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.

Reasoning to Find Area: Lesson 3
This lesson is the third of three lessons that use the following principles for reasoning about figures to find area:

If two figures can be placed one on top of the other so that they match up exactly, then they have the same area.
If a figure is composed from pieces that don't overlap, the sum of the areas of the pieces is the area of the figure. If a given figure is decomposed into pieces, then the area of the given figure is the sum of the areas of the pieces.
Following these principles, students can use several strategies to find the area of a figure. They can:

Decompose it into shapes whose areas they can calculate.
Decompose and rearrange it into shapes whose areas they can calculate.
Consider it as a shape with one or more missing pieces, calculate the area of the shape, then subtract the areas of the missing pieces.
Enclose it with a figure whose area they can calculate, consider the result as a region with missing pieces, and find its area using the previous strategy.
Use of these strategies involves looking for and making use of structure (MP7); explaining them involves constructing logical arguments (MP3). For now, rectangles are the only shapes whose areas students know how to calculate, but the strategies will become more powerful as students’ repertoires grow. This lesson includes one figure for which the “enclosing” strategy is appropriate, however, that strategy is not the main focus of the lesson and is not included in the list of strategies at the end.

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 the state it appears on the screen to make handouts. The web page has a full definition of a rectangle when the coordinates of the points defining it are known, and has links to other pages relating to coordinate geometry. 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.

Unit 9: Putting It All Together
Lesson 3: Rectangle Madness

This lesson is optional. In this exploration in pure mathematics, students tackle a series of activities that explore the relationship between the greatest common factor of two numbers and related fractions using a geometric representation. The activities in this lesson build on each other, providing students an opportunity to express the relationship between the greatest common factor of two numbers and related fractions through repeated reasoning (MP8). Thus, the activities should be done in order. Doing all of the activities would take more than a single class period—possibly as many as four. It is up to the teacher how much time to spend on this topic. It is not necessary to do the entire set of problems to get some benefit from the activities in this lesson, although more connections are made the farther one gets. As with all lessons in this unit, all related standards have been addressed in prior units; this lesson provides an optional opportunity to go more deeply and make connections between domains.

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 method for determining area and perimeter, a worked example and has links to other pages relating to coordinate geometry. 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.

This short video and interactive assessment activity is designed to teach fifth graders about identifying the properties of rectangles.

This short video and interactive assessment activity is designed to teach third graders an overview of squares and rectangles.

Unit 4: Dividing Fractions
Lesson 13: Rectangles with Fractional Side Lengths

This lesson builds on students’ work on area and fractions in grade 5. Students solve problems involving the relationship between area and side lengths of rectangles, in cases where these measurements can be fractions. Knowing that the area of a rectangle can be found by multiplying its side lengths, and knowing the relationship between multiplication and division, they use division to find an unknown side length when the other side length and the area are given.

This problem would be a good one when doing calculations with fractions. It also requires logical thinking and organizing of results. Different strategies and approaches can be taken: knowledge of addition, or multiples, or an understanding of fractions can be used to arrive at a solution.

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 short video and interactive assessment activity is designed to give fifth graders an overview of reflective symmetry.

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.

An interactive applet and associated web page that show the relationship between the perimeter and area of a triangle. It shows that a triangle with a constant perimeter does NOT have a constant area. The applet has a triangle with one vertex draggable and a constant perimeter. As you drag the vertex, it is clear that the area varies, even though the perimeter is constant. Optionally, you can see the path traced by the dragged vertex and see that it forms an ellipse. A link takes you to a page where this effect is exploited to construct an ellipse with string and pins. The 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.

This Nrich problem offers carefully chosen examples intended to give students the opportunity to prove that any recurring decimal can be written as a fraction. Together with the problems Terminating or Not and Tiny Nines, this problem offers valuable insights into the relationship between fractional and decimal representations.

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.