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Geometry Resources & Lesson Plans7 chapters | 194 lessons

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*Juan Carlo Carvajal*

The mathematical concepts of surface area and volume are covered within this lesson for teachers to use as a foundation when instructing students in the classroom setting.

Surface area and volume are mathematical concepts that are usually taught in unison. Although they may sound abstract, instructing students on how to conceptualize them can be achieved by building the right foundation. It is best to teach these concepts using visual aids rather than formulas to help with context. The combination of visual aids with their corresponding formulas, as well as real world application, will help students gain a thorough understanding of these mathematical concepts.

Let's begin by reviewing the concepts. **Area** is defined as the size of the surface of a two-dimensional object. Students find the area of an object by multiplying the length times the width. For example, if a student's desk is three feet long and two feet wide. The area would be six feet squared.

What if the student wants to measure his pencil box? The pencil box is a three-dimensional object because it has length, height, and width. The **surface area** of a three-dimensional object is calculated by adding the areas of all sides. Therefore, if the pencil box is six inches long, four inches wide, and two inches tall, the student will use the formula 2ab + 2ac + 2bc to calculate the surface area. In other words 2(6 x 4) + 2(6 x 2) + 2(4 x 2)= 48 + 24 + 16 = 88 inches squared.

Teachers may use standard units, such as rulers, yardsticks, and tape measures, or nonstandard units, such as paperclips, pencils, and erasers to measure surface area. Everyday items, such as books, soda cases, and boxes of various sizes may be used to practice measuring surface area conceptually.

How is the strategy for teaching volume different from the way you might teach surface area? First of all, let's review the concept. **Volume** is defined as the amount of space that the object takes up. It is measured in cubic units. Let's use a cube as an example. A cube is a 3-dimensional object that has six square faces. Let us assume that each side of the cube measures to four feet.

The volume can be calculated for the cube by using the following formula: Volume (cube) = length x width x height

In this case since the length, width, and height are all four feet, the volume = 4 feet x 4 feet x 4 feet which comes out to 64 cubic feet.

Conceptually, surface area measures the outside surface of the object, while volume measures what's inside. An interesting strategy to use with students is to note the ratio between surface area and volume as the size of an object increases or decreases. Let's use our cube for example. While the length of each side of the cube is four feet, the surface area is 96 feet squared, while the volume is 64 feet cubed.

If the cube were three feet long on each side, the surface area would be 54 feet squared and the volume would be 27 feet cubed. The surface area decreased by 42% while the volume decreased by 56.25%. Graphing the patterns and discussing the findings reveals the relationship between surface area and volume.

It is important to note that volume and surface area formulas vary depending on the object. Volume and surface area formulas differ for spheres, pyramids and cylinders. It is also good practice to review areas of circles, triangles, squares and rectangles just in case they are needed to compute the area of the side of an object.

For example, a pyramid has triangular sides, so the student must know how to compute the area of a triangle, which is (1/2) X base X height. Another example is that a cylinder has a circular face on top and bottom, so students need to be able to compute the area of a circle, which is ? X r^2. ? is simply a constant that equals 3.14.

Keep your lesson simple by using a combination of visual aids and manipulatives with their corresponding formulas. Remember that **area** is the size of the surface of a two-dimensional object while **surface Area** is the area of all sides of a three-dimensional object added together. **Volume** is the amount of space that the object takes up.

Also, keep in mind that volume and surface area formulas are specific for certain objects, so care must be taken to match the object to its formulas. The relationship between surface area and volume may be observed and recorded to further a student's understanding of both concepts.

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Geometry Resources & Lesson Plans7 chapters | 194 lessons

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