Cubigami. Knicko Mojica. Mr. Acre & Mrs. Gravel GAT/IDS 9C

Transcription

1 Cubigami Knicko Mojica Mr. Acre & Mrs. Gravel GAT/IDS 9C 12 January 2016

2 Mojica 1 Cube in a Cube When creating a three-dimensional figure out of paper, it may simply be seen as an inanimate object. However, when the object is examined more closely, the volume and area of the figure can be calculated. The same can be said for the cube in a cube, which consists of two cubes where one is large, one is small, and the large cube has empty spaces on each of its six faces where the smaller cube can fit inside. The goal is to be able to create, and find the volume and area of a cube inside of a cube. <NOTE: This is a sample introduction. Please create your own > Crafting the Outer Cube <NOTE: This student decided to show how he put his cube together to tell his story.> Figure 1. Step One With two of the twelve pieces of paper that were folded as directed for the outer cube, insert one flap into one of the slots on the other paper.

3 Mojica 2 Figure 2. Step Two Continue to insert the flaps into the slots until you form a square with the four pieces. Figure 3. Step Three into the slots. Crease the four pieces in half so that they are able to stand and continue to insert the flaps

4 Mojica 3 Figure 4. Step Four Insert the flaps into the slots carefully, placing them inside the slots as tightly as possible to help keep the measurement of the cube exact. Once the final flap is reached, insert it into the final slot and ensure that the flaps are inside all of the slots are placed correctly. Figure 5. Completed Outer

5 Mojica 4 The figure above shows an example of a completed outer cube with a side length of 3 2 inches. Constructing the Inner Cube Figure 6. Step One With three pieces of paper that were already folded into the correct form, insert the flap from two pieces of paper into the slots of one piece of paper, as shown above. Figure 7. Step Two & Completed Inner Cube

6 Mojica 5 Continue to insert the flaps into the slots until the inner cube is completed and resembles the figure above. Side Lengths of the Inner and Outer Cubes Figure 8. Side Length of the Inner Cube < This image is purposely blurry. Also, you MUST detail (in your anchor) how to get from 6 inches to x inches (the inner cube side length)> Figure 9. Side Length of the Outer Cube < This image is purposely blurry. Also, you MUST detail (in your anchor) how to get from 6 inches to y & z inches (the outer hollow cube side lengths)>

7 Mojica 6 Finding the Area of the Inner and Outer Cubes Surface Area (SA) = (6) (Side)² SA = (6)(y)² Formula to Find the Surface Area of a Cube SA =??? in.² Multiplication Property Figure 10. Surface Area of the Outer Cube (Without Subtracting the Empty Space) The figure above shows how to solve for the surface area. This is the total surface area of the cube without subtracting the empty space. The surface area is??? in.² Surface Area (SA) = (6) (Side)² SA = (6)(z)² Formula to Find the Surface Area of a Cube SA =??? in.² Multiplication Property Figure 11. Surface Area of the Empty Space on the Outer Cube Above, the figure shows how to solve for the total surface area of the empty space on the outer cube. The empty space area is??? in.² SA Cube = SA Outer Cube - SA Empty Space SA Cube =??? Formula to Find SA Without Empty Space SA Cube =??? in.²??? Property Figure 12. Surface Area of the Outer Cube without the Empty Space Figure 12 shows how to solve for the surface area of the outer cube minus the empty space. The surface area of the outer cube without the empty space is??? in.²

8 Mojica 7 Surface Area (SA) = (6)(Side)² SA = (6)(x)² Formula to Find the Surface Area of a Cube SA =??? in.²??? Property Figure 13. Surface Area of the Inner Cube is 27 in.² As seen above, the solution to solving the area of the inner cube is shown. The end result Total Surface Area (TSA) = SA Outer Cube + SA Inner Cube TSA =??? TSA =??? in.² Figure 14. Total Surface Area of the Cubes Formula to Find the Total Surface Area??? Property The way to find the solution to solving the total surface area of the cube in a cube is shown above. The total surface area of the cubes is??? in.² When finding the surface area of the entire cube, it is necessary to follow the steps exactly in order to get the correct area. To start, one must find the area of the outer cube, including the empty areas, shown in Figure 10. Then, finding the area of the empty space is the next step, as shown in Figure 11. The next step, as seen in Figure 12, is to solve for the surface area of the outer cube without including the empty space. To do this, the area of the empty space is subtracted from the area of the outer cube. Now, the area of the inner cube can solved for, as demonstrated in Figure 13. The last step is to find the total surface of the cube in a cube, which can be found by adding the surface area of the outer and inner cubes. In the end, the total surface area of the cube in a cube is??? in.²

9 Mojica 8 The Volume of the Cube in a Cube Volume (V) = Side 3 V = (x) 3 Formula to Find the Volume of a Cube V = in. 3??? Property Figure 15. Volume of Inner Cube The figure above shows how to find the volume of the inner cube. To find the volume of a cube, take the measurement of one side, in this case x, and cube it [(x) 3 ]. The ending result is??? in. 3. Volume (V) = Side 3 V = (y) 3 V =??? in. 3 Figure 16. Volume of Outer Cube Formula to Find the Volume of a Cube??? Property When solving for the volume of the outer cube, the same approach is taken as seen in figure fifteen. Instead, the resulting answer is??? in. 3. Total Volume (TV) = V Outer Cube - V Inner Cube Formula to Find the Total Volume TV =??? TV = in. 3??? Property Figure 17. Total Volume of the Empty Space Figure 17 shows how to find the volume of the empty space in. The final answer is??? in. 3. Finding the volume of the cube and the empty space is simple since you only have to cube

10 Mojica 9 the side lengths to get the volume of the cube and subtract the inner cube from the outer cube. It is important to follow the steps in order to get the correct volume. The first step to finding the volume of the empty space is to find the volume of the inner cube, as seen in figure fifteen. Next, the volume of the outer cube must be found, seen in figure sixteen. Finally, to find the total volume, the volume of the inner cube must be subtracted from the volume of the outer cube, shown in figure seventeen. Cubtastrophy <NOTE: This student s conclusion has a recap of what was done, problems that were encountered, and an innovative way to wrap things up ; yours should too > While going through the process of creating, and finding the volume and surface area of the cube in a cube, some challenges occurred. Above all, the largest problem was creating the outer cube so that the creases were crisp and the folds were exact. If the folds were not exact, then the measurement of the cubes could have been completely wrong. Another problem was the paper. The paper used to create the cubes were slightly thicker than average copy paper and was also hand-cut to match the six inch by six inch requirements. Because the paper was thicker and the paper was hand-cut, the folds were probably not exact, especially when assembling the cube, which relied on almost perfectly cut and folded paper. These flaws could have made the volume and area not exact. In the end, it was found that the total surface area of the cube in a cube was??? in.², and the volume of the cube in a cube was??? in. 3 Nonetheless, it is possible to find the area and volume of a cube in a cube. It is always important to follow the correct steps in order to ensure the proper measurements and answers.

11 Mojica 10 A simple cube in a cube proves that a simple object can have more value and intricacy, be it the area and volume, than one could have thought. <NOTE: This student didn t use any unapproved resources and thus didn t need works cited section of his paper. Be sure to create one if you use any additional sources>