Group Quizzes

Every time I do a group quiz in class, it’s in the context of a “Participation Quiz.”  For a description of a participation quiz, I’ll give a pointer to the excellent description and the discussion going on in the comments at Continuous Everywhere but Differentiable Nowhere.  At some point, I’ll follow-up with a post here with some ideas I’m going to try out this coming year with regard to participation quizzes.

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Completing the Square

Some advice that was given to me when I first started teaching regular PreCalculus was to skip or deemphasize completing the square.  However, I didn’t shy away from the topic because I felt it could be approached in a way that appealed to the fact that the average person enjoys solving puzzles.  I plant the seeds by introducing “Avoid the Freshman Mistake Puzzles” as warmups or puzzles for those last few minutes of class, for about a week before we dive into the topic.  This handout that I created is the most formal thing I have written up (we normally do this more informally in class).

I’ve also covered this topic in my Intermediate Algebra class (a post-Algebra 2 or Pre-PreCalculus class) with a population of students that have generally struggled with math in previous years.  One of the struggles that we often have in teaching high school math is the need to teach or review fractions and how to do that with a population that is very fraction-phobic.  This year, “completing the square” turned into an accidental ah-ha moment for many in terms of understanding fractions.  (Working on my National Boards this year, I wish I would have had the camera rolling for this whole-group discussion…)  Using words here, I know I’m not doing a good job of capturing the discussion, but basically the problem the class was wrestling with was how to find out what half of 3/8 was.  Various students described their thinking and we ended up looking at a visual representation of 3/8 in a rectangle and cutting each shaded part in half and seeing that half of 3/8 was 3/16.  Then the excellent generalization question came from the audience:  “So can we always just double the denominator to cut a number in half?”  Those are the types of questions that make a teacher proud of their students :).  From that came the observation that that’s what happens when we multiply horizontally using fractions.  Other ideas that came up in the discussion included the way we found half of 4/9 in a previous problem and the connection between thinking of it as four “ninths” and then half of it was just two “ninths,” versus going to 4/18 and then reducing, and why the denominator changes when multiplying but not adding fractions.  And all this on a 5th period on a Friday afternoon, right after lunch and about an hour before the weekend.

Just thinking about how far this class has come from September to now makes me so proud of them!  This is precisely the sort of thing I was refering to in my previous post about beginning-of-the-year review.  Review in context, in this case not even a “real-world” context but a context of an “advanced” algebra move, is much more meaningful rather than just drilling a bunch of fraction addition and multiplication problems.

How do they do it?

Right now, the impetus for me cleaning out my drafts folder of posts I’ve begun is sitting in a West Seattle coffee shop procrastinating from doing more National Boards writing…  I was genuinely shocked to see the last time I’d posted anything was mid-August.  I’m humbled by the fact that so many other math blogger folks turn out quality, thought-provoking posts on a regular basis amidst the activity and bustle of being a teacher.  So along with the “how do they do it?”, this post is also a “thank you!” to all the other bloggers I read on a regular basis for taking the time to share your ideas and your thinking, and giving the rest of us food for thought.

Review, ad infinitum…

In elementary school, I used to get excited right around when we reached p.100 of our math books, because that’s when we would really start doing stuff.  Those triple digits meant that lessons would no longer be about practicing addition and subtraction, or rehashing things we were supposed to have learned the year before.  Maybe that first month or so of the school year taught me patience, otherwise, I would have gone mad.  Unfortunately, as a teacher, I’d fallen into the same trap of needing to do that all-important review as the first unit of the school year.  What better way to kill the potential energy built up over the summer and bring whatever momentum had built up to a grinding halt?  In retrospect, it seems that “review” seems to make students who already got it last year either check out or get lulled into a sense of complacency.  For students who didn’t “get it” last year, instead of starting with a clean slate, it instead drives home the message that this year will be more of the same.

