Last week, in Fundamental Understanding of Mathematics LV, plotted our solution to the Matt and John age problem on a graph, using Cartesian coordinates, invented by Rene Descartes in the 1630s (Renatus Cartesius in the Latin used for academic writing at the time, hence "Cartesian" coordinates instead of Descartesian coordinates.)
We talked about it some in the comments, today I will go over terminology related to this mathematical tool.
Running horizontally, colored red, is the X axis. Most graphs used in teaching will begin with zero on the other axis, in this case the blue Y axis which runs vertically. Generally the numbers will have different units, time and distance, or weight and volume, or age and height, so the two numbers really aren't subject to normal addition and subtraction, multiplication and division, because they are, literally, apples and oranges: they represent different things. Adding your age in years to your height in inches doesn't make any sense.
It's good form, when drawing a graph, to note the units where it says "Unit of measurement for..." As examples, it might read "John's age in years" "Miles traveled" "Time (minutes)" or whatever is appropriate for the problem at hand.
Here is a graph for a time and distance rate problem. This shows three data points (the red dots on the black dashed line) which are from measurements and are usually given in a problem statement. Let's show these on a T Chart:
Notice that we don't have an entry for the beginning, at time zero. Since this graph involves something moving, we assume that at the beginning time, it is at the beginning location, mile zero.
There is a dotted black line connecting the dots, which is the line that represents the equation for the rate. We can figure out what that rate is in a couple of ways.
Back in Fundamental Understanding of Mathematics XLVIII we discussed rate problems, and pointed out that a time per distance rate was also known as speed. Most rates have special names, used by people in the business. It makes them easier to talk about, but it also makes it easy to forget that rates are combinations of units.
Let's take a look at the rate shown on the graph. If we take a look at the middle red dot, it is at 10 minutes along the x axis, and at 4 miles along the y axis. So our object moved four miles in 10 minutes. or 4/10 miles/minute. If we take a look at the top red dot, that one is at 15 minutes and 6 miles, or 6/15 miles/minute. How can we compare them?
The simplest way to compare two rates is to convert them to a common denominator. In the case of speed, the common denominator is usually one hour.
To change 10 minutes into an hour (60 minutes) we multiply by 6, so 4/10 miles/minute = 24 miles/hour. To change 15 minutes into an hour, we multiply by 4, so 6/15 miles/minute = 24 miles/hour. It's the same speed.
We could also have converted the fractions into miles/minute, simply by doing the divisions:
4/10 = 0.4 miles/minute
6/15 = 0.4 miles/minute.
It's still the same speed.
The line on the graph slants upward, as we move along from left to right. We call this slant the slope of the line. It has a very interesting property.
There are two more terms I'd like to introduce: rise and run. The rise of the line is a measure of how much it goes up, and the run of the line is a measure of how much it goes to the right.
For example:
Between the first and second dot, the line goes up two miles (from 2 miles to 4 miles) and goes to the right 5 minutes (from 5 minutes to 10 minutes).
Rise = 2 miles, run = 5 minutes
Between the first and third dot, the line goes up four miles (from 2 miles to 6 miles) and goes to the right 10 minutes (from 5 minutes to 15 minutes).
Rise = 4 miles, run = 10 minutes
Between the zero mark and third dot, the line goes up six miles and goes to the right 15 minutes.
Rise = 6 miles, run = 15 minutes
The slope is defined as the rise divided by the run. Let's calculate it:
2/5 = 0.4
4/10 = 0.4
6/15 = 0.4
How about that? It's the same as the speed.
This makes sense. The slope is rise divided by run. On this graph, the rise is miles traveled, and the run is time in minutes. So rise divided by run is the same as miles divided by minutes, or miles per minute, a speed measurement.
Have fun in the comments.