Each year in the spring my pre-calc kids come to the four brief sections in our text that deal with parametric equations and vectors. (I think of this chapter as the catch-all chapter, anything we might have missed that is in the California or Texas Standards). And then I throw in about three more weeks of work about vectors that I created because I think it is a) Each year in the spring my pre-calc kids come to the four brief sections in our text that deal with parametric equations and vectors. (I think of this chapter as the catch-all chapter, anything we might have missed that is in the California or Texas Standards). And then I throw in about three more weeks of work about vectors that I created because I think it is a)beautiful and b) really important. And I share with my …
Each year in the spring my pre-calc kids come to the four brief sections in our text that deal with parametric equations and vectors. (I think of this chapter as the catch-all chapter, anything we might have missed that is in the California or Texas Standards). And then I throw in about three more weeks of work about vectors that I created because I think it is a) Each year in the spring my pre-calc kids come to the four brief sections in our text that deal with parametric equations and vectors. (I think of this chapter as the catch-all chapter, anything we might have missed that is in the California or Texas Standards). And then I throw in about three more weeks of work about vectors that I created because I think it is a)beautiful and b) really important. And I share with my students that it seems incredible to me that they, the best and brightest mathematics students in our school, are nearing the end of their high school education (many are seniors) and they can’t do one of the most simple acts of coordinate geometry; that is, "Given two points, write the equation of a line containing the two points" (How does the standard for your school read?).
They look at me in wonder, confusion, and perhaps some doubt of my sanity. After all, we have done that thousands of times. I continue to bemoan their lack of ability until finally someone will challenge me...."But, Mr. B, we CAN do that. We do it all the time." Ok, We’ll see, and I turn and write two points on the board... such as (3,1,2) and (2,4,3). They are so sure of their ability that they already have their pencils to paper when they realize they have no idea how to begin. I let them talk, explore, suggest ideas, and I wait, and I wait.... I have never had a student come up with an equation. Some will suggest it must be something like z=ax+by+c or something..... but NOT ONE ever hit upon a correct equation of the line in question.... Then we talk... Not about how to, that will come later, they will discover it on their own as a natural generalization, but about why not.. I admit to them that most of the students who graduate from high school (and in fact, many of the math teachers they have studied under) can not do this simple act of writing the equation of a line in the three space dimension that they live in. And I tell them that in the following weeks they will learn to do some of the simple geometry they know in the dimension they live in.
I walk them through a simple vector approach to lines in the coordinate plane. We take y= 3x-1 and rewrite it as (x,y)= (0,-1) + t(1,3). Within minutes every kid in the class can write the equation of a line given two points in this vector form although a few struggle with the seeming reversal of order of the "slope"(in truth, several still mess up regularly when they try to use slope intercept, yet they seem reluctant to adopt the seemingly easier point-slope form). They are quickly taking two points and talking about "point vectors" and "slope vectors" as if they had used them forever. And each year it startles me anew that after a half-hour of an alternate approach, every student will intuitively generalize the method to produce a three-space equation of a line without any help...and then with a little faltering over the "fourth" variable, they can do the same thing in the barely imaginable four-space. Later we will write the equations of planes in space given three points and do some simple analytic geometry in three space. Many of them struggle with the idea of projections of lines and minor details, but I at least feel like I have made a small step to preparing them to function mathematically in the three-space they live in. And if the string-theory guys are right, and we really have a ten-dimensional universe.... no big deal, they can extend vectors to any dimension. But I wonder each year... why are we not introducing this more at an early age (alg I?). I will talk later about some of the advantages I see, and maybe you can tell me what I missing that would make it a bad idea. (Actually in the last few years I’ve been working a unit in parametric equations and vectors into my Alg II classes with lots of Bowditch curves (Lissajous?) and three D lines and planes and three D geometry ideas.)
I got this comment from one friend...
Dave L. Renfro said...
Here’s something marginally related to your topic. In 4 dimensional space it is possible for two planes to have exactly one point in common. A very simple example is the xy-plane, namely all points of the form (x,y,0,0) where x and y vary over the real numbers, and the zw-plane, namely all points of the form (0,0,z,w) where z and w vary over the real numbers. It is easy to see that these two planes have only the origin (0,0,0,0) in common, since x = y = z = w = 0 is the only solution to (x,y,0,0) = (0,0,z,w).
For what it’s worth, I first learned about this possibility in a "side-bar diagram" in the following calculus book, which I was trying to work through during the 1974-75 school year (I managed to cover about the first 1/4 of the book), when I was in the 10th grade. Incidentally, the first college math class I took (an ODE course, about a year later) was from one of the authors (Embry) of this text.
Embry/Schell/Thomas, "Calculus and Linear Algebra. An Integrated Approach" (1972)
At this point you might try exploring other possible intersections between two four spaces, intersecting at a single point other than (0,0,0,0) , or a in a line, or a 3-plane. Send me your best ideas, especially for sharing this with students.
