Completing the Square

Toward the goal of sharing more of the humdrum, everyday business of teaching math, here are a few notes about how we do completing the square in my classes. I have no cool tricks or creative activies for this, and I would very much like to see more of yours. Nevertheless, completing the square is, inexplicably, a favorite topic of mine.

The first time I taught it I relied somewhat on the formulaic addition and subtraction of the square of half the middle coefficient, but that left the class utterly confused and frustrated. Now I rely less on this rule and more on pattern recognition and intuition. This appears to make for better retention, though the students do have trouble applying it to numerically messy cases, where a formulaic approach - if ever mastered, that is - would be safer.

We start by reviewing how to square a binomial, because while we have of course worked on multiplying binomials and using standard factoring patterns earlier, the error of squaring a binomial by squaring each term is remarkably resistant to instruction. We always need to refresh that by writing out the factors and multiplying, carefully, term by term. Indeed, any time that a review of completing the square is called for later, squaring binomials from scratch is the point I will return to, and invariably it will turn out that many students have forgotten what the squared binomial looks like. After we've done the multiplication from first principles for a handful of examples, I point out the pattern in the middle and last terms of the product and ask the students to pick up speed, which they do. I write up a few binomials where the second term is a fraction and remind the students that one advantage of fraction form over decimal form is that fractions are really easy to square.

I'll call on individual students or have the class shout out answers, and will alternate between having students suggesting problems and solving them ("J., will you give us a binomial?" "S., will you square that for us?") and I have repeatedly been surprised at how engaged the students tend to become during this exchange, since the topic, after all, isn't that inherently exciting, and we aren't doing anything particularly nifty. Part of the reason may be that it is easier than for many other topics to sense just where the students are and to tailor the next example so that it matches their readiness.

When I notice that the class is beginning to get that "now what...?" feeling, we reverse the process: I write up a perfect square trinomial and have students factor it. We keep doing this until the students again reach the point where this is too easy, and then start looking at cases where only the quadratic and linear terms are given and the students need to figure out what numbers would fit in the blank spaces in a form such as this one: Later, writing this form on the board will be sufficient to cue a large fraction of the students to what they are trying to do.

We move on to rewriting simple quadratics (where a=1 and b is an integer) in vertex form. Later I will show them that adding and subtracting the square of half of the coefficient of the linear term will give us just what we want, but at this stage we simply identify the squared binomial, multiply it out, and compare this with the original quadratic to see what we need to add or subtract. For example, to write in vertex form we will recognize that (x-4)^2 is the square term, and since expanding this gives a constant term of 16 we'll need to subtract 13 in order to ensure that we have the same quadratic that we started with: This approach seems to stick fairly well in students' memories. Many students who do not correctly add and subtract the half of the middle coefficient later (they'll insert an x in there, or halve it incorrectly, or something) will be able to rewrite simple quadratics in vertex form, and I can see from their scratch work in the margin that they're just comparing the expanded square with the original quadratic. I'm pleased with that, because the equivalence of the quadratic in its two forms is one of the big ideas I want them to take away, and the fact that we aren't dealing with different quadratics even though they do look different isn't nearly as self-evident to the young ones as it is to us.

So, that was not terribly exciting or innovative, I concede. But how do you teach completing the square?

Joke

- What's the difference between an outgoing Physicist and one who is not?
- The outgoing Physicist looks at your shoes while talking to you.

Oof.

Pi Day

What do you all do for Pi Day? In particular, what might be worth the while in an Algebra 1 class?

Would anyone with a high-traffic blog mind posting some version of this question, in order to cast a wider net? That would be nice of you...

Unequal Methods

Colleagues - I could use some advice. I just graded the Algebra 1 tests on Inequalities and Absolute Value, and they were quite awful. While some of that is due to the distractions of Spirit Week and me being sick a few days, there's more to it, and any reports on successful approaches to teaching inequalities in general and absolute value inequalities in particular would be most appreciated.

A first hurdle is to help students actually understand the number line graphs that they draw of solutions to inequalities. Many use middle school mnemonics about arrows pointing in the same direction as the inequality sign, and draw their graphs from these rules (and go wrong when the variable appears on the right hand sign of the inequality, of course). We've worked on listing a few actual numbers that are part of the solution, plotting these, and then drawing the graph afterward. That helps a good deal after a while.

