Wednesday, October 2, 2019

Revisiting forgotten Arithmetic - with a new student - Part-1

It was her first day in our class... Both of us were excited as to what would unfold.....

Readers may scroll below to the first paragraph written in Bold, if they wish to know only the Math part of this post :-)

I had come to know that she hailed from a very challenging socio-economic background and was further (helplessly) doing her schooling in a language, which was not spoken by anyone in her home / family / community (a typical migrant case). My students had spotted her in their school and brought her to meet me. Why? --  Because both of us (she and I) shared the same mother-tongue: Gujarati (which was a kind of fascinating foreign language for these Marathi-speaking students :)

"Sir..... please speak to Geeta in your language... we are very eager to listen to your conversation....", my students insisted. 

While I could understand their excitement and giggles when we spoke in our native language, I found Geeta quite puzzled & shy while responding.... why not? this time she was surrounded by a gang who were carefully watching her and her words :)

So when I asked her, "would you like to join us in our class?", she immediately agreed.

After 3 days, I find her in our class today with others....

"Sir, you are late.... We came much before you...", she complained.

I looked at my watch.... I was on time... I found others smiling at this remark and I understood the matter...  Poor girl, she didn't know that she was dragged into the class by her peers much before the class begins, as usual.

We settle down.... students start sharing their respective work with me followed by my comments.... and then comes a voice from her -- 

"Teach me some Gujarati... I have forgotten many vowels...."

I was quite surprised by this request .....

"why do you wish me to teach Gujarati ? you already know / speak that language at home.... now you should start learning Marathi well, as everything will be told and taught to you in this language here...."

"I know Marathi too.... I have studied till class-5 in Marathi.... then 1 year in Gujarati.... and then now again in Marathi....."

This was another revelation for me.... !

"But how do I teach you Gujarati now.... We do not have any Gujarati books here..."

"There is one book in the shelf, I saw.....It is mix - English and Gujarati..." - instant reply.

"Oh is it ? I am not aware of it... Can you show me?"

She gets the book and starts reading it when I tell her to do so...... Incidentally it was a short picture story book meant for young kids about Circle...  I noticed that she was quite fluent in reading.... and when I asked her to tell me what she understood by the story, l was satisfied by her comprehension ability too, given her life-realities. 

Meanwhile, other students started demanding my attention to their queries and doubts.... and noticing our enriching 'mathematical' conversations, she too probably got inspired to do and demand some 'maths' now ;-)

I did something for the first time, which I generally never do, esp. with a middle school child (Class-7) ...... I gave her a naked (context-ridden) arithmetic problem....and that too a trivial one... 14 + 39 ... 

And this is how she did it.... 

I studied this for a while...... and when I asked her for the explanation, I could see her surprised..  "Is it wrong?", she asked with a worrying tone.

"I did not say it is wrong... I just want to know how you solved it...."

She began, a bit reluctant - "4 nines are 36.... so I wrote 6 down and 3 up.... and then added 3, 1, 3 to get 7....."

"ok... so why did you do - 4 nines are 36....?"

She answered ' its tables'..... I pointed my finger to the operation here.... and she gave me an embarrassing smile...  :-)

She snatched the book from me, reworked on the problem and showed it to me...

Then I gave her a problem that involved adding more than 2 numbers. She did it this way (answer is 2428)

While I was studying all of this, she started explaining - "I had first written the 2 of 28 above... But then I thought, it should be written down itself...."

Yes, this was surprising even for me esp. because she had properly worked out the procedure of previous problem (adding 2 numbers with a carry over)..... So I just drew her attention to her previous work, and asked her what did she do after getting the sum of 4 and 9... She understood & re-worked on it.... When I saw her again resorting to the drawing of new tally marks for adding the same set of numbers that she had added before, I told her to wait... and asked her few questions like 7+8 , 9+5 , 6+8 , 5+8 , 7+9 , etc.. 

"I see that you you are able to add these numbers mentally quite well, then can you try adding these numbers without using the tally marks, this time?"

