Tuesday, August 30, 2016

"There seems to be some problem in Table of 15...."

"Estimate the length of this table"

He thought for a while observing the table and said, “More than 45 inches and less than 75 inches.”

“How did you guess?”

“Our small scale is of 15 inch. I feel around 3 and half such scales will fit along the table.”

(Can you figure out his two errors? What will YOU do in this case? Take some time to think about your strategy before your read ahead)

“Yes, I agree with you that around 3 and half scales would fit along the table. However, can you explain how you got these two figures: 45 and 75?”

“Sir, 15 x 3 = 45 and 15 x 4 = 75.”

After some thinking – “What if it were exactly 5 scales?”

"Then it would be 90 inches."

“What if it were 10 scales?

"15 x 10 = 150 inches."

“Okay…  So you mean 10 scales would mean 150 inches and 5 scales would imply 90 inches….”

While he was about to nod in agreement, he paused…. I could see him being puzzled at something….

After 3-4 seconds: “What happened?”

“Sir, wait… I think there is some problem.”

“What problem?”

“Some problem in the table of 15.”

“Oh, is it? How do you know?”   

(I also succeeded in NOT reacting to his response with my laughter/ shock/ anger)

“Because 15 x 10 = 150, then how can 15 x 5 = 90?”

“Hmm…. So then, what is the contradiction according to you?”

“If 15 x 5 = 90, then 15 x 10 should be double of 90 i.e. 180.”

“True… So then…….?  Is 15 x 10 = 150 or 180?”

“180 does not seem to be correct……150 is correct.”

“Hmmm…. So?”

“But then 15 x 5 = 90 is also correct…..Table of 15 has something special it seems…. This is surprising, I never saw this…”

 I just ensured that he found an authentic enquirer along with him in this process/ problem.

“Can we check other tables also then?”

“Yes… 12 x 10 = 120……. 12 x 5 = 60…. Here it is working properly….”

“Working means?”

“Means 60 is half of 120…”

“Okay….. So?”

“Wait Sir…. Let me check some more……   11 x 10 = 110   and 11 x 5 = 55…… Here also, it’s working well…”

“Hmm…… Should we check some more tables…?”

“No sir… I feel it should work in the table of 15 also then…. 15 x 5 should be half of 15 x 10…….? I am not getting why we are not getting so in 15?”

“Okay… Let’s recite the table of 15 together….”   (Why would I have done so? Was there any other way?)

“15x 1 = ……”


“15x 2 = ……”


“15 x 3 = ……”


“15 x 4 = …….”


“15 x 5 = ……..”


“15 x 6 = …….”


I paused, waiting for him to pick up the clue….. And Yes! He did it!

“Sir, Wait… I think I have done some mistake…   How can 15 x 5 and 15 x 6 be both 90?”

“Hmmm….. So?”

He started telling the tables again but slowly and more thoughtfully this time. I could see him making the table of 15 via repeated addition (adding 15 to the previous multiple).

“Sir, I got the mistake!!  I had missed out 60.”

The JOY of Discovery on his face was unmatchable!!

“Hmmm….  So…?  What about the mistake in the table of 15….?”  (This time, I couldn’t hide my teasing smile :-)

And even he continued to laugh -- with little shades of embarrassment  :-)

Rupesh Gesota

Tuesday, August 9, 2016

What's bigger of the two?

Objective:  Assessment

Why?  -- Not for marks, but to figure out the student's level of understanding and thus to help me plan my lesson.

Student's details: Class-6 student (Her maths teacher told me (on her own) that she is among the brilliant ones in the class)

Teacher: What's bigger?    1/8  or   1/4      (i.e. one-eighth or one-fourth)

Student:  1/4

Teacher (very happy, but emotions concealed) : Why do you feel so?

Student:  Because the fraction with smaller denominator is always bigger.

Teacher: (Highly disappointed, but continues without any such communication) Ok. How about this pair now? Which is the bigger of two ?   7/21   or   25/100 

Student:  (thinking.......for long....)

Teacher (after about half a minute):  What happened? 

Student: (no answer)

Teacher: (after another 10-15 seconds):  Tell me what you are thinking. May be I can help.

Student:  Denominator of 7/21 is smaller than that of 25/100..... (Silence)

Teacher: Ok, go ahead.... 

Student (in a confused tone):  But then...... Numerator of 25/100 is bigger than that of 7/21.. So I am confused which of the two fractions is bigger.

