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:

http://rupeshgesota.blogspot.com/2018/04/adding-fractions-through-visuals.html

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

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

www.supportmentor.weebly.com

Solving one problem - Learning many things

It didn't take much time for them to figure out, with reasoning, that the total number of numbers in a given range is 'one more than the difference of boundary numbers.'

 

However, this 'formula' (of difference +1) became visible only when I gave them some 'difficult' numbers to deal with (from 32 to 75). You may also notice how the opportunity of generalization was grabbed at this moment. It was students who made me write 'b - a +1'.

So now I encouraged them to find the number in the Center of this range 25 to 30. I did not use the words 'mean' or 'average'. 

Kajal quickly answered it as 27. While Tanushri said, there is no one number, but two numbers 27 and 28 in the centre. She figured out this by writing down all the numbers and then cancelling out numbers in pairs - one from left and one from right - while approaching the centre. And to this, Yash immediately said that the central number would be 27.5

I asked others, if they agreed with this result. One of them related this situation with the cm / mm markings of a ruler and explained to the class how 27.5 would be between 27 and 28. I ensured that also knew the meaning of 27.5 i.e. 27 and half.

I asked them to solve the next problem: Central number in the range  45 to 60. All of them quickly resorted to the previous method (cancelling numbers from both the ends) and figured out the answer as 52.5

So now I challenged them to figure out the central number in the range 32 to 75 without making the list as above. This slowed them down to think. I could see some just staring at the problem while others scribbling something on their book.

After a while, Yash came forward to show his solution:

There are 44 numbers from 32 to 75. Now half of 44 is 22. So there will be 22 numbers above and below the central number. Hence the central number is 22.5

(I want you to pause over here and think about his approach & result)

I realized that he has not yet visualized the problem. So I drew a part of open number line on his book with 32 and 75 marked on it, and asked him -

"Where would the central number lie? To the left of 32 , right of 75 or between them?", while pointing at the respective regions.

He instantly answered - in between. I now drew his attention to his answer (22.5) He laughed at it and went back to work again.

Surprisingly Aryan too came up with the same solution.  while others were yet figuring out the approach. So I decided to give them a hint / tool to think over - pictorial reprsentation.

At first I just wrote 3 numbers on the line - 25, 27.5 and 30 and asked them, How do they know if a number is in the centre?

"It should have same numbers in the left and right"

Okay... So is that the case with 27.5 ?

"Yes, its 2.5 away from from both the sides."

I showed 2.5 on the picture, & asked them, - Whats the distance between 25 & 30?

"5"

I showed 5 in the picture now and told them - "So now this a good hint for you to solve the problem in hand.."

When I saw that they were still unable to go ahead, I decided to invite everyone for a collective exploration / discussion. I intentionally picked up the (flawed) solution of those two students... However I represented their thought process using the pciture, for them (& others) to 'see' what's happening.

I began -- Yash & Aryan say that there are 44 numbers from 32 to 75. They then halved it to get 22... What do you all think about this? 

I thought some one will correct my mistake i.e. we need to find their difference (43) and not the items in the range (44). But sadly, no one did.... In fact one of them picked up from this picture saying -- "now we should go back 22 from 75 to get the central number 53..."

This remark was accepted & welcomed with such a delight by all, as if someone had demystified an age old mystery!  :))

I didnt have an option - but had to accept the verdict, but only momentarily. I then aksed them - "How do you know 53 is the central number...?"

It's at a same distance from 32 and 75..

How do you know? Did you check?

Within couple of seconds, came loud revelations --- Sir.. No... the difference is not same on both the sides.... If we add 22 to 32, we do not get 53... we get 54....!!

So then? You still want to stick to 53?

To this, one of them immediately sprang up -- Sir, the central number is between 53 and 54.. So its 53.5 

When I prodded him for the reason - he said, its just a guess. So I asked him and others, if we can verify if 53.5 is correct?

They said that we should check if it is equidistant from both the numbers.

So I drew a new number line, with 32 and 75 on the ends and 53.5 in their middle.


What next?

They did:  53.5 - 32 to get 21.5 
Actually it was more of me who did this subtraction than them. I realized this mistake when they messed up in the next subtraction. This is how I did with them --

Lets subtract 32 from only 53 first, to get 21 and then we add back the left over 1/2 to get 21.5  Everyone almost said this aloud with me, thus making me (falsely) believe that they have understood this.

