Once again, it’s been days since I’ve gotten access to the blog. But I still manage to get things done. Conics—the in-depth look at the aspects of a cone. It sounds like I’m writing a feature article on it or something. I sort of enjoyed this unit, it had its moments. It was pretty reminiscent of the days when I was master paper folder in origami club. By the end of emphasizing each shape that evolved from a cone, I simply wondered, “what happened to just making cranes?” I also commend Mr. K for his innovative [yet deadly] ways of disarticulating a cone with a shward.
This may have been a short unit but I really have to admit that I learned a lot. I didn’t know that so many shapes could come from a cone, or two cones to be precise. A “hypercone”, just kidding… The proper term of course would be the double-napped cone that gives way to a hyperbola, a parabola on steroids. Sarcasm, my native tongue.
The class learned each of the formulas and also learned how to graph the formulas. Call me juvenile, but I still have trouble with the grade 12 version of the parabola. I think the 4p always throws me off. I also feel like using the other formula from grade 11 though. But using that idea again should be avoided since thinking would not be lateral. This is why practice makes perfect. I just need to do a little adapting. Otherwise, I’d think I’m ready for the test today. Just keep my hocus pocus my focus, look it’s a locus.
Showing posts with label Conics. Show all posts
Showing posts with label Conics. Show all posts
Thursday, May 22, 2008
BOB: Conics
WOAAAHH!! I almost forgot to do my BOB again! Sorry this is really really late XD
Okay, this unit was one of my favorite units. I actually enjoyed graphing for this unit and I thought that the concepts were very simple. It was a good break from the Counting unit, which I didn't really like much. I also enjoyed doing all that folding during class, such easy homework! This unit was also full of laughs, we really couldn't seem to stay serious most of the time. As for the parts that I found the hardest for this unit, I would probably have to say that the hyperbola and ellipse were definitely the most difficult, although not by much. I got confused at first with how to find the foci and the asymptotes (hyperbola) but then it all made sense after some explaining. I also remember messing up graphing my vertical hyperbolas since I switched around the transverse and conjugate axis. All in all, tis was a good unit, now I got to get ready for school! Good luck on the test guys :P
Okay, this unit was one of my favorite units. I actually enjoyed graphing for this unit and I thought that the concepts were very simple. It was a good break from the Counting unit, which I didn't really like much. I also enjoyed doing all that folding during class, such easy homework! This unit was also full of laughs, we really couldn't seem to stay serious most of the time. As for the parts that I found the hardest for this unit, I would probably have to say that the hyperbola and ellipse were definitely the most difficult, although not by much. I got confused at first with how to find the foci and the asymptotes (hyperbola) but then it all made sense after some explaining. I also remember messing up graphing my vertical hyperbolas since I switched around the transverse and conjugate axis. All in all, tis was a good unit, now I got to get ready for school! Good luck on the test guys :P
Wednesday, May 21, 2008
BOB on Conics
Hi guys this is Richard and here is my bob on the unit of conics.
This unit was not as hard as the other units there were some parts that i got really easy like the ellipse. I probably found that this topic was easiest because i had to scribe for it or it is just the simpler out of the conic shapes. Another thing that was cool was that we got to do a lot of fun folding. That led to the geometry of the conic shapes.
There were also topics that i found difficult for example the parabola. I think that the main reason i don't understand it is because i was not present for that Pre cal class. I almost forgot that the graphing of the parabola is also hard. i also felt the same was that Francis felt on that Pre test. I was also puzzled by the transforming parabola question . Well that's just me ah ha
Richard signing off.
This unit was not as hard as the other units there were some parts that i got really easy like the ellipse. I probably found that this topic was easiest because i had to scribe for it or it is just the simpler out of the conic shapes. Another thing that was cool was that we got to do a lot of fun folding. That led to the geometry of the conic shapes.
There were also topics that i found difficult for example the parabola. I think that the main reason i don't understand it is because i was not present for that Pre cal class. I almost forgot that the graphing of the parabola is also hard. i also felt the same was that Francis felt on that Pre test. I was also puzzled by the transforming parabola question . Well that's just me ah ha
Richard signing off.
Conics Bob
Ahhh Conics. Too bad Mr.K didn't show us any more of his samurai skills with this metre-stick-katana.
To the point. I will admit, like every section in this course, I had troubles in one way or another. The origami was cool, as it went with the whole Japanese theme Mr.K was trying to capture (TOYOTA-HONDA-MITSUBISHI-MAZDA-YOKOHAMA-SONY-etc...) Like every other type of graphing, I don't like to graph. Granted it does give me a better look at whats happening and I can tell what's happening, but like everything else in life, just because you know how to do it doesn't mean you're supposed to like it. Like graphing.
I liked how parabolas, circles, ellipses and hyperbolas had their distinct way to tell each other apart without having to think hard but rather just look at what's switching to make it graph that way. But, that's probably about it. The math was pretty simple once I took the time to sit down and study it all, like I am tonight.
I didn't like word problems however, how they make it seem like it's supposed to be something it's not, but yeah, I know, I'm just over complicating it.
I will be deep in study for a while because well, exams are closing in surprisingly fast. OH and don't forget about your DEV projects.
Rence ~ Out
To the point. I will admit, like every section in this course, I had troubles in one way or another. The origami was cool, as it went with the whole Japanese theme Mr.K was trying to capture (TOYOTA-HONDA-MITSUBISHI-MAZDA-YOKOHAMA-SONY-etc...) Like every other type of graphing, I don't like to graph. Granted it does give me a better look at whats happening and I can tell what's happening, but like everything else in life, just because you know how to do it doesn't mean you're supposed to like it. Like graphing.
I liked how parabolas, circles, ellipses and hyperbolas had their distinct way to tell each other apart without having to think hard but rather just look at what's switching to make it graph that way. But, that's probably about it. The math was pretty simple once I took the time to sit down and study it all, like I am tonight.
I didn't like word problems however, how they make it seem like it's supposed to be something it's not, but yeah, I know, I'm just over complicating it.
I will be deep in study for a while because well, exams are closing in surprisingly fast. OH and don't forget about your DEV projects.