She happily agreed and after some struggle, showed me her work:

While I was still looking at this, she said - " Give me some division problems..."

So I gave her, 528 divided by 7..... First she wrote it this way, 

But then she said that "it cannot be this way", and then she interchanged the two numbers..

Ignore the cancellation mark (she did that later). While I was looking at this work, she told me that she feels it is incorrect.... I asked her why does she think so? She said that she does not remember the tables of 7 well... I told her if she would like to write the table of 7 first and then work out this problem.... She agreed....

She asked for my help in subtracting 6 from 2... I was surprised by her request because I thought she would easily write the difference as 04 after studying her previous work on the subtraction part of division (52 - 54 = 02)... I noted that she had first subtracted 5 from 5 to get its answer as 0 and then came to the units place.....

I could have built upon this, but for some reasons, I resorted to a separate problem on subtraction. 

I  told her to first solve these two problems.

a) 74 - 21
b) 74 - 25

You may notice the set of numbers I have chosen in both the problems. Why did I keep the first number same and just tweaked the second number in the 2nd problem? Only to assess her knowledge of borrowing (regrouping) in the 2nd problem? Or because of something more important? If yes, then what's that I was aiming for ? 

I wanted to see if she can think & work out the answer of (little difficult problem) 74 - 25 using the answer of previous simpler problem 74 - 21. However, having seen her number-sense, I  was a bit skeptical of this 'logical' way. But children have surprised me many times and hence I did not want to make any assumptions here.

But this is what she did: 

What she has done in the 2nd problem, is not something very strange for the people working in the space of Maths education.

If you observe carefully, she has first cancelled 7 and wrote 17 above it. Then she asked me if its correct.

I asked her, " why is she doing that?"

" Because I cannot remove 5 from 4..."

" okay... so what can you do in such case... ?"

"Borrow from the next number..."

" So are you 'borrowing' then...?"
She understood what I meant from my emphasis on the word 'borrowing'.  (Though I hate to use this word, but I chose to use it now, because she was using it and ......)

So she cancelled 17.... and said, " I will take 1 from 7.... It will become 6.... and give this 1 to 4... so it will become 5..... and now 5-5 = 0 ..... and 6 - 2 = 4 "

And then she gave the book to me asking, " Is it correct?"

Mostly my response to such a question is - What do you feel? and Why so?  However this time, I chose a different approach.... I gave her this problem:

But then she stopped after couple of seconds..." How can I solve this? Even after giving 1 from 6 to 2, I cannot remove 9 from 3...."

So then I slightly modified the second number and she comfortably solved it like last one :)

When she asked me for the validation, I decided to intervene this time.

" Is it possible for us to know if our answer of subtraction is correct or not?"

-- Blank stare --

So I wrote 7 - 2 on her book and told her to solve... She solved it immediately & correctly. 

" This is a subtraction problem.... But can you see some addition too over here.... ?"

At first she did not understand what i meant by this, but with little clue, she could see and say what i had intended.  " 5 + 2 = 7 "   She was surprised by this.... But understood when we discussed why it happens so..... 

So then I told her to check the answer of previous subtraction problem.... and she realized that her answer was incorrect.... She said that 40 and 25 should add up to 74, but its 65... and hence there is mistake in the subtraction.

She was very curious and desperate to know the method to get the correct answer now.... 

I began -  "What is 74 made up of ?"  

She said: ' 7 and 4 '

'Will 7 and 4 make 74 or 11 ?'

'11' - with confusion on her face....

"So what is 74 made up of ?"  

No answer....

"okay... what is 20 + 5?"

' 25 '

what is 30 + 7?"

' 37 '

"So now tell me what is 74 made up of ?"  

with some reluctance..... ' 70 and 4......'   

"Yes... Good...."

and we started laughing now :-)

"Okay... So let us write 74 as 70 + 4 now...."

She wrote this on the floor.... and when I asked her what next, she wrote 25 as 20 + 5 on her own below the first expression.... I told her to perform the subtraction now.... I was curious to know as to how she would deal with this format now....

