(1) All beagles work at the same rate. (If you understand beagle behavior intimately, you might question this). Seriously, it's the underlying assumption of constant "rate of work" that is so fundamental here.
(2) All holes are the same size.
Well, at least, I didn't ask the classic: "How many eggs can 1.5 hens lay in 1.5 days (my all-time favorite word problem)!
The focus of this post will be on the first two stages of concept development using a concrete numerical example. You may take strong exception to the approach below of combining both direct and inverse variation in the same lesson, but, remember, the goal here is concept development, not proficiency with an algorithm! The algebraic stage will be deferred or left to the reader. The algebraic relationships are extremely important and worthy of extended discussion but that needs to be a separate discussion.
Stage I: Building on Intuition
Before developing a strict mathematical procedure involving direct, inverse or joint variation I feel it is critical for students to trust their "math sense." Encourage this with comments like:
"Forget calculations here, boys and girls, just think about this problem, use commonsense, and you might be able to arrive at the answer in less than 10 seconds!"
Don't think they can? No harm in trying...
I believe that when we tell them to trust their intuition, some will arrive at the correct answer of 5 days. Encourage those who "see" it to share their reasoning: WHY will the number days not change! This will vary according to the ability level and confidence of individuals in the group but, even more importantly, according to the environment you create in the classroom (accepting non-judgmental climate leads to greater risk-taking).
When review of homework, content coverage and time for guided practice (before the assignment is given) are the highest priorities of our lessons, then it is natural to question the wisdom of the above strategy.
This is obvious from typical comments like:
"Very nice, Dave, but who has the time for that, it's not going to be tested on the State Test and, moreover, I'm not teaching gifted kids like you must have had."
I won't react to my own Devil's Advocate arguments. Those you who know my philosophy of education know what my response would be!
Stage II: Beyond Intuition - Developing Proportionality Concepts via a Systematic Approach
"Well, boys and girls, now that we believe the answer is still FIVE days, let's try to approach this more mathematically, that is, more logically and systematically, in case the answer cannot be 'guessed' so easily."
I have found over the years that the following TABLE or matrix approach is a powerful model for devleoping proportionality concepts before the student sees a single algebraic relationship:
|EVERY DOG HAS HIS DAY!|
Note how this approach avoid changing both the number of holes and the number of dogs in the same step! By keeping one quantity fixed, the student may better be able to focus on the relationship between the other two. Thus, in the second row I kept the number of dogs constant, changing only the number of holes:
"Boys and girls, if the number of dogs stays the same and we double the number of holes, then what will happen to the number of days ?"(they will double).
[Note that I asked for the effect on the the number of days before I asked for the actual number of days, namely 10 days.]
This approach develops the idea of direct variation before we express the relationship algebraically: As one quantity increases, so does a second quantity proportionately.
Now that we have filled in the second row (replace the ?? with 10 days), we can move on to another relationship:
"Boys and girls, look at the 3rd row. What quantity (variable) did we not change (keep constant)? What quantity did change? If we double the number of dogs, what should happen to the number of days needed to dig the same number of holes?"
(Yes, some will think 'double', since direct variation is often the initial reaction of many students).
Thus we are literally constructing direct and inverse variation via numerical computation before we develop any general relationships. Yes, this is time-consuming, but hopefully you will see the payoff in comprehension.
Stage III: Expressing Relationships Algebraically
Not in this post!
Normally, we would be very reluctant to mix both types of variation in one lesson, choosing to develop mastery of just direct variation first, then inverse much later on. Yes? Therefore you might feel that combining these will lead to confusion on the part of most students in most classes. Remember, though, the intent here was to develop a strong intuitive base for different types of variations before attempting to formalize any of this! You may not agree, but I'm proposing it anyway. I have done this with good results. Once the concept foundation is laid, students can take off with all the formulas!