Thursday, February 7, 2019

"Sir, we have done this many times! ... 1/6 is Two one-twelfths....so (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 :)
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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 one-twelfths....so (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

Wednesday, October 17, 2018

Yes Sir, this answer is correct! :)


His approach to solve the problem was perfect. After a minute, there was a need to add the two fractions:

3/4  + 1/20 

He started doing some 'cancellation' .. I could hear his mumbling - 'four fives are 20'... So after couple of seconds, he provided me his answer - 

4/9

Right Sir?

He was waiting for my response, but there was only silence for few seconds. I kept staring at his solution. He was surprised with this strange kind of response :)

I am sure this is not a new situation to many maths teachers.. We have come across many interesting inventions of fraction arithmetic procedures by our students, isn't it?  :-)

So then what happened next ? How did I respond ?

Why did you do this 'cancellation -- 4 fives are 20?'

Teacher has taught us this way...!

Okay... but why to do this way? How do you know if this answer is correct?

He stared at me with a surprising look....

My answer is not right, Sir ?

After thinking for a while, I inquired - 

Okay, How much is Half + Quarter?

Three fourths.

How is it written?

3/4 

Can you write down the entire equation?

1/2  + 1/4  = 3/4 

I am quite surprised that you did not add the two fractions here, the way you did in the first problem... You did not do any cancellation here.....?

Sir, this was very simple.. I already knew that Half & Quarter adds to Three-fourths.

Hmmm... But how do you get these numbers 3 and 4 in the answer?

He thought for a while... and responded this way --


Half is made up of 2 one-fourths. So adding one more 1/4th gives 3/4.

Great !  So Can we add the two fractions in the first problem too with such an Understanding ?

I doubt if you will believe me, but he took less than 8-10 seconds to solve this:


He reasoned it this way --

There are 5 one-twentieths in 1 one-fourth.
So 3 one-fourths will have 15  one-twentieths....
Adding 1 more one-twentieth to them, gives 16 pieces of one-twentieth.... 
So answer is 16/20

He again asked me for validation...

I asked him back - "What do you think?"

And this time, he replied with confidence and even a wide smile :)

"Yes Sir, this answer is correct!"


Thanks and Regards
Rupesh Gesota

To know more, checkwww.supportmentor.weebly.com


Tuesday, September 25, 2018

Tussle of 2/3 and 3/4, again!

Some time back, I had shared how a group of students added fractions visually, without using any procedure or rule. This is the link to that classroom experience:


The above post also points out how students (with weaker conceptual understanding of fractions) generally confuse or misinterpret 1/3 as sum of 1/4 and (Half of 1/4)... this guess probably gets triggered because the small extra amount (corresponding to 1/3 - 1/4) seems to be quite close to Half of 1/4, if the figures are drawn roughly or not so accurately.....and they dont have enough experience / exposure that looks can be deceptive over here and there can be two different fractions with very small difference......

I would suggest you to read that post (if you haven't yet) to know about the argument given by one of the students as to why this extra amount (i.e. 1/3 - 1/4) cant be Half of 1/4. It was music to a maths teacher :)

Little did I know that I would be facing a similar situation so soon... However, what motivated me to share this experience with you is - that there is some twist in this tale :-)

The problem at hand was -- (about proportional reasoning)

If  a group of workers can make 4 walls in 6 days, then how long will they take to make 1 wall?  

This is how some of them started -

6 days    -->  4 walls
3 days    -->  2 walls
1.5 days --> 1 wall
1 day      --> 3/4 wall 

I am sure you would have noticed the (common) flaw in the last step...
So I asked them how to verify this result... One of them suggested lets trace back...

1 day   -->  3/4 wall
2 days --->  1.5 walls
6 days  ---->  4.5 walls

So they realized that since we didnt get back to the given condition from 3/4 wall, it means its incorrect. 

So then they were stuck.. How to figure out?

I had realized that they were stuck because of the arrival of fraction - 1.5 days - in the second last step... Also, I was sure they generally remember more common fractions like quarter, half, 3-quarters and tend to forget the other not-so-commonly-used-fractions. So first I asked them -

What do I get if I remove some amount from half?

Most of them, as expected, shouted immediately -- Quarter...

Are you sure?

Yes.. (instant uproar)

I paused for a while...

No one has any doubt...??

To this, one of them - Yash - said --- "we can have 1/3 also...."

Yes !! I was so desperately waiting for such an un-common fraction :-))
I looked at others.... Many looked puzzled... So I asked him to come forward and explain it on the board... He drew a circle and divided it into 3 equal parts, and explained how each is 1/3 and we saw that there are 3 one-thirds in a whole... Class agreed with him....

So with this background now, I headed for the second part -- 

What do we mean by 1.5 days?

1 and half days...

Ok... So this means it has how many half days?

3 half days...