The next hurdle is to understand the difference between AND and OR inequalities. The most effective approach so far has been a combination of the above mentioned insistence on a list of actual, specific numbers that satisfy the conditions, and lots of problem quartets like the following:

x > 2 and x < 5
x > 2 or x < 5
x > 5 and x < 2
x > 5 or x < 2

This way, we always get one inequality with no solution, one satisfied by all real numbers, and a couple plain vanilla and- and or-inequalities for the same pair of numbers. It's not magic, but it does seem to help to vary one thing at a time.

Then enter the absolute value inequalities, and what a mess they are. There are so many different ways of solving them, and of talking about them, and I've made the mistake of covering several instead of sticking to one geometric approach and one algebraic approach. Now students are, quite predictably, using messy combinations of these.

Geometrically, the absolute value of x - 2 can be understood as the distance of x - 2 from zero, or the distance of x from 2. Last semester, with the Intermediate Algebra class, I relied on the former and had the students set up inequalities such as | x - 2 | > 3 by drawing a number line, placing "x - 2" more than three units away from zero on either side of zero, reading the resulting inequalities ( "x - 2 > 3" and "x - 2 < -3" ) from their sketch, and solving from there. It didn't really stick. I am not sure whether that was due to inadequate repetition or due to this approach being conceptually confusing.

Anyhow, with the Algebra 1 group this semester, I instead belabored the geometric interpretation of | x - 2 | as the distance between x and 2. I taped a large number line under the blackboard and we checked this definition by walking back and forth: -1 is three steps from 2, and sure enough, | -1 - 2 | = 3, and so forth. In order to solve inequalities such as | x - 2 | > 3 I had two students walk three steps from 2 on the number line in either direction, and we talked about what numbers were more than 3 units away from 2. This was difficult for many students (and not only the small group that always tunes out when I use any concrete representations because they think that's too middle school). My hunch is that there's some relation between this confusion and the difficulties Mr. K's students had with the meaning of "more."* Once students did pick up the idea it seemed to stick, but many never really got it. Maybe it's harder to ask when you're confused about what the walking up and down the number line is supposed to mean than when the material is more evidently academic.

I had first hoped to rely on this geometric approach to help students remember the direction of the inequality sign for the two linear inequalities in terms of which they will rewrite their absolute value inequalities, but gave up on that and introduced the approach that follows naturally from the algebraic definition of absolute value. If | x - 2 | > 3 then either x - 2 is greater than 3 or else the opposite of x - 2 is greater than 3.

Now I wish I'd used the geometric approach only for predicting and interpreting answers and stuck religiously to the algebraic approach to setting up the inequalities - because now students are using strange combinations of the two, such as x + 2 > -3 or -x + 2 > -3. In other words, complete confusion, with neither a clear concept nor a clear method to rely on. That's pretty discouraging even before thinking about the many students who did not even acknowledge the fact that there are two solutions to absolute value problems, that a distance can be in either of two directions... So, math teachers, what do I do now?

*We tend to assume too much about students' immediate grasp of the very idea of comparing quantities, let alone the isomorphism from this ranking of magnitudes to a spatial ordering along a line. Bob Moses, with his interest in pre-mathematical concepts that must be in place in order to succeed at Algebra, would presumably have a lot to say about this.

Sharing worksheets

Discovered Box quite serendipitously a few weeks ago, while figuring out something for my Midsession class - and this solved part of that problem of having multiple versions of worksheets on multiple computers, only some of which are connected to a printer. It turns out it can also solve the problem of sharing materials. My goal is still to contribute to I Love Math, but my materials are mostly written for the first time this year, and I'm constantly fixing typos, and constantly updating files once they're posted to I Love Math is just not going to happen this year. Meanwhile, the worksheet versions I'm using, warts and all, can be made available to anyone by a simple click if I store them at Box, as I had started doing anyway. Here are two assignments for my Algebra classes for this week - and when I get time to tidy up my files a little and tag them somehow, I'll put more materials in the public folder. It won't happen this week - there's an ed class deadline looming now - but hopefully sooner rather than later. Constructive criticism will be welcome.

