![]() Materials: - Make sure you have sharp scissors and clean hands. If you want to view a gallery of my snowflakes, see me on Flickr: Paper Snowflake Gallery. ![]() Some people like to use fancy paper, but I use plain white copy paper because somehow the simplicity of white is more beautiful to me. Steps 4-6 tend to be the trickiest, so step 6 has an extra image to hopefully help explain better for those who are having difficulty.Īll you will need for this is paper and scissors. doesn't always show all of the steps on one page, so hit "next" to go on to the next step. Each step is one fold, but some steps have two pictures shown to help explain things, so please read the description underneath. Real snowflakes in nature form with six points (or occasionally three if they formed weird) so I choose to make my own with six points.īe sure that you follow each step carefully. Most people make (and most how-tos teach) snowflakes with four or eight points. $ABC$ is isosceles.This step by step guide will teach you how to make SIX pointed paper snowflakes. If there is a line $L$ so that $r_L$ maps triangle $ABC$ to itself then triangle ![]() Line $L$ so that $r_L$ maps triangle $ABC$ to itself. Part (b) shows that for each isosceles triangle $ABC$ there is at least one $r_M$ preserves distances this means that $|EF| = |r_M(EG)| = |EG|$ $\triangle EFG$ fixed and it must exchange the other two vertices.įor simplicity, let's assume that $r_M(E) = E$, $r_M(F) = G$, and $r_M(G) = F$. ![]() So this means that the reflection $r_M$ must leave exactly one vertex of Third must be also as there is no freedom to send it to either of the two fixed If two of the vertices are preserved by $r_M$ then the This can not happen, however, because a reflection of the plane can not leave $\triangle EFG$ in its original position. That is $E$ maps to $E$, $F$ maps to $F$, and $G$ maps to $G$. One possibility would be that it leaves them all alone, Since $r_M$ preserves triangle $EFG$ it must interchange the vertices of We are given that $r_M$ maps triangle $EFG$ to itself. $L$ is the perpendicular bisector of segment $\overline$ of triangle $ABC$ to itself, the map interchanges vertices $B$ and Order to show that $r_L(B) = C$ and $r_L(C) = B$ we need to show that Since the reflection over line $L$ leaves each point on $L$ fixed, we have This task is mainly intended for instructional purposes: for students withĪ strong visual sense of geometry, rigid motions of the plane provide a rich source of ideas which can be used to grasp many important geometric ideas. In addition, it may be necessary to recall the definition of reflection about a line $L$: reflection about $L$ maps each point on $L$ to itself and if $P$ is not on $L$ then reflection about $L$ maps $P$ to $Q$ when $L$ is the perpendicular bisector of segment $PQ$ as pictured below: The instructor may wish to discuss with students how to draw an appropriate picture for part (b) and part (c) as this will be essential in order to make progress on these problems. The mechanism through which students do this is via congruence proofs, so the task illustrates the general relationship between congruence and rigid transformations, as fits naturally under the cluster "understand congruence in terms of rigid motions." At the high school level, students will need to use the definition of reflections and the teacher may wish to recall this before having students work on the problem. In particular, the task has students link their intuitive notions of symmetries of a triangle with statements proving that said triangle is unmoved by applying certain rigid transformations. Identify line-symmetric figures and draw lines of symmetry."), and this task is a similar exploration with a level of mathematical precision more appropriate for high schools students. Students will have previously seen similar material in earlier grades (e.g., standard 4.G.3: "Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focussing on the class of isosceles triangles.
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