In my Calculus class, recently a discussion came up about when a derivative trig function would have a midline that is not 0. We quickly discussed the algebraic result of this question, but I thought afterwards that this was just too juicy of a discussion to let slide by.
So, on our new practice quiz, I decided to throw in one such bonus problem. (Go to the end.) They were AMAZED by the resulting graph when they worked on this in class today. One algebra-whiz kid was like, "This is NOT allowed! You cannot mix trig function with other functions!!" So hilarious!! I LOVE MY KIDS. I was pleased with their sense of wonder and surprise, but even more pleased that they were still able to look at and compare features between the derivative and original function graphs, even though they thought what they were looking at was super weird and not intuitive.
At the end of class, I off-handedly asked a couple of the fastest-working kids today, "So what's the average value of the derivative function?" They said, "...-1?" And then I asked, "How does that show up inside the original graph?" Those kids' eyes got so big and they said, "It has an average derivative of -1! OMG, you can see it!!!"
Cool graphical connections!!! We'll have to revisit this bonus problem as a class tomorrow, to make sure that everyone can appreciate the juiciness of this connection before we move on to other algebra goodness.
Showing posts with label grade11. Show all posts
Showing posts with label grade11. Show all posts
Tuesday, November 5, 2013
Exploring Sine and Cosine Derivatives
My onwards saga with differential calculus presented via explorations! (See previous post here.)
My kids recently did an exploration of sine and cosine derivatives. It was a little long, but I think well worth it. The idea is this:
* First, we spent a class period reviewing how to find a sine or cosine equation given a graph. This way I made sure that they'd have the pre-requisite exploration algebra skills.
* On the worksheet, they were given f(x) = sin(x) and asked to sketch its derivative using their knowledge of how to sketch derivative graphs. They're quite confident with this sketch, and then they decided that f'(x) = cos(x) by looking at its shape. I circulated and made sure that each kid used the dy/dx feature of their calculator to verify that the amplitude of f' is still 1, by making sure that the steepest part of f has a derivative value of 1.
* They then are given g(x) = cos(x), and again they started by sketching its derivative graph in order to determine that g'(x) = -sin(x) by inspection.
* They then made predictions for when the amplitude of the original function is not 1. In each case, they sketched the curve and then verified the max derivative value, in order to verify the amplitude of their resulting derivative graph.
* They repeated this for simple cases when the period changes. Based on their sketch of the derivative graph, they determined that the resulting derivative function would share the same period as the original functions, so they think k(x) = sin(2x) would have k'(x) = cos(2x). But, when they then checked the max derivative value along k(x), they realized that they were not correct. --WHY?? One of the kids immediately figured it out. I overheard him saying that because "sin(2x) is squished horizontally, that middle part of maximum steepness now becomes twice as steep as before, so obviously the amplitude will change." They checked this again using L(x) = cos(2x), whose max derivative (or steepness) is again 2, not 1, due to the horizontal compression. Brilliant!!
* They then tested their hypothesis out with M(x) = 3sin(2x), whose max steepness is now even steeper than 2.
* As a class, when we came together to discuss these exploratory problems together, I pulled up GeoGebra to guide our discussion. I entered something like M(x) = 3sin(2x) and had kids hypothesize just how steep that graph gets at x=0, since not everyone had quite reached the same conclusions before. If they said that the max derivative is 1, I entered y = 1x to show them that's unfortunately less steep than our curve is at x = 0. If they said 2, I entered y = 2x to show it's still less steep than our curve. Someone then guessed that the slope of the tangent at that point is 6, and we graphed it to see that y = 6x is indeed as steep as the curve gets. We then typed in M'(x) into the GeoGebra input bar, which automatically generates the formula AND the graph for the derivative. We discussed why the amplitude changes to 6, but why the period stays the same as that of M(x).
BAM! My kids are awesome!! They're able to now differentiate basic wave equations without knowing the first thing about chain rule. (Not yet, that's next, after we reinforce some other parts of our abstract analysis.) And they can explain the outcome of these sine and cosine derivative graphs! Yeah!!
My kids recently did an exploration of sine and cosine derivatives. It was a little long, but I think well worth it. The idea is this:
* First, we spent a class period reviewing how to find a sine or cosine equation given a graph. This way I made sure that they'd have the pre-requisite exploration algebra skills.
* On the worksheet, they were given f(x) = sin(x) and asked to sketch its derivative using their knowledge of how to sketch derivative graphs. They're quite confident with this sketch, and then they decided that f'(x) = cos(x) by looking at its shape. I circulated and made sure that each kid used the dy/dx feature of their calculator to verify that the amplitude of f' is still 1, by making sure that the steepest part of f has a derivative value of 1.
* They then are given g(x) = cos(x), and again they started by sketching its derivative graph in order to determine that g'(x) = -sin(x) by inspection.
* They then made predictions for when the amplitude of the original function is not 1. In each case, they sketched the curve and then verified the max derivative value, in order to verify the amplitude of their resulting derivative graph.
* They repeated this for simple cases when the period changes. Based on their sketch of the derivative graph, they determined that the resulting derivative function would share the same period as the original functions, so they think k(x) = sin(2x) would have k'(x) = cos(2x). But, when they then checked the max derivative value along k(x), they realized that they were not correct. --WHY?? One of the kids immediately figured it out. I overheard him saying that because "sin(2x) is squished horizontally, that middle part of maximum steepness now becomes twice as steep as before, so obviously the amplitude will change." They checked this again using L(x) = cos(2x), whose max derivative (or steepness) is again 2, not 1, due to the horizontal compression. Brilliant!!
* They then tested their hypothesis out with M(x) = 3sin(2x), whose max steepness is now even steeper than 2.
* As a class, when we came together to discuss these exploratory problems together, I pulled up GeoGebra to guide our discussion. I entered something like M(x) = 3sin(2x) and had kids hypothesize just how steep that graph gets at x=0, since not everyone had quite reached the same conclusions before. If they said that the max derivative is 1, I entered y = 1x to show them that's unfortunately less steep than our curve is at x = 0. If they said 2, I entered y = 2x to show it's still less steep than our curve. Someone then guessed that the slope of the tangent at that point is 6, and we graphed it to see that y = 6x is indeed as steep as the curve gets. We then typed in M'(x) into the GeoGebra input bar, which automatically generates the formula AND the graph for the derivative. We discussed why the amplitude changes to 6, but why the period stays the same as that of M(x).
BAM! My kids are awesome!! They're able to now differentiate basic wave equations without knowing the first thing about chain rule. (Not yet, that's next, after we reinforce some other parts of our abstract analysis.) And they can explain the outcome of these sine and cosine derivative graphs! Yeah!!
Tuesday, October 8, 2013
Backwards Intro to Differential Calculus
Towards the end of the summer I was brainstorming this idea of teaching Calculus backwards, starting with applications and graphing calculators, then manual Calculus skills, then finally tying those manual Calculus skills to various limits. It is now a little more than a month in, and I have to say that although I cannot compare this approach to a traditional curriculum because I've never taught Calculus the traditional way, I love the way that I am doing it!!!
After a unit on analyzing numerical and graphical derivatives, my kids have a very strong foundation of how to read/interpret graphs visually and how to sketch related derivative graphs carefully. Some of them even have a surprising ability to draw original functions given the shapes of the derivative graphs, even though we really didn't practice that as a class (not yet... I am saving it as part of our intuitive intro to integral Calculus).
