A Quadratic Trinomial/Factoring Investigation for Algebra I/II

In my Christmas post, I raised the issue of how much time should be spent on factoring quadratic trinomials over the integers in light of the new ADP Standards for Algebra I and II. Hopefully, some of you will provide us with the benefit of your knowledge and experience. I may even make this into a poll or survey to be voted on but, in this post, I will appear to contradict myself and propose an investigation of this topic which requires some effort and time on the part of the student. The target audience would be the regular or accelerated Algebra I/II student.  

We all need to become more creative in the strategic use of time in our classrooms (I still think of myself as being in the classroom!). What are some alternatives to using class time for this? I'll suggest one approach and I'm hoping others will offer their suggestions:


Assign the following as an extra credit or "long-term" project to be due in a week or two. I would not even take classroom time to discuss it. Just hand it out or post it on your website or the department's website if it is to be given to all the Algebra classes. Students can easily download it or print directly if they wish. After they are collected, graded and returned, you may choose to discuss it briefly for about 10 minutes using an overhead transparency, opaque projector or via your computer and a projector. You can also post some  student solutions on the website.


THE INVESTIGATION/PROJECT

[OPTIONAL HINT OR CUE]
The following may require an application of the ac-method learned in class.

(1) Factor the following over the integers and show all steps used in your method of factoring:

(a) 12x² + 27x + 15

(b) 12x² + 28x + 15

(c) 12x² + 29x + 15

(2)
(i) List all positive integers values of b, including the ones from part (a), for which 12x² + bx + 15 is factorable over the integers.
(ii) For each value of b, factor the resulting trinomial.
(ii) How many of these trinomials produce a gcf ≠ 1 for 12, b and 15?

(3) If we knew in advance that 180 has 18 positive integer factors, explain how it follows that there are 9 values for b in part (2).

(4)
(i) If the "12" and "15" were interchanged, explain why this would not change the possible values for b in part (2)?
(ii)For each resulting trinomial such as 15x² + 28x +12, determine its factors and explain how they are related to the factors of the original trinomial (i.e., before interchanging the 12 and 15).


QUESTION FOR OUR DISCUSSION (No, these are not rhetorical! Some are quite knotty)

(1)  What do you see as the benefits of this investigation, if any?

(2) Do the new standards and assessments discourage us from investing time into this type of in-depth problem-solving?

(3) Do you believe this type of assignment should be reserved for the accelerated/honors Algebra I student in 7th or 8th grade or even for the stronger Algebra II student?

(4) With the new ADP Algebra standards, do you believe this type of investigation is reasonable, particularly since it is unlikely that any variation of this would appear on an End of Course Test?

(5) If you were to give this problem, how would you edit the investigation? Parts you would delete or change? Parts you would add?

(6) My goal for this blog has always been to provide you with useful and engaging examples of in-depth problems for your students that require going beyond the mechanical aspects of the course. These problems are developed for this blog -- they do not come from my notes from 30 years ago! Would you be interested in a supplementary resource of such problems for each course you teach? Do you already have one from the publisher or from another source which you really enjoy? Share it!