Rence ~ Out
BOB on Conics
This unit was quite short, but I'm not celebrating. Personally, I need a bit more time with this unit, only because I don't think we went far enough with transformations that can happen with each conic shape. On the pre-test I saw that one question where the graph of a hyperbola was stretched 4 units, and moved down 1 unit, I was sitting there staring at the page for like 5 minutes, forgot all about those transformations. Plus, I don't think I can remember all those equations such as the equation of a circle, ellipse, hyperbola and the new parabola equation. Hopefully we'll get a formula sheet, or some other type of aid. On top of that, I don't fully understand the total anatomy of all these conic shapes, such as the major, minor, semi-major, semi-minor, conjugate, and transverse axis', and some, I suppose. I still don't exactly know about graphing these guys either. Other than that, I really enjoyed folding the paper, being interactive in a class is always a plus.
Good luck on the test everyone.
Until next time,
-Francis
Good luck on the test everyone.
Until next time,
-Francis
Bobbing on Shwards and Things
So, the infamous conic sections unit was completed late last week, and this would be a bob for said unit.
Where to begin? the good? the bad? the ugly? Lets start with the ugly
THE UGLY *dun dun dun*
-GRAPHS! Yes I'm terrible with graphs but I am getting better at them, thanks to a certain "perspective." More on that later though.
The Bad
Not to much here this time, thankfully.
-One of the most difficult parts of this unit is how most of the equations, have another equation which is nearly identical. This made memorizing/remembering which was which difficult at times.
-Going from standard to general form, and remembering which is which was often a pain.
The Good
-Short unit
-Easily understood by used of "visual" tricks (what I meant in the ugly section. By thinking of what the graph of some equation looks like, the question is often made much easier.
-Overall the concepts in this unit, and the difficultly level of the questions was not very demanding.
So yeah, there is it, my small and not very impressive bob post. :P Now I'm back to studying haha.
Ciao!
Where to begin? the good? the bad? the ugly? Lets start with the ugly
THE UGLY *dun dun dun*
-GRAPHS! Yes I'm terrible with graphs but I am getting better at them, thanks to a certain "perspective." More on that later though.
The Bad
Not to much here this time, thankfully.
-One of the most difficult parts of this unit is how most of the equations, have another equation which is nearly identical. This made memorizing/remembering which was which difficult at times.
-Going from standard to general form, and remembering which is which was often a pain.
The Good
-Short unit
-Easily understood by used of "visual" tricks (what I meant in the ugly section. By thinking of what the graph of some equation looks like, the question is often made much easier.
-Overall the concepts in this unit, and the difficultly level of the questions was not very demanding.
So yeah, there is it, my small and not very impressive bob post. :P Now I'm back to studying haha.
Ciao!
BOB For Conics
This was a short unit that I was hardly there for because of the music trip to Moosejaw, which was fun, buut now it's time to get back to school and all that piled up work that I have to catch up on. Whooo.
Missing classes for a long period of time has never been a good thing, especially if it's your weakest, and most especially if it's pre-calculus. But some things can't be helped, I think. Well, either way I get in trouble soo..
The concepts were all there for me. Probably the hardest part of this unit was breaking down the question and figuring out which numbers I could use to make an equation. Of course graphing was hard (when is it not hard), but after you have that image, it makes everything so much easier. Everything could've probably been more straight forward if I was actually there (I'm not much of a self-learner when it comes to math) and that's just my fault I guess. Honestly, I'm not looking forward to this test, but I don't know, I think I've done what I can. Hope for the best I guess.
Hahah, as if I'm the very last one to BOB (I think so anyways), but better late than never. Good luck everyone.
Missing classes for a long period of time has never been a good thing, especially if it's your weakest, and most especially if it's pre-calculus. But some things can't be helped, I think. Well, either way I get in trouble soo..
The concepts were all there for me. Probably the hardest part of this unit was breaking down the question and figuring out which numbers I could use to make an equation. Of course graphing was hard (when is it not hard), but after you have that image, it makes everything so much easier. Everything could've probably been more straight forward if I was actually there (I'm not much of a self-learner when it comes to math) and that's just my fault I guess. Honestly, I'm not looking forward to this test, but I don't know, I think I've done what I can. Hope for the best I guess.
Hahah, as if I'm the very last one to BOB (I think so anyways), but better late than never. Good luck everyone.
Tuesday, May 20, 2008
BOB: Conics
Time to Bob! (:
I'm quite happy that this unit was pretty short but I'm not going lie because truth be told, I did have some trouble in a few areas. The most trouble I had in this unit was graphing the parabola and I have no idea why I find that difficult compared to the others. The circle, ellipse and the hyperbola were probably the easiest because it was sort of similar to each other. I suppose I had trouble with the parabola because the formula seemed much different compared to others because it included 4p. Hopefully, with a bit more practice I'll feel much more comfortable graphing them. (:
The best part of the unit was folding the paper and visualizing the geometry in each conic section. I would probably have a harder time understanding the whole unit if Mr. K just explained each conic section without folding the paper. Though it took a lot of time and made me quite frustrated beacause the paper wouldn't fold the way I wanted it to, it helped me a lot to understand the anatomy of each conic.
Overall, this unit was pretty straight forward and hopefully I'll do better on the test on Thursday than the pretest we had today. *cross fingers and toes* Goodluck to everyone! Ciao! (:
PS. school's almost over! yay!!
I'm quite happy that this unit was pretty short but I'm not going lie because truth be told, I did have some trouble in a few areas. The most trouble I had in this unit was graphing the parabola and I have no idea why I find that difficult compared to the others. The circle, ellipse and the hyperbola were probably the easiest because it was sort of similar to each other. I suppose I had trouble with the parabola because the formula seemed much different compared to others because it included 4p. Hopefully, with a bit more practice I'll feel much more comfortable graphing them. (:
The best part of the unit was folding the paper and visualizing the geometry in each conic section. I would probably have a harder time understanding the whole unit if Mr. K just explained each conic section without folding the paper. Though it took a lot of time and made me quite frustrated beacause the paper wouldn't fold the way I wanted it to, it helped me a lot to understand the anatomy of each conic.
