Part-2 : Re-learning and Enjoying Polynomial Division with students.

Last week I had shared my classroom experience of working with students on polynomial division, and these students were not yet taught the standard procedure of solving such problems in their school. This is the link to the blog: 

http://rupeshgesota.blogspot.com/2019/10/re-learning-and-enjoying-polynomial.html

I forgot to mention one more interesting thing that happened while working on this problem, which I think would be worth sharing - esp for in-service maths teachers.

Before the two students (mentioned in the above / previous post) shared their solutions, another one had come up with this one :

Now, this mistake would not be new to the teachers who have been teaching algebra since long. Its one of the most common mistakes which would be done by at least one student in the class every year. I don't get irritated by these mistakes. I desperately wait for such mistakes. Yes ! Because I think it is a golden opportunity for the teacher if he / she is able to spot a student thinking / working this way. It presents just the right context and time for driving an enriching mathematical conversation in the whole class -- to know what other students think about this, if such misconception is simmering in someone else's mind too (& hence it would be nipped in the bud itself though the subsequent talk) and most importantly to know how my students see this / argue about this. 

Do they say that -- 

a) This work is incorrect because it cannot be done this way. It is a rule! OR 

b) Do they really reason about it, with proper math ?

When this student (S1) solve this way on the board, I was a little surprised as to why he did not 'cancel out' another 'x' too in the numerator (there was one in the term 3x too)  and why he did not work with 10 and 2 in the same way?  :)

I was about to argue (confuse) him by asking these, and when another student (S2) stood up - It is not possible to do this way. 

On one side, I was a bit disappointed as he had foiled my plan (of confusing A), but on the other side, I was also happy that there was another student in the class who could spot some 'non-sense math' and object about it :)  S1 was surprised by this remark of S2. 

S2 went to the board and argued - How can we divide x^2 alone by x.... We have to first add the terms 3x - 10 to x^2  and then divide this sum by the expression in denominator. 

Pause for a while and think what's your take on this argument.

---

While I was glad that he had noted and argued well about one aspect, I was not sure if he has missed or overlooked another important fact  --- that it was not just a single 'x' term but a binomial 'x - 2'  in the denominator (divisor). I was for sure going to delve into this matter in some time, but first I was curious to know how other students react to this argument of S2.

Almost all of them understood what he said, except couple of them. So he gave this example - 

     12 + 5 

   ------------      

        3

12 / 3 = 4 . So can we say the answer of above expression is 4+5 = 9?  

No, they replied. It is 17/ 3 which is 5 point something.....

And this was just enough to convince the ones who had not understood.

Wasn't this a fantastic creation?  and that too made spontaneously ;)

--------

And now, it was my turn -- to play villain ;-))

Remember I stated my concerns above - I was not sure if S2 is also aware that S1's simplification was incorrect because of one more reason. (there is -2 sitting with x in the denominator)

So I asked them - But what if the question did not have more terms in the numerator? means, what if the question had just x^2 term. 

I intentionally looked at S2 - Now, it would be okay to 'cancel out x' right ?

He was perplexed.... I was right when I was doubting about this part.... He had not considered this aspect yet...

But interestingly another student S3 jumped in and said - No !  We cannot do this.

Why?

Because we have to divide x^2 by (x-2)  &  in the above case, we are dividing x^2 by only x. 

It is like -- The question is to divide 20 by (6-2) , and we are dividing 20 by 6 itself.... So the answer obtained will not be correct. 

Fantastic !!

But I cannot give in so easily ;)

Is there any other way to prove that this is incorrect?

S2 bounced back.  -  'Let us multiply the Numerator and Denominator of the answer by 'x'  and see if we are getting the given fraction back.'

Since we are not getting the initial (given) fraction back, it means we have solved it wrong way. 

He left me speechless. This thought deserved appreciation, isn't it?  

But I responded to it with (an imp) finishing question - So will we ever be able to do this type of division?

Yes, we can do it - but only when there is no addition, subtraction in the numerator and denominator. Only when there is one term on both the sides. And while saying this, he happily showed did this on the board !

