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Topology – It has two sides but only one!

June21

Students may not have seen much about Topology while their heads are deep inside equations and trig functions. However, Mobius strips and Klein bottles are intriguing aspects of the area of Mathematics called Topology (literally, “the study of shapes”). Here is a Mobius Strip:
The fun part of Mobius Strips is that students can make them quite easily and it leads to looking at the Mathematics of 3D and 4D and surfaces, leading to some interesting perspectives on 3D functions as well. Get the students to makes a Mobius Strip (simply one twist in the length of paper and taped together so that the back of one side joins the other side). Now, what do you think will happen when you cut this strip down the middle (length-wise)? Aha, you will have to do this to find out!

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NCEA Level 2 Algebra Problem. Using the information given, the shaded area = 9, that is:
y(y-8) = 9 –> y.y – 8y – 9 =0
–> (y-9)(y+1) = 0, therefore y = 9 (can’t have a distance of – 1 for the other solution for y)
Using the top and bottom of the rectangle,
x = (y-8)(y+2) = (9-8)(9+2) = 11
but, the left side = (x-4) = 11-4 = 7, but rhs = y+? = 9+?, which is greater than the value of the opp. side??
[I think that the left had side was a mistake and should have read (x+4)?]

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