Recent Popular Posts

March 27, 2014

Generating intuition in algebra (2): story for x+y=10

I split each class that we use computers into a primer and an execution. In the primer I asked them for a story of x+y=10. I got some interesting replies and I wrote them down and we analyzed them together:
1) I have 2 mangoes. My mother gave me 8 mangoes. How many total mangoes do I have?
2) My sister has 5 sweets, I have 5 sweets. Together we have 10 sweets.
3) I have some mangoes. My mother gave me the same mangoes. Total mangoes are 10. How many mangoes did my mother give me?
4) I have some oranges, my sister has 10 more mangoes than me.
5) Total biscuits are 10. The dog ate 6 biscuits and the cat eats something. How many biscuits does the cat eat.
6) Ramu and Ravi collect some sticks. The total number of sticks is 10. Ravi collected 2 more sticks than Ramu.

There was generally a fixation with having one 'answer' which came from the one variable one equation situation they had encountered thus far...

I started listing the specific examples they had given 1) and 2) 
x + y  = 10   
5 + 5 = 10 (possible, but only one of the possibilities)
2 + 8 = 10 (possible, but only one of the possibilities)
this was enough to get them kick started in getting a load of other possibilities
1 + 9 = 10
3 + 7 = 10
4 + 6 = 10
0 +10= 10
and their flips.
I mentioned that there isn't just one answer like they have always been used to there are simply many possibilities. 
I also pointed out that if one value increases then the other decreases as the total is the same. We took an example, if Var received 10 chocolates on Sha's b'day and she shared it with her sister, if she decided to be a generous elder sister and gave them all to her sister, her sister would have 10 and she none, if she kept one her sister would get only 9 and so on. They seemed to think I went to too much effort to state the obvious. 

We then considered each case of the other stories they had come up with and went through what they mean and if they sufficiently cover x+y=10 or not.
3) Is like having twice the number of the mangoes indicating 5+5=10. Doesn't seem to cover everything.
4) You can't add mangoes to oranges unless you start treating them as fruit and forgetting that they are mangoes and oranges. The source of confusion was that x and y needed to be different as mangoes and oranges are! But, if they needed to add up to 10 they needed to be of the same kind. The child came back with, ok, they just need to be different numbers not different fruit. Another child corrected its possible they are different and its possible that they are the same!

By now they figured out that 5) was incomplete as I didn't give the number 6. 
This led us to 6) and that the first part made sense, but where did the second part of the story came from? Arc pointed out that that's how some puzzles were and more condition was needed for an answer. Hmm...I asked them to work with what we have right now and that the reason for the other condition was coming soon.

Back to the possibilities, I drew out the x and y axis and started putting in values for x and y as ordered pairs on the axis starting with (10,0) and (0,10). I put a few more points and then jumped the gun and stated that it turns out that all these points actually lie on just one line. Magic! The kids were impressed and pointed out that it seems believable as every time Var gets a chocolate her sister looses one so the line going down makes sense.

We then handled x-y=10. The kids had of course gotten the hang of it and made the stories. We tried to look at how we could make the stories believable! The best we could come up with were - The fruit vendor delivered some bananas and my sister ate some in the morning and left, now I see that there are 10 bananas left.

We then analyzed that how if the sister took more bananas then there should have been more to start with and how this line (they already guessed that this would be a line) would go up.

Computer assignment, plot 

No comments: