Open ended: Creating Arrays

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:

***
***

or

******

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?

Open ended investigation: Mathematical sequences as musical scores

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?