Sharing Po-Shen Loh’s new idea about the quadratic formula with kids

Yesterday thanks to a tweet from Tina Cardone I saw a neat article about a new idea about the quodratic formula from Po-Shen Loh:

I thought it would be fun to see what the boys thought about this new idea. We haven’t looked at the quadratic formula in a long time – probably at least 2 years – so I started with a review of the ideas. I asked my younger son if he remembered the formula and then my older son was able to derive it using ideas about completing the square.

Next I wanted to show some ideas about the sum and product of roots of equations. Personally, these are some of my favorite ideas from algebra as they were my high school math teacher’s favorite ideas. But, again, we haven’t talked through these ideas in a while so I wanted to review the ideas about the sum and product of roots in a quadratic equation with the boys before they watched Po-Shen Loh’s video:

Next we watched Loh’s video that introduces his idea:

Having watched Loh’s video, I asked the boys to give me two ideas that they took away from that video. We then talked through the ideas with a relatively simple quadratic equation:

Finally, we solved a general quadratic equation using the ideas from Loh’s video – the general solution requires a fair amount of algebra, but really is a fascinating way to get to the general result!

I think this is a really neat approach to solving a quadratic equation. The ideas of sum and product of roots are neat ideas and were emphasized in the Algebra book from Art of Problem Solving that my kids learned from. It is fun to see those ideas coming up again in a slightly different context as my older son is studying eigenvalues and eignevectors in his linear algebra book now. Hopefully Loh’s ideas will help lots of kids see the quadratic formula in a new and interesting way!

What a kid learning trig can look like

My younger son is studying in Art of Problem Solving’s Precalculus book this year. Right now he’s looking at some of the trig problems in

(1) The first problem asks you to prove that the area of a triangle is A*B*C / 4R, where A, B, and C are the side lengths and R is the radius of the circumscribed circle:

(2) The second problem asks you to prove that in an acute angled triangle that:

b = c Cos(A) + a Cos(C), where a, b, and c are the side lengths of the triangle and A and C are the angles opposite sides a and c.

(3) The third problem is Tan(A/2) = r / (S – A), where A is the angle opposite side A, r is the radius of the inscribed circle, and S is half the perimeter of the triangle.

(4) The final problem is pretty difficult -> you are asked to prove this identity:

final problem

It takes 10 min for my son to work through this problem, including a couple of false starts. But he gets to the end, which made me really happy: