The formula for the volume of a sphere is V = 4/3 ??r??. See the formula used in an example where we are given the diameter of the sphere. Created by Sal Khan and Monterey Institute for Technology and Education.

Students apply their knowledge of scale and geometry to design wearables that would help people in their daily lives, perhaps for medical reasons or convenience. Like engineers, student teams follow the steps of the design process, to research the wearable technology field (watching online videos and conducting online research), brainstorm a need that supports some aspect of human life, imagine their own unique designs, and then sketch prototypes (using Paint®). They compare the drawn prototype size to its intended real-life, manufactured size, determining estimated length and width dimensions, determining the scale factor, and the resulting difference in areas. After considering real-world safety concerns relevant to wearables (news article) and getting preliminary user feedback (peer critique), they adjust their drawn designs for improvement. To conclude, they recap their work in short class presentations.

In this Cyberchase video segment, Harry tries to snowboard and learns how to measure and identify many common angles.

The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence. In this problem, we considered SSA. Also insufficient is AAA, which determines a triangle up to similarity. Unlike SSA, AAS is sufficient because two pairs of congruent angles force the third pair of angles to also be congruent.

In this task students see different ways of partitioning a circle and a rectangle into two or more equal shares.

Beavers are generally known as the engineers of the animal world. In fact the beaver is MIT's mascot! But honeybees might be better engineers than beavers! And in this lesson involving geometry in interesting ways, you'll see why! Honeybees, over time, have optimized the design of their beehives. Mathematicians can do no better. In this lesson, students will learn how to find the areas of shapes (triangles, squares, hexagons) in terms of the radius of a circle drawn inside of these shapes. They will also learn to compare those shapes to see which one is the most efficient for beehives. This lesson also discusses the three-dimensional shape of the honeycomb and shows how bees have optimized that in multiple dimensions. During classroom breaks, students will do active learning around the mathematics involved in this engineering expertise of honeybees. Students should be conversant in geometry, and a little calculus and differential equations would help, but not mandatory.

The two triangles in this problem share a side so that only one rigid transformation is required to exhibit the congruence between them. In general more transformations are required and the "Why does SSS work?'' and "Why does SAS work?'' problems show how this works.

For these particular triangles, three reflections were necessary to express how to move from ABC to DEF. Sometimes, however, one reflection or two reflections will suffice. Since any rigid motion will take triangle ABC to a congruent triangle DEF, this shows the remarkable fact that any rigid motion of the plane can be expressed as one reflection, a composition of two reflections, or a composition of three reflections.

This particular sequence of transformations which exhibits a congruency between triangles ABC and DEF used one translation, one rotation, and one reflection. There are many other ways in which to exhibit the congruency and students and teachers are encouraged to explore the different possibilities.

Students learn about trigonometry, geometry and measurements while participating in a hands-on interaction with LEGO® MINDSTORMS® NXT technology. First they review fundamental geometrical and trigonometric concepts. Then, they estimate the height of various objects by using simple trigonometry. Students measure the height of the objects using the LEGO robot kit, giving them an opportunity to see how sensors and technology can be used to measure things on a larger scale. Students discover that they can use this method to estimate the height of buildings, trees or other tall objects. Finally, students synthesize their knowledge by applying it to solve similar problems. By activity end, students have a better grasp of trigonometry and its everyday applications.

--In this activity student will create an amusement park ride out of KNEX that displays a certain type of motion. They will describe their ride. They will then place it in the correct spot on the map by writing the name of it on the worksheet. After all groups have their rides labeled on the map- students should fill in the lines section of the worksheet- identifying parallel, perpendicular and intersecting lines.

--In this activity student will create an amusement park ride out of KNEX that displays a certain type of motion. They will describe their ride. They will then place it in the correct spot on the map by writing the name of it on the worksheet. After all groups have their rides labeled on the map- students should fill in the lines section of the worksheet- identifying parallel, perpendicular and intersecting lines.