Graphs of harmonic oscillation and rotation appear very similar when charted on graphs. Harmonic oscillation consists of a mass which moves back and forth in repetitive motion on one axis. Rotation consists of a mass in circular motion around a central point. On a graph, the mass appears to be moving back and forth in repetitive motion on each of two axes.

Consider the following graphs, excerpted from books by Schottenbauer Publishing.

__Discussion Questions__
- Which graphs show harmonic oscillation?
- Which graphs show circular motion?
- Describe the relationship between x and y components in circular motion.
- Using these graphs, describe at least one similarity between harmonic oscillation and rotation.
- Using these graphs, describe at least one difference between harmonic oscillation and rotation.

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.

As assessment tools, graphs can facilitate the evaluation of both academic and cognitive skills, including higher-order cognitive skills. The most basic test questions using graphs pertain to the concrete skills of reading a graph, while the more advanced and nuanced skills to be tested include those such as writing an equation, selecting graphs to answer a particular problem, comparing and contrasting graphs, and writing a scientific paper.

The list below identifies ideas for using graphs as assessment tools:

**Graph-Reading Skills**
- Identify the variables described in the graph.
- Identify the minimum and maximum values in the graph.
- Identify the shape of the graph.

**Mathematical Skills**
- Write a mathematical function describing the graph.

**Writing Skills**
- Write a meaningful description about a graph.
- Write a meaningful article about a set of graphs.

**Critical Thinking Skills**
- Compare and contrast two graphs.
- Compare a graph to a known theory.

**Concept Formation**
- What principles can be gleaned from a graph or set of graphs?
- Out of a group of graphs, is there an unexpected outcome? Is this an exception, or an error?

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.

Celebrate graphs with memorabilia from Zazzle! Colorful graphs from Schottenbauer Publishing are featured on these mugs, magnets, keychains, & postcards. A direct link is included below:

A variety of other STEM education collections are also available from Schottenbauer Publishing on Zazzle, which features regular sales on most items.

__Additional Information__

What is the meaning of the term "origin"?

According to Dictionary.Reference.com, there are up to eight meanings, two of which are mathematical. The first of these is commonly found in elementary and post-secondary graph-reading skills, while the second (not included here) pertains to more advanced math.

To answer the above question using the common math definition: Every graph in a Cartesian coordinate system has an origin, which is defined as the intersection of the axes.

Placement of the origin can dramatically affect the presentation of the graph. Consider the following two graphs, which present the same data, but with different origins.

__Discussion Questions__
- For each graph, identify the following coordinates: (a) Initial Position, (b) Final Position, (c) Minimum Position, and (d) Maximum Position.
- What is the relevance of the origin in each graph (if any)?
- Which graph is more useful for visual analysis?
- Do any additional differences exist between the graphs? If so, what difference(s) exist?
- Do the graphs contain any distortion of data, due to method of collection? If so, what is the distortion? How might the distortion be corrected?

Now, consider a different meaning of "origin." This definition is not mathematical; rather, it refers to "the first stage of existence," or the creation of an entity.

Using this definition, what are the origins of the over 8,000 graphs of sports, transportation, construction, environment, music, entertainment/toys, and general physics from Schottenbauer Publishing?

Created by M. Schottenbauer, Ph.D. in laboratory settings, these graphs represent several years of intermittent data collection and graphing activity. The objects studied in the graphs consist of the properties and motion of toys, hobby equipment, household objects, musical instruments, and sports equipment, measured and manipulated by the author. All music and sports activities were conducted by the author, including performance samples on 33 musical instruments and exercises from over 30 sports. Videos of some of these activities are available on YouTube.

**Blogs with Free Graphs**

**Additional Information**
What is more entertaining than learning science and math from the real world? Combining STEM education with fun, **The World in a Graph** by M. Schottenbauer, Ph.D., is a curiosity piece for educators and students alike. This concise anthology of 28 graphs presents a spectrum of interesting topics from sports, transportation, construction, environment, music, and entertainment. Graph topics, in the same order presented in the book, include:
- Sports
- Baseball
- Tennis
- Ice Hockey
- Snow Skiing
- Running
- Walking
- Biking
- Weightlifting
- Gymnastics
- Track & Field
- Figure Skating
- Ballet

- Transportation
- Wheels
- Cars
- Planes
- Trains
- Boats

- Construction
- Material Strength Testing
- Electricity
- Indoor Lighting

- Environment
- Natural Light
- Natural Water
- Wind

- Entertainment
- A Moving Block
- A Falling Domino
- A Rolling Marble

These topics provide a rich selection of concepts from physical science and physics education, including:
- General Mechanics
- Force
- Friction
- Material Strength
- Collisions
- Energy
- Momentum
- Free-Fall
- Projectile Motion
- Motion on a Plane
- Motion on an Inclined Plane
- Rotational Motion
- Harmonic Oscillation

- Electromagnetism
- Electric Potential
- Current
- Power
- Light

- Acoustics
- Biophysics
- Electromyogram (EMG)
- Range of Motion or Joint Angles (Goniometer)

These topics provide a variety of important concepts from math, including the following types of mathematical functions:
- Math Topics
- Straight Line
- Curve
- Parabola
- Exponential Function
- Sinusoidal Function
- Integral
- Derivative
- Damping Function
- Random Error

**The World in a Graph** by M. Schottenbauer, Ph.D. is available from multiple internet retailers, in paperback and Kindle digital editions, as well as wholesale.
Schottenbauer Publishing offers a wide selection of books, featuring over 8.000 graphs from sports, transportation, construction, environment, music, entertainment, and general physics. Additional information is available on the blog Science and Math Education.

**Additional Information**
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__
- What does the horizontal straight line represent?
- Why does the force line go down, and not up?
- What does the gradual, curved slope represent?
- What does the vertical straight line represent?
- What is the maximum force leading to breakage?
- Over what time is the force applied?

__Discussion Questions__
- What does the horizontal straight line represent?
- Why do the power and current increase?
- What does the first bump represent?
- What do the final 2 bumps represent?
- Does it take more or less power to remove the screw? Why?
- Why is the power to remove the screw not negative?
- What is the maximum real power?
- What is the maximum apparent power?
- What is the maximum current?

__Discussion Questions__
- What physical action is required to make this particular graph?
- What do the three lines represent?
- Why does the angle increase during each rotation event?
- Why does the velocity go down and up during each rotation event?
- Why does the acceleration go down, up, and then down during each rotation event?
- How many times is the screwdriver turned?
- How many angles is the screwdriver turned each time? Make a list containing the value for each event.
- What is the average angle turned by the screwdriver?
- What is the total angle turned by the screwdriver?
- Over what period of time is the screwdriver turned during each rotation event? Make a list containing the value for each event.
- What is the average length of time for a turn of the screwdriver?
- What are the minimum and maximum angular velocity values?
- 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.

**Additional Information**
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__
- Describe the graph in words.
- What phenomena is demonstrated?
- What physics theory or theories apply?
- 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__
- Describe the graph in words.
- What phenomena is demonstrated?
- Why is the graph messy?
- 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.

**Additional Information**