Thursday, December 11, 2008

Instructional Strategy Series for Middle School and Beyond: Developing Direct & Inverse Ratio Concepts

Three beagles can dig 4 holes in five days. How many days will it take 6 beagles to dig 8 holes?

Standard Assumptions

Note: It may be highly instructive to ask the students what natural assumptions (stated below) are being made here before you tell them!
(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.

Instructional Commentary

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:






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!
Important Note:
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!


Paul Carpenter said...

I'd say there are other natural assumptions being made, all amounting to an assumption that beagle quantity and hole digging speed have a linear relation.
It might seems petty at the level this is aimed at but why not also point out the assumption that the beagles are working independently?

Dave Marain said...

I agree with both of your observations. The independence of the dogs is often overlooked. The "linear" relationships are even more important.

I was attempting to build on youngsters' intuitive understanding of these kinds of relationships and I had to make some decisions regarding simplifying the problem for the grade levels involved. In most problems of this type that appears on assessments, the direct or inverse variation is often taken for granted. On a more advanced level, I would have indicated how the variables are related.
Thank you!
Dave Marain

Unknown said...

Interestingly, this exercise can also be used to introduce the students to the concept of 'dog days' :-)


Dave Marain said...

Since we're into puns, tc, just remember Darwin's 2nd Voyage on the Beagle and we all know what 'naturally evolved' from that!

On a more serious note, do you look at the digging problem more as a puzzle or more as a vehicle to develop important concepts? I see it as both. I think there are some fundamental teaching precepts here. The idea of analyzing 3 variables by keeping one fixed is important. What generally happens is that students get to see these later on and they're presented as formulas:
D = kH/B, where D = # of days, H = # of holes and B = # of beagles. In my day this was termed joint variation and the problem would be expressed something like: The number of days varies directly as the number of holes and inversely as the number of digging dogs, etc...

My intent for this post was to demonstrate how to informally introduce to middle schoolers important ideas via engaging problems/puzzles.

Anonymous said...

Good post, Dave.
I agree that developing an intuitive understanding of proportion should come before doing the algebraic formulas. Changing one variable at a time makes a lot of sense. Here is some evidence that focusing on the formulas alone generally doesn’t work. It is a post from what seems to be a physics college instructor explaining what he went through to get his students to understand proportions, at

Dave Marain said...

If these critical concepts are developed early on, high school and college instructors should have an easier time. For years I had to endure comments from my friends in the Science dept like: "Why do we end up having to teach them the math they should have learned from you!" My reply is usually: "Students often do not transfer knowledge well from discipline to discipline. They often do not relate what they learned in science to what I'm teaching them in math! We need to jointly help them to make these connections. Further, our roles are somewhat different. The scientist is generally more interested in math as a means to an end -- a tool. The mathematician is very concerned about using that tool properly: One needs to understand the restrictions on how that tool is to be used or the results may be invalid -- the theory, not just the application.
Both views are necessary for knowledge." Sorry for the rant on science vs. math instruction!

Overall, I'm hoping to identify some of the major conceptual blocks in the foundations of mathematics learning and suggest ways to overcome these. I am fully aware that the problem I posed and the methods offered are just one avenue to direct and inverse ratio. But I do know that it seemed to help many of my students to make sense of these ideas. I didn't begin on Day I of my teaching career knowing this. I had an idea of how to teach based on my understanding of math but I never learned this in a methods course. It took years of experimentation and communication with other educators to refine my techniques. To the end I was still learning how to make it better. Perhaps, most importantly, I learned to stop talking and listen to my students. What good was that perfect lesson that no one understood! This blog is all about networking and sharing ideas that might help the teacher who is going through that same process...

Agai, thanks for the kind words!

Maria Miller said...

I wanted to expand this idea into arbitrary numbers:

Solving direct and inverse variations in chart form.