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

Sum of the Numbers Trick

The video below is only first (Part-1) of the 12 videos. It captures the process that students go through while solving the following problem:

PROBLEM:  The number trick was performed before them a day before.. They were supposed to work out the reason at home... While Poonam starts sharing her views (video- part-1), Shreya claims that she has figured out something and takes over...   

TRICK: I claim that I can predict the sum of 5 numbers without knowing all the numbers, and hence I write this sum on a piece of paper.. I then write the first number on the board... Students tell me the 2nd number. I write the 3rd number. Students tell me the 4th number. I write the 5th number. And very smartly and Instantly, I also write the sum of these numbers as written on that piece of paper....Students take time to add up all these numbers and after 2 minutes get shocked as to how I could predict this sum before-hand.... I do this trick twice... i.e. with two sets of numbers... You can see these problems unerased on the white board....One has the sum 22466 and other has the answer 22541

To check all the twelve videos of this problem, click on this link:
https://www.youtube.com/playlist?list=PLZBBZFls_2nGBmbOh-0HmU5qVSNxu18Ym

Thank You...

3 Hats Logical Puzzle

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

The problem was given to them and  they were given 10 minutes to think independently..

PROBLEM: 

To check All the four videos of this problem, click on this link:
https://www.youtube.com/playlist?list=PLZBBZFls_2nFsth0xMJRYpSYQVkuz_UOx

Thank You...

"We can divide the product only with the first factor"

Hello friends, 

Our First "Maths Teachers Study Group" meet on 18th May was fantastic. I will soon share its proceedings. Thanks to all the teachers who participated in this initiative :)

Yesterday night, I saw the video of Deborah Ball where she shares some interesting insights about the competency required by the maths teachers to spot the error in their students’ work. I don’t want to spoil your excitement by narrating her video and hence would recommend you to first watch this video and then resume your reading. https://www.youtube.com/watch?v=nrwDM4ejNqs

Who is Deborah Ball?

http://soe.umich.edu/news_events/news/article/deborah_loewenberg_ball_receives_medal_for_distinguished_service_and_honora/

Deborah shows three multiplication problems to the committee members and asks them if they can figure out the errors made by the three students. You may call me crazy but, something drove me next day to pose the same challenge to my class-7 students J I wrote these 3 problems as it is on the board. Believe me, it was a real pleasure and a rich learning experience to watch and hear them analyze these problems - hunt for others’ mistakes and base reason on their arguments. I strongly felt the need to video record these conversations. We will discuss about this process at length during our next ‘Maths Teachers Study Group’ scheduled on Sunday, 31^st May.

But in this email, I want to share with you some equally (if not more) interesting observations that came out as a by-product of the discussions on the above problem.

The students were very happy (and even surprised) to come across one of the new ways of doing multiplication.  I leave this up to you now to figure out (and tell me) the reason for their excitement. You will have to watch the video for this. 

In fact Poonam was so thrilled by this experience that she wanted to test/ apply this method (‘new’ method - according to her) on a different problem now. This is how she did it.

​She asked me to check if it’s correct. (Hold on - What do you feel about her way of doing multiplication this way?) 

Now, they have heard enough from me that it is not the teacher’s business to check if the student’s work is correct. But my experiences have taught me that the Undoing and Unlearning effect takes some time J

She got the message. “I can cross-check this answer by using division. Shall I?”

“Go ahead.”

And this is how she did.

​I noticed that she has erred.

“Sir, how come the quotient is 154? Shouldn’t I get 198?”

“How do I know dear?” I see to it that my pretended innocence is not caught J

Perplexed, she thought for a while, and decides to solve the same problem using the ‘old’ multiplication method now.

​She errs again and gets the product (3770) which is different than the one she had got in the former one (4950)

“Oh sir, it means I have some done some mistake in the first multiplication process. Or, is it that the ‘new’ method doesn’t work?”

“What do you feel?”

“But we had seen earlier that we get the same and correct answers using both the methods....because they are actually similar methods.”

(We had already discussed about the use of commutative property in these two methods.)

“Hmm....So what do we do now?”

She thinks for a while.... And I was so happy to hear her say, “Shall I cross-check the product that I have got using division?”

(Of course, there were other smarter ways for verifying the correctness of product, but talking about all that at this juncture had the potential danger of disturbing her present flow. Hence I choose to just let her go - her way(struggle)).

So this is how she did it.

​“Oh my god! Sir, I am getting a remainder of 20 now!”

I also noticed that her process of division was not complete. Look at the quotient. (In fact, it’s one of the most common mistakes made by most of the students (why so?) However, I choose to ignore this error as of now. And I will surely bring this to her notice, but later, and with the help of this snap :-)  ...To continue to read, click on 'Read More' below...

"2 rupees 20 paise for 100 grams" - Maths class or Ethics class?

Hello friends,

I did not know that I will be back with the new post so soon... But then, this summer has given me a wonderful opportunity to work with the children Daily. And hence, I am so lucky to be a part of (learn from) such stories daily. 