Overall, this unit was pretty straight forward and hopefully I'll do better on the test on Thursday than the pretest we had today. *cross fingers and toes* Goodluck to everyone! Ciao! (:
PS. school's almost over! yay!!
Conic Workshops and Pre-Tests
This will be basic and short as I don't really have any time to spare.
Our first class consisted of
Our first class consisted of
- A workshop on Conic word problems
Our Second class consisted of a Pre-Test.
Our first class was pretty straight forward and Mr. K just put us in groups to solve a number of word problems. When we started on the first one. Some of us had an answer, but we didn't put it up, whether it was right or wrong. Mr. K pointed out that we were afraid to get out our Ideas because most of us, if not, all, are afraid of getting the wrong answer, and we'll be "bad people". He also pointed out that we shouldn't be scared in getting our idea's out, because it wouldn't make us "bad people" and if everyone was always right, we wouldn't need to attend the class. Because how else do you learn? By making mistakes. So take that into consideration.
Anyways, we worked on a few questions but we sometimes made it a little complicated by looking at it wrong. Remember Mr.K's block of wood. Look at it in different perspectives. I won't really go into detail, and you can more or less pick your parts and pieces from the slides.
In the afternoon class, we had a Pre-Test, and again, the answers are posted up on the slides. Our test is on THURSDAY. Do recall that Mark has kindly posted Links for us to brush up on our Conics. I know I'm gonna hit those links up later, so you should too :) Mr. K also has practice exam's and exam information in his LINKS section (because you know, the exam is in a bout two weeks roughly) Sorry this is short and not into detail but I have to go. Kthxbye!
Oh right.. Scribe.. The next scribe shall be... ROXANNE :)
BOB for CONICS
Here is my BOB for the unit Conic Sections.
What more can I say than .. "I've missed too many classes" ..
Although I've missed probably more than half the classes for this unit, it really seems like a fun unit. It seems like a unit that involves lots of algebra and equation handling. Anyways I'm hoping to participate in this last couple of days in this math class. Good luck to everyone! Especially me. I've read the blog, but it might not be enough. Hope I can keep up.
- This is Elven, cheers.
What more can I say than .. "I've missed too many classes" ..
Although I've missed probably more than half the classes for this unit, it really seems like a fun unit. It seems like a unit that involves lots of algebra and equation handling. Anyways I'm hoping to participate in this last couple of days in this math class. Good luck to everyone! Especially me. I've read the blog, but it might not be enough. Hope I can keep up.
- This is Elven, cheers.
Monday, May 19, 2008
Conics quizzes
Well, since i heard that the test for this unit is coming very soon I decided to compile a bunch of links that will help all of us to study, so without further ado here it is:
Link 1
Link 2
Link 3
Link 4
Link 5
Link 6
Link 7
Link 8
I hope that those links will be very useful to all of us on the upcoming test and especially the provincial exams which is only 20 days away. Leave a feedback if any of those links doesn't work.
-m@rk
Link 1
Link 2
Link 3
Link 4
Link 5
Link 6
Link 7
Link 8
I hope that those links will be very useful to all of us on the upcoming test and especially the provincial exams which is only 20 days away. Leave a feedback if any of those links doesn't work.
-m@rk
Bob for Conics
*Sigh* What a nice time to have a long weekend. Now I can bob early.
Anyways, this unit was really fun! Not to mention it was also short and simple. What really helped make me understand this unit more was the folding exercises with the ellipse, parabola, and hyperbola. Yes it really did help I'm not just making my bob extra fluffy this time XD. Folding was great for homework assignments too! I usually have trouble with anything that deals with graphing but not in this case. I found it really simple and *coughs* I enjoyed doing them. But what I'm worried about is mixing up all the standard formulas for the ellipse and hyperbolas because they're so similar. Other than that there wasn't anything that troubled me surprisingly and I'm really confident with this unit! Hopefully I'll still feel the same way about this unit after we get through word problems.
Anyways, this unit was really fun! Not to mention it was also short and simple. What really helped make me understand this unit more was the folding exercises with the ellipse, parabola, and hyperbola. Yes it really did help I'm not just making my bob extra fluffy this time XD. Folding was great for homework assignments too! I usually have trouble with anything that deals with graphing but not in this case. I found it really simple and *coughs* I enjoyed doing them. But what I'm worried about is mixing up all the standard formulas for the ellipse and hyperbolas because they're so similar. Other than that there wasn't anything that troubled me surprisingly and I'm really confident with this unit! Hopefully I'll still feel the same way about this unit after we get through word problems.
Sunday, May 18, 2008
BOB Version 6: Conics
Haha I like Ben's creativity... Unfortunately, time isn't a luxury for me.
Anyways, this unit was nice and short, even though the unit had an emphasis on graphing. Once I was able to visualize the graph in my mind's eye--the expression that Mr.K likes using--graphing the graph made things a lot easier, including the origami portions of the unit as well.
Along with remembering the terms, remembering the equations of the circle, ellipse, parabola, and hyperbola is a bit of a doozy, but mathematics is the science of patterns, and contrasting and comparing the formulas between the horizontal and vertical orientations really helped.
I really liked the shward fiasco, the horrible dubbing by DJ K, and his whole samurai act. We may have been fooling around excessively during that class--like what Benz says--but that's what gives this unit its uniqueness. (And maybe there's going to be more unique classes like that when we enter AP Calculus.)
There were some questions left unanswered though, like how golf is evidence that aliens visited our planet Earth and how the focus of the circle is infinity, but if time really isn't a luxury for us, then I guess it's understandable, seeing that THE PROVINCIAL EXAM IS ONLY 14 SCHOOL DAYS AWAY.
If I were to rate this unit, I would give it 5 stars, for being nice, short, unique, and fun.
P.S. Vote ZEPH for V.P.
Anyways, this unit was nice and short, even though the unit had an emphasis on graphing. Once I was able to visualize the graph in my mind's eye--the expression that Mr.K likes using--graphing the graph made things a lot easier, including the origami portions of the unit as well.
Along with remembering the terms, remembering the equations of the circle, ellipse, parabola, and hyperbola is a bit of a doozy, but mathematics is the science of patterns, and contrasting and comparing the formulas between the horizontal and vertical orientations really helped.
