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?
This is a great video of some students sharing their thinking while solving a problem posed by their teacher.
See this blog post where Deirdre Bailey and Amy Park frame the problem posed for the students (as well to get a copy of the text of the problem itself).
(Image credit: Mike Naylor)
Malke Rosenfeld shares yet another great project idea on her blog, which you can read in more detail there. I love the visual she found above, and I think that it, and the program that was used to create it, could be a start to an interesting investigation into number factors, and perhaps other patterns.