And the best part of spending the first part of the year on application is that kids are now itching for algebra, because all of the visual introduction has really primed them for more specificity. They were so, SO excited today when I set them loose on an exploration regarding the power rule of differentiation. After just observing three examples that they generated via explorations on their calculator, they were fully able to generalize the pattern in partner pairs, and by the time they got to the back side of the worksheet, they thought it was just SO COOL that they could now differentiate by hand, without a calculator. It was so AWESOME to see. I've never seen kids so excited to differentiate by hand before! And, when they weren't sure whether they were doing something by hand correctly (ie. when they needed to differentiate a term already with a negative exponent on x), I would encourage them to put in an x value into their resulting f' formula and to check it against the dy/dx value that the calculator returns. Their tech skills from Unit 1 are now becoming a powerful tool for self-monitoring their process.
And, because they've already done some mixed applications with f(x) and f'(x) analyses, I can immediately set them up tomorrow to answering a question such as, "When will the instantaneous rate reach 2 along this graph?" and the skills that we have already worked on for choosing the correct tool between f and f' will come in handy then, to smoothly transition them into the new sets of manual skills (I hope!).
I am so excited! Also, in context, we are reviewing various algebra skills incrementally. On their next quiz/practice quiz, they'll be expected to show work to find standard-form equations of polynomials by hand, and then to differentiate them in order to answer follow-up questions. My plan is to loop in all of the manual algebra modeling skills alongside the new derivative and integral skills, so that the kids would always know that they need to keep up with all of the transformational concepts from Algebra 2 and Precalc, as part of the norm for our class.
So, Calculus is pretty much my favorite class to teach right now. So incredibly fun!!!! I cannot wait until we dig into more of these derivative skills, in order for us to talk about some juicy applications that require both algebraic and geometric analysis. YES!!!!!
After a unit on analyzing numerical and graphical derivatives, my kids have a very strong foundation of how to read/interpret graphs visually and how to sketch related derivative graphs carefully. Some of them even have a surprising ability to draw original functions given the shapes of the derivative graphs, even though we really didn't practice that as a class (not yet... I am saving it as part of our intuitive intro to integral Calculus).
And the best part of spending the first part of the year on application is that kids are now itching for algebra, because all of the visual introduction has really primed them for more specificity. They were so, SO excited today when I set them loose on an exploration regarding the power rule of differentiation. After just observing three examples that they generated via explorations on their calculator, they were fully able to generalize the pattern in partner pairs, and by the time they got to the back side of the worksheet, they thought it was just SO COOL that they could now differentiate by hand, without a calculator. It was so AWESOME to see. I've never seen kids so excited to differentiate by hand before! And, when they weren't sure whether they were doing something by hand correctly (ie. when they needed to differentiate a term already with a negative exponent on x), I would encourage them to put in an x value into their resulting f' formula and to check it against the dy/dx value that the calculator returns. Their tech skills from Unit 1 are now becoming a powerful tool for self-monitoring their process.
And, because they've already done some mixed applications with f(x) and f'(x) analyses, I can immediately set them up tomorrow to answering a question such as, "When will the instantaneous rate reach 2 along this graph?" and the skills that we have already worked on for choosing the correct tool between f and f' will come in handy then, to smoothly transition them into the new sets of manual skills (I hope!).
I am so excited! Also, in context, we are reviewing various algebra skills incrementally. On their next quiz/practice quiz, they'll be expected to show work to find standard-form equations of polynomials by hand, and then to differentiate them in order to answer follow-up questions. My plan is to loop in all of the manual algebra modeling skills alongside the new derivative and integral skills, so that the kids would always know that they need to keep up with all of the transformational concepts from Algebra 2 and Precalc, as part of the norm for our class.
So, Calculus is pretty much my favorite class to teach right now. So incredibly fun!!!! I cannot wait until we dig into more of these derivative skills, in order for us to talk about some juicy applications that require both algebraic and geometric analysis. YES!!!!!
Saturday, September 28, 2013
Using Logger Pro in Quadratic Modeling!
One of the wonderful things of teaching in different schools is that you get to learn from different teachers. My current school has a site-wide license for Logger Pro, which (I know, unfortunately) is a proprietary program that allows you to import and analyze videos. It pulls the scaling information based on your definition of what 1 meter looks like in the video, and it uses the timestamps built into the video to retrieve timing info. From that, this program is able to pull both position information over time, and estimated velocity information over time. (The velocity bit is not that precise, however.)
I was playing around with this piece of software this morning because our Precalculus course team wishes to incorporate it into our Quadratics unit. I imported a video from David Cox, which can be found at http://vimeo.com/16506894,
and I got this screenshot in Logger Pro. The red is the horizontal position of the yellow ball over time, for the frames that I chose. The blue is the vertical position of the yellow ball over time, for the frames that I chose.
I was playing around with this piece of software this morning because our Precalculus course team wishes to incorporate it into our Quadratics unit. I imported a video from David Cox, which can be found at http://vimeo.com/16506894,
and I got this screenshot in Logger Pro. The red is the horizontal position of the yellow ball over time, for the frames that I chose. The blue is the vertical position of the yellow ball over time, for the frames that I chose.
I love this! I can see letting my kids do the same, picking out points from a video that includes both dimensions of movement, and then discussing why height is always quadratic and the horizontal distance is not. And then, they will do quadratic modeling both by hand (by setting up a system of equations) and on the calculator (via regression) in order to find the curve that fits this graph. LOVE IT!
PS. If you are lucky enough to work at a school that would agree to get a site-wide license, the really nice thing is that you get to install it at home completely legally, which is great for both you and the students. So, keep that in mind when you are talking to your admin!
Week 3 Teaching - Setbacks and Triumphs
We are in the thick of it now, the part of the semester when I see how kids handle setbacks and challenges. This is one of the ways I really get to know a kid, because I truly believe that how you handle setbacks defines your character. I tell the kids that they can keep reviewing and re-quizzing, or re-submitting drafts of a writing assignment, until they decide that their score is good enough to stop. No one is going to disallow them to keep working to get better, because I think that training kids to keep tackling something long after the class has "moved on" is how we can teach them to develop a persevering character.
For me personally, I've always viewed myself as a second-try kind of gal. If I weren't, I would have given up the first time a teaching program told me that I wasn't their ideal candidate, and I would never have ended up doing what I do, and loving it. Too many adults give up on their goals too easily, and let other people decide for them what they can and cannot achieve. I don't want that to happen to my students.
Anyhow, enough philosophizing. More about teaching. My Calculus kids are entering a very interesting phase of the course. We had our first quiz, which was quite tricky and conceptual, even though it did not involve many numbers. To my delight, about two-thirds of the class did quite well on this quiz, and the top three or so scores were all girls!!! I cannot help myself but feel gleeful about that, especially because 1. the girl who did the best on the quiz (missed actually none of the problems, including the bonus ones which we had only briefly seen during class) had previously said to me that she always felt a little behind in other people's math classes, and 2. our school has been having some conversations surrounding issues of diversity (mainly ethnic and socioeconomic), which has been making me wonder a bit about the role of "male privilege" in the math classroom, similar to the issues of "white privilege." Anyhow, the fraction of kids who didn't do so well on their quizzes are seeing me on Tuesday for a re-quiz, so as of now it is still too early for me to say whether they just had a bad day, or they are still learning how to study effectively, or they really need some serious intervention with the topics that we have covered. In this class, unlike the other classes, I have not been collecting/grading homework past discussing the answers as a group, since thus far we have been building up introductory concepts and there are not a lot of nitty-gritty skills checkpoints for me to look at and respond to. I think that contributed to the lack of individual feedback before the first quiz, which I will remedy in the coming unit by being more hands-on with grading their homework assignments, once we step into the realm of manual differentiation techniques. Overall, I have been very happy with the way that my kids have built their conceptual understanding around derivatives and instantaneous and average rates. We take every second Friday to review algebra skills from the past, and then I assign a review homework assignment for the weekend following. So far, we have reviewed: 1. factorization techniques, 2. how to solve for a parameter within an equation, and 3. when to use their calculators to solve complex equations; and already I can see their independence growing inside the classroom from these mini skills reviews. We just wrapped up a great worksheet (if I may say so myself), because every problem in this worksheet is anchored in something very real. Problem 1 was about investment, and I had talked to the kids about how my husband does real-estate investment and how he uses this type of math to calculate mortgage rates and monthly expenses on a rental property, in order to compare those fees with his rental income to make sure that he will clear a profit every month from his investment. (Related to this I talked to the kids about why they should invest, and why investment does not mean that they cannot be contributing productively to the society.) Problem 2 was about carbon-dating, but the kids needed to read the initial C14 levels out of a graph of atmospheric carbon levels to use in their carbon decay model. This is very realistic, and we got to talk a little bit about the science behind your body equalizing with the atmospheric carbon levels while you are breathing/alive, as well as about how cow-farming is causing historic carbon levels to rise (connecting this to what they see in the graph of historical atmospheric levels). Problem 3 on the worksheet is an ad that I actually found on the web for Gap Inc's credit card offers, so although it is another review problem for interest rates, it is steeped in real world context. I am trying to make my Calculus class as inter-curicular as possible on a day-to-day basis, so that kids can see the reason/motivation behind studying what we study. Interestingly enough, a couple of the faster-moving kids have already started on our end-of-unit economic mini-project (Part 1 and Part 2), and the first question they asked me before even starting the math was, "Why would anybody care about marginal costs?" --Aren't my kids fabulous? I want them to ask me questions like this, so that Calculus can come alive for them.