I really liked the shward fiasco, the horrible dubbing by DJ K, and his whole samurai act. We may have been fooling around excessively during that class--like what Benz says--but that's what gives this unit its uniqueness. (And maybe there's going to be more unique classes like that when we enter AP Calculus.)
There were some questions left unanswered though, like how golf is evidence that aliens visited our planet Earth and how the focus of the circle is infinity, but if time really isn't a luxury for us, then I guess it's understandable, seeing that THE PROVINCIAL EXAM IS ONLY 14 SCHOOL DAYS AWAY.
If I were to rate this unit, I would give it 5 stars, for being nice, short, unique, and fun.
P.S. Vote ZEPH for V.P.
Conically BOBING
The Conics Unit was pretty simple. The thing that really helped me in this unit was actually visualizing the geometry of each conic section. I started to see how each formula was related to the sections. Overall I'm doing fine so far in the unit. Just like what happened during the quiz. The circles and ellipses were easier to work with than the parabolas. I guess it was because that the
circle formula didn't change at all and that the ellipse formula
was very similar to the circle formula. The parabola
formula was a bit of a change. The a became 4p
and there are 2 binomials now. But it is like
what they all say, "Practice makes
perfect!" The Hyperbola
is still pretty new to
me but what I
noticed
was
that it
used a lot
of our older
analytic geometry
skills to graph the section.
It used the linear graphing skills
as well as many others. Now for my
thoughts about the class. I'm sorry if I am
being a bit mean but I would like it if we stayed
on task a bit more. It just seems like we were fooling
around quite a bit excessively, especially during the second
period. We would get a lot more done if we stayed on task. I don't
mind if we do have a little fun. But it would be great if we toned it down
just a tad. That's all I really have to say. I'm looking forward to the DEVs and
the exam. I'm not really nervous about it at all. As you can see I'm having some fun with my BOB. I am also looking forward to the word problems with conic sections as well as the test that is coming up. So until next time good luck on the test and YAY Calculus is going to work out. Calculus class of the future. Bye and TABBERNACK
circle formula didn't change at all and that the ellipse formula
was very similar to the circle formula. The parabola
formula was a bit of a change. The a became 4p
and there are 2 binomials now. But it is like
what they all say, "Practice makes
perfect!" The Hyperbola
is still pretty new to
me but what I
noticed
was
that it
used a lot
of our older
analytic geometry
skills to graph the section.
It used the linear graphing skills
as well as many others. Now for my
thoughts about the class. I'm sorry if I am
being a bit mean but I would like it if we stayed
on task a bit more. It just seems like we were fooling
around quite a bit excessively, especially during the second
period. We would get a lot more done if we stayed on task. I don't
mind if we do have a little fun. But it would be great if we toned it down
just a tad. That's all I really have to say. I'm looking forward to the DEVs and
the exam. I'm not really nervous about it at all. As you can see I'm having some fun with my BOB. I am also looking forward to the word problems with conic sections as well as the test that is coming up. So until next time good luck on the test and YAY Calculus is going to work out. Calculus class of the future. Bye and TABBERNACK
Saturday, May 17, 2008
WORKING WITH A(N) HYPERBOLA - CONCLUSION [beware shameless advertising.]
Sorry the post is late, but I guess this is why I needed the weekend. Thank you Justus. Nice doodle there, by the way. I don’t think I can incorporate images of the slides in my post, but I’ll still try to use some form of an outline and I’ll also parody a certain [we]blogging method. Yes, I said PARODY. I thank Richard for this.
On Friday’s class, our goal was to conclude our lesson about the hyperbola. We leaed to: rn
On Friday’s class, our goal was to conclude our lesson about the hyperbola. We leaed to: rn
- Write the standard forms of a hyperbola [horizontal and vertical]
- Find the similarities and differences between vertical and horizontal hyperbolas
- Find the Pythagorean theorem property in hyperbolas
- Finally, apply everything we’ve learned by graphing and finding the coordinates required in every hyperbola.
- As well as that, we also had a sort of English epiphany, but we’ll discuss that later.
STANDARD FORMS
On Thursday, the standard formulas of a hyperbola were:
HORIZONTAL: [[(x – h)2 / a2] - [(y - h)2 / b2]]
VERTICAL: [[(y – h)2 / a2] - [(x - h)2 / b2]]
As soon as the class first caught a glimpse of the equations, we wondered why it was the same as the ellipse, but as we examined closely, there were differences as well as there were similarities between them.
SIMILARITIES & DIFFERENCES BETWEEN HYPERBOLAS
SIMILARITIES:
- they have the same terms
- they both equal 1
- both have the same denominators
- both have square binomials
- (h, k) indicates centre
DIFFERENCES:
- numerators have different orientation
- numerators have opposite signs
We also emphasized the differences between the ellipse and the hyperbola by cloning the slides from the ellipse and altered them. We rewrote the formulas and redrew the diagrams, interactively of course. There was even a point when Mr. K was going to redraw a diagram that was on the other page. For the sake of elegancy that rubbed off on the class, they argued for him to clone it, cut it, paste it and shrink it…[that’s sounded like I was singing a new version of “Technologic” or something haha DAFT PUNK man. All the way.] The point is “Just Shrink It”, man. That should be the new Nike slogan. Jeebers, I shouldn’t call out on Mr. K, it’s not even June yet. By the way, it was deemed necessary to rotate the “O” in the graph.
Anyhow, we then realized that a2 [pronunciation: “ay squared”] could be smaller than b2 [bee squared] after seeing the animations that proved so. These diagrams revealed what would happen if there were variations in the lengths of a [semi-transverse] and b [semi-conjugate]. There were also additional animations for the variation of points h [ey-ch] and k [kay]. “h” moves center and hyperbolas horizontally because it is an “x” coordinate. “k” moves center and hyperbolas vertically since it is a “y” coordinate.
The class then asked themselves this question: if a2 can be greater than b2, how will a vertical graph be distinguished from a horizontal graph? The answer lies in this statement:
If the “x” coordinate is positive and the “y” coordinate is negative [y being subtracted from x], then the hyperbolas are HORIZONTAL. If the “y” coordinate happens to be positive and the “y” coordinate negative [x subtracted from y], then graphically, the hyperbola is VERTICAL.
