If a reflection has been done correctly, you can draw an imaginary line right through the middle, and the two parts will be symmetrical "mirror" images. Most commonly flipped directly to the left or right (over a "y" axis) or flipped to the top or bottom (over an "x" axis), reflections can also be done at an angle. The translation shows the geometric shape in the same alignment as the original it does not turn or flip.Ī reflection is a shape that has been flipped. These were described by Escher.Ī translation is a shape that is simply translated, or slid, across the paper and drawn again in another place. There are 4 ways of moving a motif to another position in the pattern. He adopted a highly mathematical approach with a systematic study using a notation which he invented himself.
There are 17 possible ways that a pattern can be used to tile a flat surface or 'wallpaper'.Įscher read Pólya's 1924 paper on plane symmetry groups.Escher understood the 17 plane symmetry groups described in the mathematician Pólya's paper, even though he didn't understand the abstract concept of the groups discussed in the paper.īetween 19 Escher produced 43 colored drawings with a wide variety of symmetry types while working on possible periodic tilings. One mathematical idea that can be emphasized through tessellations is symmetry. If you look at a completed tessellation, you will see the original motif repeats in a pattern. The term has become more specialised and is often used to refer to pictures or tiles, mostly in the form of animals and other life forms, which cover the surface of a plane in a symmetrical way without overlapping or leaving gaps. They were used to make up 'tessellata' - the mosaic pictures forming floors and tilings in Roman buildings The word 'tessera' in latin means a small stone cube. When you fit individual tiles together with no gaps or overlaps to fill a flat space like a ceiling, wall, or floor, you have a tiling. The formula gives the same answers as we calculated above in all of the rest of the cases as well.A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps.Īnother word for a tessellation is a tiling. To calculate the distance between two of the points, $A$ and $B$ for example, we can use the Pythagorean Theorem provided we can find a right triangle which has $\overline$, agreeing with our calculations in part (a). The three points and the segments joining them are plotted and labeled below: These calculations build on and reinforce student understanding of distance and absolute value developed in 6.NS.7 and 7.NS.2. Either is viable, since when we apply the Pythagorean theorem we will square the quantities and $3^2 + 2^2 = (-3)^2 + (-2)^2$. Looking at (-1,1) and (2,3) as an example, students may find 3 and 2 for the change in $x$ and $y$ coordinates or they may find -3 and -2. One issue which will likely come up in this task is the order of the points and the signs of their differences. They can test the formula they have found in part (c) with any of the examples from (a) and (b). Part (d) has been added to encourage students to put all of their work together and verify that it is consistent. This will help students see that the location of the points does not influence the calculations: only their relative position (found by taking differences of coordinates) is important. If students draw a picture for part (c), as is done in the solution, the teacher should show samples where the points lie in different quadrants or on one or both of the axes.
In part (c), students can make an abstract drawing similar to part (a) or they can build on the calculations in part (b). For part (b), students need to analyze carefully their work for part (a) in order to relate the side lengths of the right triangles to the coordinates of the points. The simplest and most uniform way to do this is to use coordinate grid lines which are perpendicular by construction. For each segment length calculation in (a), this will mean drawing a right triangle which has the given segment as one of its sides. Â The teacher will want to make sure that students explain their calculations in parts (a) and (b). Students should already be familiar with applying the Pythagorean in concrete situations such asÂ. The goal of this task is to establish the distance formula between two points in the plane and its relationship with the Pythagorean Theorem.
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