Final exam. One question.
How Tall Is It?
"Show me two different ways you could figure the height of the building."
"That's easy, Mr. Cox. Measure a brick and count the number of rows."
"Alright, make it three, smart guy."
Tuesday, May 19, 2009
Friday, May 15, 2009
Are Two Ways Better Than One?
Graphing parabolas is much easier when we can zero in on "key points." The CPM Algebra 2 curriculum was great about dealing with "parent graphs" and then showing students the process for translating and stretching these parents. It is easy to get away from this as we have other skills that we need to teach. But this year, I have really focused on having my students get really comfortable with y = x2 and then recognizing that all parabolas are really just different perspectives of this parent graph. Zoom out and the parabola get skinnier; zoom in and it gets fatter. If you know the vertex and stretch factor, then you are ready to do some graphing; this works for vertex or standard form.
One of the more interesting developments during this unit has been my students recognizing that the rate of change in a parabola has a rate of change. They are wrestling with the concepts behind derivatives and I want to keep them in that fight as long as possible. I usually have my students graph five points and I have always had them relate those five points back to the vertex. However, with the way they are handling rate of change, I need to rethink my process.
Simply use the stretch factor to adjust the relationships:
This year I have given them choice on this, but it has caused a few kids confusion as they end up with a hybrid process. Next year? Not so sure.
One of the more interesting developments during this unit has been my students recognizing that the rate of change in a parabola has a rate of change. They are wrestling with the concepts behind derivatives and I want to keep them in that fight as long as possible. I usually have my students graph five points and I have always had them relate those five points back to the vertex. However, with the way they are handling rate of change, I need to rethink my process.
Simply use the stretch factor to adjust the relationships: This year I have given them choice on this, but it has caused a few kids confusion as they end up with a hybrid process. Next year? Not so sure.
Thursday, May 14, 2009
Ya Think?
Teacher: "What do you think about...?"
Student: "I don't know." Translation: What do you want me to think?
A recent post by Jason Dyer regarding the findings of Piaget being re- interpreted by James McGarrigle and Margaret Donaldson has me thinking about how often I give off context clues without even thinking about it.
*nodding head, smiling* "Do you understand now?"
*raising hand* "Raise your hand if you get it."
*squinting with furrowed brow, head cocked to the side* "Can you explain how you got that answer?"
And more importantly, how I perpetuate the very thing that I beleive is wrong with education.
They cue in fast...really fast. Mine cue in faster than others 'cause I got the smart ones. But you know what, they don't think better than the others, they just figure out what the teacher wants to hear faster and at a higher accuracy rate. They play "school" better. The ones who are the real thinkers are the smart kids we call lazy. Yeah, that one-- the kid who doesn't turn stuff in or do homework but crushes every test.
I am starting to think it's not his fault. Maybe, just maybe, he's just not interested in me giving him answers to questions he doesn't care to ask.
Student: "I don't know." Translation: What do you want me to think?
A recent post by Jason Dyer regarding the findings of Piaget being re- interpreted by James McGarrigle and Margaret Donaldson has me thinking about how often I give off context clues without even thinking about it.
*nodding head, smiling* "Do you understand now?"
*raising hand* "Raise your hand if you get it."
*squinting with furrowed brow, head cocked to the side* "Can you explain how you got that answer?"
And more importantly, how I perpetuate the very thing that I beleive is wrong with education.
They cue in fast...really fast. Mine cue in faster than others 'cause I got the smart ones. But you know what, they don't think better than the others, they just figure out what the teacher wants to hear faster and at a higher accuracy rate. They play "school" better. The ones who are the real thinkers are the smart kids we call lazy. Yeah, that one-- the kid who doesn't turn stuff in or do homework but crushes every test.
I am starting to think it's not his fault. Maybe, just maybe, he's just not interested in me giving him answers to questions he doesn't care to ask.
Tuesday, May 12, 2009
Stretch Factor
What does a normal parabola look like again?
