now this was completely blowing me off. They were able to think off and even communicate clearly, almost all the wonderful strategies of multiplying 32 and 8.

"Wow... This is beautiful. What made you think of this?"

"Sir, I know that 8 is 4x2. So I first multiplied 32 by 4 and then by 2. Multiplying by 4 is easy. It's just double of double. And then finally, we just need to double the partial product to get the final product."

The math done by him was no less than a music to me. I have been into the classrooms of tens of private ('international') schools to assess their students. Hardly, I have been able to get such multiple responses and see such a computational flexibility, along with reasoning.

"Well done students. I am absolutely delighted by the three different ways by which you have solved this problem of 32x8. However, I am still looking for an approach that would attack this problem via the route of 22x8."

They smiled at me, probably realizing that their teacher was in no mood to give up. 

This time we had a longer silence. After about 20 seconds, I thought to intervene. 

"Okay... Can one of you plz write on board what is 12x8?"

I instructed them to write this equation 'above' the two equations (why?)

And in no time, one of them shouted out.

"Sir, the product of 32 and 8 would have 6 in its units place."

"Okay... and what makes you say so?"

"I saw the pattern in last digit of 12x8 and 22x8"

"Hmmm....And can you tell me the reason for this 6?"

Poonam quickly intervened -- "Sir, its very simple. 6 is because of the 16 of 8x2"

"Good...  So, let's write down this 6 in front of the problem."

(Check the image above)

This observation of theirs prodded me to push them further -- to solve this problem via pattern recognition. 

"How about any pattern in ten's place then?"

"Yes sir, we are looking for the same now...."    (Oh, they were already on that track :-)

Rajesh jumped up -- "Sir, ten's place will have 5."

"And why so?"

"Because we have 9 in the first line, 7 in the second line and hence 5 in the third line."

"Ohk.." I look towards the class and ask for their opinion.

"Yes sir", Sachin seconded his thought...." the ten's place digit decreases by 2. Also they are all odd numbers."

"Interesting.... Let's pad this digit 5 before 6 now.... And now? are we done with the answer?"

"No... the answer has to be 3 digits."

"And why so?"

"Because 12x8 = 96. How can 32x8 be 56?"

This Logic brought me some relief. And within no time again, Saif sprang up - "Sir, the hundred's place has 2."

When asked for his explanation, he promptly replied -- " I could see the pattern there as well. The first line has 0 in H's place, second line has 1 in H's place. Hence the third line will have 2 in H's place."

To this, Rajesh stood up saying -- "The same thing happens with the 1,2,3 of 12, 22 and 32 in the question."

"Hmmm... Good observation Rajesh.... So, what do you all feel? 32x8 = 256?"

"Yes sir... we feel so..."

"How do we verify?"

"Sir, shall we multiply the numbers now?"

I was amused by this appeal  :-))  Seems, I had tied up their hands since long...

They worked out using standard algorithm. And soon the room was filled with a loud Yaaayyyyyyyyyyyyyyy  !!!

"Sir the answer is correct..... It is 256!"

"Hmm... good.... But......"

Poonam could again gauge out as to what I was going to say and she was right.

"Sir, but this is not what you want..."

I asked her with a smile -- "Then tell me dear, what do I want?"

"You want us to find out 32x8 using the answer of 22x8."

And I was so so satisfied to find the students quickly getting engaged for yet 'another' approach. (what was pushing them?)

So did they finally give me what I wanted?

Could they 'discover'?

What made that happen?

How long did it take?

How did the trajectory look like?

What questions did I ask?

All these questions, will be answered in the part-2 of this post... Soon! :-)

Meanwhile, would you want to know who these students are ?

They are they children of LEAD (www.lead-foundation.org) and who go to an Airoli-based Marathi medium Municipal school... They are in class-6 and 7.... I had worked with them during diwali vacations for few days, but was unable to give them time later due to my full time engagement with students of other municipal school.... which finally shaped up into a regular and rigorous program - MENTOR

Now, What is MENTOR?

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).

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.

To know more about the objectives, activities and dreams of MENTOR and how exactly it got started, you may visit this website www.supportmentor.weebly.com

Meanwhile, how about you doing this multiplication masti with your students? :)

I would love to know as to how they responded to this type of question and even, how you guided them towards Discovery?  

See you soon, for the part-2 :-)

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Thanks & Regards

Rupesh Gesota