Last fall Po-Shen Loh shared a really interesting approach to understanding the quadratic formula. His idea got so much attention that they were written up in the New York Times::

I thought it would be fun to revisit some of the ideas today. It turned out to be a really good algebra review for my younger son, and a nice review of ideas about roots of equations for both kids.

We started talking about some general ideas about quadratic equations and a reminder of the sum of roots and product of roots ideas for quandraics:

Now we took a closer look at the sum of roots and product of roots ideas to give the boys a bit more background on the ideas in Loh’s paper. The ideas here were a little confusing for them, so it was good that we took a little time to review them before going to the next step:

With all of the background out of the way we moved on to Loh’s difference of squares idea. This idea wasn’t obvious to the boys, but once they saw it the quadratic formula appears immediately!

Finally, we finished up the project by showing how to derive the quadratic formula for a general equation:

I really like Loh’s approach and think it is a great way for kids to see the quadratic formula. This project showed me that the ideas are a bit more subtle than I thought, though, and we’ll probably have to run through them a few more times for them to really sink in.

This is the p-q formula, known in ancient Babylon. It has been taught in German schools for many years.