And the Devil in me had started cooking this type of situation now....

However, they had already gone into the celebration mood by now (winning against me), and so I thought to reserve this bouncer for the next match :)

-------

a] Do let me know your views / comments about this post. Would be glad to know 4m you...

b] And how would you respond / do you respond, when you see your students demonstrating such (common) algebraic misconceptions? 

Thanks and Regards

www.rupeshgesota.weebly.com

PS: Students belong to class-7 marathi medium government school in navi-mumbai. I work with them voluntarily after their school hours as a part of maths enrichment program. www.supportmentor.weebly.com

Re-learning and Enjoying Polynomial Division with students.

They had just learned how to multiply two linear binomial expressions like 

(x + 2).(x - 5) , (3x - 2).(5 - 4x) , etc. in two ways -  pictorially as well as symbolically (i.e. by expanding).

I now wanted to see their approach for the division problems, like for problems of the type:

(x^2 + 5x + 6) / (x + 2)

Polynomial division was not yet taught to them in their school. So I should have first given them a simpler problem like the example above (all +ve terms), however for some reason I directly pushed them into the challenging zone this time. This was the problem I gave;

(x^2 + 3x - 10) / (x - 2)

I would suggest you to pause and think for a while - to assure yourself as to why this could be a little difficult problem, esp for those who do not know the procedure to solve this... See if you can solve this problem using a way which was not taught to you :)

-----

After some time, one of them came to me and showed me his final result. He said it is 3 wholes and (x^2 - 4) / (x-2) . I must confess at this moment that I was completely surprised by such an answer. And I wonder if you too would expect or have seen the quotient of polynomial division in such a form. 

I suggested him to go and explain his approach on the board for others to know. And this is what he did -

Note his diagrammatic representation for the expression (x^2 + 3x - 10) . The ten small shaded circles represent negative ten. Then he explained -

We need to divide  (x^2 + 3x - 10)   by  the expression  (x - 2)  , which means we need to find out how many (x-2) are there in (x^2 + 3x - 10). 

Looking at the terms 3x and -6, we can see that (x - 2)  is present 3 whole number of times in this expression. So what is left now is (x^2 - 4),  which when divided by (x - 2) will give us  (x^2 - 4) / (x - 2). 

When one of the students did not understand this, he gave an explanation using a numerical example of 4 / 3 (he did not fully write the long division process till the end (remainder=0), but he explained the process well verbally).

Everyone agreed with this result. I asked them if we could verify this, to which one of them said - Yes, we can multiply and check. And this is what he did. 

It was an Aha moment for me!  What about you?  :-)

Now, there was a student who said that he had got a different answer. Others were surprised by this remark. He was asked to come forward and explain his approach.

This is his work:

- We need to find out (x-2) multiplied by what will give us (x^2 + 3x - 10)

- Since there is one x^2 term, it means that the multiplier of (x-2) should have at least one 'x' term..[ while saying this, he wrote 'x' next to (x-2) ] 

- Now, multiplying this 'x' by (x-2)  gives us x^2 - 2x. 

- But we need +3x  and not -2x.... So we need to add +5 to this multiplier 'x' so that this +5 after multiplying by 'x' of the expression (x - 2) gives +5x , which after combining with the -2x we have, will effectively yield the desired +3x  [while saying this, he wrote +5 next to 'x']

- Further, this +5 and -2 will multiply to give -10 too...So the answer is (x+5)

I looked at everyone and they were already with him. Before I could ask him about verification, he had already begun -

"So is the movie over?", I asked them with the hope that they should loudly say - NO.

And yes, they did not disappoint me :)

"Why not?", I ask them.

We now need to prove that both these answers are same.

"Oh is it? Why can't these 2 answers be different ? We have seen problems having multiple correct answers", I continue probing them as if I was unaware of whats going on in their mind.

Sir, how can division of the same set of 2 numbers give different answers?, argued one of them.

"Hmm... But these 2 expressions look completely different,", - my counter.

Yes, but then they should be equivalent....  