And what about one with a stretch factor of 7?
And how about 1/10?
Nice job folks.
Now get out some paper and get to work!
And quit smiling...math ain't that fun.
And what about one with a stretch factor of 7?
And how about 1/10?
Nice job folks.
Now get out some paper and get to work!
And quit smiling...math ain't that fun.
Monday, May 11, 2009
Chandler Saves the Day
You ever have a lesson that you thought was going to go pretty well only to have it fall flat? Yeah, that happened today.
My 7th graders have been going over quadratics for the past couple of weeks. I have been out quite a bit on school business, so the progress has been slow, but very rewarding. Last week, students discovered that if you change the value of "a", it has an effect on how fast the parabola grows. I then had them graph a bunch of parabolas whose line of symmetry was the y-axis only to have a student ask,
"Can we move the parabola left or right?"
"Well since you asked..." So I did what any responsible teacher would...I had them graph a bunch of parabolas whose vertex sat on the x-axis which led to the next question,
"Can we move them up/down and left/right?"
So Friday, we are graphing parabolas in the form y =a(x-h)^2 +k and they are getting it. This is stuff I couldn't do until Algebra II with my high school students and these 7th graders are just crushing everything I throw at them. I would even give them a vertex and a second point and they were giving me an equation because they figured the stretch factor using the second point. Things are looking good and I am thinking:
Man this is just toooo easy...
Yeah, I know, pride comes before the fall. Which is what started to happen today. I have had a planning block. Now that we have graphed a bunch of parabolas in vertex form, where do I go from there? Do I start dealing with standard form? Do I show them how to expand (x-h)^2 in order to arrive at standard form? I am still not sure what the ideal path would be. But being the "try anything once" kind of guy I am, I figured that since I have already had them:
...then I would focus in on what was necessary to graph a parabola: vertex and stretch factor. If they could identify the vertex and a stretch factor, they can graph anything, right? So today I wanted them to graph a bunch of parabolas in standard form, look for the line of symmetry and recognize the relationship between a,b and the line of symmetry. I didn't expect them to necessarily "discover" that the line of symmetry is x = -b/(2a), but I figured that if we graphed enough of them, we might start to notice some patterns. Once we have the line of symmetry down, then we could start looking at x intercepts which would lead us to factoring and completing the square as well as quadratic formula. (If my sequencing on this is bad, please save me from myself. )
This is where it started to go bad. GeoGebra is a great program, but it doesn't save a weak lesson. Kids were all over the place with their parabolas and we were getting lines of symmetry like x=.7923496, which wasn't going to help at all. I was about to put us all out of our misery and jump ship when Chandler says, "Mr. Cox come here, I think I found something."
She had about 10 parabolas that all had the same line of symmetry. She was making adjustments to the a and b values and recognized that the c value had no effect on the symmetry. So rather than aborting the mission, we just changed course.
"How about choosing a line of symmetry and keeping the vertex on that line?"
They got right to it. Tomorrow they are going to come to class with five different a,b,c values and the corresponding line of symmetry. We will see where it goes.
Note to the reader: Quadratics are not a 7th grade standard and these kids will go through it on a deeper level as 8th graders. So, I am not worried about "finishing" this with them. I have the flexibility to let concepts marinade for a while. Usually, I would have just followed the pacing of the book, but I have become very dissatisfied with that. I am pretty sure that I want to continue from linear relations right into quadratic relations and that graphing is a good gateway to all the other skills that go along with quadratics. I am just not sure how one skill will best lead into another. Any suggestions?
Note to self: Quit gettin' ahead of yourself and be sure to see the lesson through the eyes of a student rather than your own.
Oh yeah, and thank Chandler.
My 7th graders have been going over quadratics for the past couple of weeks. I have been out quite a bit on school business, so the progress has been slow, but very rewarding. Last week, students discovered that if you change the value of "a", it has an effect on how fast the parabola grows. I then had them graph a bunch of parabolas whose line of symmetry was the y-axis only to have a student ask,
"Can we move the parabola left or right?"
