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Transformations Through Flags Table of Contents • • • • • Translations Rotations Dilations Reflections Tessellations • • • • • David A. David C. Matt Dom Ethan Translations David Angione Summary • A translation is when you slide a figure on a coordinate grid without turning or flipping the figure. Vocabulary • Vector- A quantity that has both direction and magnitude, or size. • Initial Point- The starting point of the vector. • Terminal Point- The ending point of the vector. • Component Form- A format in which to describe a vector that combines the horizontal and vertical components. • Vector Form- A format in which to describe a vector by putting the change in the x-axis on the left, and the change in the y-axis on the right. Needs special parenthesis. <. Concepts • You can write component form by following this model. (x, y)=(x+a, y+b). • You can write vector form by putting the change in the x-axis on the left, and the change in the y-axis on the right. Needs special parenthesis. <. i.e. Vector <7, 2>. Mathematical Examples • The figure was translated four spaces to the right, and two spaces up. The figure on the left is the original shape, and the figure on the right is the shape after the translation. Activities 1. Graph a figure with points: A=(-4, 0), B=(-4, 4), C=(0, 0), and D=(0, 4). 2. Fill in the figure with the colors of the country that you family is from. 3. Translate the figure by using the vector <5, 2>. 4. Graph the new figure and name each point. 5. Fill in the new figure with the country that you would like to visit. Real-Life Applications • A real-life application is when someone raises a flag to the top of a flagpole, or when they lower a flag down to half staff. Key Terms • Rotation- A transformation where a figure is turned around the center of rotation as an isometry • Isometry- the figure is the same before and after the transformation • Center of Rotation- A fixed point that can be inside or outside the shape • Angle of rotation – the measure of degrees that figure is rotated about a fixed point • A rotation of the Japanese flag with the center of rotation outside of the shape Rotation About The Origin • The center of rotation, the origin, is located at (0,0) • The equations for rotations about the origin are – R90° (x,y) = (-y, x) – R180° (x,y) = (-x,-y) – R270° (x,y) = (y,-x) – R-90° (x,y) = (y,-x) Rotation About The Origin (continued) The flag is rotated 180 degrees about the origin Use R-90° (x,y) = (y,-x) Rotational Symmetry • Rotational Symmetry- A figure has rotational symmetry when the figure can be mapped onto itself by a clockwise rotation of less than 180 • When you rotate the flag of Switzerland, one of the two square flags, 90 degrees you will get the same shape which means the flag has rotational symmetry • Click to see The Angle of Rotation When an Object is Reflected Over Two Lines • When you reflect a figure over two lines that are not parallel the angle of rotation is double the angle between the two lines • For example, angle ACB is 65 degrees so when the triangle reflects over the two lines the angle of reflections is 130 degrees B C 65 degrees A Real Life Situation • The flag needs to be rotated so it can go on the pole • How many degrees counter clock-wise does the flag need to rotated about the center of the flag so it can be corrected? Answer Real Life Situation Answer • 180 degrees counter clockwise Rotation Activity • What is the angle of rotation between flag A and flag D? A D B C Answer Rotation Activity (answer) • 270 degrees Dilations Matthew Wechsler Key Definitions • Dilation- a transformation in which a polygon is enlarged or reduced by a given scale factor around a given center point • Reduction- 0 < X < 1 • Enlargement- X > 1 Reduction Enlargement Matrices [ ] 293 456 Scale factor: 3 • To get the answer multiply all of the exponents by the scale factor (click for answer) • Answer: [ 6 27 9 12 15 18 ] The Flag Situation • You have a small flag. You want it to be larger but it has to stay the same shape, what is the scale factor of the smaller flag to the larger X flag? Then find x 5 30 10 Answer Answer • Scale factor = • X = 15 1 3 Activity • Get a piece of graph paper • Draw a rectangle with the points: (-4, -1) (-1,-1) (-4, -4) (-1, -4) • Make the scale factor for the new shape 2.5 Answer • What are the new points? Answer (-10, -2.5) (-2.5, -2.5) (-10, -10) (-2.5, -10) Reflections