"Have we seen such cases earlier ?"

Yes, many times.., came their quick reply. 

I felt a sense of accomplishment with this conversation... So now their goal was - 

I doubt if we have seen a T.P.T. algebraic statement of this form in any textbook  :)

Thanks to my students, they keep offering me numerous wonderful learning opportunities !

Do not hesitate in pausing for a while, trying to prove this on your own first. 

-----

People with some algebraic knowledge will mostly and quickly factorize the expression in Numerator as (x+2)(x-2) and 'cancel off' one of the factors viz. (x+2) with the expression in Denominator...  But would this exercise leave us with (x+5) as desired  ?

And before you say yes, let me just tell you that these students are Not even aware of this identity of difference of squares [ a^ - b^2 = (a+b)(a+b) ], which we could instantly see and use. 

So then how would they go ahead??

Yes, that's the interesting part which even keeps me on toes while I work with them. 

I saw that they were just staring at these expressions for some time. I thought that I should intervene and offer them some clue and I did that. But soon I realized that I was wrong in doing so. 

I asked them "what does the expression on the left look like?"

They said, it resembles a Fraction. 

"Yes, and what about the one on right side"

They said, it looks like a Whole number.

When asked for the reason, they said - RHS expression (x+5) has the denominator=1.

"So what do you think, what should happen in the left side expression?", I ask them.

One of them quickly said, (x-2) should be factor of (x^2 - 4)... 

And why so ?

Only then its denominator will go away. 

While I was about to relish with this thought process, meanwhile one of them was already scribbling something on the board and he interrupted us ...

Sir, it is proved.... both expressions are equal !... he exclaimed in delight.

Oh wow !!  I felt a bit tempted to correct his vocab.... but that was not so important now....

Just study the left side of his work above. What he saw and said is - 

We need to prove the left expression to be equal to x+5.....Now, there is already a 3 in the left side expression.... So this means that the fractional part of this left expression should get simplified to x+2,  so that this when added to 3 wholes, will give us x+5 as desired.

He further continued.... 

So then I checked whether the fractional part is really equal to x+2 or not. 

'How?'

I multiplied (x-2) and (x+2).... and we get x^2 - 4. .. After factorizing the Nr., we can divide both Nr. and Dr. by the term (x-2)  and then we will be left with (x+2),  which when added to 3, gives us (x+5) as desired.

------------

To this, another student joined us, saying even he has proved both the expressions equal, but has used a little different method to get x+2.

I thought whether the Dr. is equal to the square of Nr. i.e. whether (x^2 - 4) = (x-2)^2  ?

But then I noticed that after multiplying (x-2) by (x-2), I will get  +4  and not -4 as desired in the expression (x^2 - 4) ..... So I changed the sign of 2 of one of the x-2, making it x+2..... And then when I multiplied these two terms (x-2) and (x+2), I got  (x^2 - 4)......  And then I did same as what he has done (above method) to get x+5.

-------------------

I could clearly see that the sense of wonder and accomplishment of proving the equivalence of two resulting expressions, obtained from their two different approaches - gave them more joy, satisfaction and confidence than the answer of the main (division) problem...

What do you think, would have happened if I had directly showed them or given them the procedure of dividing the polynomials as given in textbooks?  How would your students see / solve this problem if you don't give them the ready-made recipe ?

Will be eager to know your views and comments on this piece...

Thanks and Regards

www.rupeshgesota.weebly.com

PS: Students belong to class-7 marathi medium government school in navi-mumbai. I work with them voluntarily after their school hours as a part of maths enrichment program. www.supportmentor.weebly.com

Revisiting forgotten Arithmetic - with a new student - Part-1

It was her first day in our class... Both of us were excited as to what would unfold.....

Readers may scroll below to the first paragraph written in Bold, if they wish to know only the Math part of this post :-)

I had come to know that she hailed from a very challenging socio-economic background and was further (helplessly) doing her schooling in a language, which was not spoken by anyone in her home / family / community (a typical migrant case). My students had spotted her in their school and brought her to meet me. Why? --  Because both of us (she and I) shared the same mother-tongue: Gujarati (which was a kind of fascinating foreign language for these Marathi-speaking students :)

"Sir..... please speak to Geeta in your language... we are very eager to listen to your conversation....", my students insisted. 

