Althea and her friends (Kiara, Sofia, and Aiden) begin to explore the history and ideas of calculus, with a bit of help from Mom. The schools are online because of Covid, and they're all homeschooling for the duration. They've only taken one year of algebra, so Mom brings in lots of activities that review algebra and geometry concepts.
Those activities (and more) are found in the Activity Book that accompanies the story. They're also available here, in the "course site" Mom has set up for Althea and friends.
Limits Bonus Chapter (best after chapter 4, or even later)
Infinite Series (coming later)
Ways to Play
If you don't see your favorite math scene animated here, maybe you can do it in Desmos, and I'll add it here with your name to credit you!
Chapter 1: A tree is growing (What would make this animation more interesting?)
Chapter 1: On the Geoboard, does a triangle that stretches more seem bigger?
Chapter 2: Aiden's imaginary car drives on a curve (You can make changes - try using reverse contrast in the wrench panel.)
Chapter 3: Throwing a ball out a window
Chapter 4: Watching h approach 0 (as the secant approaches the tangent)
Topics Left Out
I left out many topics that are normally included in a conventional calculus course. I wanted the story to be short enough so that the size of the book wouldn't be intimidating (under 350 pages). Right now my current draft is too long, and I'm looking for things to cut.
Here's a list of the topics from Calc I left out of the published story:
Definitions of Limits (bonus chapter below) & Continuity
Related rates
Differentials
Mean Value Theorem
L’hopital’s rule
Integrals of Inverse Trig Functions
Volumes by ‘shell method’ (often done in Calc II)
If you'd like to write some fanfiction in which Althea and friends explore one of these topics, I'd love to include it as a "bonus chapter" here on the website. (It would need to be age-appropriate for ages 10 to 14.)
If you want to learn those topics, there are many good (free) textbooks online. Among them: OpenStax (conventional), Boelkins (project-oriented series, may not have complete proofs). MyOpenMath is a (free) online homework system.
After Calc I (after the book ends)
Calc II includes techniques of integration (so you can solve more integrals), applications (work is a cool one), and infinite series. I wanted to include infinite series in this book, but that wouldn've made it too long. There will be a few bonus chapters here for those.
In Calc III, with three dimensions and multiple variables, the connections to physics become more visible. It’s hard work to learn and it’s beautiful.
References
Calculus with Infinitesimals, by Keisler