This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.

Students find the volume and surface area of a rectangular box (e.g., a cereal box), and then figure out how to convert that box into a new, cubical box having the same volume as the original. As they construct the new, cube-shaped box from the original box material, students discover that the cubical box has less surface area than the original, and thus, a cube is a more efficient way to package things.

The lesson begins by introducing Olympics as the unit theme. The purpose of this lesson is to introduce students to the techniques of engineering problem solving. Specific techniques covered in the lesson include brainstorming and the engineering design process. The importance of thinking out of the box is also stressed to show that while some tasks seem impossible, they can be done. This introduction includes a discussion of the engineering required to build grand, often complex, Olympic event centers.

Student groups work with manipulatives—pencils and trays—to maximize various quantities of a system. They work through three linear optimization problems, each with different constraints. After arriving at a solution, they construct mathematical arguments for why their solutions are the best ones before attempting to maximize a different quantity. To conclude, students think of real-world and engineering space optimization examples—a frequently encountered situation in which the limitation is the amount of space available. It is suggested that students conduct this activity before the associated lesson, Linear Programming, although either order is acceptable.

This challenging problem and brainteaser gives students an opportunity to compose and decompose polygons to make rectangles.

This short video and interactive assessment activity is designed to give fourth graders an overview of ellipses, trapezoids, rhombi, and polygons.

This short video and interactive assessment activity is designed to give fifth graders an overview of forming figures with cutouts.

How can we alter the design of chip packaging to be more sustainable while protecting the product? Students will design a better way to package potato chips that will be less harmful to the environment and protect the product from being crushed during shipping. The potato chip. It’s crispy. It’s salty. It’s delicious. And you can’t eat just one! As consumers, we often devour a bag of potato chips without giving a second thought to what happens to the bag when we throw it away. Did you know that the average potato chip bag is nearly impossible to recycle? That’s because most bags used for this purpose contain up to 7 layers of plastic! Currently, there is no way to separate those layers, so the bags end up in landfills.

This is a 60 to 90-minute lesson that includes a self-paced interactive module and classroom activities. The teacher guide includes a challenge sequence (timeline), relevance to standards, materials list, assessment, evaluation rubric, and learning extensions.

Lesson objectives: (1) Research and analyze green materials that can be used in product packaging. (2) Investigate 3-D shapes and how they might be used to protect products from damage. (3) Create a design for green packaging that will contain food products.

The purpose of this task is to provide students an opportunity to use mathematics addressed in different standards in the same problem.

This high level task is an example of applying geometric methods to solve design problems and satisfy physical constraints. This task is accessible to all students. In this task, a typographic grid system serves as the background for a standard paper clip.

Students act as civil engineers developing safe railways as a way to strengthen their understanding of parallel and intersecting lines. Using pieces of yarn to visually represent line segments, students lay down "train tracks" on a carpeted floor, and make guesses as to whether these segments are arranged in parallel or non-parallel fashion. Students then test their tracks by running two LEGO® MINDSTORMS® NXT robots to observe the consequences of their track designs, and make safety improvements. Robots on intersecting courses face imminent collision, while robots on parallel courses travel safely.

This short video and interactive assessment activity is designed to teach third graders about fraction of a whole.

This task can be implemented in a variety of ways. For a class with previous exposure to properties of perpendicular bisectors, part (a) could be a quick exercise in geometric constructions, and an application of the result. Alternatively, this could be part of an introduction to perpendicular bisectors, culminating in a full proof that the three perpendicular bisectors are concurrent at the circumcenter of the triangle, an essentially complete proof of which is found in the solution below.

Students learn about and practice graphing, plotting points, drawing line segments, and finding the coordinates of points of intersection.

This task is part of a series presenting important foundational geometric results and constructions which are fundamental for more elaborate arguments. They are presented without a real world context so as to see the important hypotheses and logical steps involved as clearly as possible.

This KET animation illustrates positive and negative measurement on a straight line and on a flat plane using a point of origin.

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 connect algebra to geometric concepts with polygons as they explore the distance formula, slope criteria for parallel and perpendicular lines, and learn to calculate and justify the area and perimeter of polygons.

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.

Students formalize their understanding of compound probability, develop an understanding of conditional probability, and understand and calculate permutations and combinations.