While I could understand their excitement and giggles when we spoke in our native language, I found Geeta quite puzzled & shy while responding.... why not? this time she was surrounded by a gang who were carefully watching her and her words :)

So when I asked her, "would you like to join us in our class?", she immediately agreed.

After 3 days, I find her in our class today with others....

"Sir, you are late.... We came much before you...", she complained.

I looked at my watch.... I was on time... I found others smiling at this remark and I understood the matter...  Poor girl, she didn't know that she was dragged into the class by her peers much before the class begins, as usual.

We settle down.... students start sharing their respective work with me followed by my comments.... and then comes a voice from her -- 

"Teach me some Gujarati... I have forgotten many vowels...."

I was quite surprised by this request .....

"why do you wish me to teach Gujarati ? you already know / speak that language at home.... now you should start learning Marathi well, as everything will be told and taught to you in this language here...."

"I know Marathi too.... I have studied till class-5 in Marathi.... then 1 year in Gujarati.... and then now again in Marathi....."

This was another revelation for me.... !

"But how do I teach you Gujarati now.... We do not have any Gujarati books here..."

"There is one book in the shelf, I saw.....It is mix - English and Gujarati..." - instant reply.

"Oh is it ? I am not aware of it... Can you show me?"

She gets the book and starts reading it when I tell her to do so...... Incidentally it was a short picture story book meant for young kids about Circle...  I noticed that she was quite fluent in reading.... and when I asked her to tell me what she understood by the story, l was satisfied by her comprehension ability too, given her life-realities. 

Meanwhile, other students started demanding my attention to their queries and doubts.... and noticing our enriching 'mathematical' conversations, she too probably got inspired to do and demand some 'maths' now ;-)

I did something for the first time, which I generally never do, esp. with a middle school child (Class-7) ...... I gave her a naked (context-ridden) arithmetic problem....and that too a trivial one... 14 + 39 ... 

And this is how she did it.... 

I studied this for a while...... and when I asked her for the explanation, I could see her surprised..  "Is it wrong?", she asked with a worrying tone.

"I did not say it is wrong... I just want to know how you solved it...."

She began, a bit reluctant - "4 nines are 36.... so I wrote 6 down and 3 up.... and then added 3, 1, 3 to get 7....."

"ok... so why did you do - 4 nines are 36....?"

She answered ' its tables'..... I pointed my finger to the operation here.... and she gave me an embarrassing smile...  :-)

She snatched the book from me, reworked on the problem and showed it to me...

Then I gave her a problem that involved adding more than 2 numbers. She did it this way (answer is 2428)

While I was studying all of this, she started explaining - "I had first written the 2 of 28 above... But then I thought, it should be written down itself...."

Yes, this was surprising even for me esp. because she had properly worked out the procedure of previous problem (adding 2 numbers with a carry over)..... So I just drew her attention to her previous work, and asked her what did she do after getting the sum of 4 and 9... She understood & re-worked on it.... When I saw her again resorting to the drawing of new tally marks for adding the same set of numbers that she had added before, I told her to wait... and asked her few questions like 7+8 , 9+5 , 6+8 , 5+8 , 7+9 , etc.. 

"I see that you you are able to add these numbers mentally quite well, then can you try adding these numbers without using the tally marks, this time?"

She happily agreed and after some struggle, showed me her work:

While I was still looking at this, she said - " Give me some division problems..."

So I gave her, 528 divided by 7..... First she wrote it this way, 

But then she said that "it cannot be this way", and then she interchanged the two numbers..

Ignore the cancellation mark (she did that later). While I was looking at this work, she told me that she feels it is incorrect.... I asked her why does she think so? She said that she does not remember the tables of 7 well... I told her if she would like to write the table of 7 first and then work out this problem.... She agreed....

