Math-journey in the Train-journey ~ Part-2

I had observed that by now this 9-yr old was easily able to find out the results of subtraction problems where the first number is smaller than the second number.  For eg: if we ask her what would be 7 minus 10, then she would say "Minus 3".

And how did such a young learner know about this Negative number thing? This interesting journey is shared in the Part-1 of this post... If you haven't read that yet, then it is recommended to read that Part-1 first before you read this Part-2... Here is the link:

https://rupeshgesota.blogspot.com/2025/01/math-journey-in-train-journey-part-1.html

So we continue our exploration further.

I draw her attention to the same previous problem 24-7 again.. I write these two numbers one below the other & write the digits of the answer after asking her:

    24

-     7

---------

---------

'four minus seven?"

She tells "Minus 3" and I write it down

"Two minus ...........Oh, what's the digit here before 7?", while pointing my finger to the blank space before 7.

She tells, "Its zero"  

"Oh..okay"

i thought of asking her as to how does she know. But then chose to park this important question for later and go with the flow... So then I write zero before 7 and then ask her about the digit in the answer's place below 2 and 0.

She says, "Two"  and I write it over there. (check image)

I now tell her, "So this answer is 17, right?"

And she looks at me with surprise.. and then comes her "No."

'Why? Didn't you find out some time back that 24 - 7 = 17?"  while saying so, I showed her the previous work. (check image)

She looks at it. And then again looks at the strange looking new answer. But with no other response.

"So you say this newly worked out answer is not 17?"

"Yes..Its not 17"

"Then what's that number?"

"Umm... Its Minus 1"

'Oh.. and how is that -1?"

"Because 2 -3 equals -1"

"Oh..I get what you did..."   

(Frankly, I was expecting this type of response from her... Did you also see it as -1, the way she did ? And did you also wonder about this strange looking answer?)   :-)

I continue probing her..  

'But then can there be two different answers for the same subtraction problem?"

"No"

"Then how come by previous method the answer is 17 and by this method, the answer is -1 ?"

No response..

"You want to know this?"

She nodded her head (implying yes).

So I draw her attention to the number 23 that was written somewhere on the ticket and ask her to read that number. She said Twenty three.

Now I write 312 on the ticket, and ask her to read it. She said Three hundred and twelve.

I ask her that both the numbers have digits 3 in it. What made her read them differently? (as in why just 3 in 1st case & as three hundred in 2nd case)

"In the first case, 3 is in one's place, and in Hundred's place in the second case"

I ask her that both the numbers have digits 2 in it. What made her read them differently? (as in why twenty in 1st case and just 2 in 2nd case)

"In the first case, 2 is in ten's place, and in One's place in the second case"

"Yes, so the place in which the digit is written decides its value. Right?"

'Yes"

Now I ask her - So what is this number Three hundred and twelve (312) made up of?

No response

"Whats the value of 3?"

"Three hundred"

"of 1?"

"Ten"

"of 2?"

"Two"

"So 312 is made up of ____?"

"300 + 10 + 2"

"Correct! remember you have studied this in your school.. Expanded form"

"Yes"

"So when we write 2 and 3 next to each other, how do you read it?"

"Twenty Three"

"But how did you get this value or name Twenty Three/"

"Twenty plus three"

"correct.. So twenty plus three equals twenty three."

I write 85 and ask her its value.

She says," 80 plus 5 equals eighty five"

I write 79 and ask her its name.

She says," 70 plus 9 equals seventy nine"

Now I again draw her attention to that strange looking answer of 24 - 7 which she had decoded as "Minus 1"

"So now with whatever we have learned / understood, can you find out what's the value / name of this number."

"twenty minus 3"

"Yes, how much is that/"

"Seventeen"  but with a puzzled look and tone! 

I saw that she was still not so satisfied with this seventeen. So I thought to show her the previous work which showed the familiar 17. 

And then asked her if she remembered this?

"yes"

"Whats the answer of 24 minus 7 using old way?"

"seventeen"

Whats the answer of 24 minus 7 using new way?"

