Fraction Confusion - Part:1

I got a strange request few weeks back...  

Two 10th class students were brought to me.. Their parents wanted me to "clear their concepts" of maths that they had learned (I think, they actually meant rote-learned) in their lower classes.... Why?  because both of them were now entering into the most important (!) year... Yes, you guessed it right -- Tenth standard !! 

My maths class is generally full of debates, arguments, confusions, conjectures, etc. And hence More is the Merrier in our kitchen. So working with only two students, and that too with a time-bound and syllabus-bound objective was no longer my interest. However for some reasons, I accepted this odd assignment, perhaps because I saw the yearning for understanding more than the greed for score in their requirement....

I must confess, its a different experience working this way after a long time.  After my couple of sessions, I thought I would write and share the lessons, trajectories, experiences and reflections of each of these sessions. But sadly, this could not happen.. 

In fact, I soon realized that I have gradually recoiled back to my natural style - Inquiry leading to Discovery..

I did their diagnosis in the beginning so as to gauge their conceptual clarity. No wonder, it was a dismal performance (their mistake??)

It was our 8th session now and I wanted to unfold the tons of rules they had (rote-)learned in fraction arithmetic...

I told them to show me half with a diagram. Surprisingly both of them drew a horizontal bar rather than circles to show half (generally I get only pizzas from the private school students :)  

I then asked for one-fourth which was shown this way. 

So far so good. Then I asked them to show me my favorite fraction:  One-third

I pray for the moment to see the students answer this question correctly. But my prayer is hardly ever heard. And this time was no different. After thinking for almost a minute, one of them reluctantly pointed out the remaining portion of the bar above as 1/3. When I probed her for the explanation, she said --

fig - 1

While pointing at half, she described it as one-upon-two. The last small piece is one-upon-four. And the portion 'half plus quarter is one-third'.

I noted that she used the terms 'one fourth', 'quarter' and 'one-upon-four' interchangeably. 

"But what makes you feel that this portion is 1/3?", I ask her while pointing at her 1/3.

"We have divided the whole into 3 parts..and 1/3 should be bigger than 1/2.. So that's why."

I asked her peer for confirmation. She simply conformed. 

I was amused and even amazed by the fact that they wanted 1/3 to be bigger than 1/2. But at the same time, they did not see that their 1/4 was smaller than 1/3  :-)

I could have drawn her attention to this contradiction, or may be I could have even asked her the meaning of one-third and one-fourth, but rather I dragged her into another zone. 

I asked her to show me three-fourths. And she showed this...

fig - 2

When asked for the explanation,

"I have made 4 parts and taken 3 out of it... So 3 upon 4"

I thought the bulb might glow now. But it did not. So I took her deeper. I drew this on the slate -- 

fig - 3

"Okay... So going by your definition of 3 upon 4, what I have shown now is 2 upon 3, right?"

She stared at it mum for almost half a minute... I could see her eyes even roll down to her first figure at times...

"What happened?"

"No sir... this is not 2 upon 3."

"Why?  I have made 3 parts... and then took 2 out of 3... So shouldn't it be called as 2 upon 3?  That's how you argued for 3 upon 4.."

"No... because it is 1 upon 3... It is the same figure that we saw at first.."

"Oh !! Is it?  But then I just followed the way you had defined 3 upon 4.."  and while saying this I smartly drew her attention to 3/4 now  :-)

I knew this would baffle her completely....and yes, she did ! 

I made it more spicy !!

"Oh.. What is this??  Three identical portions having Three different names...? I am more confused now... What's correct?  Is it  'one third'  or  'three fourths'   or  'two thirds" ?"

Pin drop silence for almost 2-3 minutes... She was just staring at all the three diagrams... I was about to intervene, but just then ...

"Sir.. the last one is not 2/3 ... It is same as 3 upon 4...  "

"Really?"

"yes... "

"But then.. your definition......of taking 3 out of 4 parts......?"

"We have to make 4 EQUAL parts first.... and then take 3 parts....."

Yes !! I was so happy.... One milestone achieved :)

"Okay.... Would you like to rephrase the meaning of "3 upon 4" again?"

"Yes... Make 4 equal parts... and taking 3 out of those..."

This time, I could see her emphasizing on the word 'equal', along with a shy smile  :-)

But still our problem is not yet solved.....

"If both the portions in the last two diagrams show '3 upon 4', then what does the portion in the first figure show?"

She was quick this time... "Sir, that's also 3 upon 4.."

"Sure?"

"yes"

"Okay... then show me one-third now..."

I thought she would be able to pick up now... But no.... After few seconds, I asked her -

"what does 4 represent in one-fourth?", while pointing at the quarter in the diagram.

"It means we have made 4 equal parts."

