Masti with Multiplication -- Part - 2

Hello friends, 

As promised.. I am back, with the part-2  :-)

I wanted to write and share the part-2, the next day itself. However some day-long assignments used to drain me out completely - making difficult for me to sit and type the (long) conversation after reaching home in the night... However, I had made up my mind to complete this task today.... You know what motivated me to do this?

It is the honest appreciation (and even confessions) from some of you who could see the 'value' in the patience demonstrated during the previous conversation... Thank you so much for these acknowledgements,.. It tells me that I am on the right track...

By the way, could any of you try out that problem (32 x 8 using 22 x 8) with your children/ students? If yes, then please share your experiences....

If you have not yet read my previous post i.e part-1, then I would suggest you to first read that before you scroll down.. Here is the link: 
http://rupeshgesota.blogspot.in/2016/03/masti-with-multiplication-part-1.html

Hmmm... So where did we stop last time?

The students could (surprisingly and beautifully and confidently) think of and even communicate four different strategies for computing 32x8...  But as my student Poonam had (rightly) pointed out in the end, they were still far from the approach that I was specifically looking for....

Considering this scenario, I decided to walk along the same path that was traced out by them.... .of pattern recognition.

"Hmm.... I see, you have seen a pattern in the previous two problems and their solutions to the solve this third (similar) problem... Interesting.... So then, can you go ahead further?  I mean, can you predict what would be 42 x 8 ?"

"Yes sir....I am already working on this....", said Rajesh

And immediately, Saif shouted out the product - "Sir, it will be 336."

"And how did you do that? Can you please explain to us on board?"

He said that he had simply extended the pattern that he had seen in the former products. 

12 x 8 =  96

22 x 8 = 176

32 x 8 = 256.....  So,

42 x 8 = 336

" the one's place will have 6..... ten's place decreases by 2...... and hundred's place increases by 1...... You see, 12 x 8 = 96.... i.e. it also has '0' in the hundred's place...."

All fine, but his last observation about being able to see the invisible zero really delighted me! Mathematician at work !!

Masti with Multiplication -- Part - 1

They were eagerly waiting for me. I had not visited them or worked with them since long...Their teacher had told me the day before that students were complaining about my absence. "Sir, does not teach us now. He just goes to other school." This shook me & so I decided to meet them today....& make our Friday, really a Good Friday :)

As I was about to enter, I could hear their non-stop recitation of multiplication tables. The moment I entered, the tape-recorder stopped with a delightful surprise on their eyes. 

"Please continue." I suggest this to him while greeting their teacher who was managing this show.

And he started telling the table of 12, but with little hesitation this time. He completed it well and as was about to sit, 

"Hey, wait. How much did you say 12x8?"

Students generally (and unfortunately) get scared when their maths teacher responds to their solution with a question. 

"Don't worry... You were correct. I am just telling you to say again."

"12 x 8 = 96"

"True. Can you tell me how much would be 22 x 8 then? "    (I had no idea or plan to do this to them today.... but somehow, this happened :)

I could see some of them quickly resorting to pen and paper. I instruct them not to do so. Within 10-15 seconds, the same boy replied -

"It's 176"

"Okay.... and how did you do this?"

"Sir, I knew the table of 22."

Gosh!! My plan flopped. However, I didn't give up.

"Good, What if I ask you 32 x 8 now?"

"I don't know the table of 32."

"Yes, that's why I asked you so. You don't even need to know the table of 32... I was surprised when you said that you knew the table of 22. Though this pleased me, but I think, you don't need to memorize even till that."

His blank expression worried me. So I thought to reiterate the question.

"If 22x8 = 176, then what would be 32x8 ?  We need to find the product without directly multiplying 32 and 8."

To ensure that everyone understood the question, I asked one of them to share what they understood from the question.

Realizing that the entire class was still not with me, I asked the other student to write the question on the board.

And then, almost instantly, a hand went up.

"Sir, 32 x 10 = 320. Hence 32 x 8 = 320 - 64 = 256."