This was all back in about 2009 and I retired two tears later and never followed up on this idea with a class, but I have some thoughts and will share them by adding to this post as time goes on, so check back...and share your ideas> I will pass them along as I go.
I did come up with one more idea I wish I had thought of while I was teaching. Somewhere in their HS math education most students who stay at it will learn to solve systems of equations and understand how to use matrices to find the point where three planes intersect IF they intersect in a single point. (Thanks Dave!)
Planes in Space- part One
Take your basic bright kid in alg II or pre-calc, or often in calculus, and ask, "What is the intersection of two lines?"... They say, "A point."... good answer.
"Can you write the equation of a line?" Again they are on target.
"If I give you the equation of two lines on the plane; can you find their point of intersection?" The good ones can, and know they can.
Now ask the same bright kids, "What is the intersection of two planes in space?". They answer correctly again, "A line."
So far everything is great, but now we ask them to write the equation of a plane.... uhhh... gee..... and at this point, when asked about one of the fundamental structures of plane geometry, their analytic geometry skills are exhausted.
My experience was that after solving systems of three equations in three unknowns, they remembered how to do it (mostly) but didn’t know what it was they had done????
but thankfully after a brief review a light goes on, and they understand.
Still, a very few may actually be able to produce x+y+z=1 or some other for the equation of a plane. Now we ask about the intersection of two planes, and almost none of them can do it. The scary part, is that very (very) few of the teachers of alg II and above that I have questioned about this could provide an answer either .
I would begin by recalling that an equation in three variables, such as 2x+3y+z=6 can represent a plane in space. When students had three such equations that intersected in a unique point, they found the solution by one of several methods. Most students learn to solve such equations by the methods called elimination and substitution at the very least. Others may have also been introduced to Cramer’s rule for solving systems with determinants and perhaps two methods using matrices.
The most commonly taught matrix method is to write a matrix equation and then solve it using the inverse matrix method. A second, and as I would point out, more efficient and general method is the Gauss-Jordan reduced row-echelon form (RREF) of an augmented matrix.
We begin with three planes determined by the equations {x + y – 2z = 9; 2x – 3y + z = -2; and x + 3y + z = 2} This same system of equations can be expressed as the matrix equation.
Notice that the left matrix is made up of the coefficients of the three variable terms in each equation, and the right matrix contains the constant terms. We can find the intersection by taking the inverse of the left matrix and multiplying on the left of both sides of the equation. The simplified result gives
This seems to be the most commonly taught method, and the one that students and teachers seem to prefer, and yet it has two major disadvantages. The first disadvantage is that it tells you little or nothing about systems which have a solution, but not a single unique solution. In fact, it seems most students (or teachers?) can not distinguish between the cases (and there are several different ones) with no solutions from the ones with an infinite number of solutons. This same defect applies to attempts to use Cramers Rule. The second problem is that the inverse method is more computationally complex, that is, it takes more operations for the solution than the alternative RREF method, and the difference grows as problems reach higher orders of magnitude. For the problems that are generally assigned at the high school level, the difference in computability presents no real problem, but the difference in the range of applicable questions can be very significant in a students understanding of general systems of three equations.
In contrast with the Inverse method that will only work if the three planes intersect in a single point, the RREF form will allow us to work with systems which do not even have the same number of equations as unknowns. This is the type of situation created when we try to find the line of intersection of two planes.
RREF for two planes
So let’s talk about the situation where two plains intersect in a point.
We will use the equations 2x + 3y – 3z = 14 and –3x + y + 10z = -32. When we write an augmented matrix for the system of only two equations we get a 2x4 matrix, shown here:
When we reduce this system, by matrices or otherwise, we get
which is a matrix expression of x - 3z = 10 and y + z = -2.
Well, what can we do with that to help us find the line of intersection..... baby steps.... let’s find one point on the intersection.
We notice that both equations contain a z variable, it might occur to us to ask, “What happens if we substitute different values in for z?”. For example, if we try z=0 we note that from the first equation we get x=10 and from the second we get y=-2. What does this tell us about the point (10, -2, 0). If we check it against the two original equations we notice that the point makes both equations true, 2(10) + 3(-2) – 3(0) = 14 and –3(10) + (-2) + 10(0) = -32. So the point (10, -2, 0) is on both planes and therefore must lie on the line that is their intersection.
Can we find more points? What happens if we try z=1 or z=2 or other values. Using z=1 we get x – 3(1) = 10 which simplifies to x=13 ; and y +(1) = -2 which simplifies to y=-3. Checking the point (13, -3, 1) we see that it also makes both equations true, and so it must also be on the line of intersection.
