22 - 9 is 11 or 13 ?

Hello friends,

Hope you must have got and read my previous email about the starting of Maths Teachers Study Group.  If not, then plz read it and also let me know if you are coming. I will send you the exact address, etc. I am so happy that 9 Teachers have expressed their interest! It is on Monday 18th May at Airoli, Navi-Mumbai. For more details, plz check that email. 

I remember I had mentioned in that email, about sharing one of the fantastic experiences I had while working with students on fractions that day, but then I thought of first sharing with you what happened Today!   :-)

Meanwhile, the word fractions remind me of one more interesting thing. A group of students were asked what is the half of 2/8 (two upon eight)  And it was interesting to listen to the response of few students. They said it is 1/4 (one upon four)  :-)

If possible, plz share your views on reading the conversation below. Will be happy to read.

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We were solving a problem which encouraged her to make a list of all the possible numbers that would leave a reminder of 1 when divided by 3. And then, even hunt for any possible pattern in this list. So she wrote: 1, 4, 7, 10, 13, 16.....

After few seconds, I ask her,” Could you figure out anything?”

“Yes Sir. There is a common difference of 3 between any two adjacent numbers in the list. But I am wondering, why we got this difference of 3, because these numbers are not the multiples of 3.”

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And I was like, Waaaaowww! What an interesting query! I paused for a while, to think as to how I should respond to this.

“So do you mean, they should not have the common difference of 3?”

“Yes. I have added one to every multiple of 3 to get these numbers.”

“Hmmm... Can you write down all the numbers from 0?”

Once she is done, I ask her to highlight the movement of multiples of 3 from the beginning. She draws the path. Now I further ask her to repeat the process for ‘her’ list i.e. for 1, 4, 7, 10....  She does that too. But after this, I choose to stay mum for a while. I wanted the bulb to glow, but what’s fun in it if the student gets it so easily, isn't it? J

So she asks me,” Sir, now what?”

“Means? Now it’s your job dear! I have guided you enough.”

She smiles at me and turns at the board. After few seconds, she again looks at me with a smile, but also with wrinkle on her forehead. “Sir, I am not getting.”

I realized that it’s the time to roll up my sleeves now.

‘Let’s say, you (Poonam) and Yogita are standing two meters apart. How will this distance change if both of you move ahead by the same amount?”

{While narrating, I also represent this situation pictorially on the board. But now I realize that ‘she’ could have been encouraged to draw the diagram}

She responded correctly that the distance between them would still remain the same; and she could also reason for this. I further probed her for the other case viz. she travelling more than Yogita. And she again responded correctly, along with the justification. So Now, I drew her attention to our original problem.

And she was like – “Yes Sir, I got it now.” with a wide smile J

“Please explain it to me then.”

“It was so simple sir. 4 is one more than 3 and similarly 7 is also one more than 6. Difference btwn 3 and 6 is three. And hence the difference btwn 4 and 7 will also be three.”

Good Job, isn't it? – I could make her observe, relate, think (reason), use logic and finally, Discover!  J

We had just settled down and suddenly something stroke me.

“You know what Poonam, we can apply this reasoning to solve the problems of subtraction, mentally?”

“How Sir?”

I stood up again, writing the problem on the board. ‘Tell me what is 22 - 9 ?”

She knows that resorting to fingers and even pen and paper disappoints her teacher and hence, she goes for Thinking!

“Sir, 22-9 is 13”

“How did you do this?”

“I kept the 2 of 20 aside. I know 20-9 = 11. Adding back the 2 to 11 gives 13.”

Interesting! I was happy that she remembers the strategies we had discussed. But this time, I wanted to show her more beautiful way – application of a concept (same difference) that we had just discussed and discovered. I could have explored the new strategy there itself, but I thought this problem probably was not so appropriate for stimulating to think for another (new) strategy, hence I posed another problem to her.

“Good. So tell me what 22-7 now is?”

