Students learn about geometric relationships by solving real mini putt examples on paper and then using putters and golf balls to experiment with the teacher’s pre-made mini put hole(s) framed by 2 x 4s, comparing their calculated (theoretical) results to real-world results. To “solve the holes,” they find the reflections of angles and then solve for those angles. They do this for 1-, 2- and 3-banked hole-in-one shots. Next, students apply their newly learned skills to design, solve and build their own mini putt holes, also made of 2 x 4s and steel corners.

Students learn about common geometry tools and then learn to use protractors (and Miras, if available) to create and measure angles and reflections. The lesson begins with a recap of the history and modern-day use of protractors, compasses and mirrors. After seeing some class practice problems and completing a set of worksheet-prompted problems, students share their methods and work. Through the lesson, students gain an awareness of the pervasive use of angles, and these tools, for design purposes related to engineering and everyday uses. This lesson prepares students to conduct the associated activity in which they “solve the holes” for hole-in-one multiple-banked angle solutions, make their own one-hole mini-golf courses with their own geometry-based problems and solutions, and then compare their “on paper” solutions to real-world results.

Students take on the role of geographers and civil engineers and use a device enabled with the global positioning system (GPS) to locate geocache locations via a number of waypoints. Teams save their data points, upload them to geographic information systems (GIS) software, such as Google Earth, and create scale drawings of their explorations while solving problems of area, perimeter and rates. The activity is unique in its integration of technology for solving mathematical problems and asks students to relate GPS and GIS to engineering.

Students explore the concept of similar right triangles and how they apply to trigonometric ratios. Use this lesson as a refresher of what trig ratios are and how they work. In addition to trigonometry, students explore a clinometer app on an Android® or iOS® device and how it can be used to test the mathematics underpinning trigonometry. This prepares student for the associated activity, during which groups each put a clinometer through its paces to better understand trigonometry.

Students see that geometric shapes can be found in all sorts of structures as they explore the history of the Roman Empire with a focus on how engineers 2000 years ago laid the groundwork for many structures seen today. Through a short online video, brief lecture material and their own online research directed by worksheet questions, students discover how the Romans invented a structure known today as the Roman arch that enabled them to build architecture never before seen by humankind, including the amazing aqueducts. Students calculate the slope and its total drop and angle over its entire distance for an example aqueduct. Completing this lesson prepares students for the associated activity in which teams build and test model aqueducts that meet specific constraints. This lesson serves as an introduction to many other geometry—and engineering-related lessons—including statics and trusses, scale modeling, and trigonometry.

An interactive applet and associated web page that provide step-by-step animated instructions on how to measure angles using a protractor. Specifically, it uses a protractor to measure two angles that form a vertical pair, verifying they have the same measure. The animation can be run either continuously like a video, or single stepped to allow classroom discussion and thought between steps. 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 textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that demonstrate the inscribed angle of a circle - the angle subtended at the periphery by two points on the circle. The applet presents a circle with three points on it that can be dragged. The inscribed angle is shown and demonstrates that it is constant as the vertex is dragged. Links to other related topics such as Thales Theorem. 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 textbook project at http://www.mathopenref.com.

An interactive applet and associated web page that demonstrate the relationship of the interior and exterior angles of a polygon. The applet shows an irregular polygon where one vertex is draggable. As it is dragged the interior and exterior angles at that vertex are displayed, and a formula is continuously updated showing that they are supplementary. The tricky part is when the vertex is dragged inside the polygon making it concave. The applet shows how the relationship still holds provided you get the signs of the angles right. 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 textbook project at http://www.mathopenref.com.

Students learn about video motion capture technology, becoming familiar with concepts such as vector components, magnitudes and directions, position, velocity, and acceleration. They use a (free) classroom data collection and processing tool—the ARK Mirror—to visualize and record 3-D motion. The Augmented Reality Kinematics (ARK) Mirror software collects data via a motion detector. Using an Orbbec Astra Pro 3D camera or Microsoft Kinect (see note below), students can visualize and record a robust set of data and interpret them using statistical and graphical methods. This lesson introduces students to just one possible application of the ARK Mirror software—in the context of a high school physics class. Note: The ARK Mirror is ported to operate on an Orbbec platform. It may also be used with a Microsoft Kinect, although that Microsoft hardware has been discontinued. Refer to the Using ARK Mirror and Microsoft Kinect attachment for how to use the ARK MIrror software with Microsoft Kinect.

