Standard & Slope-Intercept Forms Don’t Play Nice

We have contexts for equations in slope-intercept form:

Johnny has 4 friends and makes 3 new friends every 2 weeks.
             f = 3/2w + 4

We have contexts for equations in standard form:

Adult tickets cost $5 each and student tickets cost $3 each, for a total of $150.
                                  5a + 3s = 150

And we ask students to convert from one form to another.

                f = 3/2w + 4        to        3w – 2f = -8
                                             and
                5a + 3s = 150       to        s = –5/3 a + 50

But you know what doesn’t convert very well? … the context from one form to another.

Take Johnny’s Friends for example. The 3/2 means 3 friends every 2 weeks, and the constant represents the 4 friends that he started with. After converting to standard form, what do the coefficients 3 and -2 represent? What does the constant -2 represent? Maybe the context is there, but a typical algebra student just isn’t going to take the time find it, or maybe the context was lost in the conversion. The key to the question (but not to any viable answer) is in the units. If we apply some quasi dimensional analysis here:

                  f = 3/2w + 4
                 friends = (friends/week)weeks +  friends
                 friends = friends +  friends
                friends = friends

Check. Now try our quasi dimensional analysis on the standard form conversion: 3w – 2f = -8. Say what? Exactly! We can do it with our Ticket equation, though:        

                  5a + 3s = 150
                 (dollars/ticket)tickets + (dollars/ticket)tickets = dollars
                 dollars +  dollars = dollars
                dollars = dollars

Check again. And the context here actually converts somewhat. We can say that the equation s = –5/3a + 50 tells us that the number of student tickets is equivalent to fifty total tickets minus five-thirds of every adult ticket. If you can hang with that, try it on the following context.

Buddy sells cupcakes for $2 dollars each and brownies for $1 each, for a total of $14.
2c + b = 14
            

When you convert  this equations to b = -2c + 14, the 14 miraculously changes from representing dollars to representing number of brownies.

While the context might get lost in the translation between these forms of an equation, I know from my mathematics experience that there is still usefulness in being able to move fluidly from one form to another. In fact, to a mathematician, context is often burdensome. The beauty of naked math problems is that their abstraction transcends an infinite number of contexts. I just don’t think that is where you want the typical algebra student to start.                

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