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?
My sister and I were walking our kids back from a trip to Science World, when we passed a park sparsely filled with people. My sister looked at the people sitting in the park, and wondered aloud, “I wonder if you can use mathematics to figure out how far apart people will sit on a lawn?” I looked carefully at the park too, and noticed that everyone seemed to be carefully at an maximal distance apart from anyone else on the field. I am particularly excited about my sister’s question, because she has always described herself as “not a math person.”
I decided to generalize her question, to “do groups of people follow predictable patterns?” This would allow for exploration in a wide variety of ways, for example:
- Do people tend to follow the same paths when crossing open-space, like a field or in the meeting room of a train station?
- How random is the motion of people as they sit waiting in a theatre?
- Can you track use of phrases of language through groups of people?
- What similarities exist, if any, between the networks of relationships each person has?
For this project, students collect data on a phenomena in which they are interested, with teacher guidance, and then look for patterns in this data. They can use a variety of different means to collect their data, attempt to find a model which matches the data, and then present what they have found. Ideally students would go through multiple revisions in their data collection model to collect as clean data as possible, particularly when they end up with major problems with their results.
Some specific things which students might model:
- The growth of plants over time;
- Population growth of bacteria in a petri dish;
- Temperature during the day;
- See more ideas here.
If you want more information on what mathematical modelling looks like, see this document or this document.
I love this activity by Bruce Ferrington with his students.
What Is My Area?
Wait – before you say anything – I know your body is 3-dimensional and area is a measurement of two dimensions – so I probably should say, “What is the area of my silhouette?”
What is the area of my silhouette?
In the past week we have measured height and width, talked about Vitruvian Man, combined our heights, compared them with other classes.
Time to have a look at comparing areas.
My chief interest here is to look at how the kids are going to approach this question. I want to see if they can come up with any short-cuts that will speed up the calculations, so they don’t have to cover their entire body in 1cm grid paper and count out each square.
Read more about the activity at his blog.