Anyhow, Precalc is also going swimmingly. I heard feedback from one 11th-grade advisor that her advisee loves my class, and thinks that all the math we do thus far is very clear and very understandable. The students are in the process of finishing up their first big lab writeup, which was very exciting because when I took them down to the computer lab, I got to show them how to 1. enter and format equations properly into MS Word or Google Docs, and 2. how to connect their TI-84s to the computer via TI-ScreenConnect to prove that they are doing the tests of their formulas. Some of the kids, in fact, didn't even know how to construct tables or write subscripts, so there was lots of tech education there, besides helping them out with the mathematical language. The kids thought that typing up their revisions to the rough drafts was going to be easy, but it did in fact take them two full (45-minute) class periods, and many still had to go home to take some time this weekend to re-read through it to make sure that they have hit every part of the project rubric. Anyhow, I prepared a graphical organizer template so that sometime next week, we can discuss how this idea of approaching and analyzing math sequences is going to be the big idea through the entire first Quint. (We have 5 Quints a year, as opposed to 4 Quarters.) Also, something quite cool that I tried recently was to put kids into groups and let them do mixed analysis of linear and quadratic sequences, and instead of me telling them whether they were correct, they got to check using the web interface of visualpatterns.org! The kids were super into it, and I think seeing the two types side by side really helped them to clarify mentally the different strategies for each type. Alongside the writeups, the kids have just about finished reviewing all the algebra skills for lines, so I will give another quiz next week before moving on to reviewing quadratic and transformational skills.
My Algebra 2 classes are moving pretty slowly through their regression project, because I have discovered that they have some holes in their Algebra 1 knowledge and am taking some daily class time to discuss homework problems before assigning new review assignments. In class, we are doing white-boarding practice about once a week, because it is a great time for me to make sure that everyone is doing some algebra practice together and getting instantaneous feedback/help as needed. Following our fairly difficult first quiz, which I had written about last week, lots of kids came to see me to do re-quizzes, which I loved. Although they didn't all get 100% on their re-quizzes, it started a very productive dialogue with kids about how they are studying, what study tips I can recommend, and why things always seem easier when you do a re-quiz. One international student in particular had an 180-degree shift of attitude towards me after the re-quiz. I think she's the kind of student who thrives particularly on positive feedback, so the fact that she had failed the first time but got 100% on the second try, really boosted her confidence and her feelings towards math (and I guess, me). So, although it had been a challenging/"discouraging" week last week, I think it was a necessary reality check for many kids and now they are much more focused and strategic in their learning. My 10th-graders, for example, took a lot of photos yesterday during our white-boarding practice, because the one student who had done that last time and who had practiced with those problems later at home, had done very well on the quiz and had offered that up as a study strategy to her peers. So, we are a growing community of learners, moving in the right direction, slowly but surely!
For both Algebra 2 classes, on Monday we will go and test out the kids' predictions for the bungee drop. They will build the cords, name their rubber chickens, take a photo with their chickens, and then we will go out to the balcony. It'll be a very fun day, but the hard work that is yet to come is to write the lab reports coherently. I am a little nervous about getting their rough drafts on Tuesday, and what those will look like, especially for my international kids....
But, I cannot complain. I love this time of the year!
For me personally, I've always viewed myself as a second-try kind of gal. If I weren't, I would have given up the first time a teaching program told me that I wasn't their ideal candidate, and I would never have ended up doing what I do, and loving it. Too many adults give up on their goals too easily, and let other people decide for them what they can and cannot achieve. I don't want that to happen to my students.
Anyhow, enough philosophizing. More about teaching. My Calculus kids are entering a very interesting phase of the course. We had our first quiz, which was quite tricky and conceptual, even though it did not involve many numbers. To my delight, about two-thirds of the class did quite well on this quiz, and the top three or so scores were all girls!!! I cannot help myself but feel gleeful about that, especially because 1. the girl who did the best on the quiz (missed actually none of the problems, including the bonus ones which we had only briefly seen during class) had previously said to me that she always felt a little behind in other people's math classes, and 2. our school has been having some conversations surrounding issues of diversity (mainly ethnic and socioeconomic), which has been making me wonder a bit about the role of "male privilege" in the math classroom, similar to the issues of "white privilege." Anyhow, the fraction of kids who didn't do so well on their quizzes are seeing me on Tuesday for a re-quiz, so as of now it is still too early for me to say whether they just had a bad day, or they are still learning how to study effectively, or they really need some serious intervention with the topics that we have covered. In this class, unlike the other classes, I have not been collecting/grading homework past discussing the answers as a group, since thus far we have been building up introductory concepts and there are not a lot of nitty-gritty skills checkpoints for me to look at and respond to. I think that contributed to the lack of individual feedback before the first quiz, which I will remedy in the coming unit by being more hands-on with grading their homework assignments, once we step into the realm of manual differentiation techniques. Overall, I have been very happy with the way that my kids have built their conceptual understanding around derivatives and instantaneous and average rates. We take every second Friday to review algebra skills from the past, and then I assign a review homework assignment for the weekend following. So far, we have reviewed: 1. factorization techniques, 2. how to solve for a parameter within an equation, and 3. when to use their calculators to solve complex equations; and already I can see their independence growing inside the classroom from these mini skills reviews. We just wrapped up a great worksheet (if I may say so myself), because every problem in this worksheet is anchored in something very real. Problem 1 was about investment, and I had talked to the kids about how my husband does real-estate investment and how he uses this type of math to calculate mortgage rates and monthly expenses on a rental property, in order to compare those fees with his rental income to make sure that he will clear a profit every month from his investment. (Related to this I talked to the kids about why they should invest, and why investment does not mean that they cannot be contributing productively to the society.) Problem 2 was about carbon-dating, but the kids needed to read the initial C14 levels out of a graph of atmospheric carbon levels to use in their carbon decay model. This is very realistic, and we got to talk a little bit about the science behind your body equalizing with the atmospheric carbon levels while you are breathing/alive, as well as about how cow-farming is causing historic carbon levels to rise (connecting this to what they see in the graph of historical atmospheric levels). Problem 3 on the worksheet is an ad that I actually found on the web for Gap Inc's credit card offers, so although it is another review problem for interest rates, it is steeped in real world context. I am trying to make my Calculus class as inter-curicular as possible on a day-to-day basis, so that kids can see the reason/motivation behind studying what we study. Interestingly enough, a couple of the faster-moving kids have already started on our end-of-unit economic mini-project (Part 1 and Part 2), and the first question they asked me before even starting the math was, "Why would anybody care about marginal costs?" --Aren't my kids fabulous? I want them to ask me questions like this, so that Calculus can come alive for them.
