Simplifying Algebraic Fractions : Part-1

Could you find the mistake in her first step of simplification of LHS expression?

I am sure many algebra teachers would agree with me that this is one of the most common mistakes students do while simplifying algebraic expressions. 

So, why would be they doing so? What could be the cause(s) for this effect? Why is it that this nonsense does not seem nonsense to them? 

(I am intentionally keeping these questions open now, with the hope that some of us would probably pause and try to find the answer for these, though reflection, research or discussion with our peers.)

So, this was the problem:

Actually, I had seen Poonam committing this mistake i.e. 'cancelling' the term x^2  (i.e. x squared) from Numerator (Nr) and Denominator (Dr), however I chose to not stop her to point out this mistake at that moment and I chose to keep my mouth shut and just allow her to go ahead with this mistake....  (Why?)

Soon, she found herself stuck up and asked me to help.. I don't know what struck me I rather asked her this question:

"I see you have cancelled out x^2 in Nr and Dr... Why don't you then even cancel out 'x' from Nr and Dr ?"

She replied immediately - "No sir, we cannot do this... Their signs are different..."

"oh ok... Is that the reason?"

"Yes", she answered with a pinch of reluctance this time.

I thought for a while... and again threw the ball back to her court,

"Okay... so how about canceling the two terms (1 + x^2) from Nr and Dr?"

"How can we do that? They are far away... There is 'x' in between them..."

"Oh ok.... Can we rearrange the addends in an addition expression? I mean, can we write 

(1 + x + x^2) as (1 + x^2 + x)?"

Poonam got what I meant this time and re-wrote the Nr and Dr expressions this way:

As you can see that, now she had no problem in cancelling out the two (1 + x^2)  ...Interesting, isn't it?

I just thought of pointing out one more thing to her ---

"Could we have cancelled out even the '1's while cancelling out x^2 from Nr and Dr?"

She reacted to this simple idea in such a way, as if I had drawn her attention to some Wonder..    :-) :-)

"yes sir, we could have easily simplified the expression there itself, instead of this long business of first rearranging and then cancelling (1+x^2)...."

So having stuck up in her first way (canceling just x^2), she now simplified the problem - the second way i.e. by cancelling (1+x^2) instead of just x^2   ;-)

Now, if you carefully check her new solution (on the right side), then you will find many more (serious) mistakes..

Can you count and tell me how many? Possible reasons for these mistakes?

I could have pointed out these errors to her..... However, at this moment I chose to stay 'out of this zone of new errors'.....Why?  (perhaps, addressing this set of errors calls for another round of discussion with her and hence an another post as well ;)

Meanwhile, you can think of the method(s) one can adopt to help her find these mistakes....

She again found herself stuck up, like last time, after simplifying the problem to some extent.. She cried for help....

Now I thought to withdraw myself and rather involve my another student -- to take over this game.... I intentionally picked up Kanchan, told her to have a look at both the solutions of Poonam so far..  She went through her work and then gave a big smile to me,... I knew that she will be able to recognize the trap I had laid for Poonam.... After all, she too had been dragged into such traps before :-)

Kanchan forged ahead, thought for a while and wrote this on the board:

I don't know what made her think of these four numbers (3,10,15,9)... This is how she instructed Poonam then:

"Now simplify this example using both the ways... with cancelling and without cancelling.."

Poonam just solved this 13 / 24 .... Surprisingly she was not working on 3 and 9 in Nr and Dr here, like the way she was cancelling out 'same addends' 'x' or (1+x^2) in the original problem....

While Kanchan was unable to understand this contradicting behavior of Poonam in both these problems, a teacher would probably understand this duality..... May be Poonam thought that the two addends in Nr and Dr can be cancelled out only when they are same, else not..... So here, Kanchan was expecting Poonam to simplify 3 and 9..... But Poonam was not doing so (because both addends were different).... It was interesting to watch both of them fixed up in this lock...  :-)  :-)

Kanchan then looked at me for help... I decided to give in easily this time..... Hence just told her to explain while solving.....

This is how she proved that the method used by Poonam (cancelling out / simplifying addends in Nr and Dr) changes the value of given fraction....and hence is incorrect.

While it was a delight to listen to Kanchan as to why 13 / 24 and 11 / 18 are unequal, I could see that Poonam was mostly accepting what Kanchan was firmly and quickly asserting.....

According to Kanchan, 13/24 and 11/18 were the two most simplified forms (gcd of 13 and 24 is 1   and   gcd of 11 and 18 is 1) and they were clearly not equal.... However Poonam was not much ready yet to digest this level of understanding of fractions and hence this type of reasoning was not going much deep into her.....

If Kanchan had taken either of these 2 following routes, then it would have been easier for Poonam to understand:

(a) Different reasoning:  13/24  is 1/24 more than half while 11/18 is 1/9 more than half. Thus both are unequal, hence one cannot simplify the addends.

(b) Different numbers: Choose the numbers such that it becomes easier to compare the two fractions.

So I decided to intervene now with (b) rather than (a).... and hence suggested Kanchan to make some changes in her numbers.... Replace 10 by 9 in Nr...  (What made me think of this replacement?)  Why did I choose strategy (b) over (a) ?

