What's the value of Square-root of 3? - Part-2

So they come on the 2nd day, and I ask them -

"Could you find the value of Square root of 3?"

People who have not yet read the previous part of this post (Part-1), then it is suggested to read that first, so as to get the context of how we landed up here and what are the mistakes, conversations & learnings that happened in that process. Here is the link

http://rupeshgesota.blogspot.com/2023/02/whats-value-of-square-root-of-3-part-1.html

"Could you find the value of Square root of 3?"

"We tried.. But we didn't get"

Means?

"1.73 squared gives less than 3 and 1.74 squared gives more than 3."

I was glad that they worked with numbers having 2 digits after the decimal point, rather than just stopping at 1.7 and 1.8. But then I also wondered how come they didn't go beyond that. So I asked them -

"So what does that mean?"

"It means, value of root 3 is between 1.73 and 1.74"

"And how do you find that now?"

Puzzled look.

------

I told them to vertically enlist the numbers they tried for the very first time... they wrote 1.6, 1.7, 1.8 and 1.9

"Then what did you do?"

"We understood that its between 1.7 and 1.8 and hence tried the numbers from 1.71 onwards...

"Okay.. So plz enlist these numbers too in the second column..."

They wrote 1.71, 1.72, 1.73, and stopped at 1.74, saying they did not try beyond this as 1.74 squared crossed 3... I agreed but asked them to continue enlisting beyond 1.74 & they wrote till 1.80.

" So Now?"

"We don't know what to do now..."

After a while, I drew their attention to the shorter first column & told them to complete this list too - write the numbers above 1.6.... They wrote till 1.1

"So these numbers are between ?"

"1 and 2"

"How did you get these numbers between 1 and 2?"

"By dividing the range into 10 parts"

"Ok.. And how did you get the numbers 1.71, 1.72, 1.73 etc.? I am asking this because I don't see them in the first column?

"We knew its between 1.7 & 1.8 So we divided the range 1 to 2 into 100 parts now"

I circled the two numbers 1.7 and 1.8 when they said this and told them -

"So can we say this second column is kind of Zoomed-In-picture between 1.7 and 1.8 ? ... numbers which were present but not visible earlier have become visible now because you have divided the range into smaller i.e more (100) parts.. Its like you have kept a magnifying glass now on the two numbers 1.7 and 1.8", while pointing my finger from the circled part of 1.7 and 1.8 towards the second column.

I paused to help them understand this new analogy being presented now.

"So now you say that the answer, the number, is between 1.73 and 1.74. What can we do now?"

They said - "We will divide the range further - into 1000 parts now - so that the numbers between 1.73 and 1.74 become visible" while saying this he also circled the pair 1.73 and 1.74

They started enlisting from 1.731, 1.732 and so on till 1.740. So I asked them what does 1.740 represent. They said its same as 1.74. So then I told them to include another form of 1.73 too because they have circled / zoomed this number too? 

So in that column, he wrote 1.730 above 1.731 

How about completing the first column too this way?

He checked and wrote 1.70 above 1.71

Now I drew their attention to the two circled pairs and the list of numbers next to each pair. So that they can also actually / easily see (& not just visualize) that 

(1.7, 1.8) expands to range of numbers from (1.70 to 1.80) next to it,      and 

(1.73, 1.74) expands to range of numbers from (1.730 to 1.740) next to it.

The picture started looking like this in some time....

The squares of 1.731, 1.732 and 1.733 were calculated by them manually using std. algorithm, but when it came to testing the squares of numbers in other columns  (one with more digits after the DP), then I became their assistant and helped them getting & giving the squares of numbers which they wanted, with the help of my phone calculator.  

Things had gone into auto-pilot mode now and they were sort of thrilled / enjoying this process, totally surprised as this hunting never seemed to stop, against their expectation. They said that they had never thought that square root of a number (that too such a smaller one like 3) will have so many digits :-))

I also shared with them that they don't need to enlist all the numbers in a column but can use dotted lines to indicate that. After some time, I stopped them and asked them what do they think about this process ?