Anyhow, Precalc is also going swimmingly. I heard feedback from one 11th-grade advisor that her advisee loves my class, and thinks that all the math we do thus far is very clear and very understandable. The students are in the process of finishing up their first big lab writeup, which was very exciting because when I took them down to the computer lab, I got to show them how to 1. enter and format equations properly into MS Word or Google Docs, and 2. how to connect their TI-84s to the computer via TI-ScreenConnect to prove that they are doing the tests of their formulas. Some of the kids, in fact, didn't even know how to construct tables or write subscripts, so there was lots of tech education there, besides helping them out with the mathematical language. The kids thought that typing up their revisions to the rough drafts was going to be easy, but it did in fact take them two full (45-minute) class periods, and many still had to go home to take some time this weekend to re-read through it to make sure that they have hit every part of the project rubric. Anyhow, I prepared a graphical organizer template so that sometime next week, we can discuss how this idea of approaching and analyzing math sequences is going to be the big idea through the entire first Quint. (We have 5 Quints a year, as opposed to 4 Quarters.) Also, something quite cool that I tried recently was to put kids into groups and let them do mixed analysis of linear and quadratic sequences, and instead of me telling them whether they were correct, they got to check using the web interface of visualpatterns.org! The kids were super into it, and I think seeing the two types side by side really helped them to clarify mentally the different strategies for each type. Alongside the writeups, the kids have just about finished reviewing all the algebra skills for lines, so I will give another quiz next week before moving on to reviewing quadratic and transformational skills.
My Algebra 2 classes are moving pretty slowly through their regression project, because I have discovered that they have some holes in their Algebra 1 knowledge and am taking some daily class time to discuss homework problems before assigning new review assignments. In class, we are doing white-boarding practice about once a week, because it is a great time for me to make sure that everyone is doing some algebra practice together and getting instantaneous feedback/help as needed. Following our fairly difficult first quiz, which I had written about last week, lots of kids came to see me to do re-quizzes, which I loved. Although they didn't all get 100% on their re-quizzes, it started a very productive dialogue with kids about how they are studying, what study tips I can recommend, and why things always seem easier when you do a re-quiz. One international student in particular had an 180-degree shift of attitude towards me after the re-quiz. I think she's the kind of student who thrives particularly on positive feedback, so the fact that she had failed the first time but got 100% on the second try, really boosted her confidence and her feelings towards math (and I guess, me). So, although it had been a challenging/"discouraging" week last week, I think it was a necessary reality check for many kids and now they are much more focused and strategic in their learning. My 10th-graders, for example, took a lot of photos yesterday during our white-boarding practice, because the one student who had done that last time and who had practiced with those problems later at home, had done very well on the quiz and had offered that up as a study strategy to her peers. So, we are a growing community of learners, moving in the right direction, slowly but surely!
For both Algebra 2 classes, on Monday we will go and test out the kids' predictions for the bungee drop. They will build the cords, name their rubber chickens, take a photo with their chickens, and then we will go out to the balcony. It'll be a very fun day, but the hard work that is yet to come is to write the lab reports coherently. I am a little nervous about getting their rough drafts on Tuesday, and what those will look like, especially for my international kids....
But, I cannot complain. I love this time of the year!
Saturday, September 21, 2013
Week 2 Teaching - the Gentle Push Back
The second full week of school has been a very meaty one. The kids seemed very eager to learn after the first few unstructured socializing/cohort retreat days. And I am starting to see the various personalities starting to emerge, which is both wonderful and more challenging because now it is real teaching and real learning.
In my Algebra 2 classes, we had our first quiz, which challenged all the students in different ways. My vision for the start of Algebra 2 had been to lay down a solid tech foundation alongside review or re-teaching of linearity skills, so that kids realize that a corner stone of Algebra 2 has to be using technology flexibly in order to self-monitor accuracy. I told the kids that I don't want them to ask me, "Is this right?" but I would be very happy to hear them ask, "How can I check this using the calculator?" So, on the first quiz they needed to demonstrate this skill throughout the algebra problems, in order to earn full points on reflection. (I gave separate points for: Communication, Approach, Accuracy, and Reflection.) In the end, my two groups were challenged differently; the Grade 10 native-speaker class had holes in their algebra skills and couldn't complete all problems, and the Grade 9 largely non-native speaker class generally showed stronger algebra skills (even with less in-class whiteboarding practice), but struggled with using the calculator flexibly. But, some of the kids shared their effective studying strategies in class, and some of the other kids are planning to see me on Monday for a requiz, so I feel quite hopeful that this first quiz is just the start of a learning dialogue.
By the way, my relationship with the international kids is developing in an interesting way. Because I speak Chinese, I am able to help the kids in my class without watering down the level of tasks I am asking them to complete. But, at the same time I can carry some weight when I see them being off-task and I offer to call their parents in China to have the dialogue directly about their efforts, in Chinese. What an interesting situation for me and them. Interestingly, they are better-behaved for me, and they try hard to speak English in my class, except when they need to help each other translate something. I am curious how they are going to do on their first big writing assignment, which we will be doing next week...
In my Precalculus class, kids are wrapping up the rough drafts of their first big project on special (triangular and stellar) numbers. I had to be very explicit in helping them to format their writeup. (I had to say at one point, "Take out a sheet of paper. At the top, write, 'In this task, I was asked to...' Now, complete that thought in your own words. I give you 30 seconds to do that. ...Now, write down, 'In order to accomplish that, first we had to...' and go ahead and complete that thought, make sure you insert a diagram here." But, after about 5 minutes of modeling, I think they all got the idea and were able to continue the rest at home, because all the drafts that they brought back to me the next day looked pretty coherent. So, it has been a tough project for them, for sure, but I still think it had tremendous learning value. We also had a quiz, and the kids are doing fine with function identification, interpretation of f(3)=7, and writing both recursive and explicit equations for arithmetic sequences. Some of the more clever ones were able to write formulas for quadratic sequences already, based on their learning from the Special Numbers project. So, I am pretty happy so far. As we wrap up our project (meaning, as I read over their drafts), the kids are doing mixed lines review and checking all answers via their calculator. They use either the Table or Trace to check all equations that they write from given info, and they use [2nd][Math] to verify equivalent expressions after simplifying. So far, so good, because kids in this class seem to be quite independent.
In my Calculus class, I had one student come forward to say that he really enjoys the exploratory nature of our class, and two others who came to ask me to do more examples followed by practice. I thought over this carefully and decided that although I think it is awesome that kids are being advocates for their own learning, and I really wanted to acknowledge that and to encourage that, the issue is really much more complex than their individual learning styles. I ended up describing to the class two contrasting learning models, direct instruction and inquiry-based learning. I said that in most math classes they have had, they probably experienced the former (intro, example, guided example, individual practice, closure, and eval), and that that is fine. It is comfortable, you know what to expect when you come to class. But, that way only reaches the top half of the class, the half that is fortunate enough to maintain focused attention and to comprehend at the speed of material presentation. Then I showed them a diagram of inquiry-based learning, which is a cycle of asking questions, investigation, creation of model or new knowledge, discussion, reflection, and back to asking questions. I explained to them that what we do in class is NOT true inquiry, because true inquiry would be like me saying, "Go. Find out how much universal health care is going to cost our country, both in the short run and in the long run." The problem would be entirely open-ended, complex, and vast, and we would learn all the necessary math skills as we move along. I explained to them that what we do typically in our class is a smaller version of this; within the individual topics of Calculus, I try to think about ways to structure our class so that they can create their own understanding. I ended the class with showing a little clip from Sir Ken Robinson's tedX talk (the animated one), and saying that teaching creativity is hard, and that our traditional schools have been doing a good job killing creativity. I told the kids that, yes, I think even in math there is room for creativity in the classroom, and unfortunately we don't get that by me doing an example and then handing out 25 problems that look the same. So, although I will try to find a balance between direct instruction and exploratory learning, I want the kids to keep an open mind and to appreciate opportunities for creativity in any discipline.
