"Frameworks and Models in Engineering Systems / Engineering System Design, Spring 2007"

" This class provides an introduction to quantitative models and qualitative frameworks for studying complex engineering systems. Also taught is the art of abstracting a complex system into a model for purposes of analysis and design while dealing with complexity, emergent behavior, stochasticity, non-linearities and the requirements of many stakeholders with divergent objectives. The successful completion of the class requires a semester-long class project that deals with critical contemporary

Parallel lines. (Coordinate Geometry)

An interactive applet and associated web page that demonstrate the properties of parallel lines in coordinate geometry. The applet presents two lines, each defined by a pair of draggable points. If they are parallel the slopes are the same and the applet adds the parallel arrow marks to them. The slope of each line is continuously recalculated and the case of vertical lines (with undefined slope) is handled correctly. The is a software 'detent' that snaps the lines to parallel if they are close

The Coordinate Plane

An interactive applet and associated web page that describe the concept of the coordinate plane (Cartesian Plane). The applet shows the plane, its axes, origin and related controls. The user can drag a point around and see the coordinates change, and click anywhere to create new points. The origin can be dragged to emphasize or eliminate certain quadrants. The grid, axis pointers and coordinates can be turned on and off. The coordinate display can be turned off to permit class exercises and then

Coordinates of a point

An interactive applet and associated web page that describe the concept of a point in coordinate geometry. The applet show one point initially, but clicking anywhere adds more. As the points are dragged, the coordinates are seen to change. The grid, axis pointers and coordinates can be turned on and off. The coordinate display can be turned off to permit class exercises and then turned back on the verify the answers. The applet can be printed as it appears on the screen to make handouts. The web

Distance from a point to a line

An interactive applet and associated web page that demonstrate how to find the perpendicular distance between a point and a line. The applet has a line with sliders that adjust its slope and intercept, and a draggable point. As the line is altered or the point dragged, the distance is recalculated. The page actually shows four ways to do it, with links to detail pages on each one. The grid and coordinates can be turned on and off. The distance calculation can be turned off to permit class exerci

Distance from a point to a line using trigonometry

An interactive applet and associated web page that demonstrate how to find the perpendicular distance between a point and a line using trigonometry, given the coordinates of the point and the slope/intercept of the line. The applet has a line with sliders that adjust its slope and intercept, and a draggable point. As the line is altered or the point dragged, the distance is recalculated. The grid and coordinates can be turned on and off. The distance calculation can be turned off to permit class

Ray definition (Coordinate Geometry)

An interactive applet and associated web page that show the definition of a ray in coordinate geometry. The applet has two points that the user can drag. The two points define a ray, one of them the endpoint, the other a secondary point through which the ray passes. The grid, axis pointers and coordinates can be turned on and off. The applet can be printed as it appears on the screen to make handouts. The web page has a full definition of a ray when the coordinates of the points defining it are

Rectangle Area and Perimeter. (Coordinate Geometry)

An interactive applet and associated web page that show the definition and properties of a rectangle in coordinate geometry. The applet has a rectangle with draggable vertices. As the user re-sizes the rectangle the applet continuously recalculates its width, height and diagonals from the vertex coordinates. Rectangle can be rotated on the plane to show the more complex cases. The grid, coordinates and calculations can be turned on and off for class problem solving. The applet can be printed in

Rectangle and its properties. (Coordinate Geometry)

An interactive applet and associated web page showing how to find the area and perimeter of a rectangle from the coordinates of its vertices. The rectangle can be either parallel to the axes or rotated. The grid and coordinates can be turned on and off. The area and perimeter calculation can be turned off to permit class exercises and then turned back on the verify the answers. The applet can be printed as it appears on the screen to make handouts. The web page has a full description of the meth

Square and its properties (Coordinate Geometry)