"Sir, it seems this is never going to stop....  We are just reaching closer and closer to the answer...."

How do you know this?

"Square comes out to be 2.9999.... or  3.0000 and few other digits after 9 and 0 .... And then number of 9's and 0's keep increasing...."

I asked them if they can be very sure of at least some digits in the square root of 3?

They looked for a while in all the columns and noticed the growing & unchanging section of digits. As you can see above , they have written the value of square root of 3 as 1.73205_......

I asked them if some one tells square root of 3 equals 1.732 , then is it correct?

They said - "No.. Answer is closer to 1.732, but not equal to 1.732"

Looking at their facial expressions and body language, It was clear that this exercise was no less than an adventure ride for them :-))

So now it was time to plug-in this correct value of root-3 into the expression they had arrived at (remember the previous post? :)

'x' and 'y' in the above equations represent the respective lengths of hypotenuse and side opposite to angle measuring 60 degrees in Right Angled Triangle.

They had observed that side length opposite to angle measuring 30 is half the hypotenuse and next attempt was to find the relation between hypotenuse and side opposite to angle measuring 60.

They were told to construct two Right Triangles (with angles and hypotenuse lengths given) and they had then measured the lengths of other two sides in both triangles.

They were delighted to see that their measured lengths matched the lengths given by the formula. Further, the value of Square root 3 was also figured out by them.

We also discussed here about the round-off error, construction/ measurement error, resolution of measuring devices. etc.

"Now we know why we Square root 3 is written as it is in text-books!" , they exclaimed. 

So what about square root 2 ?

They thought for a second and understood there can be list of such numbers like square root of 5, square root of 7, etc.., and I concluded this discussion saying that such numbers are called as Irrational numbers (of course with this thought in mind that this definition / explanation is not yet so precise and complete yet)

We moved to other topic, while giving them this assignment to find the value of square root of 2 (another irrational number which appears much in school mathematics) and they happily agreed to work on this. 

Thanks and Regards

Rupesh Gesota

www.rupeshgesota.weebly.com

What's the value of Square-root of 3? - Part-1

Plan was to help them see the relationships between the side lengths in 30-60-90 Triangles. But without hinting this to them, I told them to Construct couple of Right angled triangles in their note-books having other angles as 30 and 60 and with lengths of their hypotenuse as 10 cm and 6 cm 

Then I told them to measure other side lengths in both triangles, and then asked them - What do they Notice? They were quick to see and say that 

"in both the triangles, the shortest side length is always half the longest one"

I told them to construct two more Right Triangles, with hypotenuse as some whole numbers, but with angles other than 30 & 60 this time. They observed that in these cases shortest is Not half of the longest. So they said this relation holds true only when the angles are 30-60-90.

So now I drew this figure on the board and asked them if they could find which side length would be half of the hypotenuse length, without actually constructing it.

Now It took some time for them to figure out. But after carefully studying previous figures, they said that "Side opp. to 30 degrees will be half of the hypotenuse"

So now, I asked them - If there could be any relation between the side opp. to 60 degrees angle and hypotenuse? 

Their side lengths were in decimals (5.1 and 8.7), so it was naturally difficult for them to easily relate those to hypotenuse lengths. I asked them if they know any theorem that can be used in Right Angled Triangles. And they immediately said - Pythagoras theorem. And so we applied it & finally reached till this step.

Only last part of the derivation is showed above. A right angled Triangle was drawn where Hypotenuse was 'x', side opp. to 30 degrees angle = x/2 and side opposite to 60 degrees angle was shown as 'y'. Each of these steps were told by them and just written by me. I asked them how did they arrive at (1.5 x / 2) and they said 

"square root of x^2 = x , sq. root of 4 is 2 and sq. root of 3 is 1.5"

"How do you know sq. root of 3 is 1.5?"

"Sq. roots of 1 and 4 are 1 and 2 resp. So sq. root of 3 has to be between 1 and 2.. So it should be 1.5"

I asked them if they can verify it. They started multiplying 1.5 by 1.5 using standard procedure. I would have loved seeing them figure out this product mentally through reasoning (Can you try that way?) I thought we will discuss this approach once they get the answer using std. method, but then something else happened (& now I was in fact glad that they went by the std. multiplying method :-) 

They seemed puzzled / uncomfortable with their results. So I told them to show their work on the board.

