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# Surface Area, Statistics Problem Example

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When teaching students a new concept, it is important to give them a visual or real life attachment to help them learn it in a way that they will understand. In calculating surface area of a cube, presenting them with the formula length times width times six is necessary, but certainly not sufficient as it gives them no explanation for why that is the formula. Using an actual physical cube is required for providing this. To do this, one should use twelve sheets of paper that fit each cube side, with the length and width marked, then put six on the cube, one on each side. At this point the student will see that there are six sheets of paper not on the cube and six identical sheets that are on it. The use of twelve sheets is so that the student still has six free sheets, allowing them to calculate the area of these and seeing how little difference there is between calculating the surface area of a cube and calculating the area of six sheets of paper, providing that they match up. The twelve sheets allows the student to see the identical nature of the free and bound sheets. If the student has mastered the ability to calculate the area of a rectangle, he should easily be able to calculate the total area of the free six sheets and use that as the area for the six sheets attached to the cube.

At this point the students should be able to understand that the total surface area can be calculated by simply summing the area of each individual side. It requires four steps, the first tree of which are calculating the area of three sides, each of which will match the area of another side. The fourth and final step is to add each individual side together. If the three dimensions of the cube are defined as length (L), width (W), and depth (D) the formula can be produced as follows: (2*(L*W))+(2*(D*W))+(2*(L*D)).

On the most basic level, this will require the student to have an understanding of basic multiplication and addition. Multiplication is needed to calculate the area of each side, while addition is needed to sum them all for the total surface area. In the case of a perfect cube though, where each side will be identical, multiplication can be used instead of addition, as any side’s area can simply be multiplied by six. However, it is most likely that any student who has mastered multiplication previously mastered addition, and anyone who has not done so, should be given extra help as that is a serious gap in their mathematical ability.

Beyond that, the student must have a grasp of how area of a two dimensional figure, especially a four sided one, is calculated. Even if they completely know their times tables, a student must know to multiply length by width, or the multiplication knowledge will be completely useless in this case. Calculating area is likely best taught with graph paper or something that allows the student to count the area in a rectangle and confirm for themselves that a rectangle with a length of four units and width of six really does have an area of twenty-four square units.

After the method for calculating for area of two dimensional figures is known by the student, the surface area of three dimensional can easily be taught. First the student must understand the relation of quadrangle and a cube have to each other. This can be shown on paper, by drawing a three dimensional object consisting of two squares and four lines connecting them. These four lines simulate the third dimension, either the height or depth of the cube. Therefore the cube will be made up of six individual quadrangle sides. Therefore, the calculation can be made through the summation of the areas of each side. In the case of a perfect cube, only one side needs to be calculated as each side will share the exact same area. That one surface area can be multiplied by the number of sides, six, and that will be the surface area of the three dimensional figure in question. To give a more real life demonstration of the cube, neocubes can be used, as when stacked up they can form three dimensional objects, one of which can be a cube.

The calculation of surface area is the combination of a great deal of skills. The student must be able to multiply and use addition, while knowing how to apply those skills to the calculation of area. This is done through understanding the concepts of area calculation and evidence that length and width multiplied together does have significance. Finally, they must understand the relation between a rectangle and cube well enough to properly sum each side for the total surface area.

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