# Basic Graphing Skills

Related quantitative concepts: Understanding trends, Interpolation/Extrapolation, Functions, Graph Significance, Graphing confusion
Related quantitative concepts: Understanding trends, Interpola

# Basic Graphing Skills

## or skills we should have learned in high school

by Jennifer M. Wenner, Geology Department, University of Wisconsin-Oshkosh
Jump down to: Plotting | Describing Graphs | Reading data | Examples and Exercises

Introductory textbooks are filled with graphs and plots. Graphs and plots are key in introductory courses in which quantitative skills are emphasized because they are the essence of giving students multiple representations of mathematical concepts; they can be expressed numerically, visually, and symbolically. Although concepts of plotting and graphing are taught throughout the K-12 curriculum, I find that within the first few lectures many students struggle with concepts. If this is the case, it may be necessary to review the basics of graphing or plotting.

## Essential Concepts

Although they are certain to have experienced plots and graphs in high school, students often struggle with the basics of graphing. When I teach graphs, I find that there are five important concepts that I expect my students to be familiar with:

1. What is a graph?
2. constructing meaningful plots and graphs
3. plotting x-y data on a graph
4. describing plots or graphs
5. reading and interpreting data from plots

## What is a graph? Why are they so important?

Graphs play an important role in the modeling and understanding of complex natural systems and pop up in a number of places in the introductory geoscience curriculum. Although concepts of plotting and graphing are taught throughout grades K-12, I find that some students struggle - even with simple concepts that I would expect them to know. If this is the case in your course, it may be necessary to review the basics of graphing or plotting.

I begin by telling the students that graphs are visual representations of numerical systems and equations. Because I am a visual learner, graphs help me to visuallize the relationship of one bit of data to another.The relationship can also be translated into a mathematically meaningful equation. The equation for a line (y = mx + b) is one such equation. Then others can use that equation to understand the system, too, because math is a universal language. If that equation applies to many incidents of similar systems, geoscientists can use graphs to predict the behavior of simplified natural systems or to understand the relationships of variables within a system.

## Helping students to construct a meaningful plot

I find that there are a number of incidences when introductory geoscience students have difficulty in constructing graphs and plots.

• Choosing the appropriate axis for a variable is difficult for some.
• Placing ordered pairs on the plot can cause anxiety for others
• Determining scale and labeling the axes with numbers presents difficulty for many
A review of the importance of making a plot meaningful and readable is important for students who are struggling with graphing.
I commonly run into the problem of students misunderstanding the construction of a meaningful plot when I teach them about topographic profiles. When a student comes to me with questions about how to construct this type of plot, I walk them through the thought process of labeling axes and deciding on scale:
1. First, what are we trying to represent here?
2. Then, decide what the axes are...horizontal distance (in miles or km) on the x-axis and vertical relief (in feet or meters) on the y-axis.
3. Next, determine the range of elevations present along the cross-section of interest
4. Now, we need to determine the increments along the vertical scale. We can do that by thinking about several important things:
• What is the contour interval? Increments similar to contour interval may be helpful.
• How many possible increments are there on your paper?
• If the difference in elevation is 480 ft. and there are 10 increments on the plot, should you plot in increments of 48? Based on your contour interval, is this reasonable? Would it be easier to plot increments of 50?
5. Finally, plot the elevations in the appropriate places on your graph and connect the dots with a SMOOTH curve.
In the end, students seem to be able to relate to wanting something to be easy to read and, in the process, begin to learn about making realistic choices.

## Plotting data

Plotting ordered pairs on a Cartesian plot can be difficult for students even though it is likely that they have done it numerous times in their academic careers. The Teaching with Data module of Starting Point has some information (and links) describing good ways to teach students about plotting. Here are some important bits of information that can help faculty to make sure that students are up to speed on plotting:

• there are two axes - horizontal (often called the x-axis) and vertical (often called the y-axis),
• a point on the graph is denoted by an ordered pair (or coordinates (e.g., (3,8)) where:
• the first number refers to horizontal position on the x-axis,
• the second number refers to vertical position on the y-axis,
• sometimes the ordered pairs are listed in tabular format with headings that correspond to the labels on the axis
• the two axes intersect at point called the origin with coordinates (0,0),
• the reason that we plot data is so that we can more easily observe trends or behavior of the data
• (modified from Anderson and Swanson, 2005)

## Describing plots and graphs

Students struggle with the description of data on a plot. And, yet, that is generally the main reason that we use plots of data - to describe the data. Data can be described qualitatively using specific terminology:

• Often we use words that describe the curve or line made by the data: e.g., linear, exponential, asymptotic, periodic, etc.
• The strength of those relationships can also be characterized using words like strong, moderate or weak
• Sometimes we use words like increasing and decreasing or positive and negative to describe the relationship of a set of data.