"Well since you asked..." So I did what any responsible teacher would...I had them graph a bunch of parabolas whose vertex sat on the x-axis which led to the next question,
"Can we move them up/down and left/right?"
So Friday, we are graphing parabolas in the form y =a(x-h)^2 +k and they are getting it. This is stuff I couldn't do until Algebra II with my high school students and these 7th graders are just crushing everything I throw at them. I would even give them a vertex and a second point and they were giving me an equation because they figured the stretch factor using the second point. Things are looking good and I am thinking:
Man this is just toooo easy...
Yeah, I know, pride comes before the fall. Which is what started to happen today. I have had a planning block. Now that we have graphed a bunch of parabolas in vertex form, where do I go from there? Do I start dealing with standard form? Do I show them how to expand (x-h)^2 in order to arrive at standard form? I am still not sure what the ideal path would be. But being the "try anything once" kind of guy I am, I figured that since I have already had them:
- Graph quadratics organically (area vs. radius; area vs. side length)
- Graph quadratics with a not equal to 1.
- Graph quadratics with vertex on y axis.
- Graph quadratics with vertex on x axis.
- Graph quadratics in the form y = a(x-h)^2 +k
...then I would focus in on what was necessary to graph a parabola: vertex and stretch factor. If they could identify the vertex and a stretch factor, they can graph anything, right? So today I wanted them to graph a bunch of parabolas in standard form, look for the line of symmetry and recognize the relationship between a,b and the line of symmetry. I didn't expect them to necessarily "discover" that the line of symmetry is x = -b/(2a), but I figured that if we graphed enough of them, we might start to notice some patterns. Once we have the line of symmetry down, then we could start looking at x intercepts which would lead us to factoring and completing the square as well as quadratic formula. (If my sequencing on this is bad, please save me from myself. )
This is where it started to go bad. GeoGebra is a great program, but it doesn't save a weak lesson. Kids were all over the place with their parabolas and we were getting lines of symmetry like x=.7923496, which wasn't going to help at all. I was about to put us all out of our misery and jump ship when Chandler says, "Mr. Cox come here, I think I found something."
She had about 10 parabolas that all had the same line of symmetry. She was making adjustments to the a and b values and recognized that the c value had no effect on the symmetry. So rather than aborting the mission, we just changed course.
"How about choosing a line of symmetry and keeping the vertex on that line?"
They got right to it. Tomorrow they are going to come to class with five different a,b,c values and the corresponding line of symmetry. We will see where it goes.
Note to the reader: Quadratics are not a 7th grade standard and these kids will go through it on a deeper level as 8th graders. So, I am not worried about "finishing" this with them. I have the flexibility to let concepts marinade for a while. Usually, I would have just followed the pacing of the book, but I have become very dissatisfied with that. I am pretty sure that I want to continue from linear relations right into quadratic relations and that graphing is a good gateway to all the other skills that go along with quadratics. I am just not sure how one skill will best lead into another. Any suggestions?
Note to self: Quit gettin' ahead of yourself and be sure to see the lesson through the eyes of a student rather than your own.
Oh yeah, and thank Chandler.
Monday, May 4, 2009
Upgraded to Pandemic
At least in my own head. No, I can't shut it off! Not sure I would want to even if I could.
I am getting gas the other day and this shot is screaming at me:
So today, I put the slide up and the kids immediately start talking about slope. I like that they thought slope, but slope isn't really going to do much when it comes to fences. So I asked them:
"What would you want to know if you had to build the fence?"
They caught on pretty quickly that one would want to know the length of each board.
"Alright, then tell me the length of each board."
"But, Mr. Cox, we don't have enough information."
"What do you want to know?"
"We need to know how long the shortest board is."
Done.
"Alright, now tell me how long each board is."
"We can't. We need more information."