Teacher: Okay... But how come you did not face this problem in the previous problem.

Student: I did not look at their numerators then......... But anyways, they were both '1' there.

Teacher (after waiting for another minute): Okay.. How about this comparison now - Which of the two is bigger?     3.5    or     3.28 

Student  (after about 5-7 sec):    3.5

Teacher: (Excited again!)  And how do you know this?

Student: I converted both the decimal numbers in fraction form. 

Teacher: (excitement grows!)  Plz explain how

Student:    3.5 =  3/5      and       3.28  = 3/28       
Now we know the rule that smaller denominator means bigger fraction. 
So 3/5 (i.e. 3.5)  > 3/28 (i.e. 3.28) 

Teacher:  (almost wanting to bang his own head, but again - emotions concealed)
Okay.. Let's take the last problem. 
Arrange the following decimal numbers in descending order:   3.05  ,  3.5    ,   3.50

Student (after about a minute):  3.5 is biggest of the three.

Teacher: Can you plz explain how?

Student:  Again, I represented these decimal numbers in fraction form.  
(this is what she wrote and showed me (pen-work)

So according to her,  3.5    >   3.50     >      3.05

Teacher: Ok, thank you :)

There is a lot that can be said, discussed and learned from this experience...I learned few things.... But this time, I chose to stay mum and would like to hear from you, if there is anything that you could learn from this...

Please note that I could have helped her discover the 'correct' answers by asking right questions.... However, I chose to not do so in this exercise.... But how would you deal with such a situation, if you happen to teach her?

How about asking these questions to your entire class?  -- Not to the class who is being currently taught fractions and decimals..... But to the class who has already been taught these concepts in their previous years.... Worth trying (daring) ?   :-)

Thanks and Regards
Rupesh Gesota

Friday, August 5, 2016

How an interesting date can help a Maths teacher?

Yes, I am aware that I need to write the Part-3 of the series of posts on Solving Simultaneous equations Their way.... I will do that soon... But now I am more excited to share with you what happened in our class today --- 

Most of you must be aware of another interesting date we had couple of days back....  

For those who failed to notice -- So would you like to Discover it Now?

Well then, you need to honestly follow these directions (without scrolling down :)

1)  What was the date on first Thursday of August 2016?  
2)  Now write this date in the form DD-MM-YY format. 

Hmm.... Could you see something interesting pattern?



Try again..... 

Give a closer look.... I am sure, you will....  !!    :-)


Could you spot this beautiful sequence in the date 4-8-16 ?
Well, to be honest -- I too had failed to observe this remarkable date.   :(

I came to know about this only late in the evening via WhatsApp (& then Fb) - thanks to couple of maths teachers and maths lovers who keep me posted about such events  :)

So sadly, I missed talking about this with my students in the noon.... However luckily, something struck me and I could throw this ball the next day with my another group of students... 

"You know, there was something interesting yesterday and none of you have pointed that to me till now", in a complaining tone. I did not want to hint them by using the word 'date'

And I was so so so happy when most of them instantly shouted ---

"Sir, yes - yesterday was 4 - 8 - 16"

"Oh wow!  I am glad you had noticed it... So what's interesting in it?"

One of them -- "Sir, double of 4 is 8.... and double of 8 is 16...."

The other chap -- " All are powers of 2"

Third one --- "Sir, these are consecutive powers of 2"

"Fantastic !!  Good observation!   Did you share this with your school teachers?"

"Yes, we did... Even she was very happy to know this!  In fact, we saw this when she wrote the date on the board and we immediately pointed it out."

"Hmm.... Good... Well, can you tell me something more interesting in this date?"

after few seconds 

--- "all are multiples of 4..."

--- " all are even"

"Well, aren't the above facts 'obvious' when these are consecutive powers of 2?"


"Let me help you a bit"  and I write this on the board -- (complete year)

4 - 8 - 2016

And soon, some voices filled up the room --

"Again all of them 4,8,16 and 20 are multiples of 4...."

"But 2016 is not a power of 2..... ", Raju follows the former guy.

"And how do you know this?", I ask him.

".....16, 32, 64,.......128......256......512......1024...... 2048..... We should have the year as 2048 for all to be powers of 2"

"True.... Anything more?"

------ silence ------

"Ok.... let me help you further....Because this demands a much sharper observation... I too missed it.... I came to know about this because of a message sent by my friend.."