PYTHAGOREAN PROPERTY OF A(N) HYPERBOLA
This is where the English lesson came in. Mr. K insisted that it was “an” hyperbola since using an in a consonant was true in “an hour”. We realized that the English language is at times illogical and contradictory. If all of the rules were followed, we would either choose to have the silent letters or disregard them in the words: “talk”, “milk” and “walk”. Imagine what the Hulk would sound like. OMG Edward Norton as the new hulk. Unexpected, but I'm okay with it. By the way, you don't want to make Mr. K mad. He'll turn green [environmentally friendly?]
I'm such a subliminal message. But let’s not derail the train of thought here. The formation of a(n) hyperbola on a graph depends on a rectangular box, which forms a right angle triangle inside. The legs of the right triangle are composed of:
semi-conjugate + semi-transverse = c [the hypotenuse also equal to the length of the center (O) to a foci point.]
simply, this is where the Pythagorean [pie-tha-gore-eeiy-an] theorem is applied; a2 + b2 = c2
APPLICATIONS OF SKILLS ACQUIRED ON FRIDAY BY SCHWINGING THE SCHWARD.
Then we spent the remainder of the class solving a series of hyperbolas in two different perspectives: graphically and symbolically.
FOR EXAMPLE ON SLIDE 7:
To start we find the standard form by multiplying by (1/9) and (1/25) on each side
leaving us with [(x2 / 9) - (y2 / 25)] = 1
From this equation, we can derive the rest:
Center (0, 0) because nothing is being added to either of the x or y coordinates.
TRANSVERSE AXIS is found by finding the square root of 9 (a2) and multiplying by 2 (2a), in this case 2a = 6
CONJUGATE AXIS is found by finding the square root of 25 (b2) which is 5 (b) and then multiplying by 2. 2b = 10
VERTICES are the distances from the center to the vertex of both parabolas in the hyperbola. It helps if you know the length of the semi-transverse axis the coordinates of the vertices are (3, 0) and (-3, 0)
To find the FOCI, we start by recalling that they are the same distance away from the center as the hypotenuse of the right angle triangle formed by the semi-conjugate and semi-transverse axis serving as the legs of the triangle. Therefore we use a2 + b2 = c2.
In this case, 9 + 25 = c2
c2 = 34
Find the root of that and add/subtract from the x coordinate of the centre because this is a horizontal hyperbola, resulting in (root 34, 0) and (-root 34, 0)
Finally the asymptotes that help graph the two parabolas in a(n) hyperbola, it is important to recall grade 10 pre-cal where we spent time learning about oblique lines and how we find the slope. y = mx + b or simply "rise over run" and use a corner of the "box" made, meaning that we use the two legs of the triangle which happen to be the length of the semi-transverse axis and semi-conjugate axis. Rise 5, run 3 in both directions.
ASYMPTOTE = (5/3) and -(5/3)
Sorry if the post is messed up, but the library computers are timed... As for the next scribe.. I can't doodle, but nevertheless I dub with my very own schward the next scribe Rence.
PS. [post scribe, hahah] I know I said I was a hopeless apolitic when it comes to school elections and that I'd rather not vote. But I still encourage all of you to vote JOSEPH vice-prez.
Friday, May 16, 2008
Thursday, May 15, 2008
Hyperbolae Unveiled...Sorta
Alright guys, no shenanigans this time, I'm tired, just got off work, and would like to sleep. So I apologize if I'm a little bit frank, and a little less funny an animated in my scribe post compared to my usual ones. Hopefully the quality is still there though, if not, I'll fix it up tomorrow or w/e. Alrighty, here we go.
*note* links added under appropriate images for full sized goodness. Like I mean, it wont matter on the smartboard, since that things huge, but at home I doubt everyone has like 70 inch moniters, so ja. Thats why there there. Why arent they hotlinked or w/e its called? Cause its 1:24am haha.
Morning Class
So this mornings class wasn't much of anything special, which is kinda different from the norm (kinda paradoxical I know.) Anyways we started going through Mr. K's slides, and we solved the little equation for one of the friendly Ellipses which can be seen on the following slide.
As always, fully more large sized versions of these slides can be found in the slides themselves. Which Mr.K posts quite consistently :P
Moving on. After that we filled out a health survey. That basically took up the rest of the morning class. However, we did have a moment to do a quiz, which was marked the following afternoon class. If you were not there *cough cough* then talk to Mr.K, see if you can write it. The solutions are on the slides. If you were here, then you don't really need an explanation right? Cool.
Moving on.
Afternoon Class
The afternoon class was slightly more work oriented, and was not interrupted by some unnecessary forms/paperwork.
We began by correcting the quiz thinger from the first period.
Some usefull things to take out of that include the following.
How to distinguish Various conics from their equations.
-If the Equation has an x2 term but no y2 term it is a Horizontal Parabola
-If the Equation has an y2 term but no x2 term it is a Vertical Parabola
-If the Equation has an y2 AND an x2 term, with coefficients that are THE SAME, it is a Circle.
-If the Equation has an y2 AND an x2 term, with coefficients that are DIFFERENT, it is an Ellipse.
Remember those :)
Next we took out our previous nights homework (that being measuring, and finding the difference, of the lengths from one point on a branch of the hyperbola, to both foci. That will make more sense when the image comes up in a moment.)
The results of taking up that homework are shown as follows.
Okay, I'm going to try to explain this as best as I can, but like I said before, I just got off work, I'm crazy tired, not running on anywhere near enough sleep and a host of other not so great things. So if I phail (yes thats a "ph" fail. That doesn't mean an acidic fail but rather, means a really bad fail for all you en zero zero be's out there.)
So, the homework was to pick a point on one of the branches of the hyperbola, and measure that distance from the point, to both foci (that is to say, measure from say, point P, to F1, and then from point P to F2). We then too those values, and found the difference, (PF1-PF2). Voila! homework done. By now it should have come as no surprise that the result were values which should have been basically the same.
