This video lesson presents a real world problem that can be solved by using the Pythagorean theorem. The problem faces a juice seller daily. He has equilateral barrels with equal heights and he always tries to empty the juice of two barrels into a third barrel that has a volume equal to the sum of the volumes of the two barrels. This juice seller wants to find a simple way to help him select the right barrel without wasting time, and without any calculations - since he is ignorant of Mathematics. The prerequisite for this lesson includes knowledge of the following: the Pythagorean theorem; calculation of a triangles area knowing the angle between its two sides; cosine rule; calculation of a circle's area; and calculation of the areas and volumes of solids with regular bases.

This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications. It is geared toward high school Geometry students that have completed a year of Algebra.

Studies how randomization can be used to make algorithms simpler and more efficient via random sampling, random selection of witnesses, symmetry breaking, and Markov chains. Models of randomized computation. Data structures: hash tables, and skip lists. Graph algorithms: minimum spanning trees, shortest paths, and minimum cuts. Geometric algorithms: convex hulls, linear programming in fixed or arbitrary dimension. Approximate counting; parallel algorithms; online algorithms; derandomization techniques; and tools for probabilistic analysis of algorithms.

This task is a reasonably straight-forward application of rigid motion geometry, with emphasis on ruler and straightedge constructions, and would be suitable for assessment purposes.

The goal of this task is to give students an opportunity to experiment with reflections of triangles on a coordinate grid. Students are not prompted in the question to list the coordinates of the different triangle vertices but this is a natural extension of the task.

The goal of this task is to give students experience applying and reasoning about reflections of geometric figures using their growing understanding of the properties of rigid motions. In the case of reflecting a rectangle over a diagonal, the reflected image is still a rectangle and it shares two vertices with the original rectangle.

This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focusing on the class of equilaterial triangles. In particular, the task has students link their intuitive notions of symmetries of a triangle with statements proving that the said triangle is unmoved by applying certain rigid transformations.

This task examines some of the properties of reflections of the plane which preserve an equilateral triangle: these were introduced in ''Reflections and Isosceles Triangles'' and ''Reflection and Equilateral Triangles I''. The task gives students a chance to see the impact of these reflections on an explicit object and to see that the reflections do not always commute.

This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focussing on the class of isosceles triangles.

This task presents students with some creative geometric ways to represent the fraction one half. The goal is both to appeal to students' visual intuition while also providing a hands on activity to decide whether or not two areas are equal.

This task provides a context for indentifying different ways of representing half of a rectangle.

This task provides a good opportunity to use isosceles triangles and their properties to show an interesting and important result about triangles inscribed in a circle: the fact that these triangles are always right triangles is often referred to as Thales' theorem. It does not have a lot of formal prerequisites, just the knowledge that the sum of the three angles in a triangle is 180 degrees.

The result here complements the fact, presented in the task ``Right triangles inscribed in circles I,'' that any triangle inscribed in a circle with one side being a diameter of the circle is a right triangle. A second common proof of this result rotates the triangle by 180 degrees about M and then shows that the quadrilateral, obtained by taking the union of these two triangles, is a rectangle.

This task uses geometry to find the perimeter of the track. Students may be surprised when their calculation does not give 400 meters but rather a smaller number.

The goal of this task is to model a familiar object, an Olympic track, using geometric shapes. Calculations of perimeters of these shapes explain the staggered start of runners in a 400 meter race.

In this unit of five lessons from Illuminations, learners begin with a number-line model and extend it to investigate linear relationships with the Distance, Speed, and Time Simulation from NCTM's E-Examples. Students then progress to plotting points and graphing linear functions while continually learning and reinforcing basic multiplication facts. Instructional plan, questions for the students, assessment options, extensions,and teacher reflections are given for each lesson as well as links to download all student resources.

This is the first version of a task asking students to find the areas of triangles that have the same base and height, and is the most concrete.

This is the second version of a task asking students to find the areas of triangles that have the same base and height. This presentation is more abstract as students are not using physical models.

This real world word problem requires students to figure out the scale ratio and unit rate.

This task is an example of applying geometric methods to solve design problems and satisfy physical constraints. This task models a satellite orbiting the earth in communication with two control stations located miles apart on earthsŐ surface.