This is not to say that review isn’t important or necessary, but what makes the difference is the way that it is done.  This year, a colleague and I rearranged the sequence of topics in the PreCalculus courses that we taught (different schools in the same district).  Instead of first semester being largely a rehash of Algebra 2, we started with Trigonometry first.  This is the fifth year that I’ve been fortunate enough to teach PreCalculus, and I’m kicking myself for not coming to this realization sooner.  Even though we’re starting with a topic that’s new for most students, we’re still able to secretly review solving equations, working with exponents, and working with fractions, all in the context of doing trig.  Instead of two separate groups (those who got it last year and those who didn’t), everyone was learning something new, on equal footing, and with a clean slate.  I don’t think that there’s anything intrinsically different about the makeup of students this year, but with the new approach, we’ve been able to keep the momentum going from the beginning of the school year when students come back from summer, eager to learn something new.  We’ve been able to cover more material, go into more depth, and maintain a healthy pace, especially compared to previous years.

I’m slated to teach Algebra 1 next year and am hoping that this idea is something the team will be willing to run with, or at least consider.

All Seniors Take Calculus…

I have a terrible memory and so acronyms are something that I rely on…  “Every Good Boy Does Fine”, “FACE”, “Great Big Dogs Fight Alligators”, and “All Cows Eat Grass” are some that I live by.  Even “SOH CAH TOA” is something that’s helped me for nearly 20 years.  Acronyms and shortcuts are great for brute-force memorization.  However, I’m putting this post out there to maybe plead for help in getting my anger management in check.  I hate, really really hate, perhaps even despise the acronym “All Seniors Take Calculus.”  I’m not sure why it prokes such a strong negative reaction within me – I think it’s because it really strips away meaning at the cost of coming up with a cute saying.  After all, if one understands that (and why) the x-coordinate on the unit circle is the cosine of the angle, the y-coordinate on the unit circle is the sine of the angle, and y/x is the tangent of the angle, one doesn’t need the overhead of “All Seniors Take Calculus.”  Plus, how does that saying help deal with figuring out that the cosine of 180 degrees is negative, for example?  Help me not be so hateful…

Computer Science: Principles – Possible new AP course?

I came across the webcasts for AP Computer Science 10 at UC Berkeley, which seems to be a pilot course for Computer Science Principles (http://csprinciples.org/).  Althought it’s a non-major course, I think it’s an excellent idea.  I suspect that many students end up majoring in the subject because of courses like this.  Speaking from personal experience, I did end up majoring in computer science after taking the non-major introductory course (CS 3 at the time).  I think it would be wonderful for mathematics to be able to better publicize and offer a similar course for math.  Many colleges offer such a course under titles such as “Math for Liberal Arts” or “Math in Society.”  Different books aimed at such courses are also published, such as Key Curriculum Press’ “The Heart of Mathematics”, or COMAP’s “For All Practical Purposes” (http://www.comap.com/product/textbooks/index.html).  A course like this could provide a broader perspecive of applications of math and at least for me, what’s applealing about this approach is the logic and problem solving involved with more rigorous mathematics than just arithmetic.

Synthetic Division: to teach or not to teach?

I really can’t think of a good reason why synthetic division is still in the Algebra 2/PreCalculus curriculum.  When I was in high school (at my HS, I was in the last class not to use graphing calculators), synthetic division was a handy shortcut because we had to do tons (often enough for a whole page for one problem) of divisions to test potential zeros for higher-order polynomials.  However, with technology as it is, I can think of better ways to spend the 2-3 days it would take to cover synthetic division and finding rational zeros by hand.

I do still teach polynomial division, because it is useful for finding polynomial asymptotes of “improper” rational functions.  But for the synthetic division shortcut, unless I’m doing tons and tons of polynomial division problems (and only the case where we’re dividing by linear factors at that), I really can’t think of a good reason to take the time to do it.  Plus, I have a really bad memory and always have trouble remembering the proper sign for the number on the outside of the synthetic division box (r when the divisor is x-r…).

At my school, synthetic division is taught in Algebra 2, so I’ve been able to use that reason to skip it in PreCalculus.  However, our local community college lists it as a topic that is taught and covered on the final in PreCalculus.  So, if I have students who opt to do the college in the high school program, I do need to cover it in PreCalculus…

Please let me know if there’s something I’m missing, or why you do or don’t do synthetic division.

More Resources…

How could I forget:

8.  Mark Driscoll’s Fostering Algebraic Thinking and Fostering Geometric Thinking.  If you don’t have a copy, order these now, or see if your school can order these for your department.  Along the lines of the Problem of the Week (POW) problems from IMP, these books are great resources for groupworthy problems and include informative commentary about the mathematics involved.  I don’t have my copy handy, so my apologies if the following reference is too cryptic, but the number puzzles that deal with solving systems of equations and the problem with the snake rings for looking at geometric sequences are perennial favorites of mine.

