And vs. Or: Developing Deeper Understanding in Algebra

While some are awaiting the remaining chapters in Alec Klein's interview, I know the rest of my readers are wondering why I've gone into early hibernation. I'm actually doing some independent mathematical research apart from anything on this blog but I will not bore you with the details at this time. That may change (i.e., I may bore you later!).

In the meantime, since the central theme of this blog has always been developing student conceptual understanding, here are a couple of problems for you to consider giving to your Algebra 2 students (or beyond). Conjunction vs. Disjunction is often misunderstood by students and these ideas appear in so many contexts in mathematics, from absolute values to inequalities and beyond. Consider giving these as warm-ups, for review, practice for SAT's, etc. I'm not suggesting these are difficult or challenging problems. Their purpose is to promote deeper reflection on the part of the student. Students who have strong background and understanding will simply solve these quickly and not see why anyone would make a big deal over them. However, you may find other students who don't grasp the ideas as readily or have forgotten. Comparing/contrasting is a powerful heuristic when trying to develop a more profound understanding of mathematics...

1. If (a -4)² + (b+4)² = 0, what is the least possible value of a² + b²?

2. If (a-4)(b+4) = 0 , what is the least possible value of a² + b²?
(A) 0 (B) 8 (C) 16 (D) 32 (E) 64

Ask students to explain to each other, why the word 'least' is irrelevant in Question 1 but not in Question 2. Also, how does question 1 relate to circles?

Understanding Algebra Conceptually?

What do you think would be the results of giving the following Algebra 1 problems to your students before, during, and after the course?
Do you believe that either or both of these could be or have been SAT questions?
Do students normally have exposure to these kinds of problems in their regular assignments?
Do these kinds of questions require a deeper conceptual understanding of algebra?

1. Given: x² - 9 = 0
Which of the following must be true?

I. x = 3
II. x = -3
III. x² = 9



(A) I only (B) I, II only (C) I, III only (D) I, II, III

(E) none of the preceding answers is correct


2. Given: (a - b) (a² - b²) = 0
Which of the following must be true?

I. a = b
II. a = -b
III. a² = b²


(A) I only (B) I, II only (C) I, III only (D) III only (E) I, II, III