And now we have come full circle, and we are back to the starting point, and now they really write the equation of a line through two points, in pretty much and dimension. but what surprises may jump out in 4-D????? (cue dramatic music)
––(More to come)
BEWARE
Four Space Ahead
Addendum.... I first wrote this post around 2003 to 2005 (yeah, I write slowly.) and retired from formal education in 2011. By now I am sure that all the suffering through New Math, the Agenda for Action, and and Common Core, that the sad state of graduating HS seniors is no longer in the mess my generation left it, so please forgive me if this all seems unbelievably ancient to you. Think of it as a reminder of how far Math has come on the shoulders of a New generation of scholar educators. and b) really important. And I share with my students that it seems incredible to me that they, the best and brightest mathematics students in our school, are nearing the end of their high school education (many are seniors) and they can’t do one of the most simple acts of coordinate geometry; that is, "Given two points, write the equation of a line containing the two points" (How does the standard for your school read?).
They look at me in wonder, confusion, and perhaps some doubt of my sanity. After all, we have done that thousands of times. I continue to bemoan their lack of ability until finally someone will challenge me...."But, Mr. B, we CAN do that. We do it all the time." Ok, We’ll see, and I turn and write two points on the board... such as (3,1,2) and (2,4,3). They are so sure of their ability that they already have their pencils to paper when they realize they have no idea how to begin. I let them talk, explore, suggest ideas, and I wait, and I wait.... I have never had a student come up with an equation. Some will suggest it must be something like z=ax+by+c or something..... but NOT ONE ever hit upon a correct equation of the line in question.... Then we talk... Not about how to, that will come later, they will discover it on their own as a natural generalization, but about why not.. I admit to them that most of the students who graduate from high school (and in fact, many of the math teachers they have studied under) can not do this simple act of writing the equation of a line in the three space dimension that they live in. And I tell them that in the following weeks they will learn to do some of the simple geometry they know in the dimension they live in.
I walk them through a simple vector approach to lines in the coordinate plane. We take y= 3x-1 and rewrite it as (x,y)= (0,-1) + t(1,3). Within minutes every kid in the class can write the equation of a line given two points in this vector form although a few struggle with the seeming reversal of order of the "slope"(in truth, several still mess up regularly when they try to use slope intercept, yet they seem reluctant to adopt the seemingly easier point-slope form). They are quickly taking two points and talking about "point vectors" and "slope vectors" as if they had used them forever. And each year it startles me anew that after a half-hour of an alternate approach, every student will intuitively generalize the method to produce a three-space equation of a line without any help...and then with a little faltering over the "fourth" variable, they can do the same thing in the barely imaginable four-space. Later we will write the equations of planes in space given three points and do some simple analytic geometry in three space. Many of them struggle with the idea of projections of lines and minor details, but I at least feel like I have made a small step to preparing them to function mathematically in the three-space they live in. And if the string-theory guys are right, and we really have a ten-dimensional universe.... no big deal, they can extend vectors to any dimension. But I wonder each year... why are we not introducing this more at an early age (alg I?). I will talk later about some of the advantages I see, and maybe you can tell me what I missing that would make it a bad idea. (Actually in the last few years I’ve been working a unit in parametric equations and vectors into my Alg II classes with lots of Bowditch curves (Lissajous?) and three D lines and planes and three D geometry ideas.)
I got this comment from one friend...
Here’s something marginally related to your topic. In 4 dimensional space it is possible for two planes to have exactly one point in common. A very simple example is the xy-plane, namely all points of the form (x,y,0,0) where x and y vary over the real numbers, and the zw-plane, namely all points of the form (0,0,z,w) where z and w vary over the real numbers. It is easy to see that these two planes have only the origin (0,0,0,0) in common, since x = y = z = w = 0 is the only solution to (x,y,0,0) = (0,0,z,w).
For what it’s worth, I first learned about this possibility in a "side-bar diagram" in the following calculus book, which I was trying to work through during the 1974-75 school year (I managed to cover about the first 1/4 of the book), when I was in the 10th grade. Incidentally, the first college math class I took (an ODE course, about a year later) was from one of the authors (Embry) of this text.
Embry/Schell/Thomas, "Calculus and Linear Algebra. An Integrated Approach" (1972)
At this point you might try exploring other possible intersections between two four spaces, intersecting at a single point other than (0,0,0,0) , or a in a line, or a 3-plane. Send me your best ideas, especially for sharing this with students.
This was all back in about 2009 and I retired two tears later and never followed up on this idea with a class, but I have some thoughts and will share them by adding to this post as time goes on, so check back...and share your ideas> I will pass them along as I go.
I did come up with one more idea I wish I had thought of while I was teaching. Somewhere in their HS math education most students who stay at it will learn to solve systems of equations and understand how to use matrices to find the point where three planes intersect IF they intersect in a single point. (Thanks Dave!)
