Few days back I got to attend the maths teachers' education program, where the facilitator had given us few problems to solve. One of the problems motivated her to encourage the teachers to discuss with their students about the difference and equivalence between the following three expressions.
a / bc ; (a/b) / c ; a / (b/c)
While my students had already encountered, struggled & tackled these expressions, but I recalled that these experiences were within the context, and in isolation i.e. they had fought with these monsters one at a time and that too on different occasions. So I thought it would interesting to notice what happens when they are made to face these three expressions - all together, without any context and that too without any context.
So the next day itself, I challenged them with this problem..
I asked them,
"Which of the three are equal or not equal?"
They looked at it for a while and soon, one of them said,
"Sir, we have solved such expressions..."
Me: Yes, Lets do it again then.
Couple of them could not resist and they shouted out their answer. I just looked at them; and they understood their mistake (that they had disturbed their peers' thought process)
After a minute or so, I asked one of them to explain
Rohit - "Sir, 2 and 3 cannot be equal.
"Why?"
"Because in (2), a is divided by the quotient b/x and in (3), a is divided by the product bx.
I looked at the class for their views. And could see everyone agreeing with this. I wanted someone to counter this, but none did. So I had to play devil.
"Are you sure?"
He guessed something fishy it seems and soon exclaimed - ."..except for x=1, when they will be equal" :-))
We smiled at each other and then I picked up another student--
Sidharth - "(1) and (3) are equal"
"How?"
He stood up and slowly wrote this on the board -
"Sir, expression (1) is actually a/bx which is same as (3)"
"But how?"
"I multiplied numerator and denominator by b"
"So?"
"So in the numerator of the Numerator, we have a... it is now divided by b as well multiplied by b.... So there is no effect of b... Only 'a' remains..... And in the Denominator, we get bx..."
Did you notice the language? your comments?
I was confident that he had understood, but I wanted to take him deeper...
"Ok... So what?"
"Sir, then we can now see that both (1) and (3) are equal...."
"But how do you know that the given expression (1) still remains equal to what you have written? May be you would have changed its value by this manipulation?"
I wanted him to argue with me, but then he remained mum.... and kept staring at the 2 expressions.....
Perhaps, Jeetu could not take this prank I was playing... and he intervened immediately,
"Sir, they are EQUIVALENT FRACTIONS ! So shouldn't they be equal?"
I looked at Sidharth... And found him smiling at all of us :-)
In fact, I could have even probed him further as to why the values of equivalent fractions are equal? and why do you get equivalent fractions by multiplying the Nr and Dr by the same amount? But then, we had worked on these many times earlier... and hence discussing those again now might disturb the flow of the present problem solving process....
I looked around -- "any other views/method?"
Vaishnavi -- "Sir, in (1), we are first dividing a by b, and then the answer is divided by x... So it clearly means that 'a' is divided by the product of b and x..... and hence its equal to a/bx.."
Everyone agreed......
"But then I feel, if a is divided by b and then by x, then its effectively divided by (b+x).."
And immediately the whole class roared aloud -- "No !! We have already worked on this....."
Let me tell you that this remark made by me now, was the same made by them few months ago.....After working out few examples and even through logical reasoning they had proved that why (a/b)/c is not equal to a/(b+c).... I was happy they still remembered their experience......
"Ok, so can someone tell me what will (2) simplify to?"
Tanvi -- "Sir, (2) will become ax/b"
"How?"
"Similar method as what Sidharth did... "
"Okay..."
"So can one of you write the simplified forms of all the three expressions?"
One of them wrote these on the board..... I forgot to take the snap... but this is what it looked like.....
"So then what's the conclusion?"
This time it was a chorus
"expression (1) and (3) are equal and are not equal to (2)"
I just looked at Rohit again...
"Sir, they all will be equal when x=1" :-))
I wanted to take them still deeper on this, but I decided to hold for the moment --- so that we can re-visit this problem after few days, with some flavor...
What do you feel, can there be some more interesting/ important questions? What are those questions?
