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Christopher had his first assignment on translations last night. This was on rotations.
Ed asked me this morning why this is part of the curriculum --- where does it lead?
I don't know the answer!
working with rotations (Regents prep)
from David:
Besides being useful in sketching graphs of functions, transformations lead to the concept of symmetry, which is one of the great unifying ideas in mathematics.
[Catherine here: yes, symmetry is always taught as part of these lessons]
from Steve H:
A translation (simple move) is different than a rotation about some center point. Each is a type of transformation. There are more transformations, like mirroring and scaling. At this stage (and age), they might not seem so important and I can't really see the point, but transformations (using matrices) are very important in geometric modeling and computer graphics.
In looking at the Regents prep, my reaction was why did they have to make a very simple concept so difficult? Actually, it was the notation and terminology that made it so difficult.
In 2 dimensions, it's really simple. A point on the graph paper is defined by two numbers; an X (horizontal) distance from the origin, and a Y (vertical) distance from the origin. A translation gives you two values, one to add to (or subtract from) the X-value of a point, and one to add to (or subtract from) the Y-value of the point. If you don't let the notation throw you, it's very simple.
Rotation transformations require trig, but I have seen problems that require students to do the rotation graphically before they ever get to trig. Simple rotations are always about the origin. You can have rotations about other points, but that really requires a series of concatenated transformations.
I agree with instructivist. They can say that they are teaching transformations, but it's really just graphics arts manipulation. Maybe the student can learn to be a user of a program like Corel Draw, but they can't become the programmer who has to understand, create, and concatenate transformation matrices together. Then they have to know how to apply the transform to a series of points that define a 2- or 3-dimensional geometric model fast enough to allow for dynamic dragging (translation) or rotation on a computer screen.
Schools need to focus on algebra and trig. They need to move right along and keep their eye on the goal. They shouldn't waste time on the formal notation of transformations before they do the real math. Real math, please, not descriptive math. They need to teach the kids to become the programmer and not the user.
from rudbeckia hirta:
This is part of the study of the four rigid symmetries of the plane. You should also be seeing "reflections" and "glide reflections" at some point.
I love this topic because it helps build connections between algebra and geometry and also introduces some of the ideas of analytic geometry. These tranformations can also be studied from a purely algebraic point of view.
Eventually this will be used when studying functions. In order to be able to quickly manipulate functions without a graphing calculator, students need to be familiar with transformations and symmetry.
It's good stuff -- if done well.
Thanks, everyone!
11 comments:
Translations? As in translating graphs or what?
Translations are a common way of figuring out what a graph looks like by starting from a handful of known graphs. Eg,
f(x-pi) is f shifted right by pi.
earl
"Ed asked me this morning why this is part of the curriculum --- where does it lead?"
I see this as part of the trend to have visuals displace numbers. Math without numbers seems to be the goal. Then math will be largely a form of language arts and arts and crafts (visuals), in addition to being affective. OOps! I shouldn't use the phrase "in addition". Sounds too much like doing operations with numbers.
Regarding translations, rotation and translation are distinct.
Translating an object means moving it without rotating or reflecting it.
The image I put up is a translation.
When an object is rotated it's rotated around the origin, or it was in last night's homework.
Besides being useful in sketching graphs of functions, transformations lead to the concept of symmetry, which is one of the great unifying ideas in mathematics.
"working with rotations (Regents prep)"
It was really "Working With Translations"
A translation (simple move) is different than a rotation about some center point. Each is a type of transformation. There are more transformations, like mirroring and scaling. At this stage (and age), they might not seem so important and I can't really see the point, but transformations (using matrices) are very important in geometric modeling and computer graphics.
In looking at the Regents prep, my reaction was why did they have to make a very simple concept so difficult? Actually, it was the notation and terminology that made it so difficult.
In 2 dimensions, it's really simple. A point on the graph paper is defined by two numbers; an X (horizontal) distance from the origin, and a Y (vertical) distance from the origin. A translation gives you two values, one to add to (or subtract from) the X-value of a point, and one to add to (or subtract from) the Y-value of the point. If you don't let the notation throw you, it's very simple.