Yes... She surprised me.... She did this all on her own..... This was her sequence

"cannot remove 5 from borrow from 70..What should I take from 70?", she asked me..

"You decide", I said. 

" Can I take 10 from 70 ?", she asked me....

I gave her a plain 'okay',  without letting her know, how happy I was now :)

It was further worth noting that after removing this 10 from 70, she correctly added the same amount to 4 to make it 14.  Rest was easy for her....

When she asked me for the validation of her answer, I reminded her of the method we discovered to check our answer few minutes back....   And it didn't take her a minute to do this verification. 

And she got highly delighted looking at this 74 turning up at the bottom..... 

"Yes, our answer 49 is correct....." Her feeling of satisfaction  was no less than that of a person who had just conquered the peak of a mountain.... :-)

No doubt, her teacher too was equally delighted ;)

She quickly and happily copied this work from the floor to her notebook and ....

" Sir, please give me one more problem...."

Wouldn't a Maths teacher be desperate to hear this request from his/ her student? :)

"Oh yes, Sure...."

I dictated:  53 - 28

She notes the problem in her notebook and resorts to the floor for working on it as before.

I later realized that I should have rather asked her to now solve the same problem which she had claimed to be unsolvable some time back,,.... ( scroll up to see 62 - 29 )

It was a pleasure to watch her solve this confidently and then even verify it. 

Isn't this wonderful ?

She was on high and wanted to solve more subtraction problems.... However I drew her attention now to the pending Division problem, from where we had navigated into the Subtraction....    (I hope you remember :-)

This reminder surprised her and she immediately copied that problem as a fresh one on her next page..... and this is what she finally did after some struggle and scaffolding. 

If one carefully compares this piece with her first two attempts on this problem, one would be surprised to see her change in approach towards dividing the first part of the number...

Last time, she didn't mind writing 54 or 56 below 52.... but this time, she has taken a number less than 52 while looking at the 7's table....  when I asked her about this change, she argued that subtraction can be done only if the lower number is smaller than the upper number...

If you are now wondering how she carried out 52 - 49, then let me show you her work -

She wrote this 3 at the proper place, but was unaware of where to write the left out 8 after getting it down.... If you check the previous image, you will see that she has brought it down further (below 3)....  She was stuck up at this moment and asked for my help....

I chose to do some spoon-feeding this time, and told her that 8 has to be written next to that it becomes 38..... 

Interestingly she took it on from here on her own and completed the division process as shown.... 

But then... she was uneasy with this non-zero remainder....

"What to do now? I should get 00 as the remainder in the end, right?..."

I asked her , " Is any number left out to be divided in the dividend? "

' No '

" Can you divide 3 by 7 ? "

' No '

" Okay.. in such a case, the division process is over...  we can have non-zero remainder sometimes... "

I thought this instruction / rule might puzzle her.... but rather, she looked relieved ;)

" Shall I give you one more problem for practice, like we did in subtraction?"

' YES '

I gave her 725 divided by 5

And this is how she did - 

I would suggest you to study her above work carefully first....

It should surprise us that she has not divided 7 by 5 first.... as children are generally taught in the school....  She has directly taken the number 72 for dividing by 5.... ( reason?)

And this was not easy.... because like most of the students, even she believed and told me that the Multiplication table of 5 is till 50....  so how can we divide 72 by 5....

Rather than telling her to consider only the first digit 7 for division, I chose to work on the other correct and more important concept now....  

I asked her how had she made the multiplication table of 7 some time back..... She told me by successively adding 7..... So when I asked her if the addition work can continue even beyond 70, she immediately said Yes.....

"So does the multiplication table of 5 end at 50?"

'No'    and she started working on this,,,,,,

Based on her experience of previous division problem, she selected 70 to be written below 72 in the division.... and then carried out the further steps....

One thing worth sharing is that she got delighted getting 00 as the final remainder this time...  ;-)

And she wanted more division problems.... so I gave her these problems for home-work....  

Any thoughts on the choice of numbers in the problems above ?

She solved each of these problems at home & showed me the next day... 