So I had them discuss with their groups what information they had to have in order to figure out how long each board was. Once they had an exhaustive list, they were to write it on their group's easel. We quickly came up with the following:
So do we need the next board, or will any board do? We eventually settled on any other board.
It didn't take long before students had listed all the board lengths. Many used the rate of change and then added the increase to each board to find the length of the next. But it did't take much prodding for them to realize that having an equation would be nice. We came up with y = 2.5x + 36 fairly quickly. The interesting discussion came about when I asked what x represented.
"X is the number of boards."
"Okay, so go up to the board and point to board #1."
Which board do you think they pointed to? (You guessed it, the one labeled 36". )
"So, plug 1 into your equation and check it out. Tell me if your equation works."
You would have thought that I asked them to stand in the corner of a round room. But once the "but this equation haa-aas to work" wore off. They realized that it wasn't the equation's fault. It was how they defined x. Board 0 is important because we aren't actually counting boards, we are counting the number of increases.
Reflection: I think that the lesson went really well, but it was very telling how many students wanted to impress with their knowledge of the vocabulary as opposed to just looking at the problem and asking the obvious questions. They were trying to be mathematicians rather than someone who just needs to build a fence. Next year I want to do a better job of introducing concepts a bit more organically as opposed to "here are the rules, here are some examples, let's get to it." Students are much more engaged when the information is given a little at a time. It keeps them from answer chasing and allows them to think a little. It may take a bit longer to deliver the lesson, but the benefit of having kids think about the math is priceless.
Questions: What else could I have done with this image?
I am getting gas the other day and this shot is screaming at me:
So today, I put the slide up and the kids immediately start talking about slope. I like that they thought slope, but slope isn't really going to do much when it comes to fences. So I asked them:
"What would you want to know if you had to build the fence?"
They caught on pretty quickly that one would want to know the length of each board.
"Alright, then tell me the length of each board."
"But, Mr. Cox, we don't have enough information."
"What do you want to know?"
"We need to know how long the shortest board is."
Done.
"Alright, now tell me how long each board is."
"We can't. We need more information."
So I had them discuss with their groups what information they had to have in order to figure out how long each board was. Once they had an exhaustive list, they were to write it on their group's easel. We quickly came up with the following:
- The length of the next board.
- The length of one more board.
- The width of each board.
So do we need the next board, or will any board do? We eventually settled on any other board.
It didn't take long before students had listed all the board lengths. Many used the rate of change and then added the increase to each board to find the length of the next. But it did't take much prodding for them to realize that having an equation would be nice. We came up with y = 2.5x + 36 fairly quickly. The interesting discussion came about when I asked what x represented.
"X is the number of boards."
"Okay, so go up to the board and point to board #1."
Which board do you think they pointed to? (You guessed it, the one labeled 36". )
"So, plug 1 into your equation and check it out. Tell me if your equation works."
You would have thought that I asked them to stand in the corner of a round room. But once the "but this equation haa-aas to work" wore off. They realized that it wasn't the equation's fault. It was how they defined x. Board 0 is important because we aren't actually counting boards, we are counting the number of increases.
Reflection: I think that the lesson went really well, but it was very telling how many students wanted to impress with their knowledge of the vocabulary as opposed to just looking at the problem and asking the obvious questions. They were trying to be mathematicians rather than someone who just needs to build a fence. Next year I want to do a better job of introducing concepts a bit more organically as opposed to "here are the rules, here are some examples, let's get to it." Students are much more engaged when the information is given a little at a time. It keeps them from answer chasing and allows them to think a little. It may take a bit longer to deliver the lesson, but the benefit of having kids think about the math is priceless.
Questions: What else could I have done with this image?