After this talk, I heard from another student in this class who said that she really enjoys math this year, and that our chat helps her to understand and appreciate my philosophy even more. So, one point for being authentic with kids and treating them as intellectual equals.
Good second week. Our school is awesome, by the way. I love that kids clean the school three times a week, and I adore my colleagues!!!!!
In my Algebra 2 classes, we had our first quiz, which challenged all the students in different ways. My vision for the start of Algebra 2 had been to lay down a solid tech foundation alongside review or re-teaching of linearity skills, so that kids realize that a corner stone of Algebra 2 has to be using technology flexibly in order to self-monitor accuracy. I told the kids that I don't want them to ask me, "Is this right?" but I would be very happy to hear them ask, "How can I check this using the calculator?" So, on the first quiz they needed to demonstrate this skill throughout the algebra problems, in order to earn full points on reflection. (I gave separate points for: Communication, Approach, Accuracy, and Reflection.) In the end, my two groups were challenged differently; the Grade 10 native-speaker class had holes in their algebra skills and couldn't complete all problems, and the Grade 9 largely non-native speaker class generally showed stronger algebra skills (even with less in-class whiteboarding practice), but struggled with using the calculator flexibly. But, some of the kids shared their effective studying strategies in class, and some of the other kids are planning to see me on Monday for a requiz, so I feel quite hopeful that this first quiz is just the start of a learning dialogue.
By the way, my relationship with the international kids is developing in an interesting way. Because I speak Chinese, I am able to help the kids in my class without watering down the level of tasks I am asking them to complete. But, at the same time I can carry some weight when I see them being off-task and I offer to call their parents in China to have the dialogue directly about their efforts, in Chinese. What an interesting situation for me and them. Interestingly, they are better-behaved for me, and they try hard to speak English in my class, except when they need to help each other translate something. I am curious how they are going to do on their first big writing assignment, which we will be doing next week...
In my Precalculus class, kids are wrapping up the rough drafts of their first big project on special (triangular and stellar) numbers. I had to be very explicit in helping them to format their writeup. (I had to say at one point, "Take out a sheet of paper. At the top, write, 'In this task, I was asked to...' Now, complete that thought in your own words. I give you 30 seconds to do that. ...Now, write down, 'In order to accomplish that, first we had to...' and go ahead and complete that thought, make sure you insert a diagram here." But, after about 5 minutes of modeling, I think they all got the idea and were able to continue the rest at home, because all the drafts that they brought back to me the next day looked pretty coherent. So, it has been a tough project for them, for sure, but I still think it had tremendous learning value. We also had a quiz, and the kids are doing fine with function identification, interpretation of f(3)=7, and writing both recursive and explicit equations for arithmetic sequences. Some of the more clever ones were able to write formulas for quadratic sequences already, based on their learning from the Special Numbers project. So, I am pretty happy so far. As we wrap up our project (meaning, as I read over their drafts), the kids are doing mixed lines review and checking all answers via their calculator. They use either the Table or Trace to check all equations that they write from given info, and they use [2nd][Math] to verify equivalent expressions after simplifying. So far, so good, because kids in this class seem to be quite independent.
In my Calculus class, I had one student come forward to say that he really enjoys the exploratory nature of our class, and two others who came to ask me to do more examples followed by practice. I thought over this carefully and decided that although I think it is awesome that kids are being advocates for their own learning, and I really wanted to acknowledge that and to encourage that, the issue is really much more complex than their individual learning styles. I ended up describing to the class two contrasting learning models, direct instruction and inquiry-based learning. I said that in most math classes they have had, they probably experienced the former (intro, example, guided example, individual practice, closure, and eval), and that that is fine. It is comfortable, you know what to expect when you come to class. But, that way only reaches the top half of the class, the half that is fortunate enough to maintain focused attention and to comprehend at the speed of material presentation. Then I showed them a diagram of inquiry-based learning, which is a cycle of asking questions, investigation, creation of model or new knowledge, discussion, reflection, and back to asking questions. I explained to them that what we do in class is NOT true inquiry, because true inquiry would be like me saying, "Go. Find out how much universal health care is going to cost our country, both in the short run and in the long run." The problem would be entirely open-ended, complex, and vast, and we would learn all the necessary math skills as we move along. I explained to them that what we do typically in our class is a smaller version of this; within the individual topics of Calculus, I try to think about ways to structure our class so that they can create their own understanding. I ended the class with showing a little clip from Sir Ken Robinson's tedX talk (the animated one), and saying that teaching creativity is hard, and that our traditional schools have been doing a good job killing creativity. I told the kids that, yes, I think even in math there is room for creativity in the classroom, and unfortunately we don't get that by me doing an example and then handing out 25 problems that look the same. So, although I will try to find a balance between direct instruction and exploratory learning, I want the kids to keep an open mind and to appreciate opportunities for creativity in any discipline.
After this talk, I heard from another student in this class who said that she really enjoys math this year, and that our chat helps her to understand and appreciate my philosophy even more. So, one point for being authentic with kids and treating them as intellectual equals.
Good second week. Our school is awesome, by the way. I love that kids clean the school three times a week, and I adore my colleagues!!!!!
Wednesday, September 11, 2013
Week 1 Teaching
My year at school has begun, and as of today, we have had a full 5 days' worth of classes, even though lots of kids were missing class here and there for special retreat-type of activities. I feel quite settled, and I am starting to learn most of the kids' names despite having a terrible memory.
I am thrilled about how Precalculus is going! It's almost too good to be true -- the kids can build their own patterns from colored blocks; they can write recursive equations to represent those patterns; they can explain the limitations of knowing only the recursive formulas; they can write explicit equations that turn out to be linear; they can sum arithmetic elements and explain why that sum must be quadratic, both from a graphical standpoint and an algebraic standpoint. And they're ready to break down quadratic patterns into linear elements, in order to find the formula for the n-th quadratic element! ...Rock ON, 11th-graders!! They will start their first project very soon, either tomorrow (if they understand all preceding concepts) or next week. It is a project that I adopted from the IB, watered down quite a bit simply because I don't want the kids to be scared off right away.
My vision of tying linearity and quadratics neatly together for the kids with the concept of sequences is working out well so far; we'll see how they fare on the first quiz! :)
In Calculus, I discovered on Day 1 that the kids were strangers to the powers of the graphing calculator. I spent the next day walking them through basic features, and since then they've been flying through the conceptual material which I've presented through explorations. Unfortunately, this class has had the most amount of absences, since the Seniors and various special leadership folks had to miss class starting on Tuesday of this week. That has caused the pacing to be a bit chaotic. The kids who have been here everyday now have a solid grasp of the derivative: what it is, how to read it from the calculator, and how it relates to average rate. They can visualize derivative graphs when looking at an abstract original graph of f, and in general I think their calculator skills are slowly maturing. Yay!