If some of you have been unable to read the earlier post, where Poonam is intentionally allowed to make a spate of mistakes in the multiplication and division algorithms, how she arrives at a (erroneous) conjecture by observing / analyzing some patterns in these mistakes, and finally how she spots her mistake to counter/ discard her own conjecture. A beautiful cycle of discovery that probably mathematicians and scientists would go through. 

This is the link to this story: 
http://rupeshgesota.blogspot.in/2015/06/we-can-divide-product-only-with-first.html

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

It all started with just a casual inquiry to one of my students - "what do you feel how much does my bag weigh?"

From which the discussion gradually navigated to the fact that how some shopkeepers cheat the consumers by fixing the std. weights. Sachin claimed that he has seen the 'hole' on the other side of the standard 1kg weight so that no one can spot it. To which Sushma responded that in such cases, the vendor would earn more money.  This as seem did not at first augur well with Sachin because his opinion was stationed on the fact that consumer would get any quantity lesser than the weight on the other side of the balance.

It was interesting for me to learn that what is sometimes so obvious for one student (& the teacher) may not be so obvious for the other student. I allowed Sushma to explain her view and this quickly brought a smile on Sachin's face.

Sushma had already brought into this discussion her experience of purchasing 1-kg rice from the vendor. So I thought to just built upon it. I did not know that my next question would help me learn something so interesting from my students.

"If we assume that the vendor has removed the piece of weight worth 50 gms from the std. weight of 1 kg, then how much will he earn from the transaction of 1 kg of rice, sold at Rs.24 per kg?"

Of course, both the above values -- 50 gms and Rs.24 per kg were arrived at with the help of "students", they were not my (the teacher's) enforcement!  It was equally interesting to study their estimation sense and even help that develop using this real-life example) But I am not going deeper into those conversations this time. Why?  Well, Because I hardly get any acknowledgement from the readers for this effort :-)

Yes, coming back to the problem and how my students saw this and solved this.... Again, here too I am going very quick this time i.e I am not mentioning all the (beautiful) conversations that Really happened. 

Sachin finally could settle down with this argument that if 1 kg costs Rs.24, then half kg would cost Rs.12. So 50 grams would be worth Rs.2.20

It was interesting for me to see him arrive at the cost of 50 grams directly from half kg.  I would suggest you to pause for a while and answer this question. (of course, I will be much happier if you even reply to me with your answer) - 'What according to you must be the way that Sachin would have devised to arrive at Rs.2.20?'    I hope you will honestly pause and think before reading further :)

22 - 9 is 11 or 13 ?

Hello friends,

Hope you must have got and read my previous email about the starting of Maths Teachers Study Group.  If not, then plz read it and also let me know if you are coming. I will send you the exact address, etc. I am so happy that 9 Teachers have expressed their interest! It is on Monday 18th May at Airoli, Navi-Mumbai. For more details, plz check that email. 

I remember I had mentioned in that email, about sharing one of the fantastic experiences I had while working with students on fractions that day, but then I thought of first sharing with you what happened Today!   :-)

Meanwhile, the word fractions remind me of one more interesting thing. A group of students were asked what is the half of 2/8 (two upon eight)  And it was interesting to listen to the response of few students. They said it is 1/4 (one upon four)  :-)

If possible, plz share your views on reading the conversation below. Will be happy to read.

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

We were solving a problem which encouraged her to make a list of all the possible numbers that would leave a reminder of 1 when divided by 3. And then, even hunt for any possible pattern in this list. So she wrote: 1, 4, 7, 10, 13, 16.....

After few seconds, I ask her,” Could you figure out anything?”

“Yes Sir. There is a common difference of 3 between any two adjacent numbers in the list. But I am wondering, why we got this difference of 3, because these numbers are not the multiples of 3.”

​

And I was like, Waaaaowww! What an interesting query! I paused for a while, to think as to how I should respond to this.

“So do you mean, they should not have the common difference of 3?”

“Yes. I have added one to every multiple of 3 to get these numbers.”

“Hmmm... Can you write down all the numbers from 0?”

Once she is done, I ask her to highlight the movement of multiples of 3 from the beginning. She draws the path. Now I further ask her to repeat the process for ‘her’ list i.e. for 1, 4, 7, 10....  She does that too. But after this, I choose to stay mum for a while. I wanted the bulb to glow, but what’s fun in it if the student gets it so easily, isn't it? J

So she asks me,” Sir, now what?”

“Means? Now it’s your job dear! I have guided you enough.”

She smiles at me and turns at the board. After few seconds, she again looks at me with a smile, but also with wrinkle on her forehead. “Sir, I am not getting.”

I realized that it’s the time to roll up my sleeves now.

‘Let’s say, you (Poonam) and Yogita are standing two meters apart. How will this distance change if both of you move ahead by the same amount?”

{While narrating, I also represent this situation pictorially on the board. But now I realize that ‘she’ could have been encouraged to draw the diagram}

She responded correctly that the distance between them would still remain the same; and she could also reason for this. I further probed her for the other case viz. she travelling more than Yogita. And she again responded correctly, along with the justification. So Now, I drew her attention to our original problem.

