Thursday, September 27, 2012

First Exercises with Variables

I'd like to share the lecture notes for the very start of my introductory algebra classes. (This occurs in the 2nd hour, after administrative procedures and a little review on the most basic terms and the order of operations.) Here's how I introduce writing with variables:


Definitions
Variables: Letters that stand in for numbers.
Algebraic Expression: Series of math symbols with no equals.
Equals: Means "is" or "is the same as"; symbol =.

Using these, we can write precise statements about number patterns.

Ex.: Translate to math:
(a) "Any number times zero is zero" → For all x: x∙0 = 0.
(b) "Any number plus zero is the same as the original number" → For all x: x+0 = x;
(c) "Any number times one is the same as the original number" → For all x: x∙1 = x.
(d) "A negative sign is the same as multiplying by negative one" → For all x: −x = (−1)x.


Of course, there's a bit more that I say verbally, but that's what gets on the board. I prime the pump for these exercises by asking initial questions like, "What's 5 times 0? What's 8 times 0? What's 25 times 0? Who can clearly state the pattern that we're observing?", etc. We do the first two translations together, and then students do the second pair on their own.

This is a bit of an evolution on what I've done in the past. It works extremely well, and part of the reason I'm so tickled by these exercises is that they slyly manage to do at least quintuple duty. What's being accomplished here is:
  1. Getting initial practice in reading & writing variables (obviously),
  2. Demonstrating that math is about finding and communicating patterns,
  3. Generating a review of important writing rules (don't write "times one", etc.), 
  4. Emphasizing that math notation is a language that can be translated to & from English like any other – specializing in brevity & precision, and
  5. Sneaky presentation of important identities, starting with the Peano-axiom definitions of zero, and then segueing to theorems about the multiplication identity, etc.

3 comments:

  1. I have used a similar approach when teaching my son to add and subtract: focusing on patterns of numbers rather than just memorizing.

    For example, 8+8 is a particular case of adding a number to itself, or doubling. It will always give an even number. These cases seem to be easier for a child to remember.

    So a problem like 17-8 might be analyzed like this:

    What is 17-8?
    I don't know.
    Whell do you know 8+8?
    Yes its 16!
    Then what is 16 - 8?
    Oh, its 8.
    Then what's 17-8?
    Its 9!

    Likewise, multiplying a number by 5 can be seen as counting by 5.
    Or, multiplying 11* 11 = 11* 10 (easy!) + 11.

    This approach seems a little tedious at first, but a young child can learn to do this rapidly. Also it gives a better intuition, they are less likely to come up with an answer that is dramatically wrong.

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    Replies
    1. In some sense I agree. Like my philosophy on pretty much everything, I think you need both (a) broad conceptual intuition (like basic properties of numbers), and (b) detailed fact knowledge (like memorizing add & multiply tables). I teach a basic math class where I've just added direct training in rounding and estimating, since those oft-assumed skills are totally missing for that population of student.

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    2. ^ To complete the thought: It's the marriage of the two approaches, both estimation and precise writing, that gives true strength: i.e., a backup parachute.

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