By Dominick Gagliostro Key Definitions • Reflection- a transformation which uses a line that acts like a mirror, with an image reflected in the line. • Line of Reflection- the line which acts like a mirror in a reflection • Line of Symmetry- a line that divides a figure into two congruent parts, each of which is the mirror image of the other. When the figure having a line of symmetry is folded along the line of symmetry, the two parts should coincide. Normal Reflections Reflections over y-axis Reflections over x-axis Reflections when X isn’t zero Minimum distance Original Points Description • To find the minimum / distance. First reflect point A. Next draw a line from A’ to B. Then the point where that line crosses the x-axis is the minimum distance. Minimum Distance continued Real World Application • The cemetery wants to put an American flag in their cemetery for two war veterans that were recently buried there. They want it to be the minimum distance between both graves. Find the minimum distance to help the cemetery out. Real World Application Where do you put the flag? Answer What Are Tessellations • Tessellations are a repeating pattern of figures that completely covers a plane without any gaps or overlaps. Some Vocab • An edge is the intersection between two bordering tiles. • A vertex is the intersection of three or more bordering tiles. • A regular tessellation is when a tessellation uses only one type of regular polygon to fill up a plane. • A semi-regular tessellation uses more than one type of regular polygon to fill up a plane. Tessellations & Symmetry • Translational symmetry is when a translation maps the tessellation onto itself • Glide Reflectional symmetry is when a glide reflection maps the tessellation onto itself – glide reflection is when you reflect then translate an object • Rotational symmetry: when a rotation of 180 degrees or less is performed on a tessellation and the resulting image is the same as the original image • Reflection or line symmetry: when a figure is reflected across and axis and the image is the same as the original • Point symmetry: when a tessellation rotates 180 degrees and the image is the same Will It Tessellate • use the formula for the measure of an angle of a regular polygon = 180(−2) ) • substitute a number of sides for n • if the figure simplifies without a remainder into 360, it will tessellate • in other words, the product of the expression has to be a factor of 360 Flag Tessellations How To Make A Tessellation Construct Segment AB Construct Point “C” above AB Mark the Vector From A B Translate Point “C” by the Created Vector Construct the Remaining Sides of the Parallelogram Steps 4-7 Construct Irregular Segments Construct Your New Polygon’s Interior Translate the Polygon Interior Translate the Newly Created Column Bibliography • • • • • • http://www.globeslcc.com/2013/03/29/the-weekly-reel-how-the-media-can-tug-at-patrioticheartstrings/com-american-flag-pub-dom/ http://nathandahm.com/happy-flag-day/ http://sedalianewsjournal.com/2012/12/12/state-honors-fire-chief-meador/ http://language-assessment-anddevelopment.pusd.schoolfusion.us/modules/groups/group_pages.phtml?gid=942978&nid=69489 http://pantbeer.com/to-clubs-pubs.html http://www.viecoballoons.com/flpinwheels.htm Links • http://content1.riverdell.org/access/web?id=10718736173 731352748 • http://sevendesktop.com/wpcontent/uploads/2013/02/England-flag-art-background.jpg • http://www.flagwallpapers.com/bulkupload/flagwallpapers/Canada/canada -flag-wallpaper.jpg • http://flagwallpapers.com/bulkupload/flagwallpapers/Greece/greekflag.jpg • http://www.orlandoflagcenter.com/US_FLAGS.htm http://stationary.prissed.com/images/flag-backgroundpatriotic.jpg Bibliography • http://www.mapsofworld.com/flags/japan-flag.html • http://www.worldatlas.com/webimage/flags/countrys/mideast/tur key.htm • http://www.millersmotorcycles.com • http://axismonday.blogspot.com/2008/12/simple-flags.html • http://en.wikipedia.org/wiki/File:Flag_of_Canada.svgn • http://www.britannica.com/hispanic_heritage/article9093933?flagHistory=y • http://mathworld.wolfram.com/Rotation.html • http://www.smscs.com/picview.php?title=flags+wallpaper&photo= http%3A%2F%2Fwww.university500.com%2Fwpcontent%2Fuploads%2F2012%2F02%2F1288-map-of-flags-of-theworld-wallpaper-wallchan-1024x768.jpg Pictures • http://www2.waketech.edu/blogs/esl_mlbk/fi les/2012/10/us_flag.png • http://www.all-flags-world.com/countryflag/Italy/flag-italy-XL.jpg