I thought it would be fun to see what the boys thought about the rotated parabola, so this morning I showed them a few rotated parabolas and asked them what they thought:

My younger son was interested some of the pieces of area that were cut out by the rotated parabola. Funny enough, my older son was interested by a similar question a few years ago:

It was fun to explore their ideas about the different areas of the graph. We had a neat detour when my older son wondered what the graph of would look like of we were really zoomed out.

Finally, my older son was interested in what the parabola would look like under a variety of different rotations. The discussion here ended up being a neat surprise as what grabbed the boys’ attention was how to change the x- and y- coordinates following a rotation so that the graph would display correctly. I wouldn’t have thought to talk about that, but they were pretty interested in understanding how the coordinates changed. The funny thing is that walking down this path gets you really close to talking about trig functions.

So, a fun morning project even if math need to compute or calculate the rotation of the graph is obviously way, way over their heads. But, since the picture is actually pretty simple, there’s still plenty of interesting things to talk about with kids, AND plenty of stuff that kids might be interested in!