"Sir, if YOU could not notice it, then how can WE notice?"

"Why do you feel so?  You need not compare your abilities with me... Many of the times, you have made better and more observations than me.... Further, I have also learnt many new problem solving approaches from you....", While I continue to confess, some look at each other with wonder as to what am I saying, while others are still glued to the date.  :-)

"Lets focus on the factors of 2016"  I continue... "What do you think of 4 and 8 then?"

"Sir, both are factors of 2016"


"We applied divisibility tests for 4 and 8...  For 4, we checked the last 2 digits, and for 8, we checked the last 3 digits."

(Please note that these students also know the reasons behind these tests...
What about you and your students?)

"Ok.... What about 4 + 8 now?"

"That's 12..."

"I know it's 12 Sahil.... Can you now check if 12 too divides 2016 evenly?"

"Yes sir...even 12 divides 2016....."

"And how do you know this?"

"Because 2016 is divisible by 3 as well as 4..."

"Good... That's not all.... what do you feel I will ask you now...?"   


I write this on the board...   ( 4, 8, 4 + 8, 4 x 8, 48)

"Oh Sir,.... do you mean 4 x 8 and 48 also divide 2016 evenly?"

"Why don't you check?"

Some mur-mur and......

"Yes, its divisible by 32...."


"32 is fifth power of 2..... So we need to check the last 5 digits....  So we appended 2016 with a leading 0... which is 2016 itself.... and we divided 2016 by 32.... we got remainder=0"

"Okay.... Lets go ahead...."


I was very curious now -- What will they do to figure out if 2016 is divisible by 48....?

What will you do at this juncture ? What will your child / students do ?


"Sir, it's divisible by 48 also..."

"Don't tell me you have actually done the division"

"No sir... we have used the divisibility test for 48..."

I was so satisfied and relieved to hear this response --    :)

" 16 and 3 are co-primes of 48.... 2016 is divisible by 32, so it has to be divisible by 16.... and we have already checked that 2016 is divisible by 3.... So, 2016 has to be divisible by 48...."

Listening to such a Math from a student would be like a Music for a Maths teacher, isn't it? :)

And I was waiting for them to wonder at this fact that -- 
2016 is divisible by 4, 8, 4 + 8, 4 x 8 and even 48.....   But it seems their car was still in motion....

"Sir... even 8 / 4  and 8 - 4 divides 2016", Rohit points out to me.

I look at the students for their comments... and all roared happily with "Yes Sir!"


Wait... Wait.... There is more they had...  :)

"Sir, do you know we have another interesting date next month?" Sania asks me with confidence. 

And.... I am puzzled... calculating the number corresponding to September.....!!  And while I am still engrossed, Sidharth makes an entry !!!

" Yes, I know Sir.... The date is 4 - 9 - 16.... "

Rohit too joins him soon --- "Yes, and we even have our class on that day... It's Sunday!"

While many of them, and including me, were still wondering about the theater around....

But we could soon decode the code in this date...

See if you can decipher the code?   :-)  :-)

As usual, we even discussed more - relating to such types of dates...
Can you/ your children/ students make some observations?  

And as if this was not enough.... Pandurang had meanwhile scribbled something on the board. He asked us "Can you all now find out what's interesting in this next year's date?"  :-)

Incidentally, we had some engaging conversations few days back on some other interesting category of dates...  You may check out this post here:

Mathy Date (Square root day)

Mathy Date.. Extended....

"Sir, tomorrow is a Happy Date"

I am waiting ......to hear from you ---- the answers to the questions I asked you in this post   :-)


(1) If you liked this post Or if you have similar class-room stories to share, then I would love to know.. It will be great if you post your reply, stories and views, in the comments section of my blog itself, instead of replying over email.only to me.. Why? Because these could possible trigger some interesting conversations, thus helping (motivating) many other readers (teachers and parents), now & even later...
Here is the link: http://rupeshgesota.blogspot.in/2016/08/how-interesting-date-can-help-maths.html

(2) You may even subscribe to my blogs via email. There is a text box on the left hand side of the blog, where you can enter your email-id.

(3) The conversation above happened not in English, but in Marathi, because the students belong to marathi-medium municipal school with whom I work regularly. Check the website: www.supportmentor.weebly.com to know more about this program. There's a short video-clip on its homepage.

Thanks and Regards

Rupesh Gesota