under the sea a simple guide to transformations. table of contents transformations rotations by ryan...

31
Under the Sea A simple guide to transformations

Upload: claribel-stevens

Post on 25-Dec-2015

231 views

Category:

Documents


0 download

TRANSCRIPT

Under the Sea

A simple guideto 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

• Map – moving a figure

• Isometry – transformationthat preserves lengths

• Preimage – original figure

• Image – new figure

Matrices

singular – matrix

A way of organizing the coordinates of a figure.

A B C

A B C

xy

-3 -5 -2 3 -1 2

0 -2 1 2 -2 1

3 3 3-1 -1 -1

translation

Rotations

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)

RotationAn isometry that involves circular motion of a configuration about a given point or

line, without a change in shape.

* approximation

Center of rotation:(0 , 0)

Angle of rotation:180 ° clockwise

R180° (x,y) = (-x,-y)

Real-Life ApplicationThe bodies of starfish demonstrate rotations.

They also show rotational symmetry.At a 72° rotation, this starfish would map onto itself.

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)

B(-4, 1)

C ( -3, -1)

D(-2,1)

E(-2, .5)

GSP Activity

1. Create a pentagon and label the points A through E.

2. Add point P anywhere. This will be the center of rotation.

3. With the pointer, double-click point P. Then select point A.

4. Select transform and then rotate in the menu. Set the rotation to 45°.

5. Repeat for each point until the new pentagon is constructed.

D C

E B

A

P

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

Reflections

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

ReflectionAn 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

* approximation

Mirror line: y-axis

- Y values stay the same

- X values becomeopposite

Real-Life ApplicationThe shells of clams and the bodies of lobsters both demonstrate reflections.

Both halves of the animals show reflectional symmetry.They show a reflection over the y-axis.

Fishy Romance

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.

Step 4:If you’d like, you can copy your fish onto a plain piece of paper.

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!

Supplies• Graph paper• Pencil & eraser• Colored pencils• Computer

(optional)

Translations

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)

TranslationA 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

Real-life ApplicationA school of fish demonstrates translations.

As an ocean wave curls onto the shore, it can be depicted as a translation.

Tessellation Creation

Supplies• 2 sheets of

tracing paper• Ruler• Pencil & eraser• Colored pencil

Step 1:Rip one sheet of tracing paper in half. Then label one half A and the other B. Label the full page C.

Step 2:Using a pencil and ruler, draw a rectangle on A. Then trace it onto B.

Step 3:With a pencil (lightly) trace the rectangle numerous times onto C so that there is no space between them.

Tessellations are figures that can be repeated without any gaps or overlapping parts.

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.

Tessellation Creation (cont.)

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.

Step 7:Erase the rectangles on C and add color! Show it to everyone you know and make them jealous!

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!

Dilations

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

Dilations

Reduction1 > k > 0

center = (0,0)k = ½

Using Matrices

A B C

-6 -5 -4 1 3 1

A B C

-3 -2 ½ -2½ 1 ½ ½

22

* approximation

Dilations (cont.)

Enlargementk > 1

Using Matrices

A B C

A B C

-6 -5 -4 1 3 1

-3 -2 ½ -2½ 1 ½ ½

center = (0,0) k = 2

22

* approximation

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.

1. What is the scale factor from the pup to the mother?

2. Is this an example of reduction or enlargement?3. If the pup’s width is also proportional to the

mother’s, what would it be?

GSP Activity

1. Draw a line segment with endpoints A and B with midpoint M.

2. From the midpoint, create two triangles and label the points to create triangle ACM and triangle BDM.

3. Connect points C and D, making a trapezoid with visible lines.

4. Find the length of all four segments.

5. Construct the interior of the 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?

6. Mark point C as a center and then rotate the figure 180°.

7. Dilate the trapezoid by one half (½).

8. Label points I, J, and K, making trapezoid JKCI .

9. Find the lengths of the four sides of the new figure and compare them to the original trapezoid.

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>.