"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 = ……”

“15”

“15x 2 = ……”

“30”

“15 x 3 = ……”

“45”

“15 x 4 = …….”

“75”

“15 x 5 = ……..”

“90”

“15 x 6 = …….”

“90”

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
www.supportmentor.weebly.com

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

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?

.....

No..?

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?"

(laughter)

"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"

"Explain"

"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...?"   

Silence...

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...."

"Explain"

"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!"

(Laughter)

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)

http://rupeshgesota.blogspot.in/2016/04/mathy-date-square-root-day.html

Mathy Date.. Extended....

http://rupeshgesota.blogspot.in/2016/04/mathy-date-extended.html

"Sir, tomorrow is a Happy Date"

http://rupeshgesota.blogspot.in/2016/07/sir-tomorrow-is-happy-date.html

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

Solving the simultaneous equations 'Their' way - Part-2

Hello friends,

I am very happy that you liked my previous post on 'Solving Simultaneous Equation 'Their ' way" Thanks to many of you for explicitly expressing your interest and motivating me to continue with this sharing work.... 

To be honest, I was a bit reluctant to write the Part-2 now.... Why?

Because something more exciting has been happening in our class since past 2 days.... 
Yes, these government school students have surpassed my expectations, and I am so very happy that they keep jumping over the bar, whenever I raise it  for them !!

I will soon share that stuff too.... but first, lets complete the previous story :-)

For those, who have not yet read the Part-1 of this post, I would encourage you to do so, before you carry on ahead.... Here is the link to that story:
http://rupeshgesota.blogspot.in/2016/07/solving-simultaneous-equations-their.html

If you remember, they had made some interesting observations and had even used some sort of  'reasoning' before resorting to the trial and error method to find the solution....  I now wanted to see if it was possible to replace the trial and error element by the logical approach completely...

Both - Jeetu and Vaishnavi - were now working together.. I could see them struggling for the 'second' way... After allowing this struggle for some more time, I thought to intervene..... with a question....

"You have already figured out some relation between Square and Triangle (by comparing the first two figures)... Is it possible to go a little deeper?"

They listened to me with sincere attention but couldn't build upon it... 

May be my hint was not so clear... I stretched my hand further... 

"Can you figure out Triangle is bigger than Square by how much?"

Jeetu picked up !!

"Yes Sir...  It is bigger by 10. "

"Good... and how do you know this?"

And this time, I could see even Vaishnavi join him in the argument... It was a pleasure listening to them.

" If you add a square to circle, we get 10. But if we add a triangle to circle, we get 20.  This means, that extra 10 has come because of triangle."

"Ok.... And can we represent this using an equation?"

She wrote it this way:

"Ok... But what about that 10 you mentioned now?  How can you communicate about that?"

He borrowed the chalk from her and scribbled this immediately.

"Fantastic.... You have now got a new equation.....Can we use this along with the given data to solve the problem logically?"

They again seemed to be struck.... Please note that they were dealing with 'equations' for the first time.... And I was still expecting too much from them... Yes,.... too much !!  (why?)

I don't know what struck me... And I wrote a new problem besides this old one and drew their attention to this:

A + B = 50

A  - B = 18

"Looking at these two relations between the two numbers A and B, can you find out these numbers?"

Yes, this set of equations are very familiar to us (adults). We have probably studied those in our secondary school. But I would like to remind you that these students are just promoted to Class-6 and are from a municipal school. I had no idea what will unfold in next few minutes.. But one thing was sure, I trusted their instincts and abilities !! And I was also sure, that we will definitely learn something from this... 

Jeetu kept looking at these equations for a while......and......

"Sir... I got it."

"What?  You mean, you have got the two numbers? Beware! I am not going to accept your trial and error method this time."  

I tried to act smart and strict, but he continued with a confident smile.... Listen to him.....

"The two numbers differ by 18.... So first, we will remove this difference...... We will subtract this difference from their sum.....  i.e. 50 - 18 = 32......  Now since we have removed the difference between the two numbers, they have become equal... So each of the numbers is 32/2 = 16.....But both cannot be 16... as we know that they differ by 18..... So let us now add that difference to one number...... 16 + 18 = 34....... So the two numbers are 16 and 34."

