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:
- Getting initial practice in reading & writing variables (obviously),
- Demonstrating that math is about finding and communicating patterns,
- Generating a review of important writing rules (don't write "times one", etc.),
- Emphasizing that math notation is a language that can be translated to & from English like any other – specializing in brevity & precision, and
- Sneaky presentation of important identities, starting with the Peano-axiom definitions of zero, and then segueing to theorems about the multiplication identity, etc.