## Resources

###### Applied Filters

##### Resource Type

Modeling Scenario Technique Narrative Potential Scenario Article or Presentation Free Online Textbook General Resource Sample Syllabus Assessment Rubric or Guide SCUDEM EXPO

##### Differential Equation Type

ODE PDE Difference Delay Integral First Second Higher Linear Nonlinear System Constant Coeff Homogeneous Nonhomogeneous Other

##### Technique

Boundary Value Problems Eigen Methods Exact Equations Fourier Series Initial Value Problems Integrating Factor Laplace Transform Linear Algebra Matrices Numerical Methods Parameter Estimation Qualitative Behavior Separable Separation of Variables Series Substitution Methods Theory (general) Undetermined Coefficients Variation of Parameters Other

##### Qualitative Analysis

Equilibrium Stability Attractor Phase Plane Graphical analysis Eigenvalue analysis Parameters Other

##### Application Area

Chemistry Economics Engineering Humanities Life Sciences Mathematics Modeling (general) Physics Social Sciences Other

##### Course

Precalculus Calculus 1 Calculus 2 Calculus 3 (multivariable) Differential Equations Modeling Other

##### Course Level

Introductory Upper Level Graduate High School Other

##### Lesson Length

Portion of one class period One class period Multiple class periods One term (semester or quarter) One year Other

##### Technology

Derive Excel GeoGebra Maple Mathematica MathCad MatLab Octave Python R SAGE Calculator None Other Desmos

##### Approach

Directed Flipped Guided Open-ended Discussion Develop model Other

##### Skills

Data collection Data analysis Programming Statistics Other

##### Key Scientific Process Skills

Reading research papers Reviewing prior research Asking a question Formulating hypotheses Designing/conducting experiments Predicting outcomes Gathering data/making observations Analyzing data Interpreting results/data Displaying/modeling results/data Communicating results Translating into mathematics

##### Assessment Type

Assessment of individual student performance Assessment of student groups/teams Assignment Exam/quiz, in class Exam/quiz, take home Homework Answer clicker-type question(s) Answer essay question(s) Answer fill in the blank question(s) Answer multiple choice question(s) Answer short answer question(s) Answer true/false question(s) Create a concept map Create a diagram, drawing, figure, etc. Create a website Create graph, table etc. to present data Design an experiment or research study Design/present a poster Give an oral presentation Informal in-class report Interpret data Order items (e.g. strip sequence) Participate in discussion Peer evaluation Post-test Pre-test Produce a video or video response Respond to metacognition/reflection prompt Self evaluation Solve problem(s) Written assignment: One minute paper Written assignment: Brochure Written assignment: Essay Written assignment: Figure and or figure legend Written assignment: Report Written assignment: Literature review

##### Pedagogical Approaches

Think-Pair-Share Brainstorming Case Study Clicker Question Collaborative Work One Minute Paper Reflective Writing Concept Maps Strip Sequence Computer Model Physical Model Interactive Lecture Pre/Post Question Guided inquiry/investigation

##### Vision and Change Core Competencies - Ability

Create and develop models Use quantitative reasoning Design simulations Tap into interdisciplinary study Communicate and collaborate with mathematics community Communicate and collaborate with other disciplines Understand the relationship between material and society

##### Principles of How People Learn

Motivates student to learn material Focuses student on the material to be learned Develops supportive community of learners Leverages differences among learners Reveals prior knowledge Requires student to do the bulk of the work

##### Bloom's Cognitive Level

Foundational: factual knowledge & comprehension Application & Analysis Synthesis/Evaluation/Creation

##### Includes clear efforts on Issues

Diversity Equity Inclusion Enhancement of all

Technique Narrative

##### 1-015-DimensionlessVariables-TechniqueNarrative

This material introduces the idea of ``rescaling'' for ordinary differential equations (ODE's) by the use of dimensionless variables. In practice this is an extremely common and useful prelude to the analysis and solution of ODE's.
linearization
dimensional analysis
time scale
scaling
dimensionless variables
spatial scale

Technique Narrative

##### 9-001-SkinBurnModelNumericalMethods-TechniqueNarrative

The heat equation is an important partial differential equation (PDE) which describes the distribution of heat in a given region over time. Numerical methods play an important role in solving these.
Maternal-Fetal interface
heat equation
Conservation of Energyt
heat flux
Euler's forward method
central difference
skin burn
hyperthermia
thermal conductivity
layers

