To describe how oscillations are supported in systems of differential equations, we present a classic "Romeo and Juliet" picture of two-dimensional oscillations, and we analyze how trajectories change as nullclines are arranged at different angles in the phase plane. In addition to models based on traditional systems of differential equations, dynamical systems with time delays and dynamical systems with stochastic fluctuation (i.e. stochastic resonance) can also support oscillations.

The first video segment presents a canonical mathematical example from quantitative biology, in which mRNA is transcribed from a gene sequence, and protein is translated from mRNA. The second segment uses eigenvector-eigenvalue analysis to sketch the trajectories of the system in a phase portrait. Finally, the third segment generalizes the linear stability analysis used to study this example.

Students disassemble and analyze retractable pens. Through the process of "reverse engineering," they learn how the ink pens work.

This movie provides an 18-minute introduction to how NASA is observing our ever-changing planet. On the The Dynamic Earth DVD Web page you can download the video as an iPod or iPhone version, as well as an AppleTV/Full resolution version, and access a glossary and links to related Web sites.

Biological systems (e.g. cells) can make stochastic transitions between phenotypes (e.g. states of relatively increased or decreased drug resistance). This means that an initially drug-sensitive population can generate relatively drug-resistant subpopulations. This video presents a metronomogram, which is a tool for understanding whether such stochastic transitions can provide an opportunity for therapeutic treatment. Citation: Liao D, Estevez-Salmeron L, and Tlsty TD (2012) "Conceptualizing a tool to optimize therapy based on dynamic heterogeneity," Phys. Biol. 9:065005.

This Earth Observatory site contains data and images from NASA's Earth Observatory project. The purpose of NASA's Earth Observatory is to provide an accessible publication on the Internet where the public can obtain new satellite imagery and scientific information about our home planet. The focus is on Earth's climate and environmental change. The site is useful to public media and educators. Data and images can be obtained for a variety of subjects, the Earth's atmosphere, land, life on earth, oceans and heat and energy. Users can enter data on a variety of topics and produce animations depicting the data entered. The site also contains a helpful section explaining how to build animations and globe animations.

This tutorial provides the evidence that the amount of COå_ in the atmosphere has not been as high as it is currently for nearly half a million years and that this increase corresponds with data that human activity is responsible. Learners examine scientific data showing increases in both atmospheric becomes COå_ and the Earth's average temperature and analyze changes in atmospheric concentration of COå_ over time. They reflect on some of the barriers involved in teaching global climate change and how using data in the classroom may be used to overcome those barriers. Multimedia resources such as video clips, a data visualization exercise featuring digital resources on climate.nasa.gov, and an interview with NASA climate scientist, Dr. Gavin Schmidt, are included. This is the second of ten self-paced professional development modules providing opportunities for teachers to learn about climate change through first-hand data exploration. Lesson plans for middle and high school students, descriptions of data collection instruments, glossary links to vocabulary are included.

How many calories are in your favorite foods? How much exercise would you have to do to burn off these calories? What is the relationship between calories and weight? Explore these issues by choosing diet and exercise and keeping an eye on your weight.

Students are introduced to genetic techniques such as DNA electrophoresis and imaging technologies used for molecular and DNA structure visualization. In the field of molecular biology and genetics, biomedical engineering plays an increasing role in the development of new medical treatments and discoveries. Engineering applications of nanotechnology such as lab-on-a-chip and deoxyribonucleic acid (DNA) microarrays are used to study the human genome and decode the complex interactions involved in genetic processes.

Under the "The Science Behind Harry Potter" theme, a succession of diverse complex scientific topics are presented to students through direct immersive interaction. Student interest is piqued by the incorporation of popular culture into the classroom via a series of interactive, hands-on Harry Potter/movie-themed lessons and activities. They learn about the basics of acid/base chemistry (invisible ink), genetics and trait prediction (parseltongue trait in families), and force and projectile motion (motion of the thrown remembrall). In each lesson and activity, students are also made aware of the engineering connections to these fields of scientific study.

