sharing the fun of mentoring brilliant children

[mathwonk]

Hi all. I am again moderating the student online forum at epsilon camp, the non profit intervention for gifted math loving kids in Colorado Springs this year, and it is a lot of fun. I am trying to make some didactic points clear with the problems I pose, but as always I learn a lot from the kids by going with their flow of ideas.

 

Recently I tried to get them to offer a proof of the fact that a quadratic equation can only have two solutions, since like most kids today, they are focused mostly on finding answers and have very little or no exposure to logical arguments. They are really good at solving the clever word problems posed by Euler in his elementary algebra book, that turn out to be quadratic equations or systems of linear ones.

 

Instead of following my hint to factor the equation two different ways and get a contradiction by substitution of values, one child just said that the coefficients of the equation determined both the sum and the product of the roots, and that this determines the roots uniquely.

 

Since I have learned to try to understand what these bright kids are thinking, I thought about that and realized he is right, and that it can be shown graphically as follows. If we know the sum of two numbers X,Y equals b, it means the point with coefficients (X,Y) lies on the line X+Y = b. And if we know the product is c, the point (X,Y) also lies on the hyperbola XY = c.

 

Thus the point (X,Y) is on the intersection of the line and the hyperbola. That confused me because those two graphs meet ordinarily in two points not one. Then I realized that the line has slope -1 so contains symmetrically placed points. Thus if (X,Y) is one of the two intersections, then (Y,X) is the other. Thus we get the same two solutions just in the other order. This argument had never dawned on me before, and is just another benefit of teaching kids and keeping an open mind to their ideas.

 

I am sure you have stories like this often with your own children, and those moments can make this challenging job a pleasure.

 

best wishes to all.

 

Roy

[Kathy in Richmond]

Love it, Roy! What an interesting proof, & I love the graphical visualization of the solution set.

 

I miss the kids & their energy. Hope that epsilon continues to be fun for you!

[mathwonk]

I am currently pursuing a course of preparation in algebra that emphasizes understanding solutions of quadratics, and upgrades it to solution of cubics, as I did in 2011. Then I intend to introduce the algebraic approach to finding tangent lines, i.e. the algebraic approach to differential calculus, as envisioned by descartes and fermat. So far it is working for those students who are participating. Several of these ideas have already been described here on the WTM forum.