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 :)
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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
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
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