Students experience the excitement of watching live-streaming video of wild bears on Brooks River in Katmai National Park, Alaska, in order to explore science concepts in the real world. These three lessons, designed for grades 3–5, offer students the opportunity to engage in activities that focus in particular on the inheritance and variation of brown bear traits.

In the first video segment, we analyze the population dynamics for a test-tube of cells that affect each others' likelihoods of replication when they collide. The particular example we use is a prisoner's dilemma, which has the almost paradoxical property that survival of the relatively most fit leads overall fitness to decrease. In the second video segment, we suggest that the population dynamics from the first segment can be related to an analysis that uses payoff matrices found in traditional game theory.

Students are introduced to the concepts of digital organisms and digital evolution. They learn about the research that digital evolution software makes possible, and compare and contrast it with biological evolution.

Explore Learning offers 3 programs to support students learning at home:

Gizmos (Grades 3-12) Get hands-on with math and science.
Gizmos are online simulations that excite curiosity and invite interaction. We’ve got over 400 Gizmos in our library covering topics and concepts in math and science for grades 3-12. Gizmos help students dig deeper into subjects and really understand challenging concepts.

Reflex (Grades 2-6) When they use Reflex, kids love math.
Adaptive and individualized, Reflex is the most effective system for mastering basic math facts in addition, subtraction, multiplication and division.
Full of games that students love, Reflex takes students at every level and helps them quickly gain math fact fluency and confidence. And educators and parents love the powerful reporting that allows them to monitor progress and celebrate success.

Science4Us (Grades K-2) It’s never too early to learn science!
Science4Us covers Inquiry, Physical Science, Life Science and Earth & Space Science with lessons specifically designed for K-2 students. The sessions have 1000’s of online and offline activities that can be completed in as little as ten minutes, and teach students using videos, interaction, poems, songs, and digital notebooks.

In this activity, students will explore two given websites to gather information on Bone Mineral Density and how it is measured. They will also learn about X-rays in general, how they work and their different uses, along with other imaging modalities. They will answer guiding questions as they explore the websites and take a short quiz after to test the knowledge they gained while reading the articles.

This is a lesson about using analogues to look for life on other planets. Learners will use the results of previous lessons to write a scientific proposal to explore another planet or moon in our solar system for signs of life. This proposal should predict the types of energy and nutrients available to sustain life and describe equipment and instruments necessary for exploration and characterization of the target environment. This is activity 4, the capstone activity, in Exploring Deep-Subsurface Life. Earth Analogues for Possible Life on Mars: Lessons and Activities.

This book introduces students to some of the most unusual places in our galaxy outside of our solar system. Answering the question, "How many stars are in the sky?" introduces students to basic counting, tallying, and grouping techniques, as well as allowing for the use of simple proportions.

As science extension activities, this book of problems introduces students to mapping the shape of the Milky Way galaxy, and how to identify the various kinds of galaxies in our universe. Students also learn about the shapes and sizes of other galaxies in our universe as they learn how to classify them. The math problems cover basic scientific notation skills and how they apply to working with astronomically large numbers. It also provides exercises in plotting points on a Cartesian plane to map the various features of our Milky Way.

This is a lesson about extremophiles and using DNA sequences to classify them. Learners will describe the characteristics of a newly discoverd thermophyllic organism and use the DNA sequence to place that organism in the phylogenetic tree of life. Includes teacher notes, learning objectives, and assessment of prior knowledge and preconceptions. This is Lesson 3 in Exploring Deep-Subsurface Life. Earth Analogues for Possible Life on Mars: Lessons and Activities

All of us have felt sick at some point in our lives. Many times, we find ourselves asking, "What is the quickest way that I can start to feel better?" During this two-lesson unit, students study that question and determine which form of medicine delivery (pill, liquid, injection/shot) offers the fastest relief. This challenge question serves as a real-world context for learning all about flow rates. Students study how long various prescription methods take to introduce chemicals into our blood streams, as well as use flow rate to determine how increasing a person's heart rate can theoretically make medicines work more quickly. Students are introduced to engineering devices that simulate what occurs during the distribution of antibiotic cells in the body.