Packets to balance your weight

Situation: Practical assessment (grade-6 student of a government school)

Reason: To gauge his understanding of various math concepts that will serve as a vital input to my instruction plan later.

"Can you estimate the weight of this chikki packet?"

After thinking for a while --- "It must be around 30 grams."

"Let's check it."

He turns the packet around (seems he was already aware of the whereabouts of the weight). He shows the number to me with a smile -- "It's 50 grams."

"Oh.. your estimation was quite good.... Tell me Sidharth, have you seen a weighing balance?"

"yes"

"Imagine - I make you sit on one of its sides, while keeping this chikki packet on the other side....What will happen?"

While controlling his laughter, he replies -- "I will come down, and the packet will go up."

"Hmm... What if I add some more packets?"

"Then I will start moving up..."

"How much packets will be needed so as to balance you on the other side?"

He thinks for a while...and picks up the tools -- pen and paper. While I was happy to see his systematic work, I was also bit pissed off by the slower pace of the multiplication algorithms that he was using.. I wanted to stop him and ask him about the alternate approach... But doing that now would be 'harmful' to him.. I watched him work patiently, while figuring out what must be going on his head for the next set of moves..

This is how his work looked finally, before he just raised his head to tell me -- "600 packets."

​

I have labelled the work for you (from 1-6) to so that you can study his thinking process.. 

"Ok.. I watched you work Sidharth, But can you plz explain me you thought about this?"

What is 76 - 20 --------- 56, 55 or 54 ?

Two days back I was assessing one of the 6th grade students......

(Purpose? -- same as before -- to know her level of conceptual understanding).... So my role at that time was more of an examiner/ assessor than a maths teacher/ guide........but as you will find ahead, the teacher in me couldn't curb his urge to help and hence, I have eventually ended up teaching her, but fortunately, in a way that she could identify her mistakes on her own and then even correct those.. (I did not tell her that she has erred).... 

Also, I ensure that when I do such assessments, me and the student are no longer strangers to each other..... Such assessments are done only after enough contacts.... i.e. after I have conducted few 'Fun with maths' sessions with them and when I am confident that the student jells well with me and trusts me.... Also the purpose of assessment is clearly communicated to the students and hence they too are always eager to go through this diagnostics process honestly and fearlessly because they are now confident that this process is to help their favorite teacher to teach them better....

We had reached the section of 'Mental Math'

Q.1) "What is 50-20?"

"30" (in a flash)

Q.2) "What is 20-7"

No wonder, this time there was a pause. But the sadder part being, I found her using her fingers for this problem. Her lips and tongue moved, but there was no sound... After about 10-12 seconds, she said ---

"It is 11"

"Hmmm... And how did you get this?"

"I counted back from 20"

"Ok.. Is it possible to verify your answer using other method?"

Silent stare....

"Okay... Tell me what is 20-10" ....While I asked this, I also wrote this expression on the table.

​"10"  (and I wrote her answer too)

While writing further, I ask her what will be 20 - 9 then?

She continued to look at the two similar expressions for a while and then looked up with confidence -- "It is 11"

"Hmmm.... How did you do this --- without fingers this time?"

"Sir, 20 - 10 = 10, then if we remove just 9 from 20, then we will have one more left as compared to the previous case i.e. 10+1 = 11"

"Interesting. You have used the logic this time.. So tell me then.....20-9 = 11  and 20-7 will also be equal to 11 ?"

Spilled Juice (Missing digit) problems

The video below is only first (Part-1) of the Four videos. It captures the process that the two students (of classes-3 and 4) go through while solving the following problems posed to them:

Students were shown the above image and were asked to solve these problems. No time limit was set. But they took around 20 minutes to solve these 6 problems. They were also allowed to discuss among themselves in case they wanted to.... 

To check all the four videos of this problem, click on this link:
https://www.youtube.com/playlist?list=PLZBBZFls_2nEffHLet8RjYnfTPwGTYXrI

Thanks and Regards
Rupesh Gesota

Distributing money : Two methods -- Mentally v/s Long division

The other day I was assessing a middle school student one-on-one.

I asked her 'If you wish to distribute Rs.4,000 equally among 5 people, how much will each of them get?"

