Forgive the lame attempt at "punnery" in the title but I hope you will see the many layers in the following discussion about a 'typical' word problem involving people's ages. Perhaps, not so typical but I hope it will be topical!
Please bear in mind that this blog is geared essentially toward the K-12 math teacher. Most of the problems I write are not intended to stump people. Rather they are vehicles to engage a discussion of instructional practice as we move toward a leaner, deeper and more coherent curriculum in this country.
To reiterate a point made many times on this blog:
Most common teacher reaction to these problems and investigations:
"Dave, These are nice but you know we don't have time for this, particularly at the secondary level. What exactly do you want me to delete from the curriculum!"
My Response: "That is the point of why we need to reduce the number of topics covered and alter our paradigms of math education. The underlying ideas in these problems and explorations include some of the essential ideas of mathematics, algebra in particular. Who am I to make such a pretentious claim? Well, it isn't pretentious at all! Everyone knows that most students go through the motions of learning, concentrating solely on what they believe will be tested. They have little interest in anything else (yes, there are exceptions!). If assessments focus on procedures and superficial ideas, then there is no reason for students to make the effort to actually use their 'reasoning brain'!"
As teachers of mathematics we need to know and understand what the "big ideas" of math are before we can hope to convey this to our students. In the problem below, the essential idea is NOT how to solve a contrived word problem.
I. One of the big ideas here is of course to develop a problem-solving approach which can be applied to many problems.
II. Secondly, the important notion of comparing the number of unknowns (variables) to the number of relationships (equations in this case). If the number of variables exceeds the number of relationships, then we should NOT expect a unique solution to the problem. If the number of equations exceeds the number of variables then there might not be solution at all because of inconsistent conditions. These are huge themes in using algebra to model the real world. How many variables are involved in weather forecasting!
Mr. Oldman is currently five times as old as his great-grandson. In how many years will he be four times as old as his great-grandson will be? Assume ages are positive integers.
Fairly straightforward algebra problem? Easy to guess the "answer" like Mr. Oldman is currently 75 and the young man is 15. In five years, the ages will 80 and 20. Voila! You know in upper elementary or middle or high school there will always be some student who "guesses" these values and sits there complacently with the "I'm done!" look on her face.
But there is so much more here for us to discuss with the students. A much bigger picture with deeper concepts than guessing an answer or setting up algebraic equations.
I will get the ball rolling and leave the rest to my astute readers whose insights always surpass mine...
Why did my hs algebra teacher, Mrs. Hill, always require a chart or table for these problems? Even when I didn't feel I needed it, I had to use it. Decades later, I still do out of force of habit and I've been carrying on her legacy with my own students ever since! Rather than import some html table codes, I will resort to brute force for displaying the table. Pls forgive any problems with the display on your screen.
............................NOW .............. FUTURE (y yrs from now)
MR. OLDMAN......... 5x ................ 5x + y
YOUNG MAN......... x ................... x + y
Equation: 5x + y = 4(x + y) → x = 3y
Of course, I could have used different variables or more variables and produced more equations but there is an essential truth underlying all of this. As math people and from our experience we could sense from the beginning that there would be no unique solution to this problem. But how do we develop this conceptual understanding in our students? Some students I believe would intuit this, but many might not.
There's no earth-shattering revelation here! But IMO every student of algebra should experience this and engage in discussion about it.
Ok, teachers of algebra in middle or secondary grades -- how would you develop this into a meaningful structured lesson and, at the end, assess the learning which hopefully took place?
- Simply show them how to make the chart, set up the equations, solve and note that there is more than one solution?
- OR would you allow the students to discover this for themselves? Would you begin by asking them to intuitively guess if there is exactly one solution in positive integers, more than one or none?
- What questions can you compose ahead of time? What questions can you anticipate coming from the students?
- Can we make the problem into more a puzzle-solving experience. Do you believe the word "puzzle" evokes a different visceral reaction from "word problem"?
"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)
"You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught."
--from South Pacific