In this video, we use direction fields (drawn as quiver plots) to illustrate the numerical integration of differential equations. We include a heuristic example of how one might try to adapt step size by comparing different orders of approximation.

In the first video, we present a biological circuit topology from Ma, Trusina, El-Samad, Lim, and Tang, "Defining network topologies that can achieve biochemical adaptation," Cell, 138: 760-773 (2009). This topology supports adaptation, which is not the absence of change in response to stimulation/stress, but, instead, the ability to produce delayed compensation for those changes. In the second video, we summarize the method of almost linear stability analysis used to solve for the dynamics of this example system.

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.

In the five parts of this video, we define the derivative and then build a cribsheet of rules for expressing the slopes of simple functions and combinations of functions. These include the power rule, the chain rule, the product and quotient rules, and the rules for differentiating sinusoidal functions.

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.

This resource contains a series of 9 digital skill building assignments that were initially used in an online course of postsecondary students from a variety of disciplines. The assignments have a short description followed by an 'over to you' section, which asks students to either practice the skills or reflect on what the skills would mean for them. The assignments could be used in an online class, as a hands-on activity during a face-to-face course, or assigned for students to complete on their own time, outside of class.

My submission for the ISKME GoPro Learning Challenge: Draw, build and walk a labyrinth documenting the process in video and photography! Use the experience and images for self-reflection and to inspire others.. .

This OER explores the basic operations of a Duochord. It contains both an activity as well as resources for further exploration. It is a product of the OU Academy of the Lynx, developed in conjunction with the Galileo's World Exhibition at the University of Oklahoma.

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 student homework and problem set has students quantitatively earthquake hazard, shaking and damage.

(Note: this resource was added to OER Commons as part of a batch upload of over 2,200 records. If you notice an issue with the quality of the metadata, please let us know by using the 'report' button and we will flag it for consideration.)

Students, in partners, will use an online graphing program to complete this discovery lesson. Each pair will change one component of an equation at at time and then complete a worksheet to describe the effect of the change.

Complete set of mobile friendly Elementary Algebra sample exams with solutions. In addition, there are links to pdf versions and a sample final exam with answers.

This project outlines the procedures for creating and integrating environmental sensors into a study site.

We express the exponential function of an imaginary variable in terms of sine and cosine. The "complex exponentials" that result trace out a circle in the complex plane. Pointing to one of the positions in the complex plane, we obtain the identity exp(i pi) = -1.

In the first video segment, we introduce Euler's number by considering the problem of interest compounded continuously. After we obtain the power-series representation for exp(x), we explore its properties, in the next four video segments, to convince ourselves that exp(x) is literally an exponential function, meaning a number, approximately 2.71828, taken to the power x. In the final two segments, we present the natural logarithm and demonstrate that it is the anti-derivative of 1/x.

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.

This OER explores the operation of a Telescope. It combines a lesson on lenses with a lesson using a Galileoscope. It also includes resources for further exploration. It is a product of the OU Academy of the Lynx, developed in conjunction with the Galileo's World Exhibition at the University of Oklahoma.

This simulation allows you to use a slider to see how positive and negative integers add on a number line.

This resource combines several OER resources in a way to help build deeper understanding. Students start with exploring what they already know about ratios and proportions. Next they review (or learn for the first time, depending on background) with Khan Academy some videos on the topics. They are able to do some hands on exploring with sorting ratio cards by digging into relationships. Students can then work on a three act math problem, and dig into the entire process of writing and solving a proportion problem. They end by evaluating "student work" on the topic, explaining the steps and what went wrong in the problems.

Many math classes have students with varying backgrounds and levels of understanding. These activities have multiple entry points, giving more opportunities for learning. This is designed to give students the opportunities to dig in deeply to build conceptual understanding as well as procedural fluency.

According to the GED testing service, test takers struggle with “applying rules of exponents in numerical expressions with rational exponents to write equivalent expressions with rational exponents.” (https://www.gedtestingservice.com/uploads/files/09738c12fe4e4accd9a16bab7cb99a3c.pdf )
Students do “fairly well” with simple squares and square roots, but there is a “sharp drop-off” when things get more complicated.
These are questions included in the “no calculator” portion of the test.
These skills are Mathematics Standards Level D in the College and Career Readiness Standards for Adult Education (https://www.educateiowa.gov/sites/files/ed/documents/CCRStandardsAdultEd.pdf ) under “Expressions and Equations.”
This curriculum guide will offer opportunities to build the deeper understanding necessary to understand the rules of exponents such as (xm)n = xmn .
Write and evaluate numerical expressions involving whole-number exponents. (6.EE.1)
Know and apply the properties of integer exponents to generate equivalent numerical expressions. (8.EE.1)