Were all her solutions correct?  You might be / should be keen to know about her thought process on the last subtraction and division problems, I guess....

Some new, unexpected and interesting things have emerged in the way she has solved the last subtraction problem...  I think you would want me to share those with you....

So we will continue our  conversation on this matter in the next post... 

Meanwhile, please let me know your thoughts about this math-talk. 

Thanks and  Regards
Rupesh Gesota

Monday, September 23, 2019

Known to unknown - Integer to Rational exponents - Part-1

I had not yet introduced them to the numbers involving fractional or negative exponents, like 6^3.5 (6 raised to 3.5)....  So I was curious to know as to how they will solve this problem -

I would suggest you to make an attempt to solve this problem on your own first.... Don't worry, even if you don't remember any of such math that was taught to us in school :)


do not read further unless you have solved it  :)


As expected, I found all of them perplexed when this problem was posed to them.... But after some time, one of them came to me saying - "the answer is 1000"   and this was with Confidence !

I was surprised. So I asked him for the approach and rationale, to which he replied -

"All the other options are incorrect.... So it has to be 1000."

To be honest, I was least expecting this approach. I was more keen to listen to his explanation now...

The expression inside the bracket evaluates to 100....  and the exponent 3/2 means 1.5, which means more than 1.... 

-- It cannot be 100 because 100 raised to 1 (and not 1.5) is 100
-- It cannot be 10 because if 100 raised to 1 itself is 100, then how can 100 raised to 1.5 be less than 100?
-- It cannot be 1 because 100 raised to 0 is 1.
-- So it has to be 1000..... Further, as 1.5 is more than 1... hence (the answer of 100 raised to 1.5) should be bigger than the ( answer of 100 raised to 1).

I wonder if the people (like me) who already 'know how to' evaluate fractional indices would (even pause to ) think and reason this way...... Rather we would (almost instantly & probably even proudly :-) use the 'Laws of Indices' given to us (& accepted & memorized by us).

one of the conventional ways would be this - 

He asked me if his answer is correct. 

I looked at other students and asked them about their views about this. All agreed with this argument.  So I had to congratulate him, but how can I let him go away so easily :-)

"What if, I would have not given you the options? How would you solve it then?"

He looked at me with a smile, knowing very well that his teacher has caught him this time !

"It would have been difficult in that case", he said... "let me go & think about this....."

I could see couple of his friends giving him a teasing smile while he was returning back, loaded with the new challenge.......and while one of them started consoling him, "Don't worry, I will solve this part now :-)

And soon, everyone was again engrossed with this new problem now ...

How did they work on this ? Could they solve it ?  

We will see this in the part-2 of this post..... let me also say that the next post will talk about how these students looked at the negative indices.....   :-)

Meanwhile, you too can try out this with your children / students (who know how to deal with integer indices but haven't been introduced to fractional indices yet) and do let me know how they worked on this problem..... 

You may even share your views about the approach adopted by the boy in solving this problem and how did you solve it :-)

Thanks and Regards
Rupesh Gesota

PS: Students belong to class-7 marathi medium government school in navi-mumbai. I work with them voluntarily after their school hours as a part of maths enrichment program.

Thursday, February 7, 2019

"Sir, we have done this many times! ... 1/6 is Two (half)/6 is One-twelfth..."

Setting: Some 6th std government school students in an After-school Maths enrichment program

I knew that they had been taught the fraction arithmetic in their school. So when I gave them couple of problems to work out, either some of them arrived at non-sense answers (of course, not their mistake) or some of them were applying the 'bunch of rules' incorrectly and there were also some who were able to successfully recall and apply the rules to get the correct answers, but it didn't take much probing from my side to make them realize that this was mere answer-getting and not Understanding ! 

So we spent some time (few sessions - really few!) to understand fractions - through context based problems and pictures (but no manipulatives). And after that I gave few problems to them for practice. The only rule we applied was to solve without any rules! 

I would like to share with you the approach of one of these students to solve these 3 problems. I am sure, it will delight you as much as it did to me :)

This is how she had presented her work to me. I am sure, what would most probably catch your attention is her answer to the second problem (b).

