I’m Andrew Hogan. These articles begin with ordinary things: a bouncing ball, a drive, a triangle, the steps of long division. Each offers a question to try and some thoughts about where it might lead.
You can work through an exercise, follow a construction, or simply read. Bring your experience. There may be something here that looks different from where you are standing.
Start here: How long can a ball keep bouncing?
A ball bounces on a piece of stretched balloon rubber covering a glass. It rises, falls, and rises again.
Watch the height of each bounce. What changes? Where might the energy go? What would have to be different for the bouncing to continue forever?
A small experiment can give us something concrete to observe and something much larger to imagine.
A question with everyday consequences
How much room does it take to stop?
At highway speed, a car travels a considerable distance while its driver notices a hazard and responds. Braking takes more distance still.
But stopping before a stationary obstacle and following another moving car are different situations. What assumptions connect them—and where do those assumptions stop helping?
This article combines a simplified calculation, a personal experience, and time-based following guidance.
Explore the articles
Time and the numbers we live with
Your One Billionth Birth-Second
We usually count our lives in years. What becomes noticeable when we count in seconds?
Where Do You Draw the Line?
A drive, a traffic report, and a few minutes saved. What exactly is being measured when we talk about time?
Making something from a possibility
The Forbidden Triangle
Two sides, a missing length, and a tent made from unequal halves. What might we build when more than one answer is possible?
Robert’s Remainder
Long division leaves a trail of intermediate results. What happens when we collect them into a new number?
Motion, limits, and physical models
Car Lengths
Explore the distance traveled before and during braking, and why following distance is measured usefully in seconds.
Tight Turn Limit
A bicycle’s speed, the radius of a turn, and the grip between tires and road. How can a calculation help us think about a physical limit?
Balloon Over Glass
A bouncing ball opens questions about energy, observation, and the possibility of forever.
Charged Sphere Pump
Touching two spheres redistributes their charge. What can a sequence of these simple operations accomplish?
A way to explore
Most articles follow a familiar path: an exercise, an explanation, and some thoughts or further questions.
Before looking at an answer, you might make a prediction. While reading, notice which conditions are given and which are assumed. Afterward, try changing one condition. Does the conclusion still follow?
A model gives us a way to examine part of the world. Its assumptions help us understand where its conclusions apply. In the driving and cycling articles, calculated limits are learning tools; practical decisions also require attention to actual conditions and current safety guidance.
Sometimes a useful result is an answer. Sometimes it is a clearer question, a corrected calculation, or a new thing to make.
Keep the conversation going
These pages grew from teaching, conversations, personal experiences, and the contributions of students. I try to give those contributions room to travel.
If you find an error, a missing assumption, a different construction, or a question worth adding, I would like to hear about it. Pointing to a particular sentence or step helps me understand what you noticed.