Students learn about radar imaging and its various military and civilian applications that include recognition and detection of human-made targets, and the monitoring of space, deforestation and oil spills. They learn how the concepts of similarity and scaling are used in radar imaging to create three-dimensional models of various targets. Students apply the critical attributes of similar figures to create scale models of a radar imaging scenario using infrared range sensors (to emulate radar functions) and toy airplanes (to emulate targets). They use technology tools to measure angles and distances, and relate the concept of similar figures to real-world applications.

In this activity, students explore the relationship among angles formed by a transversal and a system of two lines. In particular, they consider what happens when the two lines are parallel versus when they are not.

Explore molecule shapes by building molecules in 3D! Find out how a molecule's shape changes as you add atoms to a molecule.

Students examine how the orientation of a photovoltaic (PV) panel relative to the sun affects the efficiency of the panel. Using sunshine (or a lamp) and a small PV panel connected to a digital multimeter, students vary the angle of the solar panel, record the resulting current output on a worksheet, and plot their experimental results.

An interactive applet and associated web page that demonstrate obtuse angles (those between 90 and 180 deg). The applet presents an angle (initially obtuse) that the user can adjust by dragging the end points of the line segments forming the angle. As it changes it shows the angle measure and a message that indicate which type of angle it is. There a software 'detents' that make it easy capture exact angles such as 90 degrees and 180 degrees The message and angle measures can be turned off to facilitate classroom discussion. The text on the page has links to other pages defining each angle type in depth. 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 textbook project at http://www.mathopenref.com.

Students are introduced to the physics concepts of air resistance and launch angle as they apply to catapults. This includes the basic concepts of position, velocity and acceleration and their relationships to one another. They use algebra to solve for one variable given two variables.

Students take a close look at truss structures, the geometric shapes that compose them, and the many variations seen in bridge designs in use every day. Through a guided worksheet, students draw assorted 2D and 3D polygon shapes and think through their forms and interior angles (mental “testing”) before and after load conditions are applied. They see how engineers add structural members to polygon shapes to support them under compression and tension, and how triangles provide the strongest elemental shape. A PowerPoint® presentation is provided. This lesson prepares students for two associated activities that continue the series on polygons and trusses.

Students learn about the role engineers play in designing and building truss structures. Simulating a real-world civil engineering challenge, student teams are tasked to create strong and unique truss structures for a local bridge. They design to address project constraints, including the requirement to incorporate three different polygon shapes, and follow the steps of the engineering design process. They use hot glue and Popsicle sticks to create their small-size bridge prototypes. After compressive load tests, they evaluate their results and redesign for improvement. They collect, graph and analyze before/after measurements of interior angles to investigate shape deformation. A PowerPoint® presentation, design worksheet and data collection sheet are provided. This activity is the final step in a series on polygons and trusses.

Blast a Buick out of a cannon! Learn about projectile motion by firing various objects. Set the angle, initial speed, and mass. Add air resistance. Make a game out of this simulation by trying to hit a target.

Students learn how simple machines, including wedges, were used in building both ancient pyramids and present-day skyscrapers. In a hands-on activity, students test a variety of wedges on different materials (wax, soap, clay, foam). Students gain an understanding of how simple machines are used in engineering applications to make our lives and work easier.

An interactive applet and associated web page that demonstrate a right angle (those equal to 90 deg). The applet presents an angle (initially 90 deg) that the user can adjust by dragging the end points of the line segments forming the angle. As it changes it shows the angle measure and a message that indicate which type of angle it is. There a software 'detents' that make it easy capture exact angles such as 90 degrees and 180 degrees The message and angle measures can be turned off to facilitate classroom discussion. The text on the page has links to other pages defining each angle type in depth. 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 textbook project at http://www.mathopenref.com.