Our (holy) text books don't have space for such weird-looking fractions, but my students are very comfortable understanding & playing with such creatures :)

I first spent enough time to carefully study her work, made a guess of how she might have thought and solved to get these results, and then finally called her & asked her to provide an explanation for each of these results. 

And this is what she gave me :

I would strongly suggest you to stop reading further and spare some time studying each of her 3 solutions.. Don't panic or give up if it takes more time than usual :)

This work was a delicious dish for me. So when my heart had enough of it, I decided to call her. I wanted to give her a big hug for this feat !!  But then I contained my emotions and rather asked her a (stupid?) question -

# I saw your pictorial explanation for your first solution. Good attempt. But I am unable to understand it very clearly. Can you plz explain?

"Sir, we want to remove 4 and 3/4 from 17 and 1/2. We can remove 4 from 17 but cant remove 1/2 from 3/4. So I split 17 into 2 parts: 16 +  1 1/2.  
Now 16 - 4 = 12   and   1  1/2 - 3/4  = 3/4.  So the answer is 12 and 3/4

# Okay. Well done. What about the 2nd (b) problem now... 

"We want to do 3/4 + 5/6 ..  Both 3/4 and 5/6 are bigger than Half. So we first add these two halves to get one whole.... Now what remains to be added is 1/4 and 2/6.......... Now I know that 1/4 + 1/4 = 1/2 ..... I observed that 2/6 has 1/4 included in it ........ So I....."

# Wait..Wait... How do you know that 2/6 is more than 1/4 ?

"Because 6/6 is whole.... so 3/6 is Half..... So (1.5)/6 is 1/4....... and 2/6 is more than (1.5)/ 6
So 2/6 = 1/4 + (Half)/6 ...."

# Oh..good one ! Then?

"So now we can add the previous whole and this new half to get 1 and 1/2 . And we further need to add the left over (half)/6 .... Now we know that this (half)/6  is same as 1/12..."

# How?

"Sir, we have done this many times! ... 1/6 is Two (half)/6 is One-twelfth..."

# Oh yes! 

(How dumb of me to not remember that we had discussed this many times, isn't it ? :-)

# So then?

" Now to add 1/2 and 1/12, we see that 1/2 has 6 twelfths in it.... So their sum = 7/12..  And hence final answer is 1 whole and 7/12"

# Got it...... But your earlier answer is (9 1/2) / 6. And now you have got 1 and 7/12... How come two different sums for the same pair of numbers?

I could see that she was struggling to figure out how she had got  (9.5)/6 .... So I rather asked her --

# Okay, can we check if both these answers are equivalent or not...."

"Sir, (9.5)/6 is more than 6/6 ( whole ).... whats left is 3.5 / 6 .....   3/6  is Half......  and now (0.5) / 6  which is = 1/12  .... so yes we are getting the same answer as second one...."

#  Nice...  But I am curious to know how you got (9.5)/6 .... 

"Sir, even I am unable to find out now.... "  And she started laughing aloud :)

# Hmm.... So I hope you 'now' understand the importance of 'writing' an explanation..

"Yes sir...."

So I thought of helping her now.... (Remember I had studied and guessed?)

# I think you kept your whole as 6/6 here rather than writing it as 1 whole...

I paused here.... And as expected, she picked up from here....

"Yes,..... I got it...... 3/4 + 1/4 = 6/6 .....   So now, whats left to be added is 5/6 - (1.5)/6 = (4.5)/6 .....  6/6 + (4.5)/6 = (9.5)/6 ....."   

And she again started laughing at this point :-))

I asked her to explain the 3rd solution also......

Yes, she did beautifully and confidently explained it.... But I would allow / invite you to study her work to know what she has thought and how she has solved this one.....

I am sure you will get it (after learning so much from her :), but if you can't then let me know, I will share with you her approach....

I will be glad, if you can share your views about this post, her approach, my approach etc.

Thanks and Regards

Rupesh Gesota