What ?  Did you really get him ?? 

I am scared if your answer is Yes.... Because, I did not get this.... It seemed logical... a bit sensible.... but my logical brain hinted that there is something fishy.... Perfectly correct answer, but could not comprehend his logic immediately...

So I requested him to explain it to me again.... But he could not rephrase it in a different way.... Rather, he continued with much more confidence....  So I turned to Vaishnavi..

"Did you get what he said?"

"Yes sir... I understood.....!"

What??? This was now driving me crazy........ It was not that I did not trust his mathematical ability... In fact, Jeetu had surprised me many times before, but this time, somehow I was not sure about his logic.... I felt there is something that's got entangled...... and I am unable to see through it or unwind it......

There were two options before me now:  

1) stick with this new problem.... make him make me understand his method..... and/ or even arrive at the other method (which would make sense to me ;-)

2) accept his method for the time being & use this strategy to solve the original problem....

I chose the second option (why?) 

"Fine.... Now that you know how to solve such equations... Can you solve our original problem?"

They did not get what I meant... So I pointed at their new equation again....

"Did you see a similar equation in the previous problem?"

They pointed it out:  A - B = 18"

"What else did you require to solve this problem?"

No answer.....

"What would be the values of A and B, if I give you only this much information ---  about their difference?"

"They can have many answers then......"

"True... So why were you able to identify exactly one solution of A and B?"

"Because we had one more condition.......  A + B = 50"

"Hmmmm........ So now, are you getting where am I taking you?"

Vaishnavi screamed......

"Yes Sir.... I got it..... We have a similar addition condition in this problem also..."

Phew.................Finally !!!   :-)   :-)

She drew Jeetu's attention to Figure (iii)  -- the one that shows Triangle + Square = 24.

Jeetu looked up to me with wonder....  He was overwhelmed with the striking similarity..... I think he had also understood the reason I made them solve the new problem (of A, B)  :-)

He worked out the solution to the original problem in no time...... and got extremely delighted to see the same answer, that he had obtained some time back, using trial n error..

(I should have captured the joy on his face... I also forgot to take the snap of his written work....  I will do that on Monday and upload in the comments section of this blog)

But Vaishnavi took some time to understand this... Jeetu explained her.... Strangely, she was struggling to relate the A & B problem with this picture problem.....   He had used the same logic as before..... He was happy with this revelation...... And I was happy because he had finally got the taste of 'logical' approach...... However, there was something that was still bothering me..... I was not yet able to understand his method completely ;-(

I was going to look at it more closely after the class...... Meanwhile, other students grabbed me for some clarifications in the assignment problems..... But little did I know, that the game was not yet over.....

Vaishnavi came running to me, saying that she has figured out one more way....

"Oh really? Plz explain...."

I now decided to draw the attention of the whole class to this discussion.... I felt it was getting more interesting.... I gave them the background - what had happened till now.....  We three ensured that they understood what are we doing and why are we doing...... (some time was given to them too to think about the solution, but most of them resorted to Trial and Error methods..... Jeetu and Vaishnavi looked at me, with smile....  Probably, this reminded them of their journey :-)

So the setting was made.... Everyone was excited and curious now - to know one more logical approach, that was about to be unfolded by Vaishnavi........  

And I am sure, even you too are curious  :)

So here it is.. This is what she scribbled on the board, with her voice-over in the background...  

And this just blew me away !! She had hit a Sixer !!

Take a careful look at her diagram... how beautifully she has merged the three figures in the problem into just two figures....

I would suggest you to analyze her work before reading further....

......

.....

......

What she said (while drawing) was:

" If we add first two diagrams of the problem, then we get a total of 4 shapes with their value = 30 (20+10).... Now we remove Square and Triangle from this combined set.....(and while saying this, she draws a box around the two lower figures, if you can see.....the Pictorial subtraction! ) ...... We are left with only two Circles now....  Further, we know the total value of Square + Triangle.... Its 24... So removing this pair from the complete set means subtracting 24 from 30..... Hence Two Circles = 6..... So one circle = 3..... Now, we can easily figure out that Square = 7 and Triangle = 17.... So the required total of all three is 3+7+17 = 27..... " 

Now, isn't that just beautiful ??

