Grading by Criteria

This is an excerpt from the grading section of my course syllabi this year. My new school is very good about letting each teacher choose an appropriate system for their own class. Two of my math colleagues saw this and thought it was interesting, so I thought that maybe it'd be interesting for some of you to consider as well. For me, this is similar to the grading system used in the IB Middle Years Programme. It also just generally makes more sense to me to be grading by criteria instead of by the type of assignments, especially because as a department we are trying to foster some of those important qualities in our kids.

Grading: Your grades are a direct feedback to you. They will speak to your areas of excellence, versus areas where I believe that you can still grow and develop your skills. They should, therefore, reflect the values that we hold in our course. To this end, your class grade will break down into grading criteria as follows:

Grading Criterion                       Assessed using…
Communication in Math 30%  All collected written explanations (classwork and projects)
Mathematical Approach  40% All collected “algebra work” (quizzes, tests, and projects)        
Accuracy and Precision  20%  All collected “algebra work” (quizzes, tests, and projects)
Reflection on Results      10%  On-going observation (class participation and written projects)

This might be different from what you are used to. Do not worry! You can always request a re-test or a re-quiz if there is something that you wish you had done better on. Also, grading rubrics will be provided in advance of project due dates, in order to clarify how these criteria will be applied specifically to the project.

Even though I still have a lot of things to prepare before the year starts, I am very excited! I think it's going to be a great year.  

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?

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!

Calculus Project Resources

I am combing the web for project ideas. Here is a cheat sheet for myself, and maybe for you as well:

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?

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?)

5. Related rates (we won't work on these until later in the year)