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 :)

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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

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

Getting them to Algebra via Arithmetic - 1

So by now, we (maths teachers) would most probably know many (or at least a few) ways to make students 'discover or arrive at' (& not simply teach / give away) the formula of Difference of Squares i.e. 

a^2 - b^2 = (a-b) (a+b) 

I too am aware of few interesting entry points to achieve this, but I was a bit surprised (at least in the beginning) when I realized midway during the 2nd session - while recording the 1st session's discoveries done by the students on the board - that this content can also be easily led to the above formula,.... and hence I decided to steer the car in that direction, though the plan was to take them to some other place :-))

So let me first share what happened in our 1st session:

We were playing with square numbers as in if they can find the square of a number using the square of its previous number. They did not yet have the knowledge of any of the expansion formulae like (a+b)^2  etc.

I came to know that they knew how to quickly find the squares of multiples of ten. So I asked them to first find out 21^2 and 31^2. They calculated using std. multiplication procedure, & then I wrote these on the board:

20^2 = 400

21^2 = 441 

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30^2 = 900

31^2 = 961

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I asked them if there is a way to find out the squares of 21 & 31 from those of 20 & 30 resp.?

It didn't take much time for some of them to quickly see the pattern.

"Yes, Add 20+21 to 400....."

This certainly surprised them and I gave them few more problems to play - use this pattern to find & verify. Luckily they were also aware of the 'trick' of quickly finding the square of a number ending with 5 (thanks to their teacher or youtube), So they happily & quickly calculated the squares of numbers ending with 1 and 6 quickly like 41, 76, etc.

So then came another question:

"Is it possible to find the square of 42 quickly?"

One of them said - "Sir, we can first find the square of 41 from that of 40 and then find out 42^2 from 41^2 using the same method.."

"Agreed.... But that's too long.... I need a short-cut... directly from 40^2..... Possible?"

And after some time, couple of them came up with this method:

"Add double the sum of 40 & 42 to 40^2"

Others verified this to be true. Some cheers. Some practice. And again another question:

"Now, how about 43^2 from 40^2 ?"

This time came a guess immediately - " Add triple the sum of 40 & 43 to 40^2..."

While this guess was followed by laughter of few, & some resorted to verification :)

And now the whole class was super excited by this emerging unexpected pattern..

When asked for 44^2 , ALL of them answered loudly & happily --

40^2 + 4 (40+44)

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We did some more wandering around this zone , but with some deviation, about which I will probably share in the next post...& then we were time out for the day..

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2nd session:

So today, I entered the class with a plan in my mind -- To lead them to the expansion formula (a+b)^2 through the previous session's exploration / discovery they did.

So we began with some revision:

1) I told them to find 11^2 from 10^2

They did.

2) Find 12^2 , 13^2  from 10^2 

They did these two quite quickly / easily.

3) Now they were told to find 27^2  from 10^2 

This took little more time, but then they figured out...

4) Now, time for another Change (in base)

Like 20^2 from 8^2, etc . You can check the image below for the progression.  One may also note that changes in representation are done - one step at a time and with Understanding, without any rules. 

Instance-1: Intentionally I replaced 12 by the expression 20-8 without brackets first. Few didn't find anything incorrect here, but with my pause, some could sense the flaw and they insisted for brackets with reasoning.

Instance-2: The term 7^2 was brought to the left of the equation without stating any rule like + term becomes - term when it crosses the = sign. The question asked to them was how to represent the addition statement using the subtraction statement.

Then, with appropriate explanation, it was time to generalize this arithmetic using variables but the words used were number-1 and number-2 along with their short-hand notation n1 and n2 instead of directly using 'a' and 'b' as found in most of the text-books. This made the transition to generalization (alphabetic representation) acceptable / easier for everyone to comprehend.

We decided to further short-hand this two-symbol & confusing variables (numbers) with one-symbol ones.

Though not all, but I could see some of them recall this formula that they had read / seen in their text-books & rejoice with Wonder and Joy !!

Yes, I am aware that this is not the derivation / proof of the formula, & we need to take them to that stage, but I think what is more important is that they could first arrive at this formula on their own with the knowledge/ tools that are accessible to them at their present level of ability.

Also, showing the visual representation of this formula to students is also in the mind. We might arrive at this formula using other ways too later :)

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I will hopefully soon come back with the next post on -

a) some more interesting explorations that we did in our previous session (as mentioned in the middle of this post)

b) how students went towards & reached the destination that was thought of / planned by me (the teacher) initially.  (a+b)^2 using / from their explorations.

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Meanwhile, I will be happy to hear your thoughts on this post / class and how would have you done in such case or how do you facilitate discovery of such formulae in your class.

Waiting for you :)

Thanks and Regards

Rupesh Gesota

wwww.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