                        

"What's the problem?"

"Both answers are incorrect. They are above 4. Because the product : 1.5 times 1.5 has to be between 1 and 4."

"Okay... But then how come both of you have got different results?"

So then they started comparing / studying each other's methods. And one of them tried this:

"But then this is also incorrect. It is less than 1", he said

"Okay... So what to do now? How to find what's 1.5 x 1.5?" I asked.

They looked more engaged / puzzled than earlier now. After few seconds, when I found them clueless, I decided to intervene.

Before reading further, I suggest you to pause and think for a while as to what would you do at this moment ?

....

.....

....

....

....

There can be many ways. This is what happened through me at that moment: 

"What's the other way we can write 0.5?"

And they picked up the hint and did this:

They reasoned out for 6/4 as if 6 chapatis given to 4 people, then each gets one and half, so 6/4 = 1.5 and 1/4 = 0.25 because 1 rupee = 100 paise, so quarter means 25 paise.

They were delighted with this answer 2.25 because it satisfied their expectation that it needs to be between 1 and 4. Further the boy who had worked out 1.5 x 1.5 as 0.225 on the board, something clicked to him; he went and corrected his answer -

"I had placed the decimal point incorrectly"

"But then how would you know where to place the decimal point?"

He studied the work again, and said, "Count the number of digits after the decimal point in both the numbers, add them, & then place it accordingly in the product."

"Okay.. It seems you have made a Rule. Will this always work?"

"Sir let's take one more example."  And they did this:

"Okay... but what if there is 1 digit after the dp in 1st no. and 2 digits after the dp in 2nd no. ?"

"Then we will put the dp after 3 digits in the product."

"Will that work?"

"We will take an example to check."  And they did this :

(((  Would like to share something that happened when they were framing the example for this case... One of them selected the numbers as 11.5 and 1.5   

Why do you think he would have taken such numbers? 

I would have loved to see the story that can unfold after testing with11.5 and 1.5, but since both of them were working together, so the other boy noticed this and he corrected him explaining what the rule is under test.... Later I felt, I could have suggested this boy to continue his testing independently with 11.5 and 1.5 .... )))

So now they were stuck with another problem: How to write 7/8 in decimal form?

They had been using 1 rupee = 100 paise, 1/2 rupee as 0.5 and 1/4 = 0.25 till now. So now they were struggling for 7/8.... For a while, they suggested 0.12.5 for 1/8 and we can reason out why they must have said this. But then they were also  uncomfortable with this because they had never seen such notation till now, they said. After a while, they had got stabilized at -

They noticed that the digits in this new representation were same as the ones they had got by multiplying 1.5 and 1.25 using std. method (viz. 8,7 and 5). But the different positions of decimal points in these two representations was bothering them. And it was also clear by now to me that they had never thought of / felt the need of working with 1000s when it came to decimal point notation. 

Suggesting you to Pause and Think about your approach now..

...

...

...

"Okay, so you have been dividing the whole into 100 parts to find the digits after the decimal point, right?"

"Yes.. because 1 rupee = 100 paise"

"Do these numbers represent always Money?"

"No"

But this hint did not help them... They were still stuck...

Again suggesting you to Pause and Think about your approach now..

...

...

...

...

"How do you write half?"

1/2 

"And using decimal point ?"

0.5

"How do you get this 5 in 0.5?  Why not 0.2 , 0.3 ?"

We can write 1/2 as 5/10  and Out of 10 parts, we have taken 5 parts... So 0.5

"Okay...  So you have converted 1/2 into other form 5/10 to find the decimal point form of half ? "

Yes

"But then why not write 1/2 as 3/6 and then write half as 0.3 ?  Or write 1/2 as 4/8 and write half as 0.8 ?"

No answer. Puzzled.

"You have been dividing the whole into 100 parts till now to get the digits after the decimal point... So how will you write half in such case ?"

