Packets to balance your weight

Situation: Practical assessment (grade-6 student of a government school)

Reason: To gauge his understanding of various math concepts that will serve as a vital input to my instruction plan later.

"Can you estimate the weight of this chikki packet?"

After thinking for a while --- "It must be around 30 grams."

"Let's check it."

He turns the packet around (seems he was already aware of the whereabouts of the weight). He shows the number to me with a smile -- "It's 50 grams."

"Oh.. your estimation was quite good.... Tell me Sidharth, have you seen a weighing balance?"

"yes"

"Imagine - I make you sit on one of its sides, while keeping this chikki packet on the other side....What will happen?"

While controlling his laughter, he replies -- "I will come down, and the packet will go up."

"Hmm... What if I add some more packets?"

"Then I will start moving up..."

"How much packets will be needed so as to balance you on the other side?"

He thinks for a while...and picks up the tools -- pen and paper. While I was happy to see his systematic work, I was also bit pissed off by the slower pace of the multiplication algorithms that he was using.. I wanted to stop him and ask him about the alternate approach... But doing that now would be 'harmful' to him.. I watched him work patiently, while figuring out what must be going on his head for the next set of moves..

This is how his work looked finally, before he just raised his head to tell me -- "600 packets."

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I have labelled the work for you (from 1-6) to so that you can study his thinking process.. 

"Ok.. I watched you work Sidharth, But can you plz explain me you thought about this?"

1. My weight is 30 kgs, I first converted this into grams, because packet's weight is in grams.   

2. Now, I wanted to figure out the no. of packets required to equal 30,000 gms. So I multiplied 50 with 50... I got 2,500 grams which is too small, so then I took a bigger number --- 500 to try with.

3. Multiplying 500 and 50 gave 25,000 which is just 5,000 short of the required 30,000

4. I now noticed that 50 times 50 has given me 2,500 in the 1st step.... So I took double of 50 i.e. 100 times 50 to get 5,000

5. 50 was taken ------ (500 + 100 =)  600 times to get 30,000... So 600 packets are needed to balance my weight.

Pleasantly surprised by the alertness he had demonstrated during the problem solving (step-4), I ask him ---

"Interesting... Can you tell me why did you choose to multiply the packet weight with 50 in the beginning?"

"I just took it randomly.......I took 50 because packet weight was 50...."

"Ok... and then why did you chose exactly 500 as the multiplier in your second step?"

"I needed a bigger number... So I increased my first multiplier (50) by a zero (500)"

(What I thought was that he might have used the result of 50 x 50 = 2,500 to choose 500, so that he can get the product 25,000 (closer to 30,000).... But, he had not thought this way (looked at or used this pattern)......)

"Ok... But I observe you also did some multiplication between 400 and 50 before you did 100 x 50.... What made you do that? Also, you missed out this step in your 

explanation, I think."

"Oh..yes.... I did that multiplication by mistake... I wanted to do 100 x 50, but I wrote 400 x 50 in the hurry ......when I got the answer as 20,000 I realized this mistake.... So then I also did 100 x 50 and I got the required answer 5,000 as I had thought. "

"Oh... so you were already aware that 100 x 50 would be 5,000...... Why did you do it then?"

"I wanted to verify that"

"Okay..... I am happy to see the way you have worked and even reasoned well....  But tell me Sidharth, can there be any other way of solving this problem?"

He thought for a while... and then started working on the Long Division method.....He had goofed up in this process and hence he got the answer as 6,000 instead of 600........This puzzled him and he got immersed in the fault-finding process.... 

It's a Pleasure to see a student trying to figure out his mistake and then even correct his mistake on his own.... (an imp. habit which is, unfortunately and generally, not seen in most of the students ---- ( Why?? )

I was glad to see that he was able to find and correct his mistake on his own... This is how it looked....

​

As you would notice, that he had written 30 (50 x 6) as the partial product instead of 300..which made him arrive at 6000 instead of 600... He explained this error to me clearly..

"Hmmm.... So then?"

"Sir, we got the same answer using both the methods.. So our answer is correct."

"Okay.... Can there be a third method?  :-)"

He thought for a while again...... and then replied with a smile -- "Sir, I don't know"

"Okay... no problem... we will learn that later during our session.... But tell me one thing Sidharth, which of these two methods would you prefer for such a problem?"

He replied almost instantly --- "Sir, the division method."

"Why?"

"Its faster."

"Hmmm..... But then why didn't you use this method in the beginning?"

"Sir, I recalled this approach when I was already half-the-way down in the 1st method.... So, I thought to continue with the same then...."

"Okay... Nice.... I am glad that you have solved this problem using more than one way.."

He acknowledged this compliment with a smile and we proceeded to the next problem......

{Oops, let me also tell you that he was stuck up in the beginning because he was not sure about the conversion factor between kg and grams....Seeing him waiting, I intervened and asked him about his problem.... He asked me about the relation, but as usual, I threw the ball back at him -- to guess it.... He said: 1 kg = 300 grams... I could have helped him discover the relation, but rather, I chose to give him the required figure (1000) at this juncture! }

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Thanks & Regards

Rupesh Gesota

www.about.me/rupesh.gesota

"The true value of a teacher is determined not by what he knows, not by his ability to impart what he knows, but by his ability to stimulate in others a desire to know."