She asked for my help in subtracting 6 from 2... I was surprised by her request because I thought she would easily write the difference as 04 after studying her previous work on the subtraction part of division (52 - 54 = 02)... I noted that she had first subtracted 5 from 5 to get its answer as 0 and then came to the units place.....

I could have built upon this, but for some reasons, I resorted to a separate problem on subtraction. 

I  told her to first solve these two problems.

a) 74 - 21
b) 74 - 25

You may notice the set of numbers I have chosen in both the problems. Why did I keep the first number same and just tweaked the second number in the 2nd problem? Only to assess her knowledge of borrowing (regrouping) in the 2nd problem? Or because of something more important? If yes, then what's that I was aiming for ? 

I wanted to see if she can think & work out the answer of (little difficult problem) 74 - 25 using the answer of previous simpler problem 74 - 21. However, having seen her number-sense, I  was a bit skeptical of this 'logical' way. But children have surprised me many times and hence I did not want to make any assumptions here.

But this is what she did: 

What she has done in the 2nd problem, is not something very strange for the people working in the space of Maths education.

If you observe carefully, she has first cancelled 7 and wrote 17 above it. Then she asked me if its correct.

I asked her, " why is she doing that?"

" Because I cannot remove 5 from 4..."

" okay... so what can you do in such case... ?"

"Borrow from the next number..."

" So are you 'borrowing' then...?"
  

She understood what I meant from my emphasis on the word 'borrowing'.  (Though I hate to use this word, but I chose to use it now, because she was using it and ......)

So she cancelled 17.... and said, " I will take 1 from 7.... It will become 6.... and give this 1 to 4... so it will become 5..... and now 5-5 = 0 ..... and 6 - 2 = 4 "

And then she gave the book to me asking, " Is it correct?"

Mostly my response to such a question is - What do you feel? and Why so?  However this time, I chose a different approach.... I gave her this problem:

But then she stopped after couple of seconds..." How can I solve this? Even after giving 1 from 6 to 2, I cannot remove 9 from 3...."

So then I slightly modified the second number and she comfortably solved it like last one :)

  
When she asked me for the validation, I decided to intervene this time.

" Is it possible for us to know if our answer of subtraction is correct or not?"

-- Blank stare --

So I wrote 7 - 2 on her book and told her to solve... She solved it immediately & correctly. 

" This is a subtraction problem.... But can you see some addition too over here.... ?"

At first she did not understand what i meant by this, but with little clue, she could see and say what i had intended.  " 5 + 2 = 7 "   She was surprised by this.... But understood when we discussed why it happens so..... 

So then I told her to check the answer of previous subtraction problem.... and she realized that her answer was incorrect.... She said that 40 and 25 should add up to 74, but its 65... and hence there is mistake in the subtraction.

She was very curious and desperate to know the method to get the correct answer now.... 

I began -  "What is 74 made up of ?"  

She said: ' 7 and 4 '

'Will 7 and 4 make 74 or 11 ?'

'11' - with confusion on her face....

"So what is 74 made up of ?"  

No answer....

"okay... what is 20 + 5?"

' 25 '

what is 30 + 7?"

' 37 '

"So now tell me what is 74 made up of ?"  

with some reluctance..... ' 70 and 4......'   

"Yes... Good...."

and we started laughing now :-)

"Okay... So let us write 74 as 70 + 4 now...."

She wrote this on the floor.... and when I asked her what next, she wrote 25 as 20 + 5 on her own below the first expression.... I told her to perform the subtraction now.... I was curious to know as to how she would deal with this format now....

Yes... She surprised me.... She did this all on her own..... This was her sequence

"cannot remove 5 from 4..so borrow from 70..What should I take from 70?", she asked me..

"You decide", I said. 

" Can I take 10 from 70 ?", she asked me....

I gave her a plain 'okay',  without letting her know, how happy I was now :)

It was further worth noting that after removing this 10 from 70, she correctly added the same amount to 4 to make it 14.  Rest was easy for her....