An interactive applet and associated web page that show the definition and properties of a square when applied in coordinate geometry. The applet has a square, and the user can drag any vertex to resize it. It shows how to calculate the side lengths and diagonal length given the vertex coordinates. The grid and coordinates can be turned on and off. The applet can be printed as it appears on the screen to make handouts. The web page has a full definition of a square when the coordinates of the po

Area and Perimeter of a square (Coordinate Geometry)

An interactive applet and associated web page showing how to find the area and perimeter of a square from the coordinates of its vertices. The square can be either parallel to the axes or rotated. The grid and coordinates can be turned on and off. The area and perimeter calculation can be turned off to permit class exercises and then turned back on the verify the answers. The applet can be printed as it appears on the screen to make handouts. The web page has a full description of the method for

Area of a triangle using the box method (Coordinate Geometry)

An interactive applet and associated web page that calculate the area of a triangle using the box method in coordinate geometry. The applet has a triangle with draggable vertices. As you drag them the triangle's bounding box is shown and the area recalculated by subtracting the areas of the outside triangles. The grid and coordinates can be turned on and off. The area calculation can be turned off to permit class exercises and then turned back on the verify the answers. The applet can be printed

Vertical line definition. (Coordinate Geometry)

An interactive applet and associated web page that show the definition of a vertical line in coordinate geometry. The applet has two points that the user can drag which define a line. The line flagged when it is vertical (slope=undefined). The grid, details and coordinates can be turned on and off. The applet can be printed exactly as it appears on the screen to make handouts. The web page has a discussion on how to test for vertical and has links to other pages relating to coordinate geometry.

Cube definition and properties

An interactive applet and associated web page that demonstrate the properties of a cube. A 3-D cube is shown in the applet which can be interactively manipulated using the mouse. Research has shown that some younger students have difficulty visualizing the parts of a 3D object that are hidden. To help with this, the student can rotate the cube in any axis simply by dragging it with the mouse. It can also be 'exploded' - where a slider gradually separates the faces to reveal the ones behind. The

Perimeter of a polygon

A web page and interactive applet show how to compute the perimeter of a polygon. A polygon is shown that can be resized by dragging its vertices. As you drag, the perimeter is continuously recalculated. Text on the page explains that the perimeter is the sum of the sides. For those who prefer it, in a formula that is given. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference interactive geometry reference book pr

Perimeter of a rectangle

A web page and interactive applet show how to compute the perimeter of a rectangle. A rectangle is shown that can be resized by dragging its vertices. As you drag, the perimeter is continuously recalculated. Text on the page explains that the perimeter is the sum of the sides. For those who prefer it, in a formula that is given. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference interactive geometry reference boo

Triangle similarity, ratios of area

An interactive applet and associated web page that demonstrate that in similar triangles, the ratio of their areas is the square of the ratio of the sides. As you drag one triangle to resize it, it remains similar to another and the ratios of sides and areas is calculated as you drag. One can be seen to be the square of the other at all times. A slight 'snap-to' effect allows easy selection of integer ratios (2:4 etc). Applet can be enlarged to full screen size for use with a classroom projector

Triangle similarity, ratios of parts

An interactive applet and associated web page that demonstrate that in similar triangles, ten items are always in the same ratio to each other (perimeter, three sides, three altitudes, three medians). As you drag one triangle to resize it, it remains similar to another and the ratios of sides and medians is calculated as you drag. They can be seen to be the same at all times. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math O

Diagonals of a square

A web page and interactive applet show how to compute the length of the diagonals of a square. A square is shown that can be resized by dragging its vertices. As you drag, the diagonals are continuously recalculated. Text on the page explains calculation. For those who prefer it, in a formula that is given. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference interactive geometry reference book project at http://ww

Inference by Analogy (Archaeology)

Students will use ethnohistoric sources and an archaeological site map to study ethnohistoric analogy. This unit is based on Lesson 2.11 in "Intrigue of the Past: North Carolina's First Peoples", which is part of a national curriculum-development program called "Project Archaeology":