"So then..... for one-third......?"

I paused....waiting for her to take it ahead......

".... three equal parts....?"

She uttered this reluctantly, looking at me for confirmation....

I raised my thumb and she instantly heaved a sigh of relief :)

"Wait, wait dear... Draw and show me one-third now..."

fig - 4

Just to implant this idea deeper in her memory, I asked her to show few more unit fractions like, 1/6 , 1/8 etc... and she showed each of these correctly.... 

I noted that she had also showed and reasoned for '3 upon 4' some time back...

So now I asked her to show me 3/8....

fig 5

"Can you explain?"

"I made 8 equal parts.... and then took 3 out of it..."

Did you notice the presence of word 'equal' in her explanation now?  :-)

So are you satisfied with her responses and explanations now?  

Did you say 'Yes'?

Hmmm...  But I am not yet..... And so, I asked her another question....

Can you guess what could be the next question ?? 

Waiting for your 'question'... I mean, response...... 

You need to be quick....as I have already started working on the Part-2 of this post this time :)

Regards

Rupesh Gesota

One puzzle - Many students - Many approaches

Puzzle

“My students have already solved such problems”, this is what I thought when I saw the above image and hence I was about to ignore this problem.... However, some strange words next to it motivated me to read the problem carefully and I realized that I was about to make a big mistake. 

The puzzle challenges you to find the missing shape “without” finding the value of any of the 3 shapes. Assuming that my students will like this twist, I threw this problem in their court.

While they were noting down this problem in their notebooks, one of them, Vaishnavi, sprang up – “Sir, I got it.”  And we all were like – stunned! 

"...without evaluating any of the shapes?", I probed her with some doubt evident in my tone.

"yes, of course, that's how you wanted us to do, right?", she replied with confidence.

"Ok.. explain me."

"Wait Sir. I will rather write and show it to you."

I was so happy to hear this. Probably, this reflected her growing interest and confidence in mathematical writing.

Vaishnavi's 1st attempt

It states that the 3 diff shapes in the first column add up to 12. 

So the middle column cannot have Star because it adds up to 13.

Further, inserting a Circle in the blank space would imply 3 Circles = 14 in the middle row which according to her was not possible as 14 is not a multiple of 3.

Hence the only option is 'Square'.

What do you feel about her solution? Are you satisfied? Do you want to ask her any question?

Well, I wanted to. But I wanted to see if my other students too could see what I could, and probably what Vaishnavi could not. So I told her to write her solution on the board and then invited others to comment on it. They studied it for a while and then Rohit raised his hand.

“Sir, how does she know that Circle cannot be a fraction?”

Super! He did my job. I turned to Vaishnavi for her response. She smiled at me, took the book from my hand and started working on the problem again!  Another wow moment for a teacher, isn’t it? :-)

And then, we all got engrossed in independent problem solving. I ensure that students see me - their teacher - solving the problem along with them. (Why?)

Soon, Rohit came up with his solution.

Rohit's solution

He explained this way:

His first equation involving  Circle and Square comes by observing  last row and last column. For the other two equations, he has mentioned the reasons (he has labelled the rows and columns using letters).

Finally he just verbally reasoned that, “Sum of the middle column is bigger than the first by 1. Also, both these columns have the same two elements. Hence the missing element has to be Square."

Did you get him? Any questions?

While I was happy with his observations & reasoning, I wanted him to re-think --

“Rohit, what do you feel, how many of these observations were really useful to you?”

He thought for a while and could identify that only the last equation was required.

“Then why did you write the other two equations?”

“Sir, I was not aware as to what will help me. So I was just recording all my observations.”

“Hmm…good habit. So should these ‘other’ observations also be stated in your final solution?”

“No..”

“Why?”

“Because they don’t lead to anything!”

“Hmm… Good…  Can you write your solution systematically and show it to me?”

He did not take more than 2 minutes to write this.  I would suggest you to take 5 minutes to study his solution. (Check the Red part) Also note how he has renamed the rows and columns of the grid as R1,R2…C2,C3.

Rohit re-writes it systematically

I was now more eager to solve the problem using my method. 

But I could see Jeetu approaching me with his solution. This is how he had solved it.

Jeetu's approach

Observing C1 and R3, we can conclude that, Circle = Star + 2

Using this relation in the row-2 we get,  (2 times Star) + 4 + ? = 14

So, (2 times Star) + ? = 10

But Row-3 says that (2 times Star) + Square = 10

Therefore, it means that ‘?’ = Square.

Further, he has also reasoned for the impossibility of '?' to be Star.

What do you feel about his thought process? Any comments/ questions?

I looked at Vaishnavi if she had worked out some reasoning for her claim. But she was still working on it. 