Technique Narrative

##### 7-006-LaplaceTransformBirth-TechniqueNarrative

We present a way of introducing the Laplace Transform as the continuous analogue of a power series expression of a function.
Transformations
power series

Technique Narrative

##### 5-030-LinNonHomoSystemSol-TechniqueNarrative

We offer strategies for solving linear systems of nonhomogeneous differential equations using a conjectured solution strategy for a system of constant coefficient, linear, n
strategies
discovery learning
eigenvectors
eigenvalue

Technique Narrative

##### 5-012-LinearSystemConjecture-TechniqueNarrative

Students go from the solution for y'(t) = k*y(t) to a natural extension to the solution conjecture of a system of two constant coefficient, homogeneous, linear differential equations introducing eigenvalues and eigenvectors through student...
discovery learning
conjecture
solution
substitution

Technique Narrative

##### 7-005-OverViewLaplaceTransforms-TechniqueNarrative

This is a specialized overview of Laplace Transform application to solving differential equations in Mathematica.
integration
Laplace transform
Inverse Laplace transofrm
integration by parts

Technique Narrative

##### 5-010-MatrixExponential-TechniqueNarrative

The matrix exponential is a powerful computational and conceptual tool for analyzing systems of linear, constant coefficient, ordinary differential equations (ODE's). This narrative offers a quick introduction to the technique, with examples and...
matrix esponential
Putzer's Algorithm
diagonalization
eigenvalues
eigenvectors

Technique Narrative

##### 5-005-StiffDifferentialEquations-TechniqueNarrative

This material introduces the topic of ``stiffness'' for a system of ordinary differential equations (ODE's), through a series of examples. Stiffness is a property that a system of ODE's may posses that make it difficult to solve numerically.
Euler's method
explicit methods
implicit methods
stiff
sitffness
stiff differential equation
instability

Technique Narrative

##### 3-090-ChebyshevPolynomialSolution-TechniqueNarrative

The Chebyshev equation is presented as a vehicle to view series solutions techniques for linear, second order homogeneous differential equations with non-constant coefficients.
series soluotion
polynomial solutions
Chebyshev polynomials
Chebyshev differential equations

Technique Narrative

##### 2-005-LinearizeItAll-TechniqueNarrative

Linear approximations are often used to simplify nonlinear ordinary differential equations (ODEs) for ease in analysis. The resulting linear approximation produces an ODE where closed form solutions may be obtained.
computation
error
Torricelli's Law
data fitting
linear approximation

Technique Narrative

##### 1-030-RandomPerturbation-TechniqueNarrative

After a brief historical view of this problem, we will demonstrate the derivation of first order linear differential equations with random perturbations.
random perturbation
Brownian motion
Langevin equation
Riemann-Steiltjes integral
Wiener process
Ito's calculus

Technique Narrative

##### 1-010-AtmosphericCO2Bifurcation-TechniqueNarrative

Students are introduced to the concept of a bifurcation in a first-order ordinary differential equation (ODE) through a modeling scenario involving atmospheric carbon dioxide whish is taken as a parameter and temperature is a function of time.
carbon dioxide
Surface Atmosphere Exchange
bifurcation
fold bifurcation
saddle node

Technique Narrative

##### 8-002-TrigSumRepresentation-TechniqueNarrative

Students discover how to represent functions as sums of trigonometric functions and the value of such representations in many fields. This is an introduction to the study of Fourier Series.
estimation
Trigonometry
Trigonometric Functions
sum of square errors
finite sum
Fourier series
Fouriere sum
spectrum
approximatiopn

Technique Narrative

##### 1-005-NavigatingNumericalMethods-TechniqueNarrative

This technique narrative is a discovery-based approach to learning the basics of numerical methods for first order differential equations, by following the graphical and analytical perspectives of the forward Euler method and second order Taylor...
Euler's method
Taylor's Method
search and rescue
Coaast Guard
lost at sea
longitude
latitude
limit definition

Technique Narrative

##### 1-001-SepartionOfVariables-TechniqueNarrative

We discuss strategies to solve first order, ordinary differential equations with mathematical models when the variables may be separated.
solutions
strategy
method
applications