How many ways can you get from the corner of Bleecker and Mercer to Broome and 6th avenue? Which of these paths is the most efficient? Is there an easier way of figuring out the most efficient path than measuring the length of each of them? How would you redesign this city to minimize distances traveled between any two blocks as much as possible? Is a grid the most efficient way to pack buildings into a city? How much total road is visible in this map? How much does that road cost to maintain?
What other questions could you ask about this map?
Passwords are something about which almost everyone needs to be better informed. As part of a unit on combinatorics (or alternatively, as a unit on passwords in a tech class), students could look at passwords and how to make passwords more secure.
To get students thinking about password strength, this interactive password haystack calculator would be useful. Students could start by trying to make some secure passwords through the interactive calculator, and then they would probably have questions (like: Why is this password so much more secure than this other password?).
This list of the 25 most commonly used passwords is also useful to start some conversation on the difference between password haystack and password strength.
Imagine you numbered each note of a scale, and then played the mathematical sequences on the notes like they were music. What would 1, 2, 3, 4, 5, 6, 7, 8, 9,… sound like? What would it sound like if you automatically jumped back down an octave every time you passed a multiple of 7? You may find this tool useful for actually listening to the sequence of numbered notes you generate.
What would the sequence of square numbers sound like? What about prime numbers? What if you kept the tone of the notes the same, but varied the length of the notes? How long would the sequence of notes that started with a half-note, but then halved the length of each subsequent note, take to play?
What would π sound like?