How did you solve this problem? With understanding or with 'rules and procedure'? :)
How about trying this problem with your students? Would love to know what happened in your class..... :)
Regards
Rupesh Gesota
PS: These students are from marathi-medium municipal school.... To know more about the math enrichment program, check the website www.supportmentor.weebly.com
a / bc ; (a/b) / c ; a / (b/c)
While my students had already encountered, struggled & tackled these expressions, but I recalled that these experiences were within the context, and in isolation i.e. they had fought with these monsters one at a time and that too on different occasions. So I thought it would interesting to notice what happens when they are made to face these three expressions - all together, without any context and that too without any context.
So the next day itself, I challenged them with this problem..
I asked them,
"Which of the three are equal or not equal?"
They looked at it for a while and soon, one of them said,
"Sir, we have solved such expressions..."
Me: Yes, Lets do it again then.
Couple of them could not resist and they shouted out their answer. I just looked at them; and they understood their mistake (that they had disturbed their peers' thought process)
After a minute or so, I asked one of them to explain
Rohit - "Sir, 2 and 3 cannot be equal.
"Why?"
"Because in (2), a is divided by the quotient b/x and in (3), a is divided by the product bx.
I looked at the class for their views. And could see everyone agreeing with this. I wanted someone to counter this, but none did. So I had to play devil.
"Are you sure?"
He guessed something fishy it seems and soon exclaimed - ."..except for x=1, when they will be equal" :-))
We smiled at each other and then I picked up another student--
Sidharth - "(1) and (3) are equal"
"How?"
He stood up and slowly wrote this on the board -
"Sir, expression (1) is actually a/bx which is same as (3)"
"But how?"
"I multiplied numerator and denominator by b"
"So?"
"So in the numerator of the Numerator, we have a... it is now divided by b as well multiplied by b.... So there is no effect of b... Only 'a' remains..... And in the Denominator, we get bx..."
Did you notice the language? your comments?
I was confident that he had understood, but I wanted to take him deeper...
"Ok... So what?"
"Sir, then we can now see that both (1) and (3) are equal...."
"But how do you know that the given expression (1) still remains equal to what you have written? May be you would have changed its value by this manipulation?"
I wanted him to argue with me, but then he remained mum.... and kept staring at the 2 expressions.....
Perhaps, Jeetu could not take this prank I was playing... and he intervened immediately,
"Sir, they are EQUIVALENT FRACTIONS ! So shouldn't they be equal?"
I looked at Sidharth... And found him smiling at all of us :-)
In fact, I could have even probed him further as to why the values of equivalent fractions are equal? and why do you get equivalent fractions by multiplying the Nr and Dr by the same amount? But then, we had worked on these many times earlier... and hence discussing those again now might disturb the flow of the present problem solving process....
I looked around -- "any other views/method?"
Vaishnavi -- "Sir, in (1), we are first dividing a by b, and then the answer is divided by x... So it clearly means that 'a' is divided by the product of b and x..... and hence its equal to a/bx.."
Everyone agreed......
"But then I feel, if a is divided by b and then by x, then its effectively divided by (b+x).."
And immediately the whole class roared aloud -- "No !! We have already worked on this....."
Let me tell you that this remark made by me now, was the same made by them few months ago.....After working out few examples and even through logical reasoning they had proved that why (a/b)/c is not equal to a/(b+c).... I was happy they still remembered their experience......
"Ok, so can someone tell me what will (2) simplify to?"
Tanvi -- "Sir, (2) will become ax/b"
"How?"
"Similar method as what Sidharth did... "
"Okay..."
"So can one of you write the simplified forms of all the three expressions?"
One of them wrote these on the board..... I forgot to take the snap... but this is what it looked like.....
"So then what's the conclusion?"
This time it was a chorus
"expression (1) and (3) are equal and are not equal to (2)"
I just looked at Rohit again...
"Sir, they all will be equal when x=1" :-))
I wanted to take them still deeper on this, but I decided to hold for the moment --- so that we can re-visit this problem after few days, with some flavor...
What do you feel, can there be some more interesting/ important questions? What are those questions?
How did you solve this problem? With understanding or with 'rules and procedure'? :)
How about trying this problem with your students? Would love to know what happened in your class..... :)
Regards
Rupesh Gesota
PS: These students are from marathi-medium municipal school.... To know more about the math enrichment program, check the website www.supportmentor.weebly.com
