And this is almost it. Our list of tessellating shapes is almost complete. It can also be shortened. Squares, rectangles, parallelograms and trapezoids all are convex quadrilaterals with various degrees of regularity.
However, no regularity is required of a quadrilateral to tessellate the plane: any simple , in particular a non convex, quadrilateral has this property. The applet below allows you to experiment with arbitrary quadrilaterals. There is one present at the outset. Its shape can be modified by dragging its vertices. The buttons "Copy" and "Copy and rotate" help create copies if that basic shape.
All so created polygons are draggable. This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. In groups, the students worked out answers by investigating particular cases. Then the teacher encouraged them to generalise the answer for all cases. She also said, "Try to explain your answer in more than one way. The diagram below shows an irregular quadrilateral which has been flipped and rotated to cover the space around a point. This group of four identical quadrilaterals can be then be replicated to cover the whole surface.
Many groups were able to reason well about the statement. You can download a Sample Response to Quadrilateral Tessellation which features a well-explained answer. This activity is suitable for years 6— It could be presented at other levels, using other propositions. You can download a list of other Reasoning Statements to use with students.
Year 8: Establish properties of quadrilaterals using congruent triangles and angle properties, and solve related numerical problems using reasoning. Big ideas. Same and different. These configurations are unique up to cyclic reordering and possibly reversing the order. For example 3,12,12 can also be written as 12,12,3 or 12,3, In the bottom row we have 4,8,8 , 3,3,4,3,4 , 4,6,12 and 3,3,3,3,6 configurations.
This means that 3 triangles and 2 squares will give us a vertex type. In this case we can arrange these polygons around the vertex in two different ways: 3,3,3,4,4 and 3,3,4,3,4.
Both of these will give rise to a semi-regular tessellation. There are only 21 combinations of regular polygons that will fit around a vertex. And of these 21 there are there are only 11 that will actually extend to a tessellation. Below are the different vertex types. An asterisk indicates that this vertex type cannot be extended to a tessellation. For some more information see Archimedean Exploration. Polygonal Tessellation Exercises. All parallelograms tessellate.
All quadrilaterals tessellate. No convex polygon with seven or more sides can tessellate. There are three regular tessellations of the plane: by triangles, by squares, by hexagons.
Theorem 1: There are 3 regular tessellations of the plane. Theorem 2: There are 8 semi-regular Archimedean tessellations of the plane. Inaugural-Disstertation, Univ. Frankfurt a. Noske, Borna and Leipzig, Freeman, For example, can you find a way to tessellate any parallellogram?
What about a kite? Or a trapezium? What do you notice about your tessellations? Do all quadrilaterals tessellate? If your answer is no, give an example of a quadrilateral which doesn't tessellate.
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