## Resources

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Resource Type (97)

Modeling Scenario(70) Technique Narrative(1) Potential Scenario(22) Article or Presentation(4)

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Differential Equation Type (97)

ODE(88) PDE(7) Delay(1) First(55) Second(30) Linear(39) Nonlinear(31) System(17) Constant Coeff(19) Homogeneous(28) Nonhomogeneous(17)

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Boundary Value Problems(6) Eigen Methods(2) Initial Value Problems(82) Integrating Factor(2) Laplace Transform(1) Linear Algebra(1) Matrices(2) Numerical Methods(8) Parameter Estimation(41) Qualitative Behavior(7) Separable(14) Separation of Variables(28) Substitution Methods(1)

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Qualitative Analysis (97)

Equilibrium(6) Stability(1) Phase Plane(7) Graphical analysis(36) Eigenvalue analysis(4) Parameters(60) Other(18)

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Application Area (97)

Chemistry(8) Economics(4) Engineering(17) Life Sciences(19) Mathematics(25) Modeling (general)(87) Physics(69) Social Sciences(4)

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Calculus 1(1) Calculus 2(46) Calculus 3 (multivariable)(1) Differential Equations(96) Modeling(89)

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Reading research papers(1) Reviewing prior research(4) Asking a question(68) Formulating hypotheses(63) Designing/conducting experiments(16) Predicting outcomes(21) Gathering data/making observations(21) Analyzing data(31) Interpreting results/data(32) Displaying/modeling results/data(32) Communicating results(33) Translating into mathematics(2)

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Assessment Type (71)

Homework(67) Solve problem(s)(1) Written assignment: Essay(1) Written assignment: Report(65) Written assignment: Literature review(1)

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Pedagogical Approaches (63)

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Vision and Change Core Competencies - Ability (72)

Create and develop models(71) Use quantitative reasoning(65) Design simulations(13) Tap into interdisciplinary study(5) Communicate and collaborate with other disciplines(5)

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Principles of How People Learn (69)

Motivates student to learn material(67) Focuses student on the material to be learned(65) Develops supportive community of learners(1) Leverages differences among learners(1) Reveals prior knowledge(2) Requires student to do the bulk of the work(1)

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Bloom's Cognitive Level (65)

Foundational: factual knowledge & comprehension(3) Application & Analysis(63)

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Includes clear efforts on Issues (1)

Equity(1)

###### Applied Filters

Modeling Scenario

##### 1-058-WaterClocks-ModelingScenario

We apply Torricelli's Law to the task of building a water clock in which the height of the water in a container falls at a constant rate when the container has a hole in the bottom to let the water flow out.Modeling Scenario

##### 1-064-TorricelliBox-ModelingScenario

The time it takes a column of water to empty and the time it takes the same volume of column of water with a box (various sizes) submerged in the column of water are compared through modeling with Torricelli's Law.Modeling Scenario

##### 1-015-Torricelli-ModelingScenario

We help students develop a model (Torricelli's Law) for the height of a falling column of water with a small hole in the container at the bottom of the column of water through which water exits the column.Potential Scenario

##### 2015-Goodnow-EtAl-Mathematical Models of Water Clocks

This is a historical tour of water clocks, known as clepsydra a Greek word meaning water thief. These devices were for telling time.Article or Presentation

##### 2020-TeachingModule-ModelingFallingColumnOfWater

We discuss how to model a falling column of water empirically and analytically from first principles in physics laws.Modeling Scenario

##### 1-073-WaterExitBottle-ModelingScenario

We offer an experiment in which data is collected to ascertain a parameter in the differential equation formulation of Torricelli's Law for water flow out of a cylindrical container.Modeling Scenario

##### 1-074-BottleWaterFlow-ModelingScenario

We offer an experiment in which data is collected to ascertain a parameter in the differential equation formulation of Torricelli's Law for water flow.Potential Scenario

##### 2012-Lensyl-Urbano-The Draining of a Plastic Bottle-Integrating a Physics Experiment into Calculus

I punched a small hole (about 1mm radius) in a one gallon plastic bottle and had my students measure the rate at which water drained.Potential Scenario

##### 2001-Knisley-Kerley-Using data to motivate the models used in introductory differential equations

In this paper, the data itself is used to motivate mathematical models in introductory mathematics courses. In doing so, various regression and optimization techniques are illustrated.Potential Scenario

##### 1999-Leonid_Hanin-Which tank empties first

Suppose that water towers like those shown in the opening are initially filled with water and have the same volume, height, and cross-sectional outlet area. Which one empties first?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.Modeling Scenario

##### 1-014-DrainingContainers

Given two rectangular circular cylinders of water with the same volume, but different radii, with a small bore hole of same radius on the center of the bottom through which water exits the cylinder, which empties faster?Potential Scenario

##### 1992-Jude_Sommerfeld-More Applied Math Problems on Vessel Draining

From a chemical engineering point of view emptying processing and storage tanks is important and this article considers many issues and shapes of containers. The main interest is in the total time it takes to empty the tank for a collection of...Modeling Scenario

##### 1-140-LeakyBucket-ModelingScenario

We seek to model the height of water in a cylindrical tank (bucket) in which water flows out the bottom of the tank through a small bore hole while we are pouring water into the tank at the top of the tank at a constant (or varying) rate.Modeling Scenario