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-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
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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