This task provides a construction of the angle bisector of an angle by reducing it to the bisection of an angle to finding the midpoint of a line segment. It is worth observing the symmetry -- for both finding midpoints and bisecting angles, the goal is to cut an object into two equal parts. The conclusion of this task is that they are, in a sense, of exactly equivalent difficulty -- bisecting a segment allows us to bisect and angle (part a) and, conversely, bisecting an angle allows us to bisect a segment (part b). In addition to seeing how these two constructions are related, the task also provides an opportunity for students to use two different triangle congruence criteria: SSS and SAS.

This short video and interactive assessment activity is designed to give fourth graders an overview of classification of angles.

In this activity, learners use a hand-made protractor to measure angles they find in playground equipment. Learners will observe that angle measurements do not change with distance, because they are distance invariant, or constant. Note: The "Pocket Protractor" activity should be done ahead as a separate activity (see related resource), but a standard protractor can be used as a substitute.

This short video and interactive assessment activity is designed to give fourth graders an overview of measurement of angles.

Find out how angles and symmetry come into play in the game of pool in this video adapted from Annenberg Learner’s Learning Math: Measurement.

This lesson unit is intended to help you assess how well students are able to use geometric properties to solve problems. In particular, it will support you in identifying and helping students who have the following difficulties: Solving problems relating to using the measures of the interior angles of polygons; and solving problems relating to using the measures of the exterior angles of polygons.

The famous story of Archimedes running through the streets of Syracuse (in Sicily during the third century bc) shouting ''Eureka!!!'' (I have found it) reportedly occurred after he solved this problem. The problem combines the ideas of ratio and proportion within the context of density of matter.

This learning video deals with a question of geometrical probability. A key idea presented is the fact that a linear equation in three dimensions produces a plane. The video focuses on random triangles that are defined by their three respective angles. These angles are chosen randomly subject to a constraint that they must sum to 180 degrees. An example of the types of in-class activities for between segments of the video is: Ask six students for numbers and make those numbers the coordinates x,y of three points. Then have the class try to figure out how to decide if the triangle with those corners is acute or obtuse.

This learning video deals with a question of geometrical probability. A key idea presented is the fact that a linear equation in three dimensions produces a plane. The video focuses on random triangles that are defined by their three respective angles. These angles are chosen randomly subject to a constraint that they must sum to 180 degrees. An example of the types of in-class activities for between segments of the video is: Ask six students for numbers and make those numbers the coordinates x,y of three points. Then have the class try to figure out how to decide if the triangle with those corners is acute or obtuse.

In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other focuses them on the work of standard G-SRT.2, using the definition of similarity in terms of similarity transformations.

This Nrich activity is particularly good in a number of mathematical aspects of learning:
Using mathematical ideas and methods to solve "real life" problems
Using and understanding vocabulary and notation related to money
Organizing and using data
Choosing and using appropriate number operations and calculation strategies
Explaining methods and reasoning
Making and investigating general statements

Math Learning Center Geoboard is a manipulative tool students use in geometry to explore the basic concepts of polygons. It also helps with the discovery of area and perimeter. It is a board that contains a certain number of nails in which rubber bands go around to create the shape.

This short video and interactive assessment activity is designed to give fourth graders an overview of composite figures composed of squares and rectangles.

An interactive applet and associated web page that demonstrate the area of a circle. A circle is shown with a point on the circumference that can be dragged to resize the circle. As the circle is resized, the radius and the area computation is shown changing in real time. The radius and formula can be hidden for class discussion. 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: Area and Surface Area
Lesson 6: Area of Parallelograms

This lesson allows students to practice using the formula for the area of parallelograms, and to choose the measurements to use as a base and a corresponding height. Through repeated reasoning, they see that some measurements are more helpful than others. For example, if a parallelogram on a grid has a vertical side or horizontal side, both the base and height can be more easily determined if the vertical or horizontal side is used as a base.

Along the way, students see that parallelograms with the same base and the same height have the same area because the products of those two numbers are equal, even if the parallelograms look very different. This gives us a way to use given dimensions to find others.

Reasoning to Find Area: Lesson 4

Students were introduced to parallel lines in grade 4. While the standards do not explicitly state that students must work with parallelograms in grades 3–5, the geometry standards in those grades invite students to learn about and explore quadrilaterals of all kinds. The K–6 Geometry Progression gives examples of the kinds of work that students can do in this domain, including work with parallelograms.

In this lesson, students analyze the defining attributes of parallelograms, observe other properties that follow from that definition, and use reasoning strategies from previous lessons to find the areas of parallelograms.

By decomposing and rearranging parallelograms into rectangles, and by enclosing a parallelogram in a rectangle and then subtracting the area of the extra regions, students begin to see that parallelograms have related rectangles that can be used to find the area.

Throughout the lesson, students encounter various parallelograms that, because of their shape, encourage the use of certain strategies. For example, some can be easily decomposed and rearranged into a rectangle. Others—such as ones that are narrow and stretched out—may encourage students to enclose them in rectangles and subtract the areas of the extra pieces (two right triangles).

After working with a series of parallelograms, students attempt to generalize (informally) the process of finding the area of any parallelogram (MP8).

Finding area of rectangular solids and cylinders by cutting them into flat pieces and adding the areas.

This short video and interactive assessment activity is designed to give fourth graders an overview of area and perimeter of rectangle.

Unit: Area and Surface Area
Lesson 8: Area of Triangles

This lesson builds on students’ earlier work decomposing and rearranging regions to find area. It leads students to see that, in addition to using area-reasoning methods from previous lessons, they can use what they know to be true about parallelograms (i.e. that the area of a parallelogram is ) to reason about the area of triangles.

Students begin to see that the area of a triangle is half of the area of the parallelogram of the same height, or that it is the same as the area of a parallelogram that is half its height. They build this intuition in several ways:

by recalling that two copies of a triangle can be composed into a parallelogram;
by recognizing that a triangle can be recomposed into a parallelogram that is half the triangle’s height; or
by reasoning indirectly, using one or more rectangles with the same height as the triangle.
They apply this insight to find the area of triangles both on and off the grid.