They said : 50/100 which is 0.50 

"So do 0.5 and 0.50 both mean half ? "

Yes

"Which one is greater?"

Both are equal. 5/ 10 can be written as 50/100 

"What about 0.500 ?"

This is also same ... Here we have done 500/1000 

I thought this discussion might help them to think, go back and divide their whole into 1000 parts now instead of 100 which they had done because of which they had got decimal point number in the Numerator (87.5) and they felt stuck... But probably this much discussion didn't click much to them either... So then I asked -

"How do you write 1/4 using decimal point ?"

0.25 

"How do you get this ?"

25/100 

"Can we divide here too into 10 parts as we had done with half ?"

Yes. It would then be 2.5 / 10

"So which of these mean quarter ?"

All of these. 

So I now wrote all of these in a single line & we discussed this for 3/4 too....

And then also drew their attention to where they were stuck earlier.. To this one of them quickly said -

"It will be 0.875"

"And How?"

"I see that the decimal point in the Numerator goes away when there is one more zero in the Denominator"

The other student also confirmed this observation. I asked them to check this with the answer they have got using std. method of multiplication. And they were highly delighted to see that it matched (1.875)

A thought came to me for a while, (based on the way student had seen and explained) that I should now ask them what would happen (how would they work), if it were 8.75 / 100 or say 13.25 / 100 or say 4.26 / 10 . But then I chose to & even we had to pause this discussion here for some reasons.

So while they were now about to happily leave (after various discoveries), I drew their attention to the problem which led us to all this exploration...

"What's the value of square root of 3 ?"

And they started laughing :)

"So Is it 1.5?"

"No"

"So then what would it be ?"

"It will be between 1.5 and 2"

"How do you know ?"

"Because we saw that 1.5 squared is 2.25 and 2 squared is 4."

"So this is your home-work now to find the value of square root of 3"

And they left happily agreeing to this challenge :) 

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

Day-2

I asked them if they could get the value?

And what do you think could have happened ? Will probably share this in the following post :-)

Thanks and Regards

Rupesh Gesota

www.rupeshgesota.weebly.com

Teachers Day 2022

Like almost every teacher,  even I received - Teachers Day Greetings - from my students, their parents and even from some friends and teachers... 

Some things really touched me deeply.... And I felt to share those with you  

Photos Link: https://photos.app.goo.gl/saKbfao8vG38V4HAA

Description of Photos: 

(1)  Letter written in English to me by a student presently in class-12 science... We are associated since she was studying in class-5 marathi medium government school..

(2) Same girl made this chevda (snacks) for the first time on her own for me....  because I had probably told her once that I like this specific food item...

(3) Message typed in English and sent by a student who is presently studying in Second Year B.A.....We are associated since she was in class-9 marathi medium government school...

(4) Same girl has made & shared a short video of some memories / snaps, mostly of our celebrations during festival, picnic, get-together etc.

(5) Message sent by a person working as an IT professional since few years... We got associated when she was in her First Year Engineering in the college where I taught somewhere in 2009-10...

(6) Message sent by another student of her class / batch...

(7) Messages sent by my school teachers who have taught me when I was in middle / secondary year school..

(8) and (9) - Few of my government school students called to meet me in our class today, where we had some quality time together

(10) I could go and meet my school teacher who had taught me Maths when I was studying in Class-6....  Saw her after more than 25 years.... Luckily she could also see me doing some Math-Magic with her grand-daughter studying in Class-3... 

End of the Day:-

And I heard someone as if calling me from behind while I had started walking after boarding down the bus. I looked back. He said --- , "Your wallet fell down from your bag."

I checked my bag and found the zip of its front side open (may be while taking the ticket).. I ran towards him and thanked him deeply for his kind gesture...

