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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.

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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.

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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.

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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.

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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…

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

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