Friday, May 1, 2009
Intro to Quadratics: 7th Grade Style
Can I just say that I love middle school kids. I mean, sometimes keeping them on the same page is like trying to herd a bunch of cats, but I love them. Now that testing is over, it is time to start preparing my 7th graders for the wonderful world of quadratics. We have been doing a bunch of activities on linear relationships and the results have been pretty good. My kids have a pretty firm grasp of the following:
They like lines; they are comfortable with lines. It is time to take them out of their comfort zone. So here is how it went down:
We had already done some activities on linear relationships like:
The latest installment was to have them bring in a few circular items and then find the relationship between radius vs. circumference. This led quite nicely into, 'Well, since you have some circles here, you may as well calculate the areas too. Graph those compared to the radius and see what you get."
It was interesting to see how many kids tried to force it into a linear relationship.
Angel recognized that "choosing a bunch of circles with around the same radius doesn't tell us much, huh, Mr. Cox."
"Nope, next time we may want to expand our sample space."
Regardless, by the end of the activity, they understood that sometimes we have relationships that are "curved" or "non-linear." Fareen recognized that we get a "half of a parabola."
So from there we do some work with the side length vs. area of a square. Hey, we may as well start at the beginning, right? But it was the simplicity of the exercise that produced the magic that I never saw in 10 years of teaching high school kids the same thing.
"Hey, Mr. Cox, it isn't a line because the slopes don't stay the same."
"Yeah, so what?"
"Well the areas increase by 3, then 5 then 7. But the increases all increase by 2."
"Okay, so what does that mean?"
"The rate of change has a rate of change."
At this point I get goosebumps.
This is when Abel, asks: "What will happen if we cube x? What happens to the rate of change then?"
Couldn't pass this one up, so we drew up a chart and did it. The kids concluded that for a cubic: the rate of change of the rate of change has a rate of change. Oh, and the number of times we have to check the rate of change tells us what the exponent is.
Moral of the story: Don't assume that your lesson objective is the right one.
or, Sometimes it is better to follow the herd of cats.
- Slope and rate of change mean the same thing.
- If the rate of change remains the same, then we have a line.
- The initial condition is the y intercept.
- If the initial condition is 0, then we have a direct variation.
They like lines; they are comfortable with lines. It is time to take them out of their comfort zone. So here is how it went down:
We had already done some activities on linear relationships like:
- Farenheit vs. Celcius
- cm vs. inches
- km vs. miles
- start height vs. rebound height for a bouncing ball on concrete between 75 and 80 degrees with no wind resistance. (Alright, we didn't control the experiment that much, but they still saw that the ball rebounded to about 70% the original height. Daniel learned that baseballs don't bounce very high when you drop them from 1 meter and the seams mess up the bounce.)
The latest installment was to have them bring in a few circular items and then find the relationship between radius vs. circumference. This led quite nicely into, 'Well, since you have some circles here, you may as well calculate the areas too. Graph those compared to the radius and see what you get."
It was interesting to see how many kids tried to force it into a linear relationship.
Angel recognized that "choosing a bunch of circles with around the same radius doesn't tell us much, huh, Mr. Cox."
"Nope, next time we may want to expand our sample space."
Regardless, by the end of the activity, they understood that sometimes we have relationships that are "curved" or "non-linear." Fareen recognized that we get a "half of a parabola."
So from there we do some work with the side length vs. area of a square. Hey, we may as well start at the beginning, right? But it was the simplicity of the exercise that produced the magic that I never saw in 10 years of teaching high school kids the same thing.
"Hey, Mr. Cox, it isn't a line because the slopes don't stay the same."
"Yeah, so what?"
"Well the areas increase by 3, then 5 then 7. But the increases all increase by 2."
"Okay, so what does that mean?"
"The rate of change has a rate of change."
At this point I get goosebumps.
This is when Abel, asks: "What will happen if we cube x? What happens to the rate of change then?"
Couldn't pass this one up, so we drew up a chart and did it. The kids concluded that for a cubic: the rate of change of the rate of change has a rate of change. Oh, and the number of times we have to check the rate of change tells us what the exponent is.
Moral of the story: Don't assume that your lesson objective is the right one.
or, Sometimes it is better to follow the herd of cats.
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