Planes in Space- part One
Take your basic bright kid in alg II or pre-calc, or often in calculus, and ask, "What is the intersection of two lines?"... They say, "A point."... good answer.
"Can you write the equation of a line?" Again they are on target. "If I give you the equation of two lines on the plane; can you find their point of intersection?" The good ones can, and know they can.
Now ask the same bright kids, "What is the intersection of two planes in space?". They answer correctly again, "A line."
So far everything is great, but now we ask them to write the equation of a plane.... uhhh... gee..... and at this point, when asked about one of the fundamental structures of plane geometry, their analytic geometry skills are exhausted.
My experience was that after solving systems of three equations in three unknowns, they remembered how to do it (mostly) but didn’t know what it was they had done????
but thankfully after a brief review a light goes on, and they understand.
Still, a very few may actually be able to produce x+y+z=1 or some other for the equation of a plane. Now we ask about the intersection of two planes, and almost none of them can do it. The scary part, is that very (very) few of the teachers of alg II and above that I have questioned about this could provide an answer either .
I would begin by recalling that an equation in three variables, such as 2x+3y+z=6 can represent a plane in space. When students had three such equations that intersected in a unique point, they found the solution by one of several methods. Most students learn to solve such equations by the methods called elimination and substitution at the very least. Others may have also been introduced to Cramer’s rule for solving systems with determinants and perhaps two methods using matrices.
The most commonly taught matrix method is to write a matrix equation and then solve it using the inverse matrix method. A second, and as I would point out, more efficient and general method is the Gauss-Jordan reduced row-echelon form (RREF) of an augmented matrix.
We begin with three planes determined by the equations {x + y – 2z = 9; 2x – 3y + z = -2; and x + 3y + z = 2} This same system of equations can be expressed as the matrix equation.
Notice that the left matrix is made up of the coefficients of the three variable terms in each equation, and the right matrix contains the constant terms. We can find the intersection by taking the inverse of the left matrix and multiplying on the left of both sides of the equation. The simplified result gives
This seems to be the most commonly taught method, and the one that students and teachers seem to prefer, and yet it has two major disadvantages. The first disadvantage is that it tells you little or nothing about systems which have a solution, but not a single unique solution. In fact, it seems most students (or teachers?) can not distinguish between the cases (and there are several different ones) with no solutions from the ones with an infinite number of solutons. This same defect applies to attempts to use Cramers Rule. The second problem is that the inverse method is more computationally complex, that is, it takes more operations for the solution than the alternative RREF method, and the difference grows as problems reach higher orders of magnitude. For the problems that are generally assigned at the high school level, the difference in computability presents no real problem, but the difference in the range of applicable questions can be very significant in a students understanding of general systems of three equations.
In contrast with the Inverse method that will only work if the three planes intersect in a single point, the RREF form will allow us to work with systems which do not even have the same number of equations as unknowns. This is the type of situation created when we try to find the line of intersection of two planes.
RREF for two planes
**So let’s talk about the situation where two plains intersect in a point. **
We will use the equations 2x + 3y – 3z = 14 and –3x + y + 10z = -32. When we write an augmented matrix for the system of only two equations we get a 2x4 matrix, shown here:
When we reduce this system, by matrices or otherwise, we get
which is a matrix expression of x - 3z = 10 and y + z = -2.
Well, what can we do with that to help us find the line of intersection..... baby steps.... let’s find one point on the intersection.
We notice that both equations contain a z variable, it might occur to us to ask, “What happens if we substitute different values in for z?”. For example, if we try z=0 we note that from the first equation we get x=10 and from the second we get y=-2. What does this tell us about the point (10, -2, 0). If we check it against the two original equations we notice that the point makes both equations true, 2(10) + 3(-2) – 3(0) = 14 and –3(10) + (-2) + 10(0) = -32. So the point (10, -2, 0) is on both planes and therefore must lie on the line that is their intersection.
Can we find more points? What happens if we try z=1 or z=2 or other values. Using z=1 we get x – 3(1) = 10 which simplifies to x=13 ; and y +(1) = -2 which simplifies to y=-3. Checking the point (13, -3, 1) we see that it also makes both equations true, and so it must also be on the line of intersection.
And now we have come full circle, and we are back to the starting point, and now they really write the equation of a line through two points, in pretty much and dimension. but what surprises may jump out in 4-D????? (cue dramatic music)
––(More to come)
BEWARE
Four Space Ahead
Addendum.... I first wrote this post around 2003 to 2005 (yeah, I write slowly.) and retired from formal education in 2011. By now I am sure that all the suffering through New Math, the Agenda for Action, and and Common Core, that the sad state of graduating HS seniors is no longer in the mess my generation left it, so please forgive me if this all seems unbelievably ancient to you. Think of it as a reminder of how far Math has come on the shoulders of a New generation of scholar educators.