It didn't take much time for them to figure out, with reasoning, that the total number of numbers in a given range is 'one more than the difference of boundary numbers.'
However, this 'formula' (of difference +1) became visible only when I gave them some 'difficult' numbers to deal with (from 32 to 75). You may also notice how the opportunity of generalization was grabbed at this moment. It was students who made me write 'b - a +1'.
So now I encouraged them to find the number in the Center of this range 25 to 30. I did not use the words 'mean' or 'average'.
Kajal quickly answered it as 27. While Tanushri said, there is no one number, but two numbers 27 and 28 in the centre. She figured out this by writing down all the numbers and then cancelling out numbers in pairs - one from left and one from right - while approaching the centre. And to this, Yash immediately said that the central number would be 27.5
I asked others, if they agreed with this result. One of them related this situation with the cm / mm markings of a ruler and explained to the class how 27.5 would be between 27 and 28. I ensured that also knew the meaning of 27.5 i.e. 27 and half.
I asked them to solve the next problem: Central number in the range 45 to 60. All of them quickly resorted to the previous method (cancelling numbers from both the ends) and figured out the answer as 52.5
So now I challenged them to figure out the central number in the range 32 to 75 without making the list as above. This slowed them down to think. I could see some just staring at the problem while others scribbling something on their book.
After a while, Yash came forward to show his solution:
There are 44 numbers from 32 to 75. Now half of 44 is 22. So there will be 22 numbers above and below the central number. Hence the central number is 22.5
(I want you to pause over here and think about his approach & result)
I realized that he has not yet visualized the problem. So I drew a part of open number line on his book with 32 and 75 marked on it, and asked him -
"Where would the central number lie? To the left of 32 , right of 75 or between them?", while pointing at the respective regions.
He instantly answered - in between. I now drew his attention to his answer (22.5) He laughed at it and went back to work again.
Surprisingly Aryan too came up with the same solution. while others were yet figuring out the approach. So I decided to give them a hint / tool to think over - pictorial reprsentation.
At first I just wrote 3 numbers on the line - 25, 27.5 and 30 and asked them, How do they know if a number is in the centre?
"It should have same numbers in the left and right"
Okay... So is that the case with 27.5 ?
"Yes, its 2.5 away from from both the sides."
I showed 2.5 on the picture, & asked them, - Whats the distance between 25 & 30?
"5"
I showed 5 in the picture now and told them - "So now this a good hint for you to solve the problem in hand.."
When I saw that they were still unable to go ahead, I decided to invite everyone for a collective exploration / discussion. I intentionally picked up the (flawed) solution of those two students... However I represented their thought process using the pciture, for them (& others) to 'see' what's happening.
I began -- Yash & Aryan say that there are 44 numbers from 32 to 75. They then halved it to get 22... What do you all think about this?
I thought some one will correct my mistake i.e. we need to find their difference (43) and not the items in the range (44). But sadly, no one did.... In fact one of them picked up from this picture saying -- "now we should go back 22 from 75 to get the central number 53..."
This remark was accepted & welcomed with such a delight by all, as if someone had demystified an age old mystery! :))
I didnt have an option - but had to accept the verdict, but only momentarily. I then aksed them - "How do you know 53 is the central number...?"
It's at a same distance from 32 and 75..
How do you know? Did you check?
So then? You still want to stick to 53?
To this, one of them immediately sprang up -- Sir, the central number is between 53 and 54.. So its 53.5
When I prodded him for the reason - he said, its just a guess. So I asked him and others, if we can verify if 53.5 is correct?
They said that we should check if it is equidistant from both the numbers.
So I drew a new number line, with 32 and 75 on the ends and 53.5 in their middle.
What next?
They did: 53.5 - 32 to get 21.5
Actually it was more of me who did this subtraction than them. I realized this mistake when they messed up in the next subtraction. This is how I did with them --
Lets subtract 32 from only 53 first, to get 21 and then we add back the left over 1/2 to get 21.5 Everyone almost said this aloud with me, thus making me (falsely) believe that they have understood this.
However, when we went to another piece: 75 - 53.5
Again, this is how led them: Lets only subtract 53 from 75, this gives us 22 , and then we add the left over 1/2 to get 22.5 (can you notice the (intentional) trap? - i hope you didnt fell into it :-)))
I was hoping that someone will point out my mistake. But no one did :-(
It's only then that I realized that I did a big mistake by rushing through the previous subtraction (53.5 - 32) If I had allowed them to work it out on their own, some of their misunderstandings would have probably cropped up then & there itself... Nevertheless, it was still smart of me to catch them red-handed in the second problem :-))
So how did they react to this two different (21.5 and 22.5) distances of 53.5 from 32 & 75?
