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?).
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?
One student expressed this as 500/1000 and 0.500. I assumed he was just extending the pattern(s). “Yeahbut where do you see the 500 and 1000?” I asked challenged. “I imagine that inside every one of these *points to a dot* there is one of these *holds up a full ten-frame*,” he explained. As his teacher and I listened to his ideas, our jaws hit the floor.
Showing 500/1000 or 0.500 using ten-frames
Have you seen examples where students come up with innovative ways of representing numbers?
Here are some questions you could use with the game.
Try solving the problems, if possible, in a variety of different ways. Which ways give you the most points? How much is each piece worth? What is a general strategy you can use to get as many points as possible? How do you know your general strategy is effective?
Note: This game does not currently work in Internet Explorer.
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?