My Algebra 2 classes are two diverse groups. One is full of international students, some of who had just stepped off the plane to arrive in America with very little English comprehension. This has made for a very interesting challenge, trying to teach those kids in the same class as the native speakers. I think it is great -- it helps to teach all kids the virtues of patience and kindness, because some of the non-native speakers are much stronger in math than some of their native-speaking peers, so in my mind at least, it's a bit of give-and-take for everyone. The other Algebra 2 class is full of bubbly tenth-graders, and most of them are my advisees. I simply love that class! They don't work as fast as the other group through the material, but they're very responsible, eager to learn, and nice. Kids from my Grade 10 class keep saying that that seems like the fastest-passing period, and they seem to genuinely enjoy their time in class even though we're still on the nitty gritties of basic algebra. We're working our way through some knowledge about how to use the graphing calculator flexibly to check our work alongside reviewing skills from Algebra 1 and wrapping our mind around visual linear patterns. The first big project will be coming up soon though (next week!), and it's the rubber chicken bungee jumping project (I think at most schools, they use barbies), which will involve a significant amount of writing and really show kids what I expect this year from their analyses. Whee! I'm excited!
How are your school years going??
I am thrilled about how Precalculus is going! It's almost too good to be true -- the kids can build their own patterns from colored blocks; they can write recursive equations to represent those patterns; they can explain the limitations of knowing only the recursive formulas; they can write explicit equations that turn out to be linear; they can sum arithmetic elements and explain why that sum must be quadratic, both from a graphical standpoint and an algebraic standpoint. And they're ready to break down quadratic patterns into linear elements, in order to find the formula for the n-th quadratic element! ...Rock ON, 11th-graders!! They will start their first project very soon, either tomorrow (if they understand all preceding concepts) or next week. It is a project that I adopted from the IB, watered down quite a bit simply because I don't want the kids to be scared off right away.
My vision of tying linearity and quadratics neatly together for the kids with the concept of sequences is working out well so far; we'll see how they fare on the first quiz! :)
In Calculus, I discovered on Day 1 that the kids were strangers to the powers of the graphing calculator. I spent the next day walking them through basic features, and since then they've been flying through the conceptual material which I've presented through explorations. Unfortunately, this class has had the most amount of absences, since the Seniors and various special leadership folks had to miss class starting on Tuesday of this week. That has caused the pacing to be a bit chaotic. The kids who have been here everyday now have a solid grasp of the derivative: what it is, how to read it from the calculator, and how it relates to average rate. They can visualize derivative graphs when looking at an abstract original graph of f, and in general I think their calculator skills are slowly maturing. Yay!
My Algebra 2 classes are two diverse groups. One is full of international students, some of who had just stepped off the plane to arrive in America with very little English comprehension. This has made for a very interesting challenge, trying to teach those kids in the same class as the native speakers. I think it is great -- it helps to teach all kids the virtues of patience and kindness, because some of the non-native speakers are much stronger in math than some of their native-speaking peers, so in my mind at least, it's a bit of give-and-take for everyone. The other Algebra 2 class is full of bubbly tenth-graders, and most of them are my advisees. I simply love that class! They don't work as fast as the other group through the material, but they're very responsible, eager to learn, and nice. Kids from my Grade 10 class keep saying that that seems like the fastest-passing period, and they seem to genuinely enjoy their time in class even though we're still on the nitty gritties of basic algebra. We're working our way through some knowledge about how to use the graphing calculator flexibly to check our work alongside reviewing skills from Algebra 1 and wrapping our mind around visual linear patterns. The first big project will be coming up soon though (next week!), and it's the rubber chicken bungee jumping project (I think at most schools, they use barbies), which will involve a significant amount of writing and really show kids what I expect this year from their analyses. Whee! I'm excited!
How are your school years going??
Friday, August 23, 2013
Precalc Breakdown
Continuing with my Precalc brainstorm, I went through my new school's textbook (Functions Modeling Change, Fourth Edition, from John Wiley & Sons, Inc.), and pulled out specific content strands to correspond to each of the core topics I will need to teach. The current Precalc teachers don't all teach sequences as a topic in Precalc, but when I took it out, I found that no matter how I sequenced the basic algebra topics, they always seemed somewhat random and disconnected. So, I will still start off with a sequences discussion, in order to tie everything together for my own students. I also brainstormed a project for each Precalc unit during the year. If I end up throwing them out, it's ok! But, hopefully this gives me ideas to fall back on throughout the year. If they don't have references, either I have that project lying around from my own creation before or I think it'll be fairly straight-forward to pull one together this year.
Thoughts??
Thoughts??
Topic
|
Key
Lessons
|
Project
or Lab
|
From Arithmetic Sequences to
Linear and Quadratic Functions |
1. Sum
of arithmetic sequences
2. Quadratic
patterns and quadratic formulas from applying sequence formulas to sum 2nd
differences
Daily openers: Practice
linear and quadratic skills
|
Triangular Numbers and Stellar Numbers (past IB Portfolio topic)
|
From Geometric Sequences to Exponential Functions
|
1. Sum
of geometric sequences
2. Exponential
formulas from applying sequence formulas to geometric sequences
3. Recursive
vs. explicit formulas
Daily openers: Practice
basic exponent and log skills
|
|
Modeling and Visualizing Graphs of Basic Forms
|
1. Interpretation
of parameters within a formula, domain, range
2. Compare
and contrast linear vs. exponential word problem setups
3. Using
GCD to analyze word problems after proper setup
Daily openers:
Practice graph prediction from formula
|
Regression lab via RC-circuit
|
Transformations of Families of Functions
|
1. Review
function notation
2. Use
the studied functions to reinforce basic transformations knowledge
Daily openers:
Practice transformations, forwards and backwards to/from formulas
|
@Samjshah’s family of transformed functions art project
|
Mini Capstone Unit: Modeling with Functions
|
1. Go
over function forms and uses (including forms not yet studied in the course)
2. Give
students guidelines on modeling assignment
3. Do
one example during class and then provide time in class to complete
individual assignments
Daily openers: Adjust
as perceiving student needs.
|
Use old IB portfolio topics
|
Trigonometry of Circles and Waves
|
1. Review
non-right triangle trig with project from @KFouss
2. Motivate
circular trig using rollercoaster problem
3. Unit
Circle
4. Graphing
and transforming waves in both modes
5. Solving
trig equations (graphical and algebraic)
6. Trigonometric
identities
7. Tangent
function and GCD solutions
Daily openers:
Unit circle memorization; from hand-drawn graphs to finding trig formulas (and
thereafter, to finding GCD solutions for intersection points)
|
Non-right triangle trig project from @KFouss
Building an animated rollercoaster in Geogebra
|
End Behaviors of Polynomial and Rational Functions
|
1. Start
with modeling problems to motivate the various forms of new functions
2. Fully
develop each function type completely, paying attention to Calculus
terminology
3. Graphical
analysis of instantaneous rates along a graph
Daily openers: Predictions
of graphs based on formulas; verify with GCD
|
Stocks modeling project
Field trip planning project / write proposal to principal
|
Logarithmic Functions
|
1. Function
inverses
2. Motivate
topic using exponential vs. log scales
3. Mixed
practice in interpreting log scale graphs
Daily openers:
Solving logarithm equations
|
Research/poster representing an exponential dataset using multiple
forms of graphs, discussing visual tradeoffs
|
Non-functions and Modeling Using Different Coordinate Systems
|
1. Circles
and Ellipses
2. Polar
coordinate system
3. Hyperbola
and the idea of Locus
Daily openers: From
Cartesian to Polar Coordinates, vice versa
|
Art project via Desmos
|
Year-long, Ongoing Review
|
Create a Precalculus Magazine summarizing each unit that we have
learned.
|
Thursday, August 1, 2013
Precalculus Retirement Project
I came up with a pretty useful retirement project idea for my Precalculus class. I say this is useful because more and more, I find that a lot of our friends are stressing over what's going to happen when our parents retire. Are they going to have enough money to get by? For how long? It seems to be a somewhat complex problem to predict accurately, because each month both the interest compounds and the principal left in the bank decreases.