When she asked me for the validation of her answer, I reminded her of the method we discovered to check our answer few minutes back....   And it didn't take her a minute to do this verification. 

And she got highly delighted looking at this 74 turning up at the bottom..... 

"Yes, our answer 49 is correct....." Her feeling of satisfaction  was no less than that of a person who had just conquered the peak of a mountain.... :-)

No doubt, her teacher too was equally delighted ;)

She quickly and happily copied this work from the floor to her notebook and ....

" Sir, please give me one more problem...."

Wouldn't a Maths teacher be desperate to hear this request from his/ her student? :)

"Oh yes, Sure...."

I dictated:  53 - 28

She notes the problem in her notebook and resorts to the floor for working on it as before.

I later realized that I should have rather asked her to now solve the same problem which she had claimed to be unsolvable some time back,,.... ( scroll up to see 62 - 29 )

It was a pleasure to watch her solve this confidently and then even verify it. 

Isn't this wonderful ?

She was on high and wanted to solve more subtraction problems.... However I drew her attention now to the pending Division problem, from where we had navigated into the Subtraction....    (I hope you remember :-)

This reminder surprised her and she immediately copied that problem as a fresh one on her next page..... and this is what she finally did after some struggle and scaffolding. 

If one carefully compares this piece with her first two attempts on this problem, one would be surprised to see her change in approach towards dividing the first part of the number...

Last time, she didn't mind writing 54 or 56 below 52.... but this time, she has taken a number less than 52 while looking at the 7's table....  when I asked her about this change, she argued that subtraction can be done only if the lower number is smaller than the upper number...

If you are now wondering how she carried out 52 - 49, then let me show you her work -

She wrote this 3 at the proper place, but was unaware of where to write the left out 8 after getting it down.... If you check the previous image, you will see that she has brought it down further (below 3)....  She was stuck up at this moment and asked for my help....

I chose to do some spoon-feeding this time, and told her that 8 has to be written next to 3....so that it becomes 38..... 

Interestingly she took it on from here on her own and completed the division process as shown.... 

But then... she was uneasy with this non-zero remainder....

"What to do now? I should get 00 as the remainder in the end, right?..."

I asked her , " Is any number left out to be divided in the dividend? "

' No '

" Can you divide 3 by 7 ? "

' No '

" Okay.. in such a case, the division process is over...  we can have non-zero remainder sometimes... "

I thought this instruction / rule might puzzle her.... but rather, she looked relieved ;)

" Shall I give you one more problem for practice, like we did in subtraction?"

' YES '

I gave her 725 divided by 5

And this is how she did - 

I would suggest you to study her above work carefully first....

It should surprise us that she has not divided 7 by 5 first.... as children are generally taught in the school....  She has directly taken the number 72 for dividing by 5.... ( reason?)

And this was not easy.... because like most of the students, even she believed and told me that the Multiplication table of 5 is till 50....  so how can we divide 72 by 5....

Rather than telling her to consider only the first digit 7 for division, I chose to work on the other correct and more important concept now....  

I asked her how had she made the multiplication table of 7 some time back..... She told me by successively adding 7..... So when I asked her if the addition work can continue even beyond 70, she immediately said Yes.....

"So does the multiplication table of 5 end at 50?"

'No'    and she started working on this,,,,,,

Based on her experience of previous division problem, she selected 70 to be written below 72 in the division.... and then carried out the further steps....

One thing worth sharing is that she got delighted getting 00 as the final remainder this time...  ;-)

And she wanted more division problems.... so I gave her these problems for home-work....  

Any thoughts on the choice of numbers in the problems above ?

She solved each of these problems at home & showed me the next day... 

Were all her solutions correct?  You might be / should be keen to know about her thought process on the last subtraction and division problems, I guess....

Some new, unexpected and interesting things have emerged in the way she has solved the last subtraction problem...  I think you would want me to share those with you....

So we will continue our  conversation on this matter in the next post... 

Meanwhile, please let me know your thoughts about this math-talk. 

Thanks and  Regards
Rupesh Gesota
www.rupeshgesota.weebly.com