Open-ended problem: How likely is it a quadratic equation will factor?

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?

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:




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?