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Under the Sea

Under the Sea. A simple guide to transformations. Table of Contents. Transformations Rotations by Ryan Reflections by Tim Translations by Katie Dilations by Joe. Transformations. Words to Know. Transformation – operation that maps the preimage onto the image

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Under the Sea

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  1. Under the Sea A simple guide to transformations

  2. Table of Contents • Transformations • Rotations by Ryan • Reflections by Tim • Translations by Katie • Dilations by Joe

  3. Transformations

  4. Words to Know • Transformation – operation that maps the preimage onto the image • Map – moving a figure • Isometry – transformation that preserves lengths • Preimage– original figure • Image– new figure

  5. Matrices singular – matrixA way of organizing the coordinates of a figure. A B C A B C translation xy -3 -5 -2 3 -1 2 3 3 3 -1 -1 -1 0 -2 1 2 -2 1

  6. Rotations

  7. Words to Know • Rotational symmetry – shape can be rotated and it still looks the same • Square at 90° • Center of rotation – point on which a figure rotates • Angle of rotation – measure of degrees that figure is rotated about a fixed point • R90° (x,y) = (-y, x) • R180° (x,y) = (-x,-y) • R270° (x,y) = (y,-x) • R-90° (x,y) = (y,-x)

  8. Rotation An isometry that involves circular motion of a configuration about a given point or line, without a change in shape. Center of rotation: (0 , 0) Angle of rotation: 180 ° clockwise * approximation • R180° (x,y) = (-x,-y)

  9. Real-Life Application The bodies of starfish demonstrate rotations. They also showrotational symmetry. At a 72° rotation, this starfish would map onto itself.

  10. Patrick’s Dilemma • A kid is watching his favorite show, SpongeBob SquarePants. Patrick the starfish is trying to rotate around his house – a rock – to get to SpongeBob. The rock is the center of rotation, (0,0). If Patrick must make a 76°rotation to reach SpongeBob, where will the starfish end up? Draw the image. A( -3, 2) D(-2,1) B(-4, 1) E(-2, .5) C ( -3, -1)

  11. GSP Activity • Create a pentagon and label the points A through E. • Add point P anywhere. This will be the center of rotation. • With the pointer, double-click point P. Then select point A. • Select transform and then rotate in the menu. Set the rotation to 45°. • Repeat for each point until the new pentagon is constructed.

  12. Drag each point and note how the shapes react. What do you notice?

  13. Reflections

  14. Words to Know • Mirror line – central line that the preimage is reflected over • Symmetry –an exact correspondence in position or form about a given point, line, or plane • Reflectional symmetry – figure contains a line of symmetry

  15. Reflection An isometry that is a flip over a line where every point is the same distance from the central , or mirror, line. Reflected over the y-axis Mirror line: y-axis - Y values stay the same - X values become opposite * approximation

  16. Real-Life Application The shells of clams and the bodies of lobsters both demonstrate reflections. Both halves of the animals showreflectional symmetry. They show a reflection over the y-axis.

  17. Fishy Romance Step 4: If you’d like, you can copy your fish onto a plain piece of paper. Supplies • Graph paper • Pencil & eraser • Colored pencils • Computer (optional) Step 1: Using points and line segments, draw a fish of your choosing on the graph paper. If you’d like, you can print a picture of your favorite fish and trace it onto the graph paper this way. Step 5: Add details and color. Behold your beautiful creation! Step 2: Create a line of reflection, or mirror line, along your fish’s mouth. Step 3: Reflect the fish over the line of reflection. Be exact.

  18. Translations

  19. Words to Know • Vector – a quantity possessing both magnitude and direction • Initial point – starting point of the vector • Terminal point – ending point of the vector • Component form – horizontal and vertical values  < a, b > • Coordinate notation– (x,y)  (x+a, y+b)

  20. Translation A transformation is a transformation that maps every two points P and Q in the plane to points P’ and Q’ Coordinate notation: (x+3, y+2) Component form: < 3 , 2 > * approximation

  21. Real-life Application A school of fish demonstrates translations. As an ocean wave curls onto the shore, it can be depicted as a translation.