“15”

“So quick?”

“Sir, we just now saw that 22-9 = 13. So 22-7 has to be equal to 15.”

What? She has used ‘logic’? Wow! But I wanted to pacify myself by ‘hearing’ it from her :-) So I tinkle her. “Please elaborate.”

“If we remove 9 from 22, we get 13. So if we remove 7 which is two less than 9, we will be left with two more than 13 i.e.15”

I was much happier now... Efforts paying back, you know? J

I too resorted to this first problem now. “So can you solve this problem 22-9 using another way?”

She thought for a while. “Yes, Instead of subtracting 9, I will subtract 10 from 22. This would leave 12. Removing that one gives 11.”

And I was now experiencing a mixed feeling. On one side, I was very impressed by her second strategy, but at the same time, I was a bit disappointed as she has erred up at one point. But I ensured that my body language doesn't hint her about the goof-up, I ask her, “Oh, It’s 11? And I thought its 13, as said by you sometime back.”

This simulated her to think again. “No no sir, it is 13.”

“What? But you just said it is 11.”

“Sir, Wait... Let me think!”

I was enjoying seeing her confused and struggled. Because, I knew that this struggle is Must! If I steal her struggle, then I will steal her learning.

And she bounced back. “Sir, I am confused. I am getting 13 by one method. And 11 by another method. How can this be possible?”

“Hmmm..So what does that mean Poonam?”

“It means that I have done some mistake in one of the methods. But I am unable to figure out any mistake in both the methods. Both seem to be perfect.”

“Ok. Both 'seem' to be correct. But what do you feel, can both be so?”

“No.. How can one subtraction problem have two different answers?”

“I am happy that you realize this.”

“Siiiiiiiiiiiiiiiiiiiiiir, plz help me....”, she insists with a pleading tone. I was actually surprised to see her melt. Because I have seen her persevering in much difficult problems.

“Do you ‘really’ want me to find out your error?”

“No sir. I will find out my own.” Her instant confident determination and willingness to persevere catapulted me back, to cloud seven J

“But can you give me a small hint?"

“Ok.” And she thought that her appeal has yielded. But it seems she forgot that I am such a tough nut to crack when it comes to revealing hints and giving answers. J

“Let’s write both the approaches so that we can observe and compare.”

She writes it down. I notice she has goofed up while expressing her mathematical thinking in the written form (she has equated the two unequal expressions 20-9 and 11+2). But as per thought process, she was in perfect synchronization and hence I decided to NOT point out to her any other mistake at this juncture (tackle only one thing at a time).

​She stares at me (waiting for me (the hint)).

While pointing at the second approach, I ask her the reason for 'subtracting one' from 12.”

“Because I had to remove only 9 from 22. But instead, I removed 10. So I will have to further subtract one from the answer because 9 is one less than 10.”

“Really? Will we have to compensate that way?” And.......I couldn’t resist my gentle laughter at this moment and this was enough to make her more impatient.

“Poonam, there is one easier way to figure out which answer is correct. Take stones and ......”

“Sir, I am not in class-1. I will do it mentally itself and I will figure out the error.”

“I am happy with your attitude, but remember, sometimes there is nothing wrong in going back to the basics.”

She looked at me without any expression.  I guess she was trying to digest my advice J

“Ok fine, tell me what 12-9 is?”

“Three.”

“And how did you do this?”

She almost jumped. “Sir, 13 is the correct answer. It cannot be 11.”

“And why so?”

“Because if 12-9 = 3, then 22-9 has to be equal to 13.”

Waaaaowww --- Another pat on the back!!

“Well, now you just need to figure out the error in 2^nd approach. Why you are getting 11 there?”

“Yes, I will think about this at home today itself and tell you for sure by tomorrow.”

“I will eagerly wait for your response, Poonam”

We rose for the national anthem and they left.... Meanwhile I started thinking -- And what about the new strategy of subtraction that I wanted to share with her?  J