For example: The data on the plot in the box below have a moderately strong negative linear relationship. Students may not have a strong picture of what those words mean, it may be necessary to explain this vocabulary (possibly using graphical and/or symbolic representation). Starting point has a discussion about how to help students with descriptions of data on graphs.

In introductory geoscience courses, there are a number of topics in which we expect students to be able to recognize linear relationships. In my course, students can also expect to be asked to draw "best-fit lines" through linear (or sometimes curved) data in order to predict behavior in other situations. When students struggle with these concepts, I try to get them to take one step back. I ask them to describe the shape of the data. Then, I say, "If you had to draw a perfectly straight line so that it went through as many points as possible, how would you draw it? Place a ruler on the paper where you would draw it." As they do this, we discuss why he/she put it there, how that line actually describes the data and I talk about making sure about the same number of points are on one side or the other. If they have done it correctly, they should be able to use their line to read/generate new data.

In addition to using qualitative terms, we can describe a plot using mathematical expressions. The most common (and one with which students are often familiar) is the equation for a line:

y = mx + b

where m = slope and b = the y-intercept.

Students who are unfamiliar with the concept of plotting data on a graph may also have a difficult time with the mathematics of describing trends and data. Once students have mastered the descriptive vocabulary trends/data, they may need additional help with the more complex concepts involved in mathematical description of trends. See the Understanding Trends page for more information about mathematical interpretation of trends.

Plots of data (and simple relationships between variables) can help geoscientists to understand and predict the physical way that the Earth works. Plotting known data can help us to visualize the behavior of systems in situations that have not been measured. Students in introductory geoscience courses are expected to be able to read data from plots of lines (or sometimes more complex mathematical relationships like curves) to predict behavior. If students have an understanding of the concepts discussed above, they should be able to generate new data from plots of known data. An example of using plots to predict behavior is presented in the forecasting sections of SERC's floods and flooding page.
Many students in my introductory courses cannot read data from a simple linear plot. Yet, I expect them to predict the discharge of a 200 year flood from a plot like the one in this box. Some students do not understand the idea of generating data from a plot - it seems too simple to them. But I explain that it is merely taking a variable that you want to know (200 years) and determining the appropriate value for another variable for which you know the relationship to the first (discharge). For students who have had a significant amount of algebra, it may be as simple as indicating that it is merely solving an equation for one variable (because you know the other variable).

Geologists use graphs in a variety of simple ways that nearly any student can comprehend. Graphs can be a visual way of predicting or forecasting geologic events. But they are also used to understand the behavior of systems, to visualize large sets of data and to help geologists understand many important systems that might not be easy to understand just by looking at a bunch of numbers. Graphs are a way of making scores of data points manageable and, often, more understandable.

The skills discussed above are basic skills that are essential for many of the applications in introductory geoscience courses. Some introductory geoscience topics require more advanced skills and may take a greater effort to get your students up to speed. There are a number of pages on the Teaching Quantitative Skills web site that are geared toward these skills:

• Understanding trends in data
• Interpolation/extrapolation of trends
• Generation of functions from graphs/data

## Teaching Examples and Exercises

• ### Plotting Data

A module at Starting Point designed to give faculty help with teaching students about plotting data on graphs. Each link gives some information about the important points for plotting points on graphs.
• ### Describing Graphs

Starting Point has this module about describing graphs with a number of good links for both faculty and students. As with the Plotting module, there are a number of useful links to web pages that address student difficulty with describing graphs.
• ### Activities designed to increase and develop quantitative skills

A number of the activities posted on the Quantitative Skills in the Geosciences site are designed to help students learn about graphing in a geoscience context. Some of these activities are designed for upper level students.
• ### Introductory level activities involving Excel

There are dozens of activities that involve the use of Excel that are available as a part of SERC's Starting Point. Each of these activities is downloadable and ready to use.
• ### ConcepTests involving intepretation of graphs

There are a number of examples of ConcepTests that can be used to break up lecture in a large classroom. Each of these ConcepTests involves the interpretation of graphs appropriate to introductory level geoscience.