They were surprised for a while. But interestingly, they were ready to disacrd away this 53.5 as the central number, as it didn't obey their rule of being equidistant !
OMG ! They were about to throw away the correct result because of some other mistake... What should I do now?
I wanted to draw their attention to their inorrect subtraction, but I realized that the numbers were too big for them to estimate & visualize the result and error. So I created a situation where they can easily analyze, learn from it so as to come back and find as well as correct their error on their own.
"Look... A student of another class explained to me this way: 37.5 is not the central number from 35 to 40 because its not equidistant from 35 and 40.... It is 2.5 away from 35 but 3.5 away from 40..."
What do you think about his argument? - I asked them.
One of them quickly said -- "Sir, 37.5 is correct... He has done the subtraction incorrectly.."
I looked at others and waited for their response.
I was so happy when one of them could relate this scene to our problem... "Sir, may be even we have done some mistake in subtraction...."
Okay.. So how do you verify if the result of our subtraction is correct or not? While asking this I wrote 10 - 3 = 6. How do you know 6 is correct or not?
Its incorrect because 6 + 3 does not make 10.
Good... What if we do not know the difference between 10 and 3 i.e. we do not know the answer... Then how can we find the answer in another way?
We should find 'what number should be added to 3 to get 10?"
Perfect... So you have converted subtraction problem to addition one...
Lets find out 75 - 53.5 using this addition method now...
I saw two interesting approaches:
Method-1: This student has written down 53.5 and 22.5 one below the other, added them to get 76. Then changed it to the desired result 75, and to achieve this she reduced the number to be subtracted by 1.
Method-2: This student didnt write the first number (blank space) in the beginning. Just wrote 53.5 and 75. He figured out 1.5 needs to be added to get 5 in the unit's place of the result. So he wrote 1.5 in the first number. Then he filled the ten's digit as 2 so as to get 2+5 = 7 :)
They realized that they had erred up in the subtraction problem and were very happy to know that this right side distance 21.5 was same as the left side distance 21.5
However before we drew our attention back to the Central number problem, I wanted them to learn how to subtract if such easier fractions are involved (also I wanted them to figure out where & why we had messed up in the subtraction 75 - 53.5)
So I took them to the board and asked them how would they solve 53 - 12 mentally, without using std. algorithm.
To this, one of them said, we would first subtract 10 and then 2.
As he was saying this, I represented his idea on the board.
Fantastic! The way you made the 2nd number easier, can you make the 1st one easier?
We do 50 - 12 and then add 3.
Great ! So if you now understand this very well.... lets do our two subtraction problems this way...
And as you see below, they could solve both the subtraction problems beautifully. We even solved couple of simpler ones (25.5 - 3 and 25 - 3.5) quickly just for practice and I must say that they super-enjoyed it !!
So then it was time now to redraw their attention to the original problem.
Whats the central number between 32 and 75?
They agreed that its 53.5 because its equidistant (21.5) from both the ends.
So I showed two times 21.5 on the picture, and then asked them whats the total distance then, from 32 to 75?
To this, one of them still said 44 hurriedly, but couple of them did spot the mistake ---
Sir, its not 44, its 43....
And then we had the most awaiting Ooooooohhhhs & Aaaaaaahs :-)))
I drew another number line... with 32, 75 and their difference 43 written on it... And asked them for the next step.... and this is how they went --
"Do half of 43 to find their centre.... its 21.5... now subtract 21.5 from 75.... (to do this, they first removed 20, then 1 and then half)... we get 53.5 ......."
Is there any other way to get 53.5?
One of them responded - Yes Sir, we can also add 21.5 to 32....
Do you get the same answer?
Yes
So what's the easier way?
We can add to the starting number instead of removing from the ending number !!
They also quickly and acurately solved the following similar problem:
Now they wanted to solve more such problems.... and hence I gave them few for home-work, including the subtraction ones that include more common unit fractions like 1/2, 1/4 etc...
-----------------------------------------
1) What are your views about this lesson?
2) How about you trying out a similar activity in your class and share its proceedings?
Thanks and Regards
Rupesh Gesota
No comments:
Post a Comment