"seventeen" with some surprise & smile on her face...

"So both the answers are same?"

"yes"

"But then why were you unable to identify this strange looking number as 17 earlier?"

"because its not one seven seventeen, its two minus 3 seventeen"

"Hmm... its not one seven seventeen... Its ten plus seven seventeen"

"yes.."

"And here?"

"here its twenty minus three seventeen.... I have not seen such 17 in school / text book.. So I got confused."

"I get that... but you have understood this now?"

Smile on her face..

"Shall I give you one more problem?"

Quick and Joyful "yes"

So I tell her to solve 42-17. But first using the old/ school / textbook way, and then using the new way she has learned. And this is what she did.

As you can see from right side image above, she has also calculated the value of strange looking number as 25.

It was a delight to see Joy on her face & Confidence in her body language when she was able to verify that both the methods gave the same answer. (an experience that's rarely given to children in classrooms).

"One more?"

And there was 'yes" again!

"But this time I will be giving you a little difficult one...Okay?"

"Okay" she said while giving a smile to her mother sitting next to her ;)

I gave her 421 - 115, but with some change this time - first using the new way and then using the old way..... And this is what she did.

This time too, if you see the left image, she has calculated the value of the result: This is what she had done: 300 + 10 - 4 = 300 + 6 = 306

But she was disappointingly surprised when she got the different answer using the old way...

"Should we get both the answers same or different?"

"Same"

"But here both are different... So lets check both one by one properly?"

She did so. And could not find a mistake in any of those.!!

There could have been a better way to help her find out the mistake, but at that time I just told her to solve the problem again using the old way... And she did this....

There was a smile on her face as now, the two answers matched!  And she also figured out the mistake in her previous work when I asked her to do so.... So I asked her -

"How come the mistake happened in the old method that you have been using since long and not in the new method that you have just learnt?"

She smiled at this...

"Which method do you like more?"

"New one"

"Really?"

"Yes"

"So when you go to your school, you will share this with your teacher and friends?"

"Yes"

I thanked her mother for being patient and allowing me to interact with her. And she thanked me for helping her child experience the Joy of Discovery..

Thanks & Regards

Rupesh Gesota

https://rupeshgesota.weebly.com

Math-journey in the Train-journey ~ Part-1

 

I often consider myself luckier -- to have at least one child near my seat during the long distance train travels..It not only kind-of shortens my journey but often such encounters / interactions have turned out to be enriching / interesting ones, and in many ways :-)

One such happened again recently. It was 13th Dec 2024, I was travelling by train to Ahmadabad from Mumbai. Just few minutes after I occupied my (window) seat, I saw a lady coming with an 8-9 yr old girl and occupying the seats in front of mine. I felt the Aha! within me, however 'controlled' my emotions for the time being :-)

After couple of hours, a girl, of age similar to that of the former one, came in our compartment to sell some key-chains. Soon she came to me and kept the tray full of key-chains on my lap suggesting me to take something. I thought for a while and then took one of those & put it in my pocket, giving her a smile. She kept looking at me with wonder for some time and then said, "Give me Rs.10"

"What for?"

"You took the key-chain."

"You gave me the tray and told me to take some.So I took."

"Give me Rs.10"

"Okay.. I will give. Tell me how much is this?" - while showing her 4 fingers,

"Four"

'Now?" - while showing 6 fingers,

"Six"

"Now?" - while showing 9 fingers,

"Seven.....No, Eight"

"Its not Eight."

She started counting one by one now... arriving at the correct answer 9.

"Good. Was there any other way?"

She thought for a while.. and then resorted to,"Give me Rs.10"

I removed the Rs.10 note from my wallet and while giving it to her asked her to show me her 10 fingers and she did so... Now, I just closed one of her fingers and asked her to tell this number.... And she said, "Nine"

"So was there another method?"

She shook her head (implying yes) and left with a smile and the 10 rupee note...

And I saw that this game (between me and this girl) had drawn the attention of some passengers around, including that of the girl sitting in front of me. I smiled at her and she shyfully pushed herself into her mom trying to hide her face :-)

Later, I noticed that her mother was telling her to sleep but she didn't want to.