Probably, you will appreciate this much more if you consider -

a) the way we are taught /  have solved such problems (dry way)

b) the age and background of this child.

I was still having conversation with Vaishnavi as to what made her think of this approach.... and she was trying to explain me....... and meanwhile, I heard another voice ---

"Sir, even I got another way !!"

This was from Payal....  I was so happy to hear from her after long time.... 

"Yes, Payal.... Plz explain to the class..."

"No sir.... I am not sure..... You plz check first....."

"Its ok... We all will try to understand it together...."

But she insisted that I should verify it first, before its discussed with the class... I agreed. And I was about to read her work, she intervened....

"Sir, I don't know the reason.....Why it works.... I have just guessed this method..... I don't know if its correct...."

"Its ok.... Let me see...."

Her work was a bit messy, so I requested her to explain.... This is what she had done:

" Half of 10 is 5 (pointing at the figure (i) of the problem)

  Half of 20 is 10 (from the fig (ii))

  Half of 24 is 12 (from fig (iii)

  If we add 5 + 10 + 12, we get 27... Its the correct answer."

And I was like....... Again, bowled !!

Did you understand this ??

I did not (at that instant).... So  I asked Payal for the reason.... And she argued --

"Sir, I told you... don''t ask me the reason.... I have simply guessed the method..... "

So I thought to share this with the whole class.... and while explaining her method to the class, something struck me ----  I started thinking while explaining... 

Can you ponder over this for a while, before reading ahead..... as to why this works? what could be the logic behind this method that makes it work? 

......

.....

.....

.....

"Payal has done half of the value in the fig (i)...  10/2 = 5... But what does this 5 signify? How can we interpret this operation?"

Sahil responded: " Sir, it means Half Circle + Half Square = 5"

ENLIGHTENMENT !!!

"Then she has halved the value in fig (ii) ... 20/2 = 10..... Means?"

Many voices join this time:   " Half Circle + Half Triangle = 10"

"And how does her last operation get interpreted?" 

" It means Half Square + Half Triangle = 12"

I had got it half the way, what's happening..... and I was so very delighted to dawn upon this beautiful approach... But I chose to stay silent and ask the students now  --

"So why does adding these three numbers give us the required solution?"

Silence for about 25-30 seconds.....and ......... Vaishanvi erupts  :-)

"Sir, I understood why it works..... Shall I explain? "

All heads turn around, looking towards her with surprise...My god, the girl is on hat-trick! :-)

I ask Payal if she has been able to figure out the reason...She gives me a shy look and a hiding smile.... I probe her....

"Sir, let Vaishnavi explain.... I told you, I don't know the reason....."

I could see students relishing this humorous moment :-)  :-)

The Girl takes the charge again... and scribbles up with the same style...

i) She first draws the two halves - of square and of circle, and equals it to 5.....  (equation (i))

ii) then she draws half of triangle, completes the circle, and writes 10 .... (eq. (ii))

iii) finally she completes the triangle and even square, and writes 12...... (eq. (iii))

iv) then she adds up this 12, 5 and 10 to get 27.... (she has erased 12 and has written 27 at its place)

So how was it ??    :-)   :-)

In case you feel that the story ends here, then yo are grossly mistaken..... 

We explored some more 'logical' methods to solve this problem..... 

What?  Some more?  

Yes.... But I am not going to share those in this post now.......  May be in Part-3   :-)

Till then, how about YOU thinking out some more approaches?   You can discuss these with your children/ students and let me know....  I would love to know some from you.....  

Let me give you some hint....... for some more approaches....

http://rupeshgesota.blogspot.in/2014/09/talking-math-in-bus-with-young-co.html

Yes, the hint is There in the above post ... Lets see if you can get it :)

Happy Mathing !! 

Regards

Rupesh Gesota