Anti-Chain Rule is going so, so well! I might never teach u-substitution again. It’s making me wonder what else we teach in calculus that we teach simply because that’s how every textbook teaches it…but it’s maybe not the best way to think about it.
With U-sub I taught it this way:
My new way goes like this (same question):
You don’t have to know anything about math to see that one way is a lot faster. I cannot even begin to explain how painless this has been compared to u-sub…and how much better my kids understand what they’re doing. Instead of me just showing them these weird steps someone made up, with a tiny bit of questioning, they came up with their own way to reverse Chain Rule, and even gave the conditions necessary for this to work:
That’s some beautiful mathematics. They’re asking deep questions like “Can I reverse this? If so, how?” and “Will my way work every time? What conditions do I need to adhere to? Why?”
I told my kids they were doing so well with this, better in two days than most cohorts do after two months.
“Well, you taught us really well.”
“It only took me a decade.”
Life-long learning. ❤️