And she was like – “Yes Sir, I got it now.” with a wide smile J

“Please explain it to me then.”

Starting the "Maths Teachers' Study Group" - at Airoli, Navi-Mumbai from 18th May

Hello friends,

It’s that time of the year again -- end of April and most of the teachers must be on the verge of finishing their academic responsibilities. Some of you might have already planned your holiday-trips, isn’t it? Hmmmm...  J

You know what? Today morning, I had a fantastic experience with my students, while working on fractions. And that’s when I decided, that I will share this story with you in the evening. So I sit down, write few lines, and all of a sudden, some other thing comes to my mind. I pause. I thought that the latter one is more important and even interesting. So, what next? I press the Ctrl+A, Ctrl+X, Ctrl+N, Ctrl+V. Yes, you guessed it right. The fraction story will come to you ‘after’ this broadcast in the next email J

A strange yet interesting thing happened last week (And I am so-so-so-happy that Finally, it did happen!) I got an email from couple of maths teachers. They wrote to me that this time, they want to make a better use of their month-long vacation - for their own professional growth.  (Waaaaaaaaow!) They wish to dive deeper into this mighty and beautiful ocean of mathematics.

Isn’t this really exciting? I really appreciate their decision – because they are embarking upon this learning journey, completely on their own, without any support (or push? :) from their respective schools. Excellent!! My dream to start a Maths Teachers’ Study Group has come to a reality, finally!!

So - to be very honest, I am personally much more excited than these couple of teachers now. To start working with them, to listen to them – about their classroom stories, their challenges, discussing about the most common maths mistakes that students make, analyze and learn from these mistakes, mathematics content and pedagogy related matters, investigation and inquiry-oriented mechanisms of teaching and learning mathematics, and of course, share with them my book-readings, experiences and even insights of senior global mathematics educationists -- their articles and host of educational videos, and ,.....and........and........ – there will be loads of FUN – like Playing mathematical games, solving ancient mathematics puzzles, and trying to unlock and appreciate the value embedded in these (often ignored) mathematically rich entities.... Oooooffff :-)  :-)

All this might remind some of you about my initiative of starting the maths-learners google group, and then even the compilation and circulation of "Math Energizer", isn't it? For the new folks in this group, this is the link that will navigate you to all the documents and maths stories that have been shared in this group till now: (you may esp. like to check out the oldest posts):  https://groups.google.com/forum/?nomobile=true#!forum/maths-learners-group/join

But then, I am sad that I am unavailable till 16^th May, because of some other prior-commitments. Hence, despite these teachers being ready to start immediately, we will be able to start only from 18^th May.

So friends, here’s an Invitation to All of you.....

If you or any of your colleagues would be interested to embark upon this thrilling-learning journey of mathematics, you are Most Welcome!! Write an email to me about your interest and participation in this "Maths Teachers’ Study Group" and you are IN! So simple, isn’t it? J

But what about the Fees, Time/ Duration, Frequency of sessions/meetings, Number of hours? 

Well, nothing is decided on these 'trivial' matters - as of now. We will work out all these details later, as the group evolves (when I say evolve, I do not mean by quantity - but by wisdom :-). We, the interested teachers and me, believe that all these details are important, but still are secondary and hence, can be easily worked out later, esp. by a person or group who is really hungry to learn!

I can personally relate to this fact because some time back, I used to travel for around 6 hours by ST buses, on the first Sunday of every month, to be a part of similar Teachers’ Study Group based at Wada Taluka of Thane district. To reach there, I had to first travel by a train, then change two buses and then again an auto. J Yes, the journey drained me out completely, but the peer-teachers and the discussions in the Study Group recharged and refreshed me, much more!

The only thing we are sure about now is that We Are Beginning – without worrying too much about the nitty-grittes. We are also not much concerned now about the destination and the challenges that we may face in this journey.

We are meeting on 18^th of May at Airoli, Navi-Mumbai at sharp 10 am – to Begin! 

I am eagerly waiting to hear those precious three words from you --  "I am IN"  :)   I will then share with you the necessary guidelines and documents. 

Imp: I am not taking any space on rent and all... So the group size will (unfortunately) be limited by the size of the room.

I never use this phrase - First come First Serve -- but sadly, it will be applied this time :-)

PS: I am also happy to share with you all an article that was recently published on Divya Bhaskar, a marathi newspaper. Though some statements and figures do need some correction, however I am thankful to the reporter for sparing few minutes to listen to me and even the editor for to share my story, work and vision with people around. People often wonder about my switch from engineering to teaching school maths. And it has been Real challenge for me to answer this question to anyone (esp. strangers) in even half an hour... So, I guess, this 2-minute quarter-page article will turn out be a good handy beginner to quench their curiosity :)

This is the link:  http://divyamarathi.bhaskar.com/news-srh/MAH-MUM-positive-monday-news-mission-for-math-4976021-NOR.html