However, when we went to another piece:  75 - 53.5 
Again, this is how led them:  Lets only subtract 53 from 75, this gives us 22 , and then we add the left over 1/2 to get 22.5 (can you notice the (intentional) trap? - i hope you didnt fell into it :-)))

I was hoping that someone will point out my mistake. But no one did :-( 

It's only then that I realized that I did a big mistake by rushing through the previous subtraction (53.5 - 32) If I had allowed them to work it out on their own, some of their misunderstandings would have probably cropped up then & there itself... Nevertheless, it was still smart of me to catch them red-handed in the second problem :-))

 So how did they react to this two different (21.5 and 22.5) distances of 53.5 from 32 & 75?

They were surprised for a while. But interestingly, they were ready to disacrd away this 53.5 as the central number, as it didn't obey their rule of being equidistant ! 


OMG ! They were about to throw away the correct result because of some other mistake... What should I do now? 

I wanted to draw their attention to their inorrect subtraction, but I realized that the numbers were too big for them to estimate & visualize the result and error. So I created a situation where they can easily analyze, learn from it so as to come back and find  as well as correct their error on their own.

"Look... A student of another class explained to me this way: 37.5 is not the central number from 35 to 40 because its not equidistant from 35 and 40.... It is 2.5 away from 35 but 3.5 away from 40..."

What do you think about his argument? - I asked them.

One of them quickly said -- "Sir, 37.5 is correct... He has done the subtraction incorrectly.."

I looked at others and waited for their response. 

I was so happy when one of them could relate this scene to our problem... "Sir, may be even we have done some mistake in subtraction...."

Okay.. So how do you verify if the result of our subtraction is correct or not? While asking this I wrote 10 - 3 = 6. How do you know 6 is correct or not?  

Its incorrect because 6 + 3 does not make 10.

Good... What if we do not know the difference between 10 and 3 i.e. we do not know the answer... Then how can we find the answer in another way? 

We should find 'what number should be added to 3 to get 10?"

Perfect... So you have converted subtraction problem to addition one... 

Lets find out 75 - 53.5 using this addition method now...

I saw two interesting approaches:

Method-1:  This student has written down 53.5 and 22.5 one below the other, added them to get 76. Then changed it to the desired result 75, and to achieve this she reduced the number to be subtracted by 1. 

Method-2:  This student didnt write the first number (blank space) in the beginning. Just wrote 53.5 and 75. He figured out 1.5 needs to be added to get 5 in the unit's place of the result. So he wrote 1.5 in the first number. Then he filled the ten's digit as 2 so as to get 2+5 = 7  :)

They realized that they had erred up in the subtraction problem and were very happy to know that this right side distance 21.5 was same as the left side distance 21.5

However before we drew our attention back to the Central number problem, I wanted them to learn how to subtract if such easier fractions are involved (also I wanted them to figure out where & why we had messed up in the subtraction 75 - 53.5)

So I took them to the board and asked them how would they solve 53 - 12 mentally, without using std. algorithm.

To this, one of them said, we would first subtract 10 and then 2.

As he was saying this, I represented his idea on the board.

Fantastic! The way you made the 2nd number easier, can you make the 1st one easier?

We do 50 - 12 and then add 3.

Great ! So if you now understand this very well.... lets do our two subtraction problems this way...

And as you see below, they could solve both the subtraction problems beautifully. We even solved couple of simpler ones (25.5 - 3  and  25 - 3.5) quickly just for practice and I must say that they super-enjoyed it !!

So then it was time now to redraw their attention to the original problem.

Whats the central number between 32 and 75?

They agreed that its 53.5 because its equidistant (21.5) from both the ends.

So I showed two times 21.5 on the picture, and then asked them whats the total distance then, from 32 to 75?

To this, one of them still said 44 hurriedly, but couple of them did spot the mistake --- 

Sir, its not 44, its 43.... 

And then we had the most awaiting Ooooooohhhhs & Aaaaaaahs :-)))

I drew another number line... with 32, 75 and their difference 43 written on it... And asked them for the next step.... and this is how they went --

"Do half of 43 to find their centre.... its 21.5... now subtract 21.5 from 75.... (to do this, they first removed 20, then 1 and then half)... we get 53.5 ......."

Is there any other way to get 53.5? 

One of them responded - Yes Sir, we can also add 21.5 to 32.... 

Do you get the same answer?

Yes

So what's the easier way?

We can add to the starting number instead of removing from the ending number !!

They also quickly and acurately solved the following similar problem:

Now they wanted to solve more such problems.... and hence I gave them few for home-work, including the subtraction ones that include more common unit fractions like 1/2, 1/4 etc... 

-----------------------------------------

1) What are your views about this lesson?

2) How about you trying out a similar activity in your class and share its proceedings?

Thanks and Regards

Rupesh Gesota