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 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!

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!!!!!

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??

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??

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	Retirement Project / writing letter to parents
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.

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).

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?