So, to that effect, I plan to do a retirement project at the end of studying sequences and recursive/explicit formulas, and the kids are going to write a letter to their parents to make recommendations on why planning for a continued source of income post-retirement is really essential. Hopefully, this will help their parents talk to them a bit about money and financial planning, which I find that in some families is not as open a topic as it ought to be.
Check it out: http://bit.ly/retirementProj
It's still early in my planning of the course, and I would love any suggestions you may have! Geoff recommended that I could teach the kids to cross-check the explicit formula results from their calculator tables against the results from Excel (which are essentially iterations of the recursion from row to row, when you drag the formula downwards).
So, to that effect, I plan to do a retirement project at the end of studying sequences and recursive/explicit formulas, and the kids are going to write a letter to their parents to make recommendations on why planning for a continued source of income post-retirement is really essential. Hopefully, this will help their parents talk to them a bit about money and financial planning, which I find that in some families is not as open a topic as it ought to be.
Check it out: http://bit.ly/retirementProj
It's still early in my planning of the course, and I would love any suggestions you may have! Geoff recommended that I could teach the kids to cross-check the explicit formula results from their calculator tables against the results from Excel (which are essentially iterations of the recursion from row to row, when you drag the formula downwards).
Monday, July 29, 2013
Precalculus Brainstorm
I was brainstorming for my Precalc class this morning, and I realized that my perception of the Precalc topics has changed since I began teaching IB two years ago! The IB equivalent of this course heavily emphasizes the interconnectedness in applying them, which has in turn changed the way I wish to organize my Precalc class next year. I'd like to organize my Precalc class this year as a sequence of topics that each arises naturally from the previous topic, with the entire course anchored upon modeling as the core skill and purpose for building further functional knowledge.
See flow chart below. (The bold parts are what I think are the more important concepts from the course.) If my new colleagues would support my decision in organizing my course this way, then I hope to start with sequences as a way of re-introducing linear, quadratic, and exponential forms. The kids will see, for example, that if you use summation formulas to capture the sum of second differences, then the resulting n-th element will have a quadratic form in terms of n. At the end of those basic functions re-introduction, my hope is that the kids can do a written project analyzing triangular and stellar numbers, similar to the old IB portfolio task from a few years back. (I've misplaced that prompt now, so I'll have to create one that is similar.)
Then, using their knowledge of these basic functional forms as a basis, we will examine the graphs formed by these basic forms and use that to re-introduce the core concept of transformations. We will learn the other functional types only as necessitated by modeling of different types of data, so that the kids can always remember the importance of contextual analysis and interpretation. Eventually, at the end of the course, each student will do two modeling projects:
1. a project using GeoGebra or Desmos in order to create a picture with functions and to practice basic functions modeling and specifying domain restrictions.
2. a real-world modeling project of their choice, in order to practice asymptotic analysis and written communication. In this final project, we can add additional requirements such as analyzing the rates of change, in order to preview some introductory concepts from Calculus.
Thoughts? Do you think this organization would make sense to students?
See flow chart below. (The bold parts are what I think are the more important concepts from the course.) If my new colleagues would support my decision in organizing my course this way, then I hope to start with sequences as a way of re-introducing linear, quadratic, and exponential forms. The kids will see, for example, that if you use summation formulas to capture the sum of second differences, then the resulting n-th element will have a quadratic form in terms of n. At the end of those basic functions re-introduction, my hope is that the kids can do a written project analyzing triangular and stellar numbers, similar to the old IB portfolio task from a few years back. (I've misplaced that prompt now, so I'll have to create one that is similar.)
Then, using their knowledge of these basic functional forms as a basis, we will examine the graphs formed by these basic forms and use that to re-introduce the core concept of transformations. We will learn the other functional types only as necessitated by modeling of different types of data, so that the kids can always remember the importance of contextual analysis and interpretation. Eventually, at the end of the course, each student will do two modeling projects:
1. a project using GeoGebra or Desmos in order to create a picture with functions and to practice basic functions modeling and specifying domain restrictions.
2. a real-world modeling project of their choice, in order to practice asymptotic analysis and written communication. In this final project, we can add additional requirements such as analyzing the rates of change, in order to preview some introductory concepts from Calculus.
Thoughts? Do you think this organization would make sense to students?
Saturday, July 27, 2013
Differential Calculus Intro - Feedback Please!
I find it difficult to brainstorm without laying down some specifics, so I pulled together a potential Unit 1 for introducing differential Calculus without actually introducing differentiation rules.
Please take a look, Calculus teachers, and let me know what you think! http://bit.ly/differentialCalculusIntro. I'll be trying to meet up with one of my new colleagues soon to discuss some of this material, since she is trying to pull together a projects-based Calculus class (somewhat more slow-paced than the regular Calculus class that I'll be teaching) and she's also working on the explicit incorporation of Habits of Mind this year into all of our curricula. But, in the mean time, I'd love any feedback on whether: 1. you think this is paced too slowly for a "regular" (non-AP) calculus class? 2. Are the worksheets relevant and appropriate for the level of course? Our classes are about 45 minutes each.
Thanks!
After this intro unit, depending on how my colleagues respond to my idea of "teaching backwards", I'll either be planning a similar "integral Calculus intro" unit that focuses on application and skips over the algebra skills, or we'll be doing manual differentiation skills as the next unit...
Addendum 7/31/2013: Thanks to Sam Shah, I will be sharing this link on the Calculus of saying I love you with my students during this first introductory unit to derivatives!
Please take a look, Calculus teachers, and let me know what you think! http://bit.ly/differentialCalculusIntro. I'll be trying to meet up with one of my new colleagues soon to discuss some of this material, since she is trying to pull together a projects-based Calculus class (somewhat more slow-paced than the regular Calculus class that I'll be teaching) and she's also working on the explicit incorporation of Habits of Mind this year into all of our curricula. But, in the mean time, I'd love any feedback on whether: 1. you think this is paced too slowly for a "regular" (non-AP) calculus class? 2. Are the worksheets relevant and appropriate for the level of course? Our classes are about 45 minutes each.
Thanks!
After this intro unit, depending on how my colleagues respond to my idea of "teaching backwards", I'll either be planning a similar "integral Calculus intro" unit that focuses on application and skips over the algebra skills, or we'll be doing manual differentiation skills as the next unit...
Addendum 7/31/2013: Thanks to Sam Shah, I will be sharing this link on the Calculus of saying I love you with my students during this first introductory unit to derivatives!
Wednesday, July 24, 2013
Calculus Project Resources
I am combing the web for project ideas. Here is a cheat sheet for myself, and maybe for you as well:
Designing a smooth-riding rollercoaster. I know that there are different versions of this project out there, and when the time comes, I'll be doing some shopping around to make sure I'm using the most kid-friendly version. I think my new school already has a version of this project floating around somewhere in the shared files.
Some sciency projects for Calculus.
Some nice activity sheets to spark discussion. (I know, they're not projects, but they're nice, no?)
Also, as a bonus find, I came across an easy-to-read explanation of what makes good math writing , and why writing is necessary in a math class.
Ah, I'm glad it's still just July. Even though I am keeping busy, summer is still cool and bright for another few weeks.
I like these Calculus writing projects because they're based in realistic context, and the kids have to
write a letter to explain their recommendations and to justify them with
math explanations.
More of the same type of writing projects, but extra nicely formatted.
Art, technology, and Calculus - what could be better? Designing a smooth-riding rollercoaster. I know that there are different versions of this project out there, and when the time comes, I'll be doing some shopping around to make sure I'm using the most kid-friendly version. I think my new school already has a version of this project floating around somewhere in the shared files.
