Jordan Massoud: Blog #4: Teaching linear functions

Hello class, my name is Professor Massoud and today we will learn about linear functions. To begin with, don’t get intimidated by the terms linear or function, I assure you each one of you has real world experience with both of these concepts.

First, in order to grasp the concept of a “linear function,” an explanation of a function is in order. For example, take an automated baseball-pitching machine. This simple, yet illustrative example is a function. How you might ask? Well, the machine pitches a ball for every ball that is fed into it. The machine in this example is a function. From this example we can label a few terms. The input of a function is the value you insert into a function. The answer or value that pops out, in the example it would be the ball flying you at 90 mph, is the corresponding output. Just look at the beginning of both words to keep these two concepts straight—input is what you put in, and output is what a function spits out. The formal definition of a function is a relationship in which one output is paired with exactly one input. It is also important to know that a function can be represented in four different ways: a graph, table, formula, or words.

Why learn about input/output or even functions for that matter? Well for mathematics of course, but also it is important to be able to explain the world around you in a systematic and consistent way. These concepts are also the building blocks for many mathematics courses you all will have to take.

Now that we have a basic understanding of the second part of the phrase “linear function” let’s examine the term linear. To do so, let’s think of running on a treadmill. When you get on the treadmill you speed up to your desired speed and let’s say you run for 25 minutes at that pace. In this scenario we have two variables: time spent running and miles ran. To show this, let’s look at a table.

          Time Spent Running (x)                                               Miles Ran (y)

5 minutes	.5 miles
10 minutes	1.0 miles
15 minutes	1.5 miles
20 minutes	2.0 miles
25 minutes	2.5 miles

Here, our input is time spent running and our output is the number of miles ran. As you can see, at 5 minutes John Doe has ran .5 miles, and at 10 minutes he has ran 1.0 miles, and so on.

Now I’d like to introduce another concept—rate of change, or more commonly known as slope. Rate of change simply measures the changes in the y values relative to the x-values. The rate of change or slope is a crucial tool in mathematics. It is used extensively in finding the formula to be able to graph a line. Just from looking at a graph, one can tell if the slope is positive, negative, less than one or greater than one. The slope is the single most defining feature of any graph.

 In the table above, we have a set of x values and a set of y-values. The formula for the rate of change is:

Change in y-values/Change in x values

                        Or

(Y2-Y1)/(X2-X1)

In the second equation, Y2/Y1 and X2/X1 represent an ordered pair of x,y values. So, in our example we could have the ordered pairs: (10,1) and (25, 2.5). Let’s practice calculating the rate of change between these two points.

(2.5-1)/(25-10)= 1.5/15=   .1 miles/minute

So, you might be thinking this is great, but what does this have to do with linear functions? Well, a linear function is a function where the rate of change at every interval is the same. An interval is two sets of points. In other words, for our example, this function can only be linear if we continue to get .1 miles/minute when we compare different sets of x and y values. Let’s see:

(1.5-1)/(15-10)= .5/5= .1 miles/minute

(2.0-.5)/(20-5)= 1.5/15= .1 miles/minute

As you can see, we continually get .1 miles/minute for the rate of change. This tells us that the example we started class with is indeed a linear function.

Now, try this problem and determine whether this function is linear or not and explain why/why not.

**Remember: functions can be represented as words, graphs, tables, or formulas

Let’s say that you and your parents go out for an ice cream cone… aw so cute. From start to finish, the trip takes you 20 minutes. The ice cream parlor is a few miles away so you decide to drive. Below is a table representing your trip.

         Time in minutes (x)                                   Speed of car in mph (y)

5 minutes	25 mph
10 minutes	42 mph
15 minutes	21 mph
20 minutes	0 mph

As you should have discovered this function is not linear because the rate of change at every interval is changing. This is because the car is not traveling at the same speed the entire length of the trip, due to stop signs, speed limits, and other factors. As a result, this table still models a function where for every output there is one corresponding input, that function is just not linear.