So now, if 3 half days correspond to 1 wall... then 1 half day corresponds to.... ?? (I paused)

Silence for @ 5-6 seconds... And again, the same voice --- "Sir, it will be 2/3 wall"

I was so glad..... but the class was still wondering.... I invited him to again come & explain with the help of pictures.....


This is what he did on the board:


He didn't draw both the steps as I showed above.... He erased one part from both the sides in the 1st step (equation) so that he can find the result corresponding to 1 day (2 halves).. which is 2/3 wall.....

Not everyone in the class understood him well.. But some did.... And so, I called up one among these -- Sania - some to come and explain.... 

She rephrased it very well.... and I could see a sense of satisfation on everyone's face :)


But then as we were celebrating this understanding, Sania popped up again, but with a tone of surprise this time ----

"Sir, if we remove half from these 2/3, then the remaining two small pieces will join together to make a quarter...."

Oh !! Her remark took me to Past...... Because exactly same comment / guess was made by a student in another class, few months back (this incident is described in the previous post whose link is shared above)

I didnt do the mistake of losing this golden opportunity - I just grabbed it !!

So it means 2/3 is same as 3-quarters, isn't it ?, I asked her with confident tone :-)

And she agreed to this...

I drew the attention of class to this unfolding interesting conversation.... To this, most of them agreed like her...  (see the diagrams / scribbling above...)... but Adarsh argued --

Sir, how can be 2/3 same as 3/4 ?  We just saw that 3/4 wall was an incorrect result.... It gave us 4.5 walls and does not take us back to the given condition i.e. 4 walls....

Again, there were some toggles.... rest got this point when he pointed out to the matter written on the board.... 


Though this was a pretty good counter to why 2/3 and 3/4 are not same, however I realized that we are missing something very important at this junction of misunderstanding....

So I again sparked off the debate --

Luckily, we had this problem where there was reference to 3/4.... and we had worked with 3/4 to conclude its incorrect...... But what if we didn't have this problem to refer to?  If we were in some other context..... How would we then find out if 2/3 and 3/4 are same or different ?

They got my point.....and some of them agai started drawing the pictures of 2/3, 1/2, 3/4 etc....   After some time, I saw that they were unable to find a lead, I invited them for a whole class collctive disussion.....  We started drawing the pictures and discussing about it...

Yash again bounced back after some time,

"We know that double of 2/3 is 4/3 .... and double of 3/4 is one-and-half...... (pictures of 4/3 and 1.5 were drawn while this was said) ... and because 4/3 is less than one-and- half..... we can say that 2/3 is less than 3/4...."

I hope you will pause and think about his arguement for a while..... 

Isn't this beautiful ?? 

He had compared the doubles of quantities to find the relation between original quantities...

I looked at others for their views, and almost everyone bought this idea !!
One of them even appreciated him :-))

So now I turned to the person who had sparked this exploration - Sania... 

What do you feel now about these two smaller quantities? Do they add up to Quarter?

No sir..

Why?

Because we saw that 2/3 is smaller than 3/4....  So if the 2 pieces add up to 1/4, then 2/3 will become equal to 3/4 .... she said this to me with a confident smile... :)

Students were about to disperse now.... but how could I let them go without a germ of thought again? :)

So whats the relation between 2/3 and 3/4...?

"2/3 is smaller than 3/4....."

Correct... So now my doubt is -- Its smaller than 3/4, by how much??

    


Few again didnt get what I asked.... Those who got explained others..... 

Again, there were some quick guesses -- "Its smaller by half of quarter...."

Do you see ? They still rush to interpret such a piece (shape) as 'half of quarter'  :-)

Just for my satisfaction, I probed them -- Do you mean that this missing piece is same as that extra piece (1/3 - 1/4)?

No.. No.. Sir.... we have just now proved that those two extra pieces are not halves of 1/4...... 

They didnt know that the answer to the question was Yes.... that the size of this missing piece is same as that of that extra piece :)

I told them to figure out this at home-work, which they happily agreed ! We will be discussing this in our next class.....  

I am pretty excited, what will unfold now.....  What about you?  :-)

Thanks and Regards 
Rupesh Gesota

PS: These students are from grade-7 and 8 Marathi medium government school and are part of a maths enrichment program- MENTOR. To know more, check www.supportmentor.weebly.com

Friday, September 21, 2018

Revisitng the forgotten lesson.. (on mental math)

The other day I did some mental maths with a new bunch of students. I noticed that they were highly dependent on the standard algorithms for basic arithmetic like addition and subtraction. Yet, it was interesting to note that most of them were already aware of 'many informal' ways of doing this manipulation. However they never used these (more sensible ways) during their school maths (why?)

