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Mensuration
Published in W. Bolton, Mathematics for Engineering, 2012
This can be demonstrated by first considering a triangular-based prism (Figure 2.19). Diagonal planes can be used to divide the prism into three equal volume triangular-based pyramids ABCE, DEFC and BCDE. Thus, since the volume of the prism is the base area multiplied by the perpendicular height, the volume of the triangular prism is one third the base area multiplied by the perpendicular height. This argument applies to other prisms.
Bridging to mathematical modelling: Vietnamese students’ response to different levels of authenticity in contextualized tasks
Published in International Journal of Mathematical Education in Science and Technology, 2020
Dung Tran, Duyen Thi Nguyen, An Thi Tan Nguyen, Giang-Nguyen Thi Nguyen, Phuong Minh Ta
G1 generated two real models by adding real-life considerations (e.g. types of materials, labour cost) as they progressed through the modelling process and revised their models. At first, they read the task for comprehension and came up with the first real model [S]: How could a farmer buy metal sheets and other materials to make a goat cage with 1,500,000 (VND)? This group, using a trial-and-error approach [WM], decided to buy 16 metal sheets, which cost 16*61,000 = 976,000 (VND) and use the remaining money, 1,500,000–976,000 = 524,000 (VND), to buy other materials, including 400,000 (VND) for wood, and 124,000 (VND) left over. They included another contextual consideration about the labour and decided that the farmer would build the cage himself. They assumed about labour cost explicit and proceeded with their real model. Using a similar mathematical approach [WM], the group then decided to buy 15 metal sheets costing 915,000 (VND) and to use 400,000 (VND) for wood and pondered over the remaining money for nails and other materials as they felt the remaining money after buying 16 metal sheets was not enough for other materials. The students considered constructing a feeding trough and subsequently changed their minds about buying 14 metal sheets and ended purchasing only 13 to save money for other expenses. They discussed the amount of money spent on the materials and how the materials would be used for the cage. In the end, they used 793,000 (VND) for metal sheets, 350,000 (VND) for wood, 100,000 (VND) for nails, and 257,000 (VND) for a feeding trough. After the materials were finalized, the students discussed how to design the goat cage and formed the second real model [V, S]: With 13 metal sheets, 350,000 (VND) for wood, 100,000 (VND) for nails and 257,000 (VND) for a feeding trough, how could a farmer design a cage? They decided to make a rectangular cage using wood for fencing, 12 metal sheets cut into halves for the roofs and one metal sheet for the water drainage system. They designed the ‘roof as a triangle’ [triangular prism with six sheets on each side] (Figure 3a).