Feeling grateful 

Thank you friends, for reading this message & checking the photos 

Rupesh Gesota

Demo-class to Teachers (on Fractions)

With children of migrant laborers working in IIT Gandhinagar campus, while a group of teachers sat around for observation (6-7 children of @ 13-15 yrs)

Children were asked – If there are 6 Rotis to be equally distributed among 4 of you, then how much will each get? They thought for a while and answered ‘Dedh’ (means 1 and half) , supported by reasoning. This solution was written on the board in picture and words. Similar representations were done on board for the next problem: 9 Rotis among 4 people, for which their answer was ‘Savaa Do’ (means 2 and a quarter). Now the challenge for them was to how to write these two fractions using numerals. Of the two numbers ‘one and half’ and ‘two and quarter’ they could write 1 and 2 , but how to write half and quarter.

So when the desire to know was created & expressed, they were given one proposal. I told them – I will tell you how to write half, but then you will have to find out how to write quarter. When they agreed to this condition, the notation was half was written on the board, but followed by a question – Why is half written as that way? And to this they reasoned – Because 1 Roti has been divided into 2 parts. So then they were asked how to denote Quarter, and they immediately suggested its correct notation. In fact they also figured out the notations for other smaller pieces like one-eighth (they called it as ‘aadhaa-paav’) and one-sixteenth. It was evident from their facial expressions that some of them had seen the pattern.

Now they were asked if half roti can be showed/ given in any other way. They were given paper plates to work and demonstrate. It took couple of mins and some rephrasing to understand this problem like if we cut the full roti (paper plate) in more than 2 pieces, and then can we make half.  And this was responded very well. They gave me 2 quarters. I kept one single half piece and two quarters next to each other, and asked them which one is half? And they said – Both. So then the problem posed was – there is one more name for this new representation of half. What it could be? After few seconds, this difficult to find answer was written on the board: 2/4 But then students were asked to reason about this and they did quite well with little help. Now they were asked to cut the plate into more pieces and make half and then express it symbolically. So they came up with the notations 3/6, 4/8 on their own with reasoning.  These fractions were written on the board side by side and they were asked to guess the next possible one... And one of them said 5/10.

Now they were asked which of the two is bigger 2/4 or 5/10, by writing one of these signs <, > , = between them. (I had to remind them of these signs they had used in their early school years like 43 > 17, 26 = 26, etc.)  And then one of them came forward and wrote ‘=’ between the two fractions 2/4 and 5/10 saying both are equal. They represent half. They also solved 1/2 is bigger than 1/8 with proper reasons (without any rules).

I now wanted them to see if they can find out the notation of another commonly used fraction ‘three-fourths’. So I asked them such question. How much Roti would one get if he is given half a roti followed by a quarter piece? This was backed by the use of paper plate (3 pieces of quarters). And they said it well – “pounaa’ (which means three-fourths).  When asked how would you write this using numerals, one of them said – 1/3, to which others agreed and one argued that its three pieces and one is taken away (because they saw three pieces of quarters, with one missing to form the whole).

This response was not new for me, as many other students have responded this way to three-fourths in the past. So now the teacher had two goals: 

(1) To help them discover that this is not the notation for ‘pounaa’

I knew that they had figured out the notations of other unit-fractions like half, etc. So strategy was to use this knowledge.  I asked them to draw the whole roti on the board, cut it into 2 equal pieces & write the notation of each piece. They did. Now repeat the similar process with cutting into 4 parts. They did. Now repeat the process with 3 parts. And they did this too. They drew the Y inside the circle and wrote 1/3 next to it.  Though the three parts were not precise but I choose to not correct their diagram at this moment because it did make sense to them.

So then I drew their attention to their previous guess/ solution for ‘pounaa’. And Wow!  They said that their previous guess/ solution was incorrect. So I cancelled it.

(2) To make them discover the correct notation of ‘pounaa’

I asked them – So what is the correct notation of this fraction ‘pounaa’? And they kept staring at the picture with wonder. Since they had figured out notations of different representations of halves few minutes back, so I thought to use their previous knowledge. I showed them the paper plate halves they had made & asked them what made them give those names to each of those. This conversation was enough for them to figure out the name of ‘pounaa’ (three-fourths) as 3/4.

And then finally, I wrote this notation of ‘pounaa’ next to its picture on the board J

Context : How this class shaped up? 