Some sciency projects for Calculus.
Some nice activity sheets to spark discussion. (I know, they're not projects, but they're nice, no?)
Also, as a bonus find, I came across an easy-to-read explanation of what makes good math writing , and why writing is necessary in a math class.
Ah, I'm glad it's still just July. Even though I am keeping busy, summer is still cool and bright for another few weeks.
Calculus Brainstorm
I will be teaching Calculus for the first time next year. Yay! So excited about that. In the IB course, I had taught Calculus as a topic, but not as a full year-long course. I am very excited about the prospects of teaching an end-to-end Calculus class and having time to delve into all the nitty gritties. I am spending part of the summer sorting out the overall organization of the course, although I am sure it is to be changed once I start the year and chat with my future colleagues to get feedback.
In my mind, I picture two contrasting ways of structuring a Calculus course (and a hybrid way of doing something in between):
1. The traditional way: You introduce limits/infinity, followed by derivatives and their applications, followed by integrals and their applications, followed by mixed Calculus applications practice. Maybe you'd have a research component in the start of the year, in which kids do some research on all the modern applications of Calculus. From the various areas of applications, each student would choose one area to complete a capstone project on. This gives you chance to give them written feedback on their capstone project throughout the year, and their final paper to be submitted at the end of the year would have to contain an explanation of the relevant Calculus concepts, show example calculations that are applied within their area of research, incorporate technology as a way of enhancing/verifying their results, and they would present their projects to their peers as a way of solidifying the entire class's understanding of the applicability of Calculus.
2. Working backwards: You first spend some weeks taking the kids through the many ways of analyzing real-world problems using basic differential and integral concepts. They do everything on their calculator only, and they don't learn any of the algebra skills until after they have mastered the interpretation/application of results in the setup and analysis of a problem. Then, you peel away one layer of the magic and you work on all the core algebra skills of differentiation and integration without calculator, while consistently referring back to the applications that they have already seen, so that they don't forget the point of these algebra skills. Then, after they master doing all the same contextual applications manually and verifying their differentiated/integrated results via graphing calculators, you go on to more advanced concepts such as related rates problems, which cannot be explored solely on a calculator. Eventually, once the students have learned all the ins and outs of the core Calculus skills, you finally peel away the last layer of the magic and tie those "algebra shortcut" skills to the ideas of limits and infinity. The advantage of leaving limits to last is that the kids would already know what the derivative and integral formulas should look like by this point, and in taking the limits, they can self-monitor correctness. This "backwards" structure provides them with the full picture by the end of the course, of where everything comes from, but hopefully you have provided so much reinforcement of the applications that no kid will walk away from your class with only vague ideas of where Calculus is useful or how all the pieces are interconnected.
I talked briefly to one of my department chairs, and he is very supportive of the latter approach as an experiment. That makes me pretty excited! I will wait until I have chatted with the other Calculus teachers to start specific planning, since I have never taught Calculus as a full course before, but I think that our school culture is very supportive of teachers trying new things. So, I am going to try to wiggle my way towards a somewhat backwards organization of the course, because that makes the most sense to me.
I did some quick research, and these are the Calculus applications that I would like to introduce, either at the beginning of the year or at the start of each unit prior to developing those specific skills. Are there other ones that I am leaving out, but that are very common/useful for students to know?
In my mind, I picture two contrasting ways of structuring a Calculus course (and a hybrid way of doing something in between):
1. The traditional way: You introduce limits/infinity, followed by derivatives and their applications, followed by integrals and their applications, followed by mixed Calculus applications practice. Maybe you'd have a research component in the start of the year, in which kids do some research on all the modern applications of Calculus. From the various areas of applications, each student would choose one area to complete a capstone project on. This gives you chance to give them written feedback on their capstone project throughout the year, and their final paper to be submitted at the end of the year would have to contain an explanation of the relevant Calculus concepts, show example calculations that are applied within their area of research, incorporate technology as a way of enhancing/verifying their results, and they would present their projects to their peers as a way of solidifying the entire class's understanding of the applicability of Calculus.
2. Working backwards: You first spend some weeks taking the kids through the many ways of analyzing real-world problems using basic differential and integral concepts. They do everything on their calculator only, and they don't learn any of the algebra skills until after they have mastered the interpretation/application of results in the setup and analysis of a problem. Then, you peel away one layer of the magic and you work on all the core algebra skills of differentiation and integration without calculator, while consistently referring back to the applications that they have already seen, so that they don't forget the point of these algebra skills. Then, after they master doing all the same contextual applications manually and verifying their differentiated/integrated results via graphing calculators, you go on to more advanced concepts such as related rates problems, which cannot be explored solely on a calculator. Eventually, once the students have learned all the ins and outs of the core Calculus skills, you finally peel away the last layer of the magic and tie those "algebra shortcut" skills to the ideas of limits and infinity. The advantage of leaving limits to last is that the kids would already know what the derivative and integral formulas should look like by this point, and in taking the limits, they can self-monitor correctness. This "backwards" structure provides them with the full picture by the end of the course, of where everything comes from, but hopefully you have provided so much reinforcement of the applications that no kid will walk away from your class with only vague ideas of where Calculus is useful or how all the pieces are interconnected.
I talked briefly to one of my department chairs, and he is very supportive of the latter approach as an experiment. That makes me pretty excited! I will wait until I have chatted with the other Calculus teachers to start specific planning, since I have never taught Calculus as a full course before, but I think that our school culture is very supportive of teachers trying new things. So, I am going to try to wiggle my way towards a somewhat backwards organization of the course, because that makes the most sense to me.
I did some quick research, and these are the Calculus applications that I would like to introduce, either at the beginning of the year or at the start of each unit prior to developing those specific skills. Are there other ones that I am leaving out, but that are very common/useful for students to know?
Calculus applications
1. Optimization
* Maximum enclosed area
* Build a box (hands-on activity)
* Max/min rotating distances from a fixed point (incorporate technology)
2. Instantaneous rates / steepness
* Falling ladder analysis (exploratory starting with diagrams/tables)
* Learning curve, length of list vs. memorization time (lead-in with experiment)
* Marginal cost / marginal revenue / marginal profit
* Incremental effect of education/experience on income
* Rates of change within different savings account setups after fixed time
3. Summation of variable quantity
* Physics applications (distance, velocity, acceleration)
* Total work required/done in stretching a spring
* Total interests paid on a loan over time
* Supply and demand analysis – total opportunity cost for producer
* Supply and demand analysis – total consumer surplus, total producer surplus
* Architectural integration for accurate surface area
* Center of mass (difficult)
4. Average of a function
* Average velocity
* Average day length over a season
* Average power of an AC circuit
* Moving average of stocks (mini-project?)
* Maximum enclosed area
* Build a box (hands-on activity)
* Max/min rotating distances from a fixed point (incorporate technology)
2. Instantaneous rates / steepness
* Falling ladder analysis (exploratory starting with diagrams/tables)
* Learning curve, length of list vs. memorization time (lead-in with experiment)
* Marginal cost / marginal revenue / marginal profit
* Incremental effect of education/experience on income
* Rates of change within different savings account setups after fixed time
3. Summation of variable quantity
* Physics applications (distance, velocity, acceleration)
* Total work required/done in stretching a spring
* Total interests paid on a loan over time
* Supply and demand analysis – total opportunity cost for producer
* Supply and demand analysis – total consumer surplus, total producer surplus
* Architectural integration for accurate surface area
* Center of mass (difficult)
4. Average of a function
* Average velocity
* Average day length over a season
* Average power of an AC circuit
* Moving average of stocks (mini-project?)
5. Related rates (we won't work on these until later in the year)
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