The purpose of this task is to engage students in an open-ended modeling task that uses similarity of right triangles, and also requires the use of technology.

7.EE.B.3  Solve real-life mathematical problems using numerical and algebraic expressions and equations.8.EE.A  Work with radicals and integer exponents.8.G  Understand and apply the Pythagorean TheoremG-GMD  Visualize relationships between two-dimensional and three-dimensional objectsG-MG  Apply geometric concepts in modeling situations  

In this lesson, students will learn that math is important in navigation and engineering. Ancient land and sea navigators started with the most basic of navigation equations (Speed x Time = Distance). Today, navigational satellites use equations that take into account the relative effects of space and time. However, even these high-tech wonders cannot be built without pure and simple math concepts basic geometry and trigonometry that have been used for thousands of years. In this lesson, these basic concepts are discussed and illustrated in the associated activities.

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.

Do art and math have anything in common? How do artists and architects use math to create their works? In these lessons, students will explore the intersection of math and art in the works of two artists and one architect for whom mathematical concepts (lines, angles, two-dimensional shapes and three-dimensional polyhedra, fractions, ratios, and permutations) and geometric forms were fundamental.

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 lesson unit is intended to help sixth grade teachers assess how well students are able to: Analyze a realistic situation mathematically; construct sight lines to decide which areas of a room are visible or hidden from a camera; find and compare areas of triangles and quadrilaterals; and calculate and compare percentages and/or fractions of areas.

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

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.

In this unit, students investigate fractional parts of the whole and use translations, reflections, rotations, and line symmetry to make four-part quilt squares. Students have a practical context for using the mathematical terms associated with divisions of the square, transformations, and symmetries. Suggestions for implementation and extensions are included.

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 interactive applet lets students create quilting type patterns and explore tessellation possibilities. The applet gives learners a choice of six polygons that can be turned or flipped and fitted together. Suggestions for exploration and five challenge patterns are provided. The tool supports a number of lessons and units including What's So Special About Triangles and Virtual Pattern Blocks (cataloged separately).

This five lesson Illuminations unit gives students an opportunity to create and analyze numeric and geometric patterns with particular emphasis on growing patterns. The lesson titles are respectively, What's Next?, Patterns On Charts, Growing Patterns, Exploring Other Number Patterns and Looking Back And Moving Forward. Each lesson includes learning objectives, material list, instructional plan, linked resources and assessment options.

"Each student creates parallelograms from square sheets of paper and connects them to form an octagon. During the construction, students consider angle measures, segment lengths, and areas in terms of the original square" (from NCTM's Illuminations).

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.

In this interactive Flash game learners use their knowledge of coordinates on a grid to move Plotter the penguin and avoid trouble. The game includes three levels of difficulty, a story mode, and a choice of five different games to play. Each game provides instruction as well as in-depth help in linked PDF documents.

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