  22. Tessellation Creation Step 3: With a pencil (lightly) trace the rectangle numerous times onto C so that there is no space between them. Step 1: Rip one sheet of tracing paper in half. Then label one half A and the other B. Label the full page C. Supplies • 2 sheets of tracing paper • Ruler • Pencil & eraser • Colored pencil Step 4: Add a squiggly line to the left of the rectangle on A, being sure to transfer it onto the left and right sides of B’s rectangle. Step 2: Using a pencil and ruler, draw a rectangle on A. Then trace it onto B. • Tessellations are figures that can be repeated without any gaps or overlapping parts.

  23. Tessellation Creation (cont.) Step 5: Repeat step 4 for the top and bottom of the rectangles, being sure that the left and right as well as the top and bottom look EXACTLY the same! Step 7: Erase the rectangles on C and add color! Show it to everyone you know and make them jealous! Step 6: Fill in the parts of the rectangle that do NOT contain your shape (shown here in blue). Then trace your shape onto C.

  24. Dilations

  25. Words to Know • Reduction – shape gets smaller • Enlargement– shape gets bigger • Scale factor– ratio of corresponding sides of image over preimage (k) • Center of dilation – fixed point about which all points are dilated

  26. Dilations Using Matrices Reduction 1 > k > 0 A B C center = (0,0) k = ½ -6 -5 -4 1 3 1 2 2 A B C -3 -2 ½ -2 ½ 1 ½ ½ * approximation

  27. Dilations (cont.) Enlargement k > 1 Using Matrices A B C center = (0,0) k = 2 2 2 -3 -2 ½ -2 ½ 1 ½ ½ A B C -6 -5 -4 1 3 1 * approximation

  28. Real-Life Application Scientists have studied the length and width of a parent shark compared to its pup and found that the sides were proportional. This relationship was considered to be a dilation. The mother, from snout to tail, was 20 feet and had a width of 6 feet. The pup’s length was 5 feet. • What is the scale factor from the pup to the mother? • Is this an example of reduction or enlargement? • If the pup’s width is also proportional to the mother’s, what would it be?

  29. GSP Activity • Draw a line segment with endpoints A and B with midpoint M. • From the midpoint, create two triangles and label the points to create triangle ACM and triangle BDM. • Connect points C and D, making a trapezoid with visible lines. • Find the length of all four segments. • Construct the interior of the trapezoid.

  30. Mark point C as a center and then rotate the figure 180°. • Dilate the trapezoid by one half (½). • Label points I, J, and K, making trapezoid JKCI . • Find the lengths of the four sides of the new figure and compare them to the original trapezoid. THINK IT OVER: 1a.) What is the scale factor of the two figures? 1b.) What happens if the side of the original trapezoid is changed?

  31. Bibliography • "Define Symmetry." Dictionary.com. Ask.com, 2011. Web. 14 Apr. 2011.      <http://dictionary.reference.com/browse/symmetry>. • "Dilations." Regents Exam Prep Center. Donna Roberts, 2011. Web. 14 Apr. 2011.      <http://regentsprep.org/Regents/math/geometry/GT3/Ldilate2.htm>. • "Dilations in Math." Math Warehouse. Math Warehouse, n.d. Web. 14 Apr. 2011.      <http://www.mathwarehouse.com/transformations/dilations/      dilations-in-math.php>. • "Geometry and the Ocean." Foundations: Geometry. Chrissi Von Renesse, 22 Apr.      2009. Web. 14 Apr. 2011. <http://biology.wsc.ma.edu/Math251/node/55>. • "Scale Factor - Geometry." iCoachMath.com. HighPoints Learning Inc., 2011. Web.      14 Apr. 2011. <http://dictionary.reference.com/browse/symmetry>. • "Tessellation Do It Yourself: Easy Pencil and Paper Method." Tesselations.org. Seth Bareiss, n.d. Web. 14 Apr. 2011.      <http://www.tessellations.org/diy-paper-a.htm>.

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