The Context Of Equations

Last Thursday, I covered direct and inverse variations.  As I wrote the equations on the board for each one, I realized the direct variation formula of y = kx is similar to a linear equation once a value for k is calculated while the inverse variation is similar to the 1/x equation.  As I’m writing them on the board, I’m talking about mathematical context helping to determine the proper choice for interpreting the equation.

I recently reported on how being able to read well helps in math.  Although it didn’t address context, good readers understand context and how it is used to define certain words such as product.  

In one context, a product is something that is sold by stores while in another context, it is the result of multiplying two numbers together.  It might also mean a result of a process or situation such as the child is a product of a marriage.

Thus being able to understand context in math is extremely important. As noted earlier if I see the equation y= 1.5x without any context, I don’t know if it is an arithmetic sequence, a linear equation, a direct variation, or anything else.  We tend to teach the equations in isolation rather than referencing having used them before in this situation or that situation.  

In one pre-calculus textbook I use, it does have a nice link from arithmetic sequencing to linear equations.  It took time to show how finding the distance between terms uses the slope formula with the number of the term and it’s value.  It also took it a step further by having students find the linear equation by starting with the nth term = the first term + (n-1)d and rewriting it.

This made me wonder if one reason students have difficulty transferring knowledge is simply that we do not take time to show how the arithmetic sequence relates to a linear equation and how the difference relates to the slope.  If you look at direct variation, you have y = kx.  Normally, they give you the value for X and Y at a certain point and ask you to find the value of k.  When you divide y by x, it is like having the delta y/delta x or the slope.  

If you look up mathematics and context, you’ll find an abundance of articles which talk about relating the math learned in the classroom with real world situations.  This is nice but it is also important to show students how certain equations seem to be related in different situations based on context. 

This is one way to show mathematical connections.  In the next entry, I’ll talk more about mathematical connections.  Let me know what you think, I’d love to hear. Have a good day.