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
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10 comments:
What's with the South Pacific?
Find 3 consecutive integers such that twice the largest is three more than the sum of the first two...
or such that the square of the middle number is one more than the product of the first and last...
but those make a different point, don't they?
Jonathan
On the contrary, Jonathan, your examples are exactly what I intended! Fantastic!
Even though you are demonstrating cleverly worded "identities", the point is the same. We must provide a richer environment for our students where posed questions and sample problems deal with more profound ideas.
My example dealt with a comparison of the number of variables with the number of relationships among them, but, in both of our samples, we are exposing students to a deeper view of modeling problem situations mathematically.
The issue for most people reading this is:
WHEN DO I INTRODUCE THESE 'EXCEPTIONAL' PROBLEMS? AFTER THEY DEMONSTRATE MASTERY OF THE MECHANICS OF EQUATION-SOLVING OR SOLVING SYSTEMS?
My response is that EVERY lesson can be made even richer than it is currently. By combining skills, procedures and higher-order examples we will train our students to think more conceptually and learn to question/challenge simply stated rules. This applies to ALL students not just the Honors classes!
In fact, I would argue that teaching exceptional cases is crucial to developing conceptual understanding! The oft-quoted cliche, "It's the exception that proves the rule" can be modified to:
IT'S THE EXCEPTION THAT MAKES THE MATHEMATICS MEANINGFUL!
Thank you, Jonathan! I miss these exchanges...
Here are some more ways to develop your students' intuition:
1) Take the problem in the original post and take the first equation therein. Graph it; it's a line. Then, take the second equation and graph it. It's another line.
Now, when do two lines meet? When they aren't parallel, and then you get only one solution. What else can happen?
The lines might be parallel, and then one has no solutions. The coefficients do not have to be equal however; one equation may be a multiple of the other. What else can happen? The lines might be identical, and then there's a line worth of solutions. Note that the case of no solutions and of a line's worth of solutions are much closer to each other than to the single solution case.
2) Now, do this for problems with three variables. Each equation leads to a plane, two planes generically intersect in a line, and a line and plane generically intersect in a point. Then, think about what else can happen.
3) This is taken from "Concrete Mathematics", by Graham, Knuth, and Patashnik, I think. Suppose you have a pizza, and you want to make slices through it and count how many pieces you can get with n cuts. With no cuts, you get 1 piece. With one cut, you get 2 pieces. With 2 cuts, you get 4 pieces. With n cuts, what do you get?
No, the answer is not 2^n pieces.Suppose n lines give p(n) pieces. Add line n+1. It intersects each line in a single point, generically. Draw the line and the n points of intersection. There are now 2 rays and n-1 line segments each of which is slicing through one of the p(n) pieces. So, p(n+1) = p(n) + (n+1). Find p(3): p(3) = p(2) + 3 = p(1) + 2 + 3 = p(0) + 1 + 2 + 3 = 7. Now use induction to show that p(n) = nC2 + 1.
Actually, before you start analyzing this, ask the students to guess the type of function p will be. Explain that if p is linear, you'll be getting a fixed number of new pieces with each additional line. Is that realistic? If p is exponential, you would expect each line to slice every region in two, even though it's likely to miss most of them.
Now, for advanced students, ask how many times a line can hit a conic. Try all three kinds. Then, ask how many times two conics can intersect. Note that the answer is different for two circles than for two ellipses. Why, circles are a little two special and symmetrical. Work your way to Bezout's theorem.
Eric,
I know you probably contribute to many blogs but what you add to this blog is inestimable. It isn't only your breadth and depth of knowledge that leaves me in awe. It's your ability to communicate complex ideas in a way that even I can understand. Your students were very fortunate and i can't help but think that you miss that interaction as much as i do.
The "cuts in the circle" problem is famous and still entices students and teachers alike. I'm not sure about your formula however. I get (n+1)C2 + 1 for the maximum number of "pieces" with n cuts. Would you check that for me?
Oh dear. You're right. p(n) = n(n+1)/2 + 1 = 1 + 2 + 3 + ... + n + 1.
