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April 13, 2014

Generating intuition in algebra(3): story of ax+by=10

The kids had gotten comfortable with a story of 
and what it looks like.

It was time to move to the story for:
Once we cleared the confusion with 
x+2 = 10 and remembered that 2y = 2*y=y+y
The story that Sub came up with was
I have some oranges, my mother gave me some oranges and my father gave me the same number of oranges as my mother. Now I have 10 oranges.

How would this extend to
Now the extended family was joining in and the kids felt that the stories were making less and less sense and I should now start contributing to the discussion. 

I reminded them of a simple multiplication story which Arc used to come up with the following
I don't know the price of a liter of milk or the price of 1 liter of oil, but the cost of one liter of milk and 5 liter of oil is 10. 
Though we all agreed that given the price of milk and oil the story was difficult to believe it made sense in principle. As in case of addition of variables if one went up the other went down, we talked about the relative steepness of the curves without entirely venturing into slope and that we should try different curves in geogebra to get a handle of what happens if these numbers change.

Using geogebra we concluded that what mattered is the relative index of x vs y. The larger the index of x vs y the flatter the curve.

We then used the method of putting x zero to get y (y intercept) and putting y zero to get x (x intercept) to draw a few lines. Once we drew a line we used the intuition from the stories to check if the line made sense. This helped catch some mistakes when trying to find x and y intercepts.

We then pushed into freeing up the limits of what indices and constants were including 
that involves division of two negative numbers and if something went wrong with the intercepts looked at the final direction of the line constructed.

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