Note: This is a break from the triangle area stuff I was talking about yesterday. I have full intention of returning to that sometime very soon.
I was reading through some math stuff yesterday and I came across a discussion of a grade-four question in Singapore math that requires a solution of only pseudo-algebra, thereby promoting an intuitive understanding of algebraic substitution.
It reminded me of one of my own favorite activities when I used to teach middle-schoolers, in introducing simultaneous equations and the idea of substitution. To preface this, I should state that my proudest accomplishment in creating these materials is that basically 80% or 90% of my (regular Bronx) 8th-graders would, within a few days, begin to realize that they can partially solve a simple system like this in their heads:
3x + 7y = 65
3x + 5y = -15
Essentially, they got so good with visualizing linear systems using shapes (triangles, circles, stars) that they can literally "see" that because the first equation has an overall right-hand-side value that is 80 greater than the second equation, and the only contributing factor is that the first equation has 2 extra y's, then that must mean that each y is "responsible" for 40, or y = 40. --To you guys, this may seem trivial, but it is not at all trivial to kids who, in many cases, have a natural fear of symbolic representation!
I presented this topic using a series of increasingly-difficult visual puzzles that you can find on the internet, and I let the kids work in groups of 3 or 4. I didn't introduce the idea of substitution, linear combination, or simultaneous systems, but I let them discover it for themselves through some scaffolded questions. Once they were able to solve the puzzles (WITHOUT guessing/checking), I then gave them algebra sheets and asked them to individually draw / "solve" the equations using shapes only. We didn't even begin to use algebra symbols until a couple of days down the road, at which point the transition from shapes to algebra was seamless for most kids. (I did this for two years in a row, and with other teachers at my school. The results are very duplicate-able.)
Anyway, here are the worksheets I have used in the past (that worked well in these first couple of days of group/individual exploration). I also modified and used them with my 10th-grade Algebra 2 Honors class during our intro to systems this year, and even those kids loved it!
Let me know what you think. I think it's similar to the Singapore method, but presented in a way that might be a bit more intuitive and generalized, supporting the learning of algebra. --And yes, sometimes I do miss teaching middle school (ie. having more time to focus on building introductory concepts), but whenever I feel nostalgic, I also hope that understanding middle-school approaches/foundation would help me be a better high-school teacher, in the long run. :)