So on my emphasis, this is how they solved some of the problems:


Watching 'their' methods getting space (time, ear, respect) in the classroom surely delighted them... So delighted, that they asked me to give some more problems for home-work. :-)

We had another session after couple of days, but only 4 (out of 15) were present (thanks to the festival vacation). We decided to discuss how they had solved these problems.

Out of these 4, one of them - Vaibhav - had solved all the given addition problems using the standard procedure. When asked for the reason, he didnt say anything.

So I asked other three to share their approaches. The board got virtually divided into 3 parts and they started writing on it parallely, while Vaibhav and I watched them work.


Plz take some time to notice and think about each of these solutions.

I asked them to look at each others' solutions and share their views.
I was amazed by the fact that the only thing they noticed / checked was the subtraction... So Kiran and Rama did their corrections...


Fine... I now had a better opportunity... I told them -

"Oh... All of your answers are different. I am confused now... How do we go ahead...?"

 Suraj (the one with the correct solution) was about to comment, but I stopped him and asked Kiran to respond... She looked at Rama's work first -

"She has converted 329 to 400 by adding 1... How is that possible?"

Rama got this and she did the correction:


She said, she has written 330 itself in her notebook.. but wrote 400 on board by mistake.

So now we had 2 different answers: Kiran's 448 and (Suraj and Rama)'s 450

I again asked Kiran to comment about this different answers. She said she too is puzzled and feels both answers are correct. 

I asked Rama to notice Kiran's answer... She could not take it ahead...  Vaibhav too was mum. 

So I finally released Suraj ... and this is how he argued --

'Kiran has added 1 to 329 to make it 330..... And then she has kept 1 of 121 aside.... and has added 120 to 330 to get 450.... Now she has removed 2 from 450.... but she should remove only 1 from 450..... to get 449 .... and then add that 1 (that was kept aside) to get 450....."

He further continued ---

"She can also give that 1 of 121 to 329 to make it 330..... and then 330 + 120 = 450.... then she need not do any addition / subtraction...."

Since I knew that we had discussed such an approach in the last class, so I asked him to write and explain. This is what he did -

 
I asked others if they agree with this and they did. Finally, I thought to wrap up this problem by --
a) 'showing' them what Suraj said in the beginning..
b) how even 400 would work if worked well..



We also discussed why 330 is a better option than 400...

Next problem:  703 + 58


 Got two approaches:

  Next problem:  254 + 258

solved by Suraj...  He solved it one way and showed me... When I asked him for the 2nd method, he did it as shown above and was surprised to see a different answer :-)

Incidentally, Rama and Kiran too had errred up in the doubling of 260 as 420...   So they were rather surprised with Suraj's 512...  However Suraj was sure about 512 and was surprised with 412....  It was a scene worth watching....    :-))

After struggling for some time, Suraj resorted to the standard method to see what turns up over there....


The other two girls were watching what Suraj was doing.... They saw this 512 and immediately figured out the flaw in their addition process....


Now, Suraj noticed their correction and understood his mistake.... They all had a good laughter at this moment of revelation !!  And I was enjoying learning from them -- how they were learning from each other's mistakes and corrections..

I had thought that someone would add the two 250s together first and then add 12 (4+8).. but it seems they were carried away with the 'rounding up' strategy....  So I thought to draw their attention to even this approach...asking them what would they do if I ask them 26 + 27... It didnt take much time for them to relate this problem with the 3-digit case.... They loved it !

Next problem: 538 + 25

Suraj and Rama solved it this way:   540 + 30 = 570 ; 570 - 7 = 563

Following was other two students' work:

Hope you will spend some time analyzing these solutions....

So then I asked four of them to figure out the correct solution....

Other 3 students could immediately point out the flaw in Kiran's method. They argued -

"She has placed 7 at the ten's place...and it should be at the one's place..."

Kiran understood this and corrected her answer... I could also see her hiding her face from me this time :))

Vaibhav was not yet able to arrive at the process..... So the whole gang went to him / his solution to help him -


And I was so happy to note that this time it was Kiran who took the lead in explaining the logic to him, who had done the same mistake in the very first problem ---

"SInce you are keeping aside the 5 of 25, we should not subtract this 5 from the answer, but add it back to the answer ... We should subtract when we have added someting extra to make a bigger number....so we will remove 2 here because you have converted 538 to 540"

And while she was explaining this, Suraj wrote represented this thought on the board......

538 + 25

540 + 20 = 560
560 - 2 = 558 
558 + 5 = 563

I asked them -- "Should we add first or subtrat first in this case?"

Suraj responded to this by working out this on the board:

560 + 5 = 565
565 - 2 = 563

And they all were surprised to note this that the order does not matter .....  

I realized that it was a good opportunity to give them this problem: 150 - 73 + 24  

But never mind, I will catch them in our next session! Meanwhile, how about you asking this problem to your students? :-)

Thanks and Regards
Rupesh Gesota