Dan Meyer recently posted an article entitled The Chinese Room. The moral of the story is: if I give you something written in Chinese and a manual for writing down other Chinese characters, are you literate in Chinese? Now, you have to write the actual response (assuming the manual is correct), and to someone who is actually literate the answers must be correct. So…are you?
It’s an interesting debate, and one I’ve seen many time working with students. The idea is a bit abstract at first, but this post is dedicated to the same idea, outlined in a particular example: Linear Equations
Consider the two equations:
(1) y = -(3/4)x + 24
(2) 3x + 4y = 24
Mathematically, they’re equivalent. But do they mean the same thing? Same slope, same intercept…same interpretation?
When I’ve seen it taught, one of the fundamentals is changing between the 2 primary forms of equations (slope-intercept, standard form). But why?
Equation (1), how we often write them, to me signifies change over a given interval. You start with 24, and every 4 units it decreases by 3. Dollars per hour? Cookies per day? The graph ultimately shows how much you have (or owe, if you like negatives). I think of it as filling in the blank “after ___ days you will have ___ dollars” (change units as you please). There is an answer to be found, depending on what you’re looking for. Traditional algebra.
Equation (2), though, is something totally different. I see that, usually, as comparing costs. I sold x-ray goggles and yams, making $24 total. The goggles cost $3, and the yams $4. So, how much did each cost? Here, the 2 variables each represent their own object. There is no change over time. So, what does the slope mean? For every goggle, the price of yams goes down? Not quite. It shows how much each could cost. Each ordered pair is a possibility. There is no one answer you’re seeking. Uncertainty, conditional information, no answer.
Reflecting on my own experience, I have basically taught students to turn all equations into form (1) and go from there (unless you’re graphing using intercepts – but that’s another show). So, I’ve been teaching them to read and write Chinese according to the post title. I haven’t promoted true understanding.
Have you done what I’ve done, or how have you been able to get students to the heart of it?
Equation (1) can also use, with reason, the negative values in quadrants II – IV. -1 for x means the day before. -1 for y means you owe someone money. Negative = directionality.
Equation (2), though, in this case does not allow for negatives. Unless buying x-ray goggles entitles you to a refund.
Another bit of algebra caught up in this…has anyone used these as a motivator for discussion about different ways to use negatives?