My eighth graders in honors Algebra 1 would love this problem. We primarily focused on systems in two variables, but this could be a good challenge for them. They are very inquisitive. They have asked me how to graph something in three variables, so they would probably find this problem very interesting. Especially, if I relate it back to graphing, like Eric suggests. Thank you for the idea!
jopie--
The inseparable relationships among the verbal (the statement of the word problem), the numerical (plugging in values for one of the variables or guessing a solution), the algebraic system and the visualization (graphing of the resulting lines or planes suggested by Eric) is one of the "big ideas" to which I was alluding. I have found that my students could see the bigger picture so much clearer when I used multiple representations aka "The Rule of Four."
Let me know how your 8th graders react to the problem if you are able to give it in class. I would also strongly recommend Jonathan's "word problem identities." They should prove intriguing for students!
I would also add that this "basic" problem can even be used at more advanced levels. The variable y can be thought of as a parameter. The current and future ages can be represented parametrically. For example, if we allowed ages to be non-integers, we can let y = 1/3 yr, then x = 1 yr and the current ages are 1 and 5. In one-third of a year, the ages would be 4/3 and 16/3 --- there's your 4 to 1 ratio! For your 8th graders, they can also be introduced to functions here since all four ages can be thought of as functions of the variable y if we so choose.
I would also be interested in a non-honors class reaction to this word problem. I would definitely encourage students in both classes to "guess" some possible solutions first. This process does not waste time IMO. It often helps students to "see" and then formulate the algebraic relationships. The idera that there can be many solutions to the age problem should not be that surprising to students. They should come to see that there simply wasn't sufficient data to guarantee a unique solution.
By the way, after is said and done, I can't escape the truth that Mrs. Hill's method taught to me over 50 years ago is simple and effective and students will still be using it for another 50 or 100 years at least!
Dave,
You're right. It is n(n+1)/2 + 1. Darn!
One thing you should watch out for in algebra courses like this or in more advanced courses like multivariable calculus is that there are some techniques that work almost by accident for problems with two variables, but give the wrong generalization to more complex problems.
Once, I was teaching multivariable calculus, and the textbook I was forced to use gave a particularly misleading criterion on extrema of doubly-differentiable functions of two variables: 1) find the critical points, find the Hessian, check that is determinant is positive, and look at the upper-left entry. It's positive at minima, and negative at maxima.
Unfortunately, this is precisely the wrong thing to generalize from. The real question is, is the Hessian positive or negative definite. I told my students that they should burn that portion of their book, and I taught the more useful result.
Eric,
You're raising an interesting pedagogical issue.
Knowing that a theorem, rule, method, etc., is valid in two dimensions but not in general for n dimensions, is it acceptable to have students learn the easily applied special case for now as long as you tell them that for higher dimensions the criteria for max, min, saddle is more sophisticated, requiring more machinery from linear algebra. If the students already have a good understanding of conjugate transpose and related topics then you could develop the higher-dimensional case afterwards but I personally wouldn't be comfortable starting off with that. OR did I miss your point?
Dave, that's what I meant. One can state a correct result, but provide a shortcut to understanding it that hurts the student's understanding. I disin't want to cripple my students' mathematical development by simply accepting the book's 2-dimensional focus. The book did state that the techniques wrer invalid for higher dimensions, but it said nothing about the real motivations and results. Additionally, at Stony Brook then, Multi=variable calculus followed linear algebra in the syllabus. Students could eb expected to know about eigenvalues and definiteness, although only one student in my class had heard of Sylvester's Law of Inertia.
Take the geometry and linear algebra problem we were originally discussing. If the teacher isn't careful, the student will think that any linear equation is associated to a line in the ambient space. One equation, one dimension.
In fact, the teacher should point out that a linear equation leads (generically) to a codimension 1 space. In dimension 2, that's a line. In dimension 3, that's a plane. In dimension 4, that's a 3-space.
Incidentally, anyone who immediately refers to 'time' as the fourth dimension should be told that one can think about spaces and dimensions without relating them to physical space, and even in relativity theory, the association of 'time' to a single direction is precisely wrong.
Back to geometry, one then should point out that k dimensions leads (generically) to a codeminsion k space. Two equations on 2-space lead to a point, 2 equations in 3-space lead to a line, and 2 equations in 4-space lead to a plane.
And, where could those counting arguments go wrong? Redundant equations, equations that have 'common intersections', and 'parallel' equations. As I suggested once, a discussion of projective geometry may prove illuminating.
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