"May be she has already slept for 20 hours!" I said this while looking at her.

She shook her head (implying no) with a smile...

"Oh.. you don't sleep much? How many hours do you sleep then --- 2 hrs, 3 hrs?"

"More than that..".

Phew!! finally she spoke! ;-)    

"Oh okay... What time do you sleep?"

She looked at her mom for the answer who told her "around 11 o' clock"

"And you get up at?"

Again she looked at her mom for the answer -- "6 o' clock"

"Hmm... So how long do you sleep then?"

I see her thinking and then it comes from her -- "Seven hours"

"Yes...correct.. so then how many hours are you awake?"

She gave me a puzzled look...

"Why? Is it difficult to find?"

No response for few seconds..

"Do you know how many hours in a day?"

"24 hrs."

"So can you find now?"

I see a smile on her face and her fingers start moving in the air now..   :-)

"18 hrs." she said.

"What???" her mother reacted loudly to this (wrong answer!)..And I had to request her to hold / hide her reactions for a while..... Luckily, she understood and kindly stayed away from the rest of the game, though she paid attention to our conversations probably to learn something from this....

I asked the girl how did she calculate 18 hrs... Her explanation implied the vertical orientation of numbers - (standard) subtraction algorithm. 

"Hmm OK.. Do you have pen and paper?"

"No"

The passenger next to me who was listening to all this kindly & immediately offered his pen and one of the old train tickets from my wallet became our note-pad. 

"Can you plz show your work on this paper?"

And this is what she did (check image)

She was surprised with this answer, as it was different than the one she had said.

She gave the paper back to me with a shy smile.

"So how many hours do you sleep?"  :-)

"17 hrs."  with a smile & pinch of embarrassment.

I decide to play devil now... "I think there is some mistake in this work. . Let me show you where.".... And I scribble the numbers on paper while trying to explain that 4 minus 7 equals 3 and then 2 comes down as it is. . So the answer is 23.  (check image)

And then she gave me a strange look while shouting -- "No, its wrong"

"Whats wrong? Isn't 7-4 equals 3?"

"It is.. but here it is 4-7"

"So?"

"That's not possible... So we borrow from 2."

"Oh okay.. I got it..."

She looks relieved now :)

"But you know what? you said that 4-7 is not possible... But actually, its possible!"

Again a strange look :-)

"You want to see?"

Curiosity is high by now!!

"Take your mom's phone.. Do you know calculator?" 

She takes her phone, starts the App and waits for me.

"See what happens when you type 4-7 in it"

And guess what -- it happened what was expected -- puzzled look on her face again! And I could see her mother hiding her laughter :-)

"What's the answer?"

She said "3"

"Ummm.. its not 3.... Plz read properly"

She looked down and with little hesitation came..... "Minus 3"

"Yes,. 4 - 7 is minus 3....So its possible or not?"

She was smiling but I could also see some dissatisfaction on her face.

"What do you think 8-10 is possible?"

"No"

I was surprised by this. But then told her "Lets check on the calculator."

And soon came "Minus 2" with a surprised smile from her...

"So now what do you think, will 10-40 be possible?"

She was about to plug these symbols on the calci and I stopped her!!

"Wait... I want you to guess first now.. What would be the answer?"

And within couple of seconds she said "Minus 30"

"lets check this on calci now"

And when HER Answer appeared on calci, Joy too appeared on her face!

I saw there was a smile on mother's face too :)

We played this game with few more pairs, like 40-90, 30-25, etc in the Same order - first Guess & then Check. It was a delight to see her play & enjoy this (new) math! 

Well, if you think that the game was over... then the answer is NO! . Something more was simmering in the teacher's mind... in fact, the real game was about to begin :-)

And you might wonder what's that? This adventurous journey is shared in the Part-2 of this post... Link below :-)

https://rupeshgesota.blogspot.com/2025/01/math-journey-in-train-journey-part-2.html

Thanks and Regards

Rupesh Gesota

https://rupeshgesota.weebly.com