Update (August 2): Moved everything except tests over to a public Box at after some cleanup this week. Not that rummaging around in other people's files is generally very edifying, but making decisions worksheet by worksheet about whether to share it or not has ended up meaning that it just hasn't gotten done, and besides the very idea that my files are public just might make me keep them a little more organized. Maybe. Without guarantees (of anything), though, they're at http://public.box.net/hmath

Pink Dragons and other Real World Applications

In order to determine what would make science classes appear relevant to learners, researchers of the ROSE project actually asked the students. What they found was that teenagers care little for learning about plants in their local area, how car engines function, and how chemicals interact. They are significantly more interested in learning about how atomic bombs function, why stars twinkle in the night sky, and phenomena that science still can not explain. And the item of most interest to the young learners was the possibility of life outside earth.

So much for the mantra that science must be made "relevant to students' daily lives."

I should not be surprised. I can dutifully and with some determination work up a bit of interest for the functioning of car engines, but only a little bit. I majored in Physics.

Why do we think that math problems will be more engaging to students if they are about bake sales, CD shopping, and other real world applications? And those little vignettes in the textbook that purport to explain how useful and applicable all this math will be - why do they always seem so contrived? Who thinks that a note in the margin stating that "If you become an ornithologist, you may use polynomial functions to study the flight patterns of birds!" will be more convincing to the kids than it is to us? And if the value of high school math for students' daily living were so clear cut, why isn't the case made more forcefully after so many years of textbooks?

Svein Sjøberg of the ROSE project argues that the reason why all students should learn science is not primarily that this knowledge will be so useful to them in their daily lives, nor should it be society's need for a sufficient supply of engineers and technicians. He instead emphasizes 1) the cultural argument and 2) the democratic argument. All citizens need to learn science because science, like arts and history and poetry, is a part of our common human heritage. Also, political decisions about issues involving science ought to be made by an informed electorate.

By the same line of reasoning, primary rationales for learning math could also be the cultural and political weight that this subject carries. Humans have calculated, devised and solved puzzles, and developed multiplicities of algorithms in all kinds of cultures throughout thousands of years. Accessing some of this heritage is part of the enculturation of a person in today's world - it is a privilege, not something we need to excuse or justify with awkwardly implausible future employment scenarios. As for the democratic significance of math, must not an informed electorate be able to interpret data displays and ask critical questions about statistical statements?*

There are times when I feel that my subjects are gatekeeper courses rather then essential components of a well-rounded education, as when I see a student aspiring to be a nurse struggling with logarithmic functions, and I wonder who ordered this, who has an interest in setting up this barrier between a dedicated and in many ways talented student and her choice of profession? On the other hand thinking of math in other terms than job training makes teaching it so much more interesting. I can happily create ridiculous word problems about pink dragons and syrup fountains, and remember that "relevance" for a teenager need not have much to do with usefulness in some narrow technical sense. The "relevance" of a math problem may have to do with the investment in completing it faster than the neighboring team, the joy of working together with a classmate on it, or the beauty of the graph when it is done in colored pencil.

*If we take the democratic argument seriously, maybe we should consider replacing most of Geometry with Applied Statistics as a graduation requirement and make formal, proof-based Geometry a college prep class rather than a course mandated for all citizens.

Emergency Math

Sarah at Mathalogical has suddenly gotten her course load increased to four preps (General Math being the latest addition) with little curriculum attached, and she's asking for suggestions. I'm responding here because the comment got too long.

First, four preps without textbooks or curriculum is rough. I did that last year, am veryvery glad it's over, and wasn't proud of the results. On the positive side, it gives you exposure to a large range of typical conceptual hurdles in a short amount of time, and your toolkit will grow very quickly. You'll know a lot more about just what your students in later courses aren't getting due to your experience with this course. In order not to get too discouraged it may sometimes be necessary to remind yourself of how much you're learning when you don't get enough time to prepare what it takes to have the students learning enough, selfish and futile as that may sound. And starting this marathon now rather than in August means you can try things out knowing that you can start over again in just one semester.