8 is a good place to pause.  Now I’ll get some sleep.

Resources for Good Problems

The excellent question that comes up often seems to be, “Where can I find problems that are rich, groupworthy, and with multiple entry points to use in my classes?”  Being a packrat, I keep accumulating stuff, so I figure I can kill 2 birds with one stone – partly as an organizational tool for myself and to share what I’ve accumulated, I’ll periodically post some pointers to resources that I find useful.  If you have other favorite sources of problems, I’d also love to get pointers to those items in the comments section.

Here’s a quick list of some resources I go to the most.  As time goes on, I hope to post links to more resources and point to some specific activities that I’ve used from the links below.

1.  Henri Picciotto’s Math Ed Page is one I visit often.  His “Make these Lines” and “Make these Parabolas” are ones I’ve used at many different levels.  I also have a copy of his “Geometry Labs” book and find the “Polyominos” activites good for both algebra and geometry.

2.  The Dana Center at UT Austin used to have a bunch of free activities for download from their assessment books.  Click on a link, such as “Algebra II” – the section headings are actually links (it wasn’t obvious to me) – the assessment links have good samples.  There used to be more free content available for download, it looks like they’re now available for purchase.

3.  One of my favorite series is Key Curriculum Press’ Interactive Mathematics Program.  I student-taught parts of Years 1 and 2.  Used copies are available for a reasonable price on Amazon.com.  I often use their Problem-of-the-Week problems for group tasks (typically lasts anywhere from 1-4 class days).  I switch seats every 2 weeks and in my Intermediate Algebra class, I usually had them work through a .5 or 1 day group task when they were placed in new groups.  I also love the way the “Shadows” unit in Year 1 approaches the idea of similarity, and they way that they approach solving proportions.

4.  My go-to resource for PreCalculus is Foerster’s PreCalculus, also from Key Curriculum Press.  He has lots of great word problems, especially for Trig.  The way that he develops the Law of Cosines and Law of Sines is also great.  On my to-do list for next year is to look through his Calculus materials.

5.  Another resource I use heavily for PreCalculus is the University of Washington’s Math 120 materials.  There are lots of challenging problems and interesting application problems.

6.  On the subject of PreCalc, Connally, Hughes-Hallett, and Gleason’s Functions Modeling Change is another book I’d be happy to use as a course text.  Again, there are a number of excellent problems that elicit deep thinking.

7.  Also on my to-do list is to become more familiar with the CME Project materials.  What I have seen so far has been excellent.  At PCMI, I am in the functions working group and the project my partners and I are working on are lessons to help students understand general function transformations.  There is a particular method I used over the last 4 years, which I’ve never seen in any textbook.  That is until I had a chance to chat with someone on the CME Project development team.  It turns out that’s the approach they’re using to function transformations in Algebra 1 and 2, but in a better way.  When I get a chance, I’ll post more details.

7 seems like a good place to pause.  I should go to sleep soon anyway…

AP Precalculus?

Recently, there have been discussions on the AP Calculus mailing list about whether or not there should be an AP Precalculus course. If done well, I think this is a unique opportunity for the College Board. I feel that one of the strengths of the AP program is the reputation it maintains for high standards, thereby allowing many 4-year universities to award credit based on student performance on the AP tests. Precalculus is a course that sits in that grey area. Many 4-year universities do not consider Precalculus to be a credit-bearing course if they offer it at all (evidence needed to support this assertion – I could just be talking out of a body part other than my mouth here). However, Precalculus is a credit-bearing course at community colleges. Oddly enough, I don’t believe that many community colleges award credit for a passing score on the AP Calculus exam. It seems like community colleges have been left out of the conversation at the College Board – is that true? Are 2-year institutions represented in those that read the AP exams? Do they have representation on the committees that determine the curriculum or write the AP test questions? Maybe this is an opportunity to bring the 2-year colleges into the AP conversation. It might also provide options or alternatives for “calculus readiness” tests that many 4-year institutions spend resources on administering.  Perhaps a passing AP Precalc score could count for credit at a 2-year college and count as a prerequisite for Calc I at a 4-year college?

These are the kinds of thoughts that literally keep me up at night.