Describe two different ways you could determine whether a function is linear. Describe how these two methods relate to one another, and a possible classroom activity that would help students to see this connection

Question

What will be an ideal response?

Answer 1

There are a variety of possible methods. A few include the following:
Graphing the data points of the function to see whether they form a straight line
Examining a table of the data sets and looking at whether the increase from each x and y value to the next is proportional
Example: The ratio of the change in the x to that of y is always 2/5.
x 2 4 6
y 5 10 15
Examining the function in equation form; if it is a first-degree equation (independent and dependent variables have only an understood exponent of 1), it is linear.
There are a number of different ways to show how these methods are related and classroom activities that could be used to illustrate. Those that would be most effective should be in a context. One possible example could involve helping students see the connection between the table and the graph of a linear function. For instance, students could use the situation at the top of the text page 283, which describes Brian's hotdog cart. They could create a table of values for hotdogs sold and profits, similar to table 14.2. By examining the ratio of the changes between the independent and dependent variables and then graphing the values, they will see a connection between the fact that proportional changes in the independent and dependent variables results in a consistently linear slope in the graph. This could also lead to a more in-depth discussion of how proportional situations will always result in a linear graph and pass through the origin.