Rotation transformations require trig, but I have seen problems that require students to do the rotation graphically before they ever get to trig. Simple rotations are always about the origin. You can have rotations about other points, but that really requires a series of concatenated transformations.
I agree with instructivist. They can say that they are teaching transformations, but it's really just graphics arts manipulation. Maybe the student can learn to be a user of a program like Corel Draw, but they can't become the programmer who has to understand, create, and concatenate transformation matrices together. Then they have to know how to apply the transform to a series of points that define a 2- or 3-dimensional geometric model fast enough to allow for dynamic dragging (translation) or rotation on a computer screen.
Schools need to focus on algebra and trig. They need to move right along and keep their eye on the goal. They shouldn't waste time on the formal notation of transformations before they do the real math. Real math, please, not descriptive math. They need to teach the kids to become the programmer and not the user.
This is part of the study of the four rigid symmetries of the plane. You should also be seeing "reflections" and "glide reflections" at some point.
I love this topic because it helps build connections between algebra and geometry and also introduces some of the ideas of analytic geometry. These tranformations can also be studied from a purely algebraic point of view.
Eventually this will be used when studying functions. In order to be able to quickly manipulate functions without a graphing calculator, students need to be familiar with transformations and symmetry.
It's good stuff -- if done well.
Thanks!
I'll get all these comments up front.
My rule of thumb is: if Singapore Math is teaching it, then it should be taught.
(Don't laugh! It works!)
Because Singapore Math teaches all of these topics, I've assumed it's good to learn them.
Also, as I've said more than a few times, "visual" representations are incredibly useful for me, and if they're useful for me they're going to be useful for a lot of students.
Speaking of Singapore Math, SM also teaches nets and we don't seem to.
What do you think about that?
I assume, with all of these visual/geometric topics, that the more the better.
I assume this purely based on findings that people who excel at math are good at doing things like rotating figures in their heads, something I can't do at all -- and something that, like anything else, can be learned with practice.
When I do lessons on translations I can feel my brain expanding. I'm serious. It really is almost like a muscle being used for the first time.
It's cool.
btw, I realize that practicing the skill of rotating figures inside your head may have no causal relationship to the ability to learn and do math whatsoever.
Correlation isn't cause, etc.
Still, I'm not going to bet my money that correlation isn't cause.
"Also, as I've said more than a few times, "visual" representations are incredibly useful for me, and if they're useful for me they're going to be useful for a lot of students."
I love visual representations myself and find them very helful. But I like to see them in support of some mathematical problem, the way, say, SM is doing with bars. My concern is that a type of visual is driving out math more and more. Look at the ISAT tests. You can do no-math visuals and still pass in the lower grades.
My one quibble is with the notion of something "driving out math." While some of the things done in these classes is a waste of time, some of it is real math. Something doesn't have to be computational to be real math. (The canonical example is geometry proofs.)
These geometric transformations are widely applicable in mathematics. Both in school mathematics (such as a pre-calculus course) and in higher mathematics. The symmetries preserving a planar pattern ("isometries") form a group, and we can prove that there are only seven frieze groups and 17 crystallographic groups. While no one is planning on teaching group theory to a seventh-grader, for some technical professionals, these ideas are used more often than calculus.
Just because a bit of mathematics is not an essential prerequisite for calculus doesn't make it worthless. There is a lot of math out there, and a lot of it is worth knowing. We can't teach it all, but it's also a mistake to only teach the math that leads to calculus and to leave out everything else.
Rud, I'm right there with you. And in fact the idea of "there's more to math than the race to calculus" is about all I blog about these days.
As far as I can tell in many cases these topics are never going to get developed. It's like the waqy they throw in Venn diagrams without ever making it to set theory. It seems completely clear that there are many non-calculus topics that can be taught, and taught well, but in many texts they pop up, are never taught in any depth and then disappear under the surface.
What would be very useful would be someone who is interested in teaching something other than calculus in high school to blog and post about resources available to do things like teach abstract algebra, linear algebra, number theory, combinatorics, etc with high school students...oh yes, and proofs in areas other than Euclidean geometry would be awesome. Proofs are considered an illicit substance for anyone under the age of 21 to possess unless it's in the diluted form of derivation.
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