The three resources I found of most use last year were

1. I Love Math
2. The Math Worksheet Site (this costs $20 per year), and
   
3. The National Library of Virtual Manipulatives

I don't know that these are better than anything else out there, but these are the ones I returned to again and again, and where whatever did work usually came from. With only that basis for recommending the following, here's what I did:

There was no time for dreaming up a coherent curriculum with much by way of unifying themes or red threads, so in the General Math type courses I prioritized according to what skills I thought were hindering students the most in accessing more math. Some areas I focused on were

1. Integers on the number line. The Math Worksheet Site has neat pages of number lines with addition and subtraction problems that the students solve by diagramming the problem on the number line. A large number of 10th graders could not deal with negative integers, and in most cases these number line problems helped. The very idea of associating the numerical operations of addition and subtraction with the geometrical idea of motion along a line is the Big Idea that students just have to get in place, it's much less obvious than we like to think, and missing skills in this area really holds the students back.
   
   
2. Place value, and decimal numbers on the number line. First, placing these on the number line was a priority - though in many cases I did not succeed in teaching this. Dan Greene has great stuff on it (as you would already know) - but teaching place value just is not easy. It's awfully important, though, as the kids trip badly over this missing skill when they attempt to do more advanced stuff, so if you can do anything for them in this area, you're helping, even if it sucks up quite a bit of time. The Math Worksheet Site has lots of practice sheets for translating between Decimals, Percents and Fractions, and they're tidy and neat for what they do. As for resources for placing the numbers on the number line, the worksheets at this site aren't that satisfying. There must be animations out there that let you zoom in on a piece of the number line to study place value - but I haven't found anything great, and spent quite some time searching for it last year.
   
   
3. Solving simple linear equations. The common student error that bothered me the most was students' insistence on subtracting the coefficient of the variable instead of dividing by it - my explanations just did not work, and they were inelegantly wordy. What did work for many students was practicing with the Algebra Scale Balance at the National Library of Virtual Manipulatives. After working on this site the incidence of that error went down very noticeably, and it's the concrete representation that does the trick - doing a verbal version of this lesson, well, good luck. For practice problems, the "Partner Problems" worksheet for equations at I Love Math is great. It has two columns of problems of increasing difficulty, and horizontally aligned problems have identical solutions, so that the students can get near immediate feedback on their solutions. The students liked that sheet, and would gladly redo it if I photocopied it onto paper of a different color (and yes, they did need the repetition).
   
   
4. The basic operations. Many kids were more likely to settle down and do something when their assignment was a boring worksheet on practicing multi-digit multiplication, a fact that always puzzled me - my "interesting" discovery activities were much less likely to elicit absorbed concentration (they would involve reading a line or two of directions for each task - bad, bad idea :) The Math Worksheet Site has lots of practice worksheets, at various levels of difficulty, and the card game Top Deck at I Love Math (in the Middle School Folder) is a lot of fun. (Digression: The card games for practicing skills with fractions worked less well, because students tended to devise their own rules that defeated the purpose of the activity: for example, they'd agree to match denominators of different fractions rather than matching fractions for equivalence, as I wanted them to do!)
   
   
5. Area and Perimeter. If students can just get the difference between the two, nevermind formulas for calculating anything, that helps - it was a defining moment for me that October day when I realized that the students truly were unable to distinguish the two - that was when my ideologically rigid commitment to grade level standards started to give. A hands-on activity (measure the area of your desk in terms of number of colored paper squares you need to cover it; measure the perimeter of your desk in terms of number of standardized pieces of string you need to reach around it) did some good, but only some. A worksheet from a colleague, which involved drawing rectangles on a grid that all had the same area but different perimeters, or the same perimeter but different areas, did more good. There were still plenty of students who had plenty of trouble with just counting up line segments to find a perimeter of an irregular shape, though, and - well, I don't know what to do about that.

That was a sort of braindump of what for me emerged as priorities in a general math type course for underperforming students last year, without any authoritative pretenses. Never took a math ed class (and wonder whether they'd ever deal with 10th graders enrolled in Geometry who add 5 and 3 on their fingers and don't know multiplication from addition). If you are able to post about upcoming topics a week or two ahead of time ("We're doing the Pythagorean Theorem next week - what are the students going to struggle with?") you just might avoid some of my unpleasant surprises ("Squares? Square roots? What's that?") and do something relevant to what the students actually need.