CCL team of IIT Gandhinagar ( https://ccl.iitgn.ac.in ) had invited me to interact with some KV and JNV school teachers who had come there for their residential training program. I was wondering as to what ‘sustainable’ could be done with teachers in just 2-3 hours, as I have now mostly restricted myself with long-term interventions with learners. So I thought of exposing them to reading & discussing of couple of thoughtful articles on maths education instead of doing some hands-on activities. 

My friend Swati https://azimpremjiuniversity.edu.in/people/swati-sircar suggested me to  demonstrate a session before reading a class-room story. She told me to do a session where few teachers can act / ask like children, but somehow I wasn’t much happier with having adults (teachers) as children (students) as I wondered if this conversation would be as natural/ spontaneous with simulated children :) Accidentally & luckily, just a day before my session with teachers, I found 3 children with their bags by the road-side, seemingly waiting for someone. On inquiring, they told me about their Nyasa School inside the campus. Then CCL team told me more about this social responsibility program run by IIT Gandinagar https://initaitives.iitgn.ac.in/nyasa/index.php 

So I couldn’t resist and found myself playing with these children, rather than attending the teachers training program J It was super-fun spending couple of hours with them (outdoor games & magic tricks).

While departing, I shared with few older children what was going on in my mind, if they would like to come for more playing (I meant playing maths now) the next day too (It was Sunday, a day-off for them). And they immediately & happily said YES! I guided them about the timings and venue. I got a bit worried the next day, when we (I and group of teachers) were ready for our session and they had not yet arrived. One of the CCL team members kindly waited outside the hall, looking for these children while I engaged teachers with some introductory fun activity. And soon, this bunch of children entered into the spacious hall full of teachers. I was thinking what must be going on in their heads while entering as they were not given any clue by me the previous day. So I oriented them about this, told them to relax & be as they were on the previous day ignoring other things / people around them. There was some hesitation in the beginning, but luckily the ice broke in few minutes and what emerged is as mentioned in the post above.

After the class, children left for outdoor play and teachers shared their observations & reflections. Some glimpses of this class and some bites of what teachers shared can be checked on links below. We then read and discussed an article that was published in this maths magazine. https://azimpremjiuniversity.edu.in/at-right-angles

-- Link of the article we read (classroom story having shades of Inquiry & Discovery:  https://tinyurl.com/3bvx36pn

-- Demonstrating a class with children : https://youtu.be/eIh4_OPVolM

-- Teachers sharing their observations: https://youtu.be/uEcshqAzBn

Thank you

Rupesh Gesota

www.rupeshgesota.weebly.com

Onset of out-station 5-day Maths Camps..

Sharing with you some thoughts and snaps of the lovely moments I experienced during my past couple of trips, to conduct 5-day summer Maths camps outside my hometown (Mumbai).

 

 

My immediate 2nd trip in May, after the 1st trip in April, became a possibility mainly because of the love and support of my friends & their families - Parin’s (at Bangalore) and Lalitha’s (at Tumkur) - who not only hosted me in their homes for 5 days each, but also opened up the doors of their hearts and homes to offer the access to learning (growing) opportunities to many unknown people in their cities. Lalitha even took me to a few government schools, some much away from her place so that those children too can get a flavor of Maths, which is generally missing in our modern racing systems.

 

Photos:

 

Bangalore Maths camp 2 

https://photos.app.goo.gl/ddwGPWtSQMor5gUR6

 

with govt school students at Tumkur

https://photos.app.goo.gl/Nysds6ZUQ18fzjWC8

 

with school maths teachers at Tumkur

https://photos.app.goo.gl/1bBZgUkjMuPyuwWKA

 

Similar generosity & care for the community was also expressed during my 1st trip in April, by the families of my 2 students with whom I had worked when they were in Mumbai, but then shifted to Bangalore. They talked around to form a group of parents for the maths program, and offered me their house to stay with them as long as I wished. And what better, I could even share my time doing fun maths with a few city children going through some physical & socio-economic challenges..

