Welcome to the world of mathematical modeling. The objective of this course is to provide students with the experience of harnessing the power of mathematical reasoning to the solution of real-world problems. Your students will investigate real problems to a depth that far exceeds that commonly encountered in a classroom. This will be a very active experience for your students. It will be an interesting and engaging course. The cognitive demand will be both high and continuous, and the resulting growth in your students’ abilities will be commensurate. The students’ understanding of the problems being modeled will be continuously developed as simple models are tested and found wanting and more sophisticated models are developed. Your students will develop thought patterns and reasoning processes that are necessary to apply those mathematical tools that they already possess to produce useful results.

A Path Through the Course

This flowchart describes a recommended path through the science section of the Mathematical Models with Applications course. The goal for this part of the course is for you to teach five units: Three are recommended, and two can be selected based on student career and subject interests, access to technology, and mathematics level. To help you determine which of these units might be appropriate for your classroom, here is some information on the unit choices.

After teaching Prediction and Rain, you should teach one of the following units.

After teaching Oscillation, you should teach one of the following units.

Important note: These are recommendations! Every class is different, and teachers should use materials in a manner consistent with the unique abilities and interests of their students.

TEKS-Related Content by Unit

I. Bivariate data analysis

A. Proportional relationships

B. The meaning of slope

II. Linear relationships

A. Least-square line fitting

B. Variability of data and residual plots

C. The limited accuracy of predictions

D. Effect of outliers

I. Weighted averages

II. Centers of influence, distance, and Voronoi diagrams

A. Geometric construction of perpendicular bisectors

B. Algebraic analysis of perpendicular bisectors

C. Systems of linear equations

III. Areas of irregular polygons

A. Break complex shapes into a sum of simple shapes

B. Pick’s formula

C. Heron’s formula

D. Monte Carlo simulation

IV. Sensitivity of a product to errors in either factor

I. Linear equations as models of one-dimensional motion of a point

II. Parametric equations as models for two-dimensional motion of a point

III. Matrix description of the motion of multiple points

A. Addressing elements of a matrix

B. Matrix addition

IV. Evolution of closed form descriptions from recursive forms

I. Velocity and acceleration

II. Local linearity and instantaneous rate of change

III. Linear and quadratic regression

IV. Parametric equations using technology

I. Data analysis using least-squares regression

A. Linear, quadratic, and exponential regression

B. Use of residual patterns to determine "goodness of fit"

II. Probability area analysis development of a quadratic model

III. Vertex and standard forms of the quadratic equation

IV. Transformations of quadratics

V. Solution of quadratic equations

A. Graphical

B. Completing the square

C. The quadratic formula

I. Periodic functions

II. Radian measure

III. Sinusoidal functions

IV Transformations of sinusoidal functions

V. Sinusoidal fitting and sinusoidal regression

I. Linear relationships

A. Recursive forms characterized by addition

B. Linear equations as the closed form

II. Exponential relationships

A. Recursive forms characterized by multiplication

B. Closed form of y_n = P₀kⁿ

I. Sequential use of linear and exponential models

II. Sequence notation

III. Partial sums and limiting values of series

IV. Solving exponential equations using logarithms and technology