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

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).

Images of rectangle on coordinate plane, one with easily identifiable area, other image with same rectangle rotated. Distance formula shown to calculate same area.

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.

Definition of area of triangle, image of triangle on coordinate grid with height constructed. Area of triangle calculated using distance formula.

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.

This problem is part of a very rich tradition of problems looking to maximize the area enclosed by a shape with fixed perimeter. Only three shapes are considered here because the problem is difficult for more irregular shapes.

The purpose of this task is primarily assessment-oriented, asking students to demonstrate knowledge of how to determine the congruency of triangles.

The purpose of this task is to provide students with a multi-step problem involving volume and to give them a chance to discuss the difference between exact calculations and their meaning in a context.

This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot.

The purpose of this task is to help students understand what is meant by a base and its corresponding height in a triangle and to be able to correctly identify all three base-height pairs.

Unit: Area and Surface Area
Lesson 10: Bases and Heights of Triangles

This lesson furthers students’ ability to identify and work with a base and height in a triangle in two ways:

By learning to draw (not just to recognize) a segment to show the corresponding height for any given base, and

By learning to choose appropriate base-height pairs to enable area calculations.

Students have seen that the area of a triangle can be determined in multiple ways. Using the base and height measurements and the formula is a handy approach, but because there are three possible pairs of bases and heights, some care is needed in identifying the right combination of measurements. Some base-height pairs may be more practical or efficient to use than others, so it helps to be strategic in choosing a side to use as a base.

This short video and interactive assessment activity is designed to teach third graders an overview of lines, curves, vertices, and sides of 2d shapes.

This short video and interactive assessment activity is designed to teach second graders an overview of lines, curves, vertices, and sides of 2d shapes.

This short video and interactive assessment activity is designed to give fifth graders an overview of simple figure patterns.

This short video and interactive assessment activity is designed to teach second graders an overview of simple figure patterns.