Students deal with a different ‘types’ of equations: equations in one unknown, equations in two unknowns, and the equation representations of function. There are others like the parametric equations but let’s talk about the first three I just enumerated. Is there a connection among all these three apart from being equations?
Let’s take for example the equation 4x – 1 = 3x + 2. To solve the equation, students learnt to use the properties of equality. When the topic gets to equation in two unknowns, this equation is learned independently of the equation in one unknown especially in finding the solutions. When the topic gets to solving systems of equation say 3x+y = 4 and x-y = 5, the methods for solving the system of linear equation – substitution, elimination, graphing – are also learned without making the connection to the methods of solving equation they already know. Then, function comes in the scene; the y‘s disappeared and out of nowhere comes f(x). Most times we assume the students will make the connection themselves.
How can we help students make connection among these three? To solve equation in one unknown, I think we should not rush to teaching them how to solve it using the properties of equality. There are other ways of solving these equations one of which is generating values which I’m sure you use in introducing equations in two variables. Using the example earlier, students can generate the values of 4x - 1 and then 3x+2. This way, the question “What is x so that 4x-1 = 3x+2 is true?” will make sense to students. They will have to find the value of x that belongs to the group of numbers generated by 4x-1 as well as to the group of numbers generated by 3x+2.
Now, why go through all these? Two reasons: 1) to reinforce the notion that algebraic expressions is a generalized expression representing a group of numbers/values and 2) to plant the seed of the notion of function and equations in two unknowns which students will meet later. Of course this does not mean we should not teach how to solve equation using the properties of equality. I just mean we should teach them other solutions that will help students make the connection when they meet the other types of equations.
Another popular tool is the input-output machine which is the same really as the table of values. For some reason they are used mostly to introduce equations in two unknowns or to introduce function. Why not introduce it early with equations in one unknown? Of course you need a second machine for the other expression. The challenge for the students is to find what they need to input in both machine so they will have the same output. The outputs can be represented by the expressions on each side of the equal sign but later you get to the study of function you may introduce y provided that y = 4x-1. Students need to see that this equation does not just mean equality but that it also means the value of y depends on x according to the rule 4x-1. Since every x value generates a unique y value, y is said to be a function of x, in symbol, y=f(x). Since y = f(x), we can also write f(x) = 4x – 1.
In most curricula, the formal study of function comes after systems of linear equation so there’s no hurry with the f(x) thing. The use of the form y = 4x-1 would be enough. If students understand equations this way I think they can figure out the substitution method for solving systems of linear equations by themselves. Graphing would therefore also be a natural solution students can think of. Equation Solver is a simple GeoGebra applet I made to help students make the connection.
Wouldn’t it be nice if students see 4x-1=3x+2 not just a simple equation in one unknown where they need to find x but also as two functions who might share the same (x,y) pair? This will really come in handy later. Solutions #2 and #3 of solving problems by equations and graphs are examples of problems where this knowledge will be needed.
To understand is to make connections. This has become a mantra in this blog. Students will not make the connection unless you make it explicit in the design of the lesson.