She thought for almost 1-2 minutes. I guess, she was working out mentally. I generally don't open my mouth (interfere) when students are thinking. Then I noticed that she resorted to pen and paper. I could see her work and she was trying out this problem using long division method. 

"Sir, each one will get Rs.800."

"Ok. I saw that you gave enough time to think before using pen and paper. Can you please share what were you thinking?"

She replied with a smile -- "Sir, I was trying mentally but I messed up somewhere."

"Oh... and Where?"

"I first did half of 4000, gave 2000 to two of them, further halved this 2000 and now 4 people got 10000. I then took 200 from each of the 4 people and gave the collected amount to the 5th person. So each got 800."

"Wow !! You have worked out correctly. What made you feel that you have messed up?"

"Yes, now I did correctly. But in the beginning, I had, by mistake, taken 300 from one of the 4 persons, and hence, had given 900 to the last person. So four of them got 800 and the 5th person got 900."

I paused for a second. 

"Would each of the 4 guys get 800, if the 5th one has got 900?"

She thinks for a while. " No..No... I took away 200 from the 4 guys but I did a mistake in adding four 200s... I calculated the sum as 900 instead of 800... So I thought I have done a mistake as each of them have not got the same amount. So I switched to pen and paper (log division method.)"

"Okay.."

"Shall we try one more similar problem?"

"Yes Sir", she happily agreed. :)

"How will you distribute Rs.2,500 equally among 4 people?"

I wanted her to do this mentally, but she directly jumped to pen/paper. I did not interrupt her (will push her for mental work after this).

"Sir, each person gets Rs.435." 

I was happy with her confidence, but was also disappointed to note that she was victimized!  This is how she did.

​I would request you to hold on for a while and..... Reflect...... Think what will be your reaction to such a situation -- i.e. if possible, write down sequence of steps -- statements/ questions/ actions -- you would take when your student (s) err out this way. 

ABCDE x 4 = EDCBA

The video below is only first (Part-1) of the two videos. It captures the process that students go through while solving the following logical puzzle:

PROBLEM:  Students were given this problem a day before.... They have to find out a non-zero 5-digit number ABCDE such that its fourth multiple is the reverse of itself i.e. EDCBA.... 

To check Both the videos of this problem, click on this link:
https://www.youtube.com/playlist?list=PLZBBZFls_2nG-hl3pxr7jK3FotkPGMBNv

Fractions Laddoos Problem

The video below is only first (Part-1) of the three videos. It captures the process that students go through while solving the following problem:

PROBLEM:  Four friends take away the laddoos from the basket, but only one by one (i.e. one after the another). The 1st person takes one fourth of the total quantity and leaves. The 2nd person takes one-third of the remaining quantity and leaves. The 3rd person takes half of what was left and the 4th person takes away the remaining 2 laddoos, thus emptying the basket. The question is to figure out the total number of laddoos and the amount each friend takes.

To check All the Three videos of this problem, click on this link:
https://www.youtube.com/playlist?list=PLZBBZFls_2nE6opV47C367_aV_SDi5z5Y

ABCD x 4 = BEEB

The video below is only first (Part-1) of the five videos. It captures the process that students go through while solving the following problem:

PROBLEM: This problem was given to them a day before so that they can work on this at home... They need to find out a Four digit number ABCD such that when it is added to itself 4 times, the answer is BEEB..... 

To check All the five videos of this problem, click on this link:
https://www.youtube.com/playlist?list=PLZBBZFls_2nH4ET0kvN8ENwlek7LkMoeD

Chocolates problem

The video below is only first (Part-1) of the two videos. It captures the process that students go through while solving the following problem:

PROBLEM: Its a problem involving two types of chocolates. An Eclair costs one rupee while a mango bite toffee costs 50 paise... If 10 chocolates are purchased in all using Rs.6, then figure out the quantity of each of these chocolates purchased... 

To check both the videos of this problem, click on this link:
https://www.youtube.com/playlist?list=PLZBBZFls_2nGhGIc6lnL23IsTXFQNK4HI