I think almost the whole world must be aware, by now, about the interesting fact about 15/08/17 —
I had received this message over WhatsApp couple of days before this date itself… Like for anyone, it was a delightful surprise for me too to face & digest this fantastic fact.. And I did not miss this opportunity to share this message with almost everyone in my circle esp. students, teachers and parents on our Independence Day.…
While many replied back with the Ooohs and Aaaahs, but there was one guy who did something more…. He is aware of the crazy math experiments that I do with my bunch of government school students… So he asked me “Do your kids know how to manufacture pythagorean triplets (PT) thereby proving that they are infinite in number?”
I replied – “No, I haven’t worked on Pythagoras Theorem with them yet… But I am planning to do that today..”
He — “My son stumbled upon Euclid’s proof of ‘infinite triplets’ in 8th class.. Try to make that happen with your kids (students)” And while saying this he also shared the link to his blogpost:
I was planning to read this post later, but luckily I clicked on this link – only to get highly motivated.. I now wanted to try this out with my students immediately… I was confident that my students would love to crack this code….So I scrapped my plan A i.e. to first work on the geometric interpretation and different proofs of Pythagoras theorem….and zoomed into my class with plan B…
After our daily ritual (meditation and mental maths), I shared with them this interesting fact of the date and how it has beautifully coincided with our Independence day this year…I was little surprised and even disappointed when I could not see much expressions of wonder on their faces as I had expected ! However, I forged ahead, sharing with them the details of —
1) pythagoras theorem (the right angled triangle, hypotenuse, etc..)
2) with the conventional example of 3,4,5
3) idea of pythagorean triplets — how (8, 15, 17) is one of those..
4) and finally the story of Euclid – as how he could prove there are infinitely many PTs…
To this, one of my students exclaimed – ‘Yes, there will be infinitely many such triplets because there are infinitely many natural numbers….’
‘True.. But mathematics demands proof ! How can you do so?’
They look puzzled to this. So, I asked them –
“Ok, you all seem to believe that there will be infinite PTs… Can you give some more other than (3,4,5) and (8,15,17)?” Now I could see them trapped…. Some of them started scribbling on their books, while others were still staring at these two sets, probably to get some pattern…I could not resist but teased them after 2-3 mins…. “So? Infinite, right?” 🙂
And now I could see couple of them trying to escape my sight.. 🙂 And soon, there was a guess –
“Sir, what about 30, 40, 50 ?”
“Are you sure or are you guessing this?”
“I am not sure…”
“How about verifying this?”
And in another moment, the other student reasoned — “Sir, it will obviously work !!”
I found the other students relieved with this solution…
“Sir, then even 80, 150, 170 will work…”
I wrote this triplet on the board… “More?”
One of them reluctantly guessed — “How about multiplying 3,4,5 with 2?”
“I am guessing 6,8,10?”
“Okay… Can we all check this case….?”
The class verified this and by now, all of them could figure out the game….
“Sir, we can now multiply 3,4,5 and 8,15,17 with any number and get more PTs.…”
“Good…. and how do we represent this fact?”
I wrote on the board what he told me — (3x, 4x, 5x)
“what is x?”
“So can I put 0?”
He understood what I meant….
“No Sir…. x is a natural number…”
Now, this realization made me share with them about “Primitive Pythagorean Triplet” For those unaware –
A Pythagorean Triplet (a,b,c) is said to be Primitive Pythagorean Triplet if the GCD of (a,b,c) = 1 means there is no other common factor among them other than 1.…
for example: (3,4,5) , (8, 15,17) are PPTs but (30,40,60) and (6,8,10) are Non-primitive PTs
But you know what? I did not give this ‘explanation’ to my students… I just told them,
“Look, I got to know from you that we can make Infinite PTs from (3,4,5) and (8,15,17). However, these new sets of triplets formed are not the Primitive Pythagorean Triplets. But (3,4,5) and (8,15,17) are PPTs, so can you tell me now, what do I mean by PPT?” And I was so happy that one of them could say this
“The numbers in the triplet should be co-prime!”
So then I asked others – Whats co-prime? And they responded very well
“Numbers with gcd=1” … “numbers with only one common factor i.e.1”
They thought that their job was done since they could make infinite PTs from the two given PTs, but soon, I played the Devil, “Hey wait guys…. what Euclid proved is that there are infinitely many PPTs and not just PTs… So can you give me some more PPTs….?” And it was worth watching their faces again !! 🙂
There was silence for almost 5 minutes, all engrossed in research. It was extremely difficult for me to Not give them any hint, but when I saw that some of them were not able to go ahead at all I thought to intervene, I wrote the equation 3^2 + 4^2 = 5^2 on the board and asked them,
“What type of numbers are related by this equation?”
“Natural numbers!” came an instant answer… 🙂 🙂
“ohk true…. but are these numbers 9, 16, 25 something more?”
“yes, they are square numbers….”
“Yes, True….. So do we know anything more about square numbers that can help us here?”
I was unaware that Kanchan had already made this observation, I understood this when she answered to this question instantly,
“Sir, their differences are odd and hence we can now look for..…”
Her enthusiasm conveyed to me that she had made that ‘million dollar observation’ and so I stopped her immediately, so as to not reveal more clue to other students. I just gave her thumbs up, signaling her to go ahead, while others were completely puzzled with our exchange of expressions and sign language. I was pretty sure by now that – not only will Kanchan crack this in next few minutes, but even others will pick up the clue to get some breakthrough. So I waited for 5 more minutes.
I found Kanchan engrossed in her work, while I found others still struggling. So I decided to write down the list of square numbers on the board in this manner, 1, 4, 9, 16, 25, 36, & so on till 121. I asked them, “Do you remember we had done many observations relating to this list of square numbers?”