Once we did that, things started to get tricky. The next thing we did, was find the Vertices (sp?) of the hyperbola. To do this, you take your ruler, and line it up so it goes through both focus points. The POINTS at which the line made by the ruler intersects with the curve of the hyperbola make up it's vertices.
Okay, so if you followed my last paragraph's directions correctly, you should end up with two points at what are the vertices of the hyperbola. The next step is to connect those two points with your handy dandy ultra straight ruler. This creates what we called the TRANSVERSE AXIS. This axis can be described as the line which connects the two vertices of a hyperbola, and is seen AND labeled in green on the slide crop up there. Got it? I hope so. Moving along.
Next, we bisected this transverse axis.
"How do we do that Mr.k?"
"Well I'll show you."
Thats a rough translation of what probably everyone was thinking. Except the Ogre 1 and 2 of our class. They knew the answer already, (Outside joke.)
So to bisect the transverse axis, we folded our papers for one more time. This time, we folded one focus point, directly on top of the other. Thus we obtained a crease which defined not only the bisector, but the perpendicular Bisector of the Transverse Axis. We see this as a vertical red line on the image above. Lets step things up a notch (I know Mr. K did.)
So, do you remember the Transverse axis? and how we just cut it in two? Were going to take this line, and modify it a little bit. Using our rulers again, we measured the line from one of the FOCI (in this case F2) to that vertical line (which we now call the CONJUGATE AXIS.) This gave us some length.
Now this next bit is tricky, we took the length we measured from the PREVIOUS STEP (that being from the conjugate axis, to Focal point F2) and moved it so that one end of it was on the (whatever the singular of vertices is. Verticii?) of the nearest branch, and so that the other end of the length lies on the Conjugate axis. Since I bet your saying "what the frig" I made a quick little paint mockup to get the point across, or rather, to help you "see" what I mean.
http://i24.photobucket.com/albums/c21/skyL193/Slide3.jpg
Okay, so as I wrote on the picture, the light green is the length Conjugate Axis to F2. Then by moving it like I mentioned before, we get the makings of a triangle.
Step 3. Measure the distance from the point where the light green line meets the conjugate axis, to where the dark green axis meets the conjugate axis. This we can call length b for explanations sake. The next step is to take this length, and go from the intersection of the Transverse axis and the conjugate axis, (aka the cross in the centre) and "b" units down. I'll show you what I mean again.
http://i24.photobucket.com/albums/c21/skyL193/Slide4.jpg
So, as I wrote again, the light blue, and dark blue lines are the ones I was talking about before, and they are of equal length. Thus we get those points G1 and G2. This brings us to our next step.
By drawing perfectly 90 degree lines from the vertices (going through them, vertically) and from the end points of the light and dark blue lines (going through points G1 and G2, horizontally) we can construct a box of sorts. This is seen once again on the slides.
http://i24.photobucket.com/albums/c21/skyL193/Slide5.jpg
So you see the box right? The black dotted lines there are actually it :P So yeah after that we took our ruler again, and drew that orange cross that goes far off the page kinda. We drew that cross by going from one corner of the box to the other (and beyond.) After we had done this Mr.K revealed to us that those orange lines, are actually asymptotes for the graph. In other words, neither branch of the hyperbola will actually ever touch those orange crossey lines.
The final slide here, is the equation of the hyperbola, along with some similarities, and differences, which we came up with together, to help us remember the equation, and keep it distinct from the others.
http://i24.photobucket.com/albums/c21/skyL193/FinalSlide.jpg
To Conclude: The basic parts within the anatomy of a hyperbola
- Transverse Axis.
- Conjugate Axis
- Focus Points
- The Asymptotes
- And finally, you should know the equation, which is shown in the above slide image.
Alrighty, I think that about wraps this blog post up. I'm not entirely sure, because at 1:00am I kinda start to get a bit loopy and my thoughts aren't so coherent (thats probably evident based on how my writing progressed.) If you guys have any questions or whatever lemme know. I'll try to answer them, and post em up. Or if you find any errors, or know something I dont/forgot to add, lemme know so I can add it.
Well I'm off, but of course, not without letting you know who the next scribe poster is :)
http://i24.photobucket.com/albums/c21/skyL193/JamieSlide.jpg
Remember, vote for Zeph for VP.
K? Kthxbai
Justus- FINALLY out.
*note* links added under appropriate images for full sized goodness. Like I mean, it wont matter on the smartboard, since that things huge, but at home I doubt everyone has like 70 inch moniters, so ja. Thats why there there. Why arent they hotlinked or w/e its called? Cause its 1:24am haha.
Morning Class
So this mornings class wasn't much of anything special, which is kinda different from the norm (kinda paradoxical I know.) Anyways we started going through Mr. K's slides, and we solved the little equation for one of the friendly Ellipses which can be seen on the following slide.
As always, fully more large sized versions of these slides can be found in the slides themselves. Which Mr.K posts quite consistently :PMoving on. After that we filled out a health survey. That basically took up the rest of the morning class. However, we did have a moment to do a quiz, which was marked the following afternoon class. If you were not there *cough cough* then talk to Mr.K, see if you can write it. The solutions are on the slides. If you were here, then you don't really need an explanation right? Cool.
Moving on.
Afternoon Class
The afternoon class was slightly more work oriented, and was not interrupted by some unnecessary forms/paperwork.
We began by correcting the quiz thinger from the first period.
Some usefull things to take out of that include the following.
How to distinguish Various conics from their equations.
-If the Equation has an x2 term but no y2 term it is a Horizontal Parabola
-If the Equation has an y2 term but no x2 term it is a Vertical Parabola
-If the Equation has an y2 AND an x2 term, with coefficients that are THE SAME, it is a Circle.
-If the Equation has an y2 AND an x2 term, with coefficients that are DIFFERENT, it is an Ellipse.
Remember those :)
Next we took out our previous nights homework (that being measuring, and finding the difference, of the lengths from one point on a branch of the hyperbola, to both foci. That will make more sense when the image comes up in a moment.)
The results of taking up that homework are shown as follows.
