- Why should I bother learning this?
Many events in every day life, as well as in scientific inquiry, are related more or less linearly. It is a good idea to be comfortable with linear equations and graphs in order to make convincing arguments about these relationships and to evaluate the arguments others make. Have students collect examples of line graphs from newspapers and magazines. Even though many of these graphs do not show straight lines, they show a series of segments, each of which can be evaluated based on direction and steepness (slope).
- How can I tell by looking at an equation whether it will graph as a line?
The clues are the variables.
- If there is one variable, it will graph as a horizontal or vertical line or as a single point on a number line.
- If there are two variables with no exponents and no variable is the denominator of a fraction, the equation will graph as a line.
- If there are more than two variables, the equation will not graph as a line on a coordinate plane, but may graph as a line on a three-dimensional grid.
- Why can't the variable be in the denominator of a linear equation?
If your students are familiar with negative exponents and how they describe fractions, use the pattern of exponents to answer this question. If you have not spent time with negative exponents, have students try to graph the equation, y = 1/x. They'll see that each time x increases by one, y increases, but not by the same amount each time, so the graph curves. Relate this discovery to the definition of fraction.
Since division by 0 is undefined, there is no point with an x coordinate of 0, but there can be points with x coordinates less than 0. That's why this graph has two parts. The x and y axes are the asymptotes for this graph: They provide the boundaries across which the graph cannot go.]
Two-step Linear Equations
with Rational Numbers.