It's not just he way that you teach or the way that you demonstrate how you think, but also the way that you interact with your students when they ask questions.
I like the idea of using a graph to verify an algebraic result, but to me it should be just that: the graph pictorially verifies what the algebra is trying to show. x^2 = 1 was used to demonstrate the effectiveness of a graphical approach. While I certainly think it is good to think in a graphical setting, a minor tweak to the system reveals a pitfall in thinking primarily in terms of graphs: consider x^2 = -1. A quick graph sketch might have you believe that there are no solutions, but +i and -i are both solutions. The algebra had to be done to find those solutions, and then the resulting graph pictorially verifies this by showing that there were no REAL solutions. In addition, if the student knows that x^2 = -1 has complex solutions and yet sees no intersection, it is not a logical leap to conclude that the linear system as written on the board might ALSO contain complex solutions; as we have seen from the previous graph, no intersection does not imply no solutions! While the geometric picture leaves open this ambiguity, the algebraic argument is quite clear: trying to find a solution leads to a logical contradiction, thus there will be no solutions, not even possibly complex solutions. I think that this is important if students go on to solve problems in higher dimensions or even 1D problems that are sufficiently complicated - while it is great to visualize things when you can, it is not always (I would say rarely) possible to - and in such cases you must "fly by your instruments".
I prefer how my math teacher did it. No offense to Eddie, he's a brilliant mathematical mind on top of being a consummate educator. But in this case I think he could have done it better. She was maybe one of the brilliant few who even taught things to this depth, most secondary school teachers in SG would simply be happy if you knew how to compute the algebra and come up with no solution. But while Eddie conducted this like a lecture, she made us sit in our seats and think. She got people to come up and draw the graphs on the left and right hand side, superimposed on the same chart, and drawn with a measured scale. While maybe only one or two of us were brilliant enough to see this on our own, and maybe only half of us could draw the graphs ourselves without thinking too hard (esp after including the X^2 terms), all of us walked away with this deep understanding. It took a little more time but I think it was well worth the spent time.
It's not just he way that you teach or the way that you demonstrate how you think, but also the way that you interact with your students when they ask questions.
You are a good teacher!
Never understood 'No Solution' until now. Thanks
You should do a collaboration with 3blue1brown
I like the idea of using a graph to verify an algebraic result, but to me it should be just that: the graph pictorially verifies what the algebra is trying to show.
x^2 = 1 was used to demonstrate the effectiveness of a graphical approach. While I certainly think it is good to think in a graphical setting, a minor tweak to the system reveals a pitfall in thinking primarily in terms of graphs:
consider x^2 = -1. A quick graph sketch might have you believe that there are no solutions, but +i and -i are both solutions. The algebra had to be done to find those solutions, and then the resulting graph pictorially verifies this by showing that there were no REAL solutions. In addition, if the student knows that x^2 = -1 has complex solutions and yet sees no intersection, it is not a logical leap to conclude that the linear system as written on the board might ALSO contain complex solutions; as we have seen from the previous graph, no intersection does not imply no solutions!
While the geometric picture leaves open this ambiguity, the algebraic argument is quite clear: trying to find a solution leads to a logical contradiction, thus there will be no solutions, not even possibly complex solutions.
I think that this is important if students go on to solve problems in higher dimensions or even 1D problems that are sufficiently complicated - while it is great to visualize things when you can, it is not always (I would say rarely) possible to - and in such cases you must "fly by your instruments".
Nathaniel Kroeger you know you could’ve written that in a way simpler and shorter way..
Teachers don't teach imaginary numbers at that stage, and so anything not on the graph is irrelevant
Yoav Mor you usually learn about imaginary numbers before learning about parabolas though
Interesting!
This isn't roblox cheats
It’s better
Nope, it's school cheats ;)
I prefer how my math teacher did it. No offense to Eddie, he's a brilliant mathematical mind on top of being a consummate educator. But in this case I think he could have done it better.
She was maybe one of the brilliant few who even taught things to this depth, most secondary school teachers in SG would simply be happy if you knew how to compute the algebra and come up with no solution. But while Eddie conducted this like a lecture, she made us sit in our seats and think. She got people to come up and draw the graphs on the left and right hand side, superimposed on the same chart, and drawn with a measured scale.
While maybe only one or two of us were brilliant enough to see this on our own, and maybe only half of us could draw the graphs ourselves without thinking too hard (esp after including the X^2 terms), all of us walked away with this deep understanding. It took a little more time but I think it was well worth the spent time.
Could the lines intersect in non-Euclidean space?
My guy thought of drawing a non-Euclidean graph