Okay, I'm going to try to explain this as best as I can, but like I said before, I just got off work, I'm crazy tired, not running on anywhere near enough sleep and a host of other not so great things. So if I phail (yes thats a "ph" fail. That doesn't mean an acidic fail but rather, means a really bad fail for all you en zero zero be's out there.)So, the homework was to pick a point on one of the branches of the hyperbola, and measure that distance from the point, to both foci (that is to say, measure from say, point P, to F1, and then from point P to F2). We then too those values, and found the difference, (PF1-PF2). Voila! homework done. By now it should have come as no surprise that the result were values which should have been basically the same.
Once we did that, things started to get tricky. The next thing we did, was find the Vertices (sp?) of the hyperbola. To do this, you take your ruler, and line it up so it goes through both focus points. The POINTS at which the line made by the ruler intersects with the curve of the hyperbola make up it's vertices.
Okay, so if you followed my last paragraph's directions correctly, you should end up with two points at what are the vertices of the hyperbola. The next step is to connect those two points with your handy dandy ultra straight ruler. This creates what we called the TRANSVERSE AXIS. This axis can be described as the line which connects the two vertices of a hyperbola, and is seen AND labeled in green on the slide crop up there. Got it? I hope so. Moving along.Next, we bisected this transverse axis.
"How do we do that Mr.k?"
"Well I'll show you."
Thats a rough translation of what probably everyone was thinking. Except the Ogre 1 and 2 of our class. They knew the answer already, (Outside joke.)
So to bisect the transverse axis, we folded our papers for one more time. This time, we folded one focus point, directly on top of the other. Thus we obtained a crease which defined not only the bisector, but the perpendicular Bisector of the Transverse Axis. We see this as a vertical red line on the image above. Lets step things up a notch (I know Mr. K did.)
So, do you remember the Transverse axis? and how we just cut it in two? Were going to take this line, and modify it a little bit. Using our rulers again, we measured the line from one of the FOCI (in this case F2) to that vertical line (which we now call the CONJUGATE AXIS.) This gave us some length.
Now this next bit is tricky, we took the length we measured from the PREVIOUS STEP (that being from the conjugate axis, to Focal point F2) and moved it so that one end of it was on the (whatever the singular of vertices is. Verticii?) of the nearest branch, and so that the other end of the length lies on the Conjugate axis. Since I bet your saying "what the frig" I made a quick little paint mockup to get the point across, or rather, to help you "see" what I mean.
http://i24.photobucket.com/albums/c21/skyL193/Slide3.jpg
Okay, so as I wrote on the picture, the light green is the length Conjugate Axis to F2. Then by moving it like I mentioned before, we get the makings of a triangle.
Step 3. Measure the distance from the point where the light green line meets the conjugate axis, to where the dark green axis meets the conjugate axis. This we can call length b for explanations sake. The next step is to take this length, and go from the intersection of the Transverse axis and the conjugate axis, (aka the cross in the centre) and "b" units down. I'll show you what I mean again.
http://i24.photobucket.com/albums/c21/skyL193/Slide4.jpg
So, as I wrote again, the light blue, and dark blue lines are the ones I was talking about before, and they are of equal length. Thus we get those points G1 and G2. This brings us to our next step.
By drawing perfectly 90 degree lines from the vertices (going through them, vertically) and from the end points of the light and dark blue lines (going through points G1 and G2, horizontally) we can construct a box of sorts. This is seen once again on the slides.
http://i24.photobucket.com/albums/c21/skyL193/Slide5.jpg
So you see the box right? The black dotted lines there are actually it :P So yeah after that we took our ruler again, and drew that orange cross that goes far off the page kinda. We drew that cross by going from one corner of the box to the other (and beyond.) After we had done this Mr.K revealed to us that those orange lines, are actually asymptotes for the graph. In other words, neither branch of the hyperbola will actually ever touch those orange crossey lines.
The final slide here, is the equation of the hyperbola, along with some similarities, and differences, which we came up with together, to help us remember the equation, and keep it distinct from the others.
http://i24.photobucket.com/albums/c21/skyL193/FinalSlide.jpg
To Conclude: The basic parts within the anatomy of a hyperbola
- Transverse Axis.
- Conjugate Axis
- Focus Points
- The Asymptotes
- And finally, you should know the equation, which is shown in the above slide image.
Alrighty, I think that about wraps this blog post up. I'm not entirely sure, because at 1:00am I kinda start to get a bit loopy and my thoughts aren't so coherent (thats probably evident based on how my writing progressed.) If you guys have any questions or whatever lemme know. I'll try to answer them, and post em up. Or if you find any errors, or know something I dont/forgot to add, lemme know so I can add it.
Well I'm off, but of course, not without letting you know who the next scribe poster is :)
http://i24.photobucket.com/albums/c21/skyL193/JamieSlide.jpg
Remember, vote for Zeph for VP.
K? Kthxbai
Justus- FINALLY out.
Wednesday, May 14, 2008
Hyperbolas
Today's class we didn't do a whole lot of anything other then hyperbolas. We recapped on the equation of an ellipse, refer to last class: Richard's scribe post.
We learned about how to put subscripts and super scripts, that are used to right those tiny numbers, or letters, that are used to show various things, such as an exponent on a power
Example: 21 - The 1 in this equation would be the super scripted number.
Use: This is given with <.sup> and end with <./sup> Get rid of the periods.I put them in so it wouldn't actually work.
Example2: log2 - The 2 in this log would be the subscripted number.
Use: This is given with <.sub> and end with <./sub> Get rid of the periods. I put them in so it wouldn't actually work.
This is a pretty useful html code to use. Remember you can only use these codes in the blog if you are in the "Edit Html" tab, shown beside the "Compose" tab at the top right, just above the various posting tools (spell check, insert picture, etc.).
We were given 1 choice of 3 different questions that we were asked to solve. We started by evaluating this equation: 16x2 + 9y2 = 144
The first part of the question was to change it to standard form, which was quite easy. To do this, you would have to reduce the coefficients so there wouldn't be any on the x or y values. This could be done by multiplying each side by (1/(16)9). By doing this, it would reduce the coefficients, giving us the standard form of the equation: (x2)/9 + (y2)/16 = 1.
For the 2nd part of the question, we were asked to find the centre, major axis, minor axis, vertices and the focii points.