“That’s all then, its just one of those important observations that’s going to help you solve this problem at hand.”
Kanchan got in, “Sir, you have given them a good hint now…”
“True.. Lets see what they do now…..”
I found few of them interacting with others after this interaction and then, she raised her hand…
“yes Kanchan, give me the Pyth Triplet you found?”
” (5, 12, 13) ”
“Can others verify this?”
And they agreed for its correctness.
“Can you find one more?”
Silence dominated the class again and finally she broke it with her another announcement, ” (7, 24, 25) ”
I verified this on the board with the class…. the result surprising other students.
“Kanchan, can you make the formula now, that will give you the list of all the Pts?”
This was the first time when I was not asking my student to explain her thought process before asking her to work on the extension of the problem. The reason for this change in approach was my awareness that she was on the ‘right track’ and I just wanted her to leverage on her buzzing flow of thoughts/ strategy. I had decided to interrogate her after her stint on the formula (generalization) part, in case she gets stuck, then I will grab that opportunity to probe her about her thought process…
While she was again engrossed in her book, I decided to throw some more hint to other students, “Can you notice something in the two triplets she has shared?”
Almost all of them replied, ” two out of three are consecutive numbers.”
Kanchan looked up at me with a smile. Probably she knew what I was doing. By this, I found one of them, Rohit staring at the chart of first 100 square numbers, that we have pinned up on our board since long….
Its amazing that we all have used this chart so many times till now in various situations. I never knew during my school days, that square numbers are so resourceful. It was difficult for me to wait. But Patience has always yielded more beautiful results and this time was no different. Kanchan approached me with her work after 10 minutes.
Of course, I made her explain/ reason in detail what she had done and how her line of thinking about the previous two pythagorean triplets and further how she constructed the formula, the terms that she required, etc. I choose not to share these interesting interactions with you directly, as I first want you to think about this on your own. (the above hint is anyways there with you now) But as you see, we realized that the formula has some bug and it does not yield the desired triplet for y=2. So she went back and started working on this. Meanwhile, Rohit raised his hand. He had got this triplet…
” (9, 40, 41) ”
The class verified and it was correct.
“Good one! How about one more? But this time, don’t tell the answer aloud, just write and show it to me.”
Soon he came to me with this — (11, 60, 61)
“Hmm…. correct…. So it seems you too have made some observations now….”
He nodded with a smile, “Formula??”
He went and Kanchan came back with the second version of her formula, which she was confident of being correct this time.
The most satisfactory part for me as a Maths teacher was when she told me that she could “find out the mistake” she had done in her previous work (formula) and that she could also successfully correct it with the associated reasoning. While I was congratulating her for this effort, she exclaimed,
“Sir, but this does not give us all the PPTs! The (8, 15, 17) is left out!”
Isn’t this observation worth appreciating again?
“True! So what will you do now to include even such triplets?”
“Hmmm, may be there will be another formula to include the family of such triplets.”
“May be you are right, so what do we do?”
“I will work on it.”
“Excellent! I will wait for your work.”
And meanwhile, Rohit had already queued up,
“Have you verified it?”
I asked him for the explanation and he too gave a very satisfactory one. The mathematical thinking was evident. I called Kanchan to take a look at this and she quickly inferred, “Its same! He has just used a = 2x +1 in the representations and I have used the term 2x+1 everywhere.”
“Correct! Any advantage?”
“This looks neater.”, she smiled & further added that, “he could have also written c = b + 1 instead of writing the whole expression for b again in the equation for c…”
I looked at Rohit and he agreed.
” We need to get the other PPT included which are missed out.”
“yes. May be both of you can work together if you wish.”
After few minutes, Jeetu too was ready with his work. He directly came up with the formula instead of sharing few PPTs like Rohit and Kanchan. But while explaining that, he figured out there is some mistake in representation, and corrected it on the spot. And this was a delight to watch!!
I had already told them to take a break and continue after that. However there were some who did not move at all and one of them was Sahil. This is what he came up with.
What’s interesting about his work is he arrived at the formula for PPT through the patterns and not through logic (like other three).
So I asked him “what would you have done if you were not given the data of (7,24,25), (9,40,41) ?
“Sir that’s when I was not able to do anything with just (3,4,5) and (8,15, 17). It was possible for me to find the formula only when I got more data.”
“Hmmm, but then how come your other three peers have constructed the formula for PT. They did not have the privilege that you had (more data/ pattern observation)”
He was silent. I think he was getting what I meant.I continued, “They were able to do so, because they have used Logic along with some observations in square numbers and not Pattern.”
“Okay Sir, I will try to find out using the other way now.”
And while going, he further added, “Sir, even the set (8, 15, 17) does not come out from this formula!” 🙂 🙂
Some students had not yet arrived at this, though they had made some observations and hence we decided to not discuss this exploration today but in the next session, thus giving them some more time to research at home.
So probably you might hear again from me, on this post 🙂
1) How about trying this out with your students? Do share your experiences if you do so.. Would love to study…
2) What are your views about the exploration that happened and the way it happened?
3) Could you decipher the rationale behind the approaches of Kanchan/ Rohit and Jeetu?
4) How can you include the missed out PPTs like (8,15,17) ? any change in this formula or new formula? Will that solve the problem of including all the possible PPTs, or still some might be missed out? How do you know?
5) Any more questions ? 🙂
Waiting for your responses/ comments.…
PS: These students are from Marathi-medium Navi Mumbai Municipal School Number 55, Ambedkar Nagar, Class-7 and 8.…
How can students be guided towards discovery by asking the right questions, so that they are able to discover and correct mistakes on their own? Some answers may be found in the various stories and Maths conversations recorded in my blog.
Blog & photos: Rupesh Gesota, Maths Educator, Mumbai for Comet Media Foundation