This post by Michael Pershan has an excellent way to give students opportunities to play around with exponents and problem solve with them.
The basic idea is, give students a bunch of dots in an array, and ask them to find ways to change the picture so that it represents a power.
A related, and more generic, question is:
How do I change this diagram/image/equation/formula so that it represents <something else>?
Use the numbers 2 and 3 only once (but any combination of mathematical symbols you know), make the largest possible number you can. Make the smallest possible number you can. Make a number as close to zero as you can.
Bonus, use any numbers you like, and then look for patterns between the numbers. In general, which collection of operations will produce the largest number for any two given initial numbers?
Thanks to Michael Pershan for the idea.
A Fermi problem is an estimation problem, where the objective is to answer a question that requires a number of assumptions be made, and for which the final answer can usually be best given as a range of possible values.
Some examples of Fermi problems are:
Could the forests of North America serve as the lungs of the world?
How many phone books are delivered in NYC?
How many piano tuners are there in the United States?
How many different civilizations like ours exist in our galaxy?
An activity that students could do would be to brainstorm their own Fermi problems, and then choose one of the problems to attempt to solve in a small group.
Here is a sample outline for what a project about facial recognition algorithms might look like.
- Investigation phase:
What is automated facial recognition? How does it work? What does a computer do to look at two different photos of someone and determine if they are the same person? What algorithms are in use? How successful are these algorithms? Where is facial recognition used?
- Problem posing:
What factors affect the success of automated facial recognition? How efficient are these algorithms? What are the problems with these algorithms?
- Share results
This question comes from Professor Wright via Twitter:
When Professor Wright means factor, he means, be written in the form (dx + e)(fx + g) with d, e, f, and g all being rational numbers.
Here are some questions to extend this investigation:
How often does ax2+bx+c factor with rational values of a, b, and c? Does the range of numbers we choose to look at matter? What if we allow d, e, f, and g to be irrational numbers? How often does this factor if a is 1? What if a is 2? Is there a relationship between the value of a, and the likelihood the quadratic function factors? Can you write a computer program to test out your hypothesis?
In this project, students attempt to create all of the possible array representations for each of the numbers smaller than a given number(which may depend on the amount of time they have to devote to this project).
For example 6 = 2 x 3 and 6 = 1 x 6, so two possible arrays for 6 are:
The objective after they have these arrays is to classify the numbers into groups based on whatever patterns they see in the structure of the arrays.
What observations do you think students might have when doing this project?
A number is called perfect if the sum of the proper divisors is equal to the number. 8 is not a perfect number because the proper divisors of 8 are 1, 2, and 4, and 1 + 2 + 4 = 7, which is not 8.
Can you find a perfect number? Can you find all of the perfect numbers less than 1000? Could you write a program to find all of the perfect numbers smaller than a million? Is there a largest possible perfect number?
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?