Centre:
The centre being (h,k) but because there is no h and k values in this equation, the centre is at (0,0).
Major Axis:
the major axis being 8, because 16 = b2, and b is the length of the semi-major axis from the centre to one vertices, so square root 16, to get a semi-major length, which is 4, then multiply that by 2, to get the other semi-major axis, which gives us the major axis(two semi-majors = 1 major), which is 8.
MinorAxis:
Same idea with the minor axis, but using the a2 value, which is 9, square rooted giving us 3, which is the semi-minor axis, then multiplying by 2, to get the minor axis, which is 6. the vertices would be the endpoint of the semi-minor axes, and semi-major axes, starting at the centre (0,0).
Vertices:
The vertices of the minor-axis spanning a value of 6, from the centre and outwards horizontally, because it's a vertical ellipse, due to the fact that b2 is below the y-value (need more info?, refer to Richard's post). So if a value of 6 was spread outward horizontally, from the centre (0,0), it would give the vertices a value of (3,0) and (-3,0). The other vertices would come from the value, of 8, spanning outward vertically from the centre (0,0) giving the other vertices points at (0,4) and (0,-4).
The 3rd part of the question was to graph it: (I'll edit this in later, when the slides come out, so you can see the actual graph)
Hyperbola:
Finally, is the hyperbola, the last section of todays class, we made these hyperbola by folding a piece of paper. It started off by having a circle that was off centre on the paper, and the circle had a given centre point. We were asked to make a point about 2 cm off, out of the circle. If you were to draw a line perpendicular to the edge of the paper in landscape view that bisected this centre point, the point out of the circle should be close to touching this line. We were then asked to draw atleast 25 dots with 5 of the dots bunched in the area on the edge of the circle closest to the outside point. These points should be touching the edge of the circle. We then folded all these points, onto the points outside of the circle. This gave us an outline of a hyperbola, with the centre point of the circle, and the outside point the focus points of the hyperbola. Voila a hyperbola.
For homework, we were asked to pick a point on the left branch of the hyperbola and 2 points on the right branch of the hyperbola, after that, we measure the distance from this point to both of the focal points, then find the difference. What do you get? Find out.
That was pretty much everything we did for today's pre-cal class, and now I feel like a world-class math shaped origami teacher. I''m sure everyone feels this way, after we folded 3 different different shapes. I can't complain though, I found it quite fun. I propose the next scribe will be: eeeny meeny miineyy, Justus. Surprised?.. I thought so.
Until next time,
Francis
We learned about how to put subscripts and super scripts, that are used to right those tiny numbers, or letters, that are used to show various things, such as an exponent on a power
Example: 21 - The 1 in this equation would be the super scripted number.
Use: This is given with <.sup> and end with <./sup> Get rid of the periods.I put them in so it wouldn't actually work.
Example2: log2 - The 2 in this log would be the subscripted number.
Use: This is given with <.sub> and end with <./sub> Get rid of the periods. I put them in so it wouldn't actually work.
This is a pretty useful html code to use. Remember you can only use these codes in the blog if you are in the "Edit Html" tab, shown beside the "Compose" tab at the top right, just above the various posting tools (spell check, insert picture, etc.).
We were given 1 choice of 3 different questions that we were asked to solve. We started by evaluating this equation: 16x2 + 9y2 = 144
The first part of the question was to change it to standard form, which was quite easy. To do this, you would have to reduce the coefficients so there wouldn't be any on the x or y values. This could be done by multiplying each side by (1/(16)9). By doing this, it would reduce the coefficients, giving us the standard form of the equation: (x2)/9 + (y2)/16 = 1.
For the 2nd part of the question, we were asked to find the centre, major axis, minor axis, vertices and the focii points.
Centre:
The centre being (h,k) but because there is no h and k values in this equation, the centre is at (0,0).
Major Axis:
the major axis being 8, because 16 = b2, and b is the length of the semi-major axis from the centre to one vertices, so square root 16, to get a semi-major length, which is 4, then multiply that by 2, to get the other semi-major axis, which gives us the major axis(two semi-majors = 1 major), which is 8.
MinorAxis:
Same idea with the minor axis, but using the a2 value, which is 9, square rooted giving us 3, which is the semi-minor axis, then multiplying by 2, to get the minor axis, which is 6. the vertices would be the endpoint of the semi-minor axes, and semi-major axes, starting at the centre (0,0).
Vertices:
The vertices of the minor-axis spanning a value of 6, from the centre and outwards horizontally, because it's a vertical ellipse, due to the fact that b2 is below the y-value (need more info?, refer to Richard's post). So if a value of 6 was spread outward horizontally, from the centre (0,0), it would give the vertices a value of (3,0) and (-3,0). The other vertices would come from the value, of 8, spanning outward vertically from the centre (0,0) giving the other vertices points at (0,4) and (0,-4).
The 3rd part of the question was to graph it: (I'll edit this in later, when the slides come out, so you can see the actual graph)
Hyperbola:
Finally, is the hyperbola, the last section of todays class, we made these hyperbola by folding a piece of paper. It started off by having a circle that was off centre on the paper, and the circle had a given centre point. We were asked to make a point about 2 cm off, out of the circle. If you were to draw a line perpendicular to the edge of the paper in landscape view that bisected this centre point, the point out of the circle should be close to touching this line. We were then asked to draw atleast 25 dots with 5 of the dots bunched in the area on the edge of the circle closest to the outside point. These points should be touching the edge of the circle. We then folded all these points, onto the points outside of the circle. This gave us an outline of a hyperbola, with the centre point of the circle, and the outside point the focus points of the hyperbola. Voila a hyperbola.
For homework, we were asked to pick a point on the left branch of the hyperbola and 2 points on the right branch of the hyperbola, after that, we measure the distance from this point to both of the focal points, then find the difference. What do you get? Find out.
That was pretty much everything we did for today's pre-cal class, and now I feel like a world-class math shaped origami teacher. I''m sure everyone feels this way, after we folded 3 different different shapes. I can't complain though, I found it quite fun. I propose the next scribe will be: eeeny meeny miineyy, Justus. Surprised?.. I thought so.
Until next time,
Francis
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