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