Meanwhile Kanchan had placed her book on my table. This is how she had reasoned. 

Kanchan's solution

2 Stars and a Square adds up to 10. So Square must take an even value.

Similarly, 2 Stars and Circle adds up to 11. So Circle must take an odd value.

Now, the middle row wants the missing element to be such that it adds up with 2 Stars to give 14. It means the missing element should take an even value.

Further, Star + Square + Circle = 12 (even)

Because, Square = even and Circle = odd, so Star should be odd. 

Hence the only shape that is even out of the three is Square.  Hence the missing element which takes an even value is Square.

What do you think about this method? Any questions?

Kanchan also solved this problem by evaluating each of the shapes. This is her 2^nd solution:

Kanchan's 2nd solution (by evaluating)

She has proved that the three shapes represent three consecutive numbers., with ‘Square’ being their middle one.  Since they add up to 12, this should be thrice the value of Square, thus evaluating it to 4 and then even other shapes.

I was really getting overwhelmed by their creative thinking by now. But seems, there was more to this. Sahil was ready with his thought process. This is how he presented his solution to me.

  

Sahil's solution

A good deal of work, isn’t it?

While he has neatly written down the relations in the form of equations, the problem with this solution was it lacks justification/ reasoning alongside the equations. It was tough time for me to find out how he was re-framing his equations.  But he confidently and clearly reasoned out all the steps when probed. So I asked him to re-write this solution systematically with some guidelines. And I was so happy to see his re-work. Check this –

Sahil re-writes his solution systematically

I think his solution and reasoning is clear, however for language reasons I will translate it below:

1) Square + 1 = Circle (using R3 and C3)

2) Circle + Circle + ? = 14 (as given in R2)

3) Hence, (2 times Square) + 2 + ? = 14 ….. (substituting value of circle from eq.1)

4) So, (2 times Square) + ? = 12

5) Now, (2 times Square) + 1 + Star = 12 .. (substituting value of Circle from eq. 1 in C1)

6) So, (2 times Square) + Star = 11

7) Now, (2 times Star) + Circle = 11 ….. from C3

8) So, (2 times Star) + Square + 1 = 11 …… substituting value of Circle from eq.1

9) R.H.S. of equations 6 and 8 are same, hence their L.H.S. should be equal.

So, Star + 1 = Square
10) Substituting this value of (Star +1) in equation 5, we get (3 times Square) = 12
11) So, (3 times Square) + 2 = 14 …… adding 2 on both sides

12) Comparing equations 11 and 3 with equal R.H.S., we get ‘?’ = Square.

 

Finally the last student Tanvi too shared her solution.

Tanvi's solution

She too had used the properties of even & odd numbers, like Kanchan, to solve the problem. 

When I told her that one of her peers too has used the same approach, she said --

“Yes sir. I know that Kanchan has used the same approach. I overheard the words – Odd/ Even – while she was explaining you her solution.”

“Oh.. Is it? Have you looked at her solution as well?”

“No Sir… I have solved this completely on my own.”

I can trust her completely. I was just amused by the fact that observations and justifications made by both of them were exactly same.

It was finally Vaishnavi's turn now -- remember the first one to solve - but stuck up because of the assumption that Circle cannot be a fraction ?

This is how she could beautifully prove it.

Vaishnavi's solution

I was also delighted by the way she has neatly and systematically written down all the steps with associated reasoning.

I think the written solution is self-explanatory. 

First she has found the relations between Square and Star and that of Circle and Star. Then she has formed the equation in R1 using only Stars which leads to the conclusion that Star is an integer. This further proves that other two shapes are also integers.

Now, replacing '?' by Circle would imply (3 times Circle = 14) which means Circle is a fraction. But Circle is an integer. 

Further, Star cannot fit in the C2 equation, as the three shapes add up to 12 as given in R1. 

Therefore the only shape left is Square.

I could not resist and just declared – 

“Wow !! Six students and Five approaches. I am so happy.”

Jeetu asked me -- "Sir, which solution is the best?” :-)

I could see them eagerly wanting to hear one of their names from me. But then they also knew that they would not get the answer to such a question. I would only help them do the analysis.

But let me ask this to you. 

Which solution or approach you liked the most? 2nd most? Why? 

What are your views about this type of approach – allowing the students to devise their own ways to solve the problems and then facilitating the discussion among them ? 

Did you solve this puzzle in  a different way?  If so, then do share your solution. My students would be happy to study one more way :-)

Waiting to hear from you....

PS: These students belong to Marathi medium government school based at Navi-Mumbai. To know more about MENTOR  - a special maths enrichment program for students from such challenging backgrounds, visit the website www.supportmentor.weebly.com

Regards
Rupesh Gesota