Graph Lines Up & Down: The Meaning of Change

Line graphs go up and down. What is the meaning of change? Consider the following three graphs, excerpted from Volumes 1 & 2 of The Science of Home Construction from Schottenbauer Publsihing:

Discussion Questions

1. What does the horizontal straight line represent?
2. Why does the force line go down, and not up?
3. What does the gradual, curved slope represent?
4. What does the vertical straight line represent?
5. What is the maximum force leading to breakage?
6. Over what time is the force applied?

Discussion Questions

1. What does the horizontal straight line represent?
2. Why do the power and current increase?
3. What does the first bump represent?
4. What do the final 2 bumps represent?
5. Does it take more or less power to remove the screw? Why?
6. Why is the power to remove the screw not negative?
7. What is the maximum real power? 
8. What is the maximum apparent power? 
9. What is the maximum current?

Discussion Questions

1. What physical action is required to make this particular graph? 
2. What do the three lines represent? 
3. Why does the angle increase during each rotation event? 
4. Why does the velocity go down and up during each rotation event? 
5. Why does the acceleration go down, up, and then down during each rotation event?
6. How many times is the screwdriver turned?
7. How many angles is the screwdriver turned each time? Make a list containing the value for each event.
8. What is the average angle turned by the screwdriver?
9. What is the total angle turned by the screwdriver?
10. Over what period of time is the screwdriver turned during each rotation event? Make a list containing the value for each event.
11. What is the average length of time for a turn of the screwdriver?
12. What are the minimum and maximum angular velocity values?
13. What are the minimum and maximum angular acceleration values?

Over 8,000 graphs from Schottenbauer Publishing provide real-life topics for student learning, including sports, transportation, construction, environment, music, entertainment/toys, and general physics. 

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Physics Theory vs. Physical Reality

The theories of physics consist of neat, clean, mathematical formulas, derived from laboratory conditions. Although these formulas apply to all real life, the data collected from the real world demonstrates much more complexity, resulting in messy graphs.

Consider the following two neat, simple graphs, taken from simple laboratory experiments:

Discussion Questions for Each Graph

1. Describe the graph in words. 
2. What phenomena is demonstrated? 
3. What physics theory or theories apply? 
4. Is the graph consistent with theory?

Now, consider the two graphs below. These graphs show real-life conditions which are mathematically messy.

Discussion Questions for Each Graph

1. Describe the graph in words. 
2. What phenomena is demonstrated? 
3. Why is the graph messy? 
4. How can the graph be analyzed with theories from physics and/or math?

In elementary and high school classes, emphasis is often on the clean, neat graphs which perfectly illustrate theory. Including messy graphs from real-life conditions can provide a segue into the advanced physics and mathematics topics which are required for working in Science, Technology, Engineering, and Mathematics (STEM) fields as a career scientist.

Over 8,000 graphs from Schottenbauer Publishing provide real-life topics for student learning, including sports, transportation, construction, environment, music, entertainment/toys, and general physics. 


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Estimating Averages from Graphs

Learning to draw visual estimates of averages is an essential graph-reading skill. Consider the graph below:

This graph shows a jagged line. This line seems to increase over time, although the progress has many "ups" and "downs." One way to understand the graph is to draw a line which shows the approximate shape of the graph, averaged over time. In the graph below, this approximate line has been drawn in red.

Notice how the red line shows the approximate slope, or trend, of the overall line. This trend line can be used to understand the progress of the variable over time.

Now, consider the following two graphs, excerpted from Fluid Dynamics & The Science of Natural Waterways: Volume 1:

Discussion Questions

1. Do these graphs consist of smooth curves or short, connected, straight lines?
2. Estimate the data collection rate in these graphs by counting the points of change in a 50-second segment, and dividing by time. What does this information imply? 
3. Is there a problem with the way data has been collected in this graph?
4. Using a visual estimate, determine which graph shows higher water flow overall, as averaged over the entire graph.
5. Visually estimate the shape of the first graph, averaging between the high and low points. Describe the shape in words, then draw the averaged line on the graph.
6. Visually estimate the shape of the second graph, averaging between the high and low points. Describe the shape in words, then draw the averaged line on the graph.
7. Compare and contrast water flow in the two graphs, in a short paragraph.
8. Write a hypothesis about the cause of water flow differences in the two graphs.

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Considering Sources of Error in Graphs

The topic of error is essential when teaching graphs. Error describes inaccurate data, resulting in a graph which does not match the real-life phenomena it intends to measure. Error may occur at one or more points on the graph, which may be found in one or more segments of the graph.

Graphing errors may occur when the measurement device is not accurate, or when data is not properly recorded. Not all errors may be what they seem, however. Sometimes graphs accurately portray an inaccurate performance. Is it possible to determine the source(s) of error from a graph alone? 

Consider the following two graphs, excerpted from the book series The Science of Home Construction from Schottenbauer Publishing:

Discussion Questions

1. Does either graph contain error? Why or why not?
2. Does either graph show error due to the measurement device? Why or why not?
3. In either graph, does the real phenomena provide an unexpected contour? Why or why not?
4. For either graph, would changing the perspective increase/decrease the apparent error? Why or why not?
5. Redraw these two graphs, eliminating the likely error(s).

Schottenbauer Publishing offers over 100 books with graphs of popular topics, including sports, transportation, construction, environment, music, entertainment/toys, and physics. Free samples from these books are featured in the following free blogs, which contain graphs, videos, and discussion questions:


Blogs with Free Graphs

• Sport Science

• Athletic Training
• Physical Fitness
• Yoga, Pilates, Dance, & Ballet

• Summer Sports
• Archery & Shooting Sports
• Ball Sports
• Fencing
• Gymnastics
• Track & Field

• Winter Sports
• Snow Sports
• Ice Skating

• Music

• Toys

• Transportation

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