How Mathematics Works
Bridge the gap between mechanical calculations and deep mathematical intuition. Explore functions, calculus, vectors, and formal proofs to see how the mathematical toolkit fits together.
Like Duolingo, but for How Mathematics Works. Tomo turns the whole topic into a game you play five minutes a day, until it actually sticks.
24 levels across 4 sections, about 48 minutes end to end, roughly 10 days at five minutes a day. It moves through Translating the World into Functions; The Calculus of Motion and Accumulation; Space, Vectors, and Linear Systems; and Uncertainty and Logical Structure. It assumes you already know the basics.
Free forever · No credit card · iPhone & Android
Key ideas in How Mathematics Works
- A relation is a function if every distinct input matches to exactly one output value.
- Multiple different inputs can legally produce the exact same output in a valid function.
- A relation fails to be a function only when a single input yields multiple conflicting outputs.
- The independent variable is the input lever that causes a change in the output response.
- The dependent variable measures the resulting outcome, such as consumed or remaining charge.
- Identifying which quantity moves freely helps assign the input axis on a coordinate plane.
- Linear models add or subtract a fixed amount per unit of time or input.
- Exponential models multiply the current value by a constant factor across equal intervals.
- A constant percentage change indicates an exponential function rather than a linear slope.
- The domain represents all input values for which a real-world scenario physically exists.
- Negative time values precede the event and fall outside the scenario's valid domain.
- Outputs falling below ground level mark an endpoint where the flight model ceases to apply.
- Piecewise functions switch mathematical rules depending on which interval the input enters.
- Boundary thresholds require adjusting the variable so added rates apply only to excess units.
- The baseline flat rate must be added to the overage rate calculated past the cutoff point.
- Subtracting inside the input shifts the graph horizontally to delay the timing
You've tried the other tabs
Thirty open tabs. Four facts you actually kept.
You watched. You nodded. By Sunday it was gone.
One answer, then back to scrolling.
Eight weeks. You meant to finish. You didn't.
Tomo gives How Mathematics Works the Duolingo treatment: levels, streaks, and quick quizzes that test what you just learned. That game loop is what the tabs above never had, so it's the one you actually finish.
Here's what playing it feels like
A real question from this course. Take your best guess.
A weather station records 18°C at 2:00 PM and 18°C again at 3:00 PM. Does this still qualify as a function?
Get it right to open this lesson and 23 more in the app.
Where How Mathematics Works takes you
- 1
Translating the World into Functions
- Modeling Real Relations with Functions
- Manipulating and Inverting Relationships
- 2
The Calculus of Motion and Accumulation
- Finding Rates of Change with Derivatives
- Measuring Totals with Integration
- 3
Space, Vectors, and Linear Systems
- Navigating Space with Vectors
- Transforming Space with Matrices
- 4
Uncertainty and Logical Structure
- Reasoning Under Uncertainty
- Constructing Mathematical Arguments
4 sections · 8 units · 24 levels. Built to play, not to enroll.
Tomo is in ChatGPT
Tomo is also a ChatGPT plugin: the same playable courses, built and played inside your chat.
Add Tomo to ChatGPT- What to ask
- “Make me a Tomo course on black holes.” “Turn this chat into a Tomo course.”
- What you get
- Quick questions with instant feedback, hearts and stars, ready to play in about a minute.
- Where it goes
- No sign-up to start. Connect your Tomo account and your streak carries on in the app.
You pick the voice
How Mathematics Works is taught in the Bestie style: your friend who just gets it. Want a different feel? In the app you can spin up the same topic in any of Tomo's teaching styles. Same facts, totally different vibe.
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Start How Mathematics Works today.
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