This replacement made Kanchan's job easier:

So this is the way she explained to Poonam now:

If you simplify the fraction (3+9)/(15+9) without any modifications, then it is equal to 12/24 which is 1/2 i.e Half

While, if you simplify the two addends 3 and 9 in Nr and Dr as 1 and 3 respectively, then the fraction becomes equal to 10/18 = 5/9 which is more than half...

Hence you cannot simplify the addends in the Nr and Dr of a fraction.

This was pretty easier than the previous one... However I wanted to ensure if Poonam had understood this.... 

"So did you get this now?"

She replied - "Sir, how come Kanchan knows 5/9 is more than half?"

I looked at Kanchan.... She began -

"Look Poonam.... Why do you say 12/24 is half?"

"because 12 is half of 24..."

"So similarly if we have 9 in the Dr then whats required in the Nr to make it Half?'

She thought for a while and answered in a low-confident tone --- " 4.5 "

"yes...correct.... So if 4.5 / 9 is Half then, what about 5 / 9?"

There was a Big Smile on Poonam's face now.... and so on Kanchan's  :-) :-)

---- 

Little did they realize that their game was not yet over....

"Wait Kanchan.... You have showed her that she cannot cancel or simplify the "addends" in the the Nr and Dr of a fraction... But what about the reason for why we CAN cancel or simplify the "factors" in the Nr and Dr of a fraction?"

She understood what I meant.... and hence again made another example to make this point:

I was very happy with the way she achieved this...  She explained how the same factors in the Nr and Dr.  (x^2) come together to equal to 1, and hence the simplified fraction formed (5/4) by the remaining factors multiplied by this 1, does not change the value of simplified fraction (1 x  5/4  =  5/4 )..... and thus we casually say that we are 'cancelling out' the same factors..... whereas actually or mathematically, we are multiplying and dividing by the same factor, thus multiplying by 1..........

I thought to reinforce this idea using only numbers now..... So I gave her this example:

I further asked her, what if the problem were (18 x 5) / ( 9 x 4)?

This is how she simplified it:

(18 / 9) x (5 / 4) = 2 x  (5 / 4) = 10 / 4

I could feel the joy of understanding radiated from her face.......

Do you or your students simplify the fractions with such deeper understanding or by simply cancelling out the factors in top and bottom using the 'times tables'? 

How about you triggering such a math discussion in your class, may be with few students, if not all?

Both of them had got so engrossed in this fault finding that they had forgotten that the actual problem (solving for x) however still remained unsolved...   :-)

So now, Can you solve the main problem and share your solution?

But more importantly, if you are a teacher or teacher educator or researcher, I would love to hear from you, your views about this post........your responses to many reflective questions raised in this post (in blue font)....

Regards

Rupesh Gesota

PS: Both the students hail from marathi medium government school, based at Navi-Mumbai. To know more about the maths enrichment program run for them, check the website www.supportmentor.weebly.com

Relooking at stack of Tables (Triangle Numbers)

The  manner  in which tables were stacked up in our classroom that day , it suddenly caught my attention.. I counted them 4+5+6= 15 and Aha ! It's a triangular number..  I was somehow amused by the fact that how come 4+5+6=15? because the triangular expression for 15 is 1+2+3+4+5 

I usually include my students too in such investigations when they are around.. and hence this seemingly trivial question was posed to them as well... 

They started staring at this structure ... And soon, one of them - Vaishnavi responded: 

"Yes, it's easy," she said, " I can visualize this.."

I asked her to explain and this is how she had restructured the given structure in her mind... 

As you can see , she had formed the triangle arrangement of 1+2+3 in the top 3 layers.. and then shifted one and two blocks from the second last layer to its upper and lower layers respectively to form 4 and 5 respectively, thus getting the common expression for 15 = 1+2+3+4+5

Her spatial flexibility was worth appreciating.. but what absolutely delighted me was the visualization of my other student - Kanchan.. This is the way she had imagined:

She noticed that the rightmost column had 6 tables and it's previous two columns had 4 and 5 tables... So she rearranged 6 tables mentally as 1+2+3 before the stacks of 4 and 5 tables , so as to form the triangular equation of 15= 1+2+3+4+5

I asked her how did she know that the last column of 6 could be arranged as 1+2+3, then she immediately replied that 6 is a Triangular number...! 

So I asked her with curiousity that will we able to do this restructuring for every 3 consecutive numbers, say for 5+6+7 also?

She thought for a while and said: No, it's only possible when the last number is Triangle number...

"Why do you say so?"

"Because only then we will be able to rearrange the last column (number) in terms of triangular form before it's previous columns.."

"Okay... So can you tell me what would be the next possible case of consecutive numbers?"

It didn't take much time for her to work out that the expression would be 5,6,7,8,9,10.. 

"Can you explain how?"

"Sir, 10 is the triangle number and hence can be expressed as 1,2,3,4 before 5,6,7,8,9...."

"Hmm... Good thought.... Can we generalize this then?"

She looked at me for more clarity...

"Means... If the number in the last column is Nth triangular number, then what should be the sequence of consecutive numbers , how many numbers, and what triangle number will be eventually formed?"

I knew she would have understood my query... After about half a minute, she said this...

"If the last Nth Triangle number is K, then the sequence should start from N+1 and go on till K.. This will add up to (K-1)th Triangle number..."

And this was just awesome for me....

Do let me know your views about this exploration which was triggered by a casual observation of stack of tables :)

Thank you...