 

Bangalore Maths camp 1 (April)

https://photos.app.goo.gl/YrfHfHaSjoGh9Fpy8

 

Volunteering activities

https://photos.app.goo.gl/Qo78mdUoMQCMdogLA

 

It was nostalgic visiting those lanes, places, meeting friends and colleagues after more than a decade.. And all of this became much easier because both the families had immediately & graciously handed over the keys of their vehicles to me during my stay.. 

 

And finally, I cannot forget my friend Bijal who had instantly & voluntarily offered her younger daughter's train seat to me while travelling along to Bangalore from Mumbai, because I didn't have a confirmed seat..

.

What a blessing it is to be able to share what you are learning in your journey with people around and receive so much trust, support & wishes... There is so much love and goodness in this world, and there are times when one can feel it...

 

I have felt it more while reaching out to others because people do wish to take part in such positive initiatives everywhere which it why I firmly believe in the adage of *Vasudhaiva Kutumbakam*

 

Rupesh Gesota

www.rupeshgesota.weebly.com

Students make / extend the squaring trick :)

While playing with the square numbers, as to how can we find the square of numbers using the square of other numbers in various ways, we figured out the extension / modification of a well-known trick, something which was completely new for the teacher too :)

But before we go ahead, I would like to thank you all who read & even responded to my previous post with your lovely comments & thoughts. That was quite encouraging. In case you are the one who has not yet read the previous post, then I would recommend reading that first before this 

http://rupeshgesota.blogspot.com/2022/02/getting-into-algebra-through-arithmetic.html

Almost all students were aware of the 'trick' of finding the square of number ending with 5, thanks(?) to their teacher who had directly fed them this technique.

In case you are unaware of this technique then this is a good opportunity for you to figure out on your own. I have seen even few grade-4 students been able to do so, and pretty quickly :-))

Check the image given in the end of this post for your help !!

So after allowing them to impress me with this trick for few such numbers, I challenged them for the square of little different numbers like 48. And most of them, as I had expected or rather wanted, said:2064  (Do you get this, how they guessed this number?)

But when I asked them to verify their guess, they realized that its incorrect & they soon concluded (after trying for few other numbers like 29, 63, etc.) that the trick (quick method for finding the square of no. ending with 5), does not work with all the numbers.

So after this discussion / conclusion we had then moved to the other exploration (mentioned in detail in the previous blog post) and once we were done with that, one of the students told me that we can find the square of number ending with 6 by modifying the trick for the one ending with 5. And I was like highly. surprised with this claim. . Was she thinking over that one for this whole span?  And secondly I was also very curious now. Because I had never thought of / was unaware of this 'modification' till now.

She said, " We need to do some addition after applying the same method as that of 5."

She explained this with an example...

Let me share an image with you, allowing you to figure out what she did. Would suggest you to study this before you read the explanation below.

                                       

So yes.. This did intrigue me very much.... And hence we all tried our hands with various numbers. As you can see below: ..

                                        

Students also got excited looking at this method. And they quickly started trying numbers ending in 8 & 9 too... And their guesses to these did work. 

I hope some queries must have come to your mind by now :)

1) What about numbers ending with digits less than 5?

2) What is the explanation for this trick / method / algorithm ? [proof]

Well, these students did work on the 1st question and could crack it. However second question was just posed to them as of now so that they become aware of this possibility or rather necessity in Mathematics. 

In fact some of them became more curious to know the explanation now :)

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

Some questions I wish to ask you:

1) Were you too aware of this particular trick (esp. the modification/ extension) ?

2) If yes, then nice.... If no, then what was your reaction to this one, and esp knowing that it got discovered by a student :) 

I would be happy to know your response to these questions and any other comments / thoughts on this post.

And yes, as mentioned in the beginning of this post, here is the image to help those who wish to find the trick for squaring the numbers ending with 5.

                                      

Thanks & Regards

Rupesh Gesota

www.rupeshgesota.weebly.com

PS: These session are with a bunch of government school students from disadvantaged economic background, as a part of maths enrichment program MENTOR run with them. More details can be found here: www.supportmentor.weebly.com