Math: Graph translations and rotations

Math: Graph translations and rotations.

Science: How the tilt of the Earth on it’s axis as it rotates and revolves around the sun causes changes in seasons and the length of a day.

This lesson was created by Alfonso Salinas, Oralia Fernandez , and Diana De Alba at Texas State University

Summer 2008

Pre-requisites:

•Use angle measurement to classify pairs of angles as complementary or supplementary.Including acute, obtuse and right angles.

•Locate and name points on a coordinate plane using ordered pairs of integers.

Organize class into groups. Distribute: a) Day and night spinnerb) Pinc) Flash lightb) Earth and Sun place mat c) Coordinate plane (Rhombus)d) Coordinate plane including: pre-image and image ordered pairs

1. Choose a point on your earth spinner (use the pin to locate it)2. Observe your position in relation to the sun located in the place

Mat (Use the flash light to represent the sun) 3. Spin your spinner(s) on your place mat4. Observe what would be needed to have the spinner going

through each position identified on the place mat?

What did you see happen to the point you chose in reference to the sun light ?

What do you think is needed to have the spinner going through each position identified on the place mat?

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A transformation is a general term to manipulate the position of a point, a line, or shape.

The original shape of the object is called the pre-image and the final shape and position of the object

is the image under the transformation.

Choose a point in your earth spinner, Did it face the sun at all times?

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Choose a point in your earth spinner, Did it face the sun at all times?

http://upload.wikimedia.org/wikipedia/commons/2/2c/Rotating_earth_%28large%29.gif

What could be the reason for your answers?

Earth Rotation:• The Earth's rotation is the rotation of the solid earth around its own

axis, which is called Earth's axis or rotation axis. • The time it takes for the Earth to make a complete rotation is

approximately 24 hours. This rotation results in daytime when an area is facing the Sun and night time when an area is facing away from the Sun.

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Find and connect the ordered pairs indicated in the table.

a) A-B b) A’-B’

A (0,0)B (4,2)

A’ (0,0)B’ (-4,2)

2. Find the distance from the pre-image point B to the origin (vertex of rotation) and the distance from the origin (vertex of rotation) to the image point B’.

What do you observe?

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Find and connect the ordered pairs indicated in the table.

a) A-B

b) A’-B’

A (0,0)

B (4,2)

A’ (0,0)B’ (-4,2)

A

B

A’

B’

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Pre-Image Image

A(2,2) A’ (-2,2)

B (4,5) B’ (-5,4)

C (4,9) C’ (-9,4)

D (2,6) D’ (-6,2)

1. Find and connect the ordered pairs indicated in the table.

a) A-B-C-D

b) A’-B’-C’-D’

2. What do you observe among the distance from the pre-image point to the origin (vertex of rotation) and the distance from the origin (vertex of rotation) to the image point?

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A B

CD

Properties of rotation:a)_______________________________________________________________________________________________________b)______________________________________________________________________________________________________

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A

B

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A’

B’C’

D’

Properties of Rotation:• The distance from the pre-image point to the vertex of rotation is equal to the

distance from the vertex of rotation to the image point.• The angles formed by the pre-image point to the vertex of rotation and it’s

image point remains constant.

MATHRotation:A transformation in which the coordinate axes are rotated by a fixed angle about the origin.

MATH

http://nlvm.usu.edu/en/nav/frames_asid_294_g_3_t_3.html?open=activities&from=applets/controller/query/query.htm?qt=rotation&lang=en

What was needed to have the spinner going through each position identified on the place mat?

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MATH

Slide a Figure….. Translate a Figure.

F a l l (September)

S p r I n g (March)

What do you observe in your place mat in reference to the axis of the Earth with respect to the orbit around the Sun?

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The Earth revolves around the Sun

TranslationRotation

The Earth revolves (Orbits) around the Sun • The seasons are caused by the tilt of the earth (23.5 degrees).

• In the winter, the sun is lower in the sky. The light is indirect because the light strikes the earth from a lower angle. Because of this lower angle, the same amount of sunlight is spread out.

• Another result, of the sun being lower in the sky, is that the days are shorter. Sunrise is later, and sunset is earlier. This gives the sun less time to warm the earth.

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B (10,0) B’ (-2,4)

Pre Image point Image point

A (7,0)

C(10,4)D (7,4)

A’ (-5,4)

D’ (-5,8)C’(-2,8)

1. Find and connect the following ordered pairs.

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A B

CDA’ B’

C’D’

Pre Image point Image point

A (7,0)B (10,0)C(10,4)D (7,4)

A’ (-6,4)B’ (-2,4)

D’ (-6,8)C’(-2,8)

b) How can you go from point A to A’? __________________________________________

c) How can you go from point B to B’ ? _______________________________________

d) Find the distance between points A and A’ (You may use your ruler)

Connect those two point._____________________________________________________

e) Find the distance between points B and B’ (You may use your ruler)

Connect those two point._____________________________________________________

Properties of Translation

a) What did you observe when comparing the distance between the pre-image and image points? ____________________________________________________________________________________________________________________________________________

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A B

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C’D’

Properties of Translation:a)_______________________________________________________________________________________________________b)______________________________________________________________________________________________________

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C’D’

Properties of Translationa) Find and compare the distance between the pre-image and it’s image point for A and A’, B and B’, C and C’, D and D’

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A B

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C’D’

Properties of Translation:a) The distance between the pre-image points and their image points is constant.b) Pre-image points move to their image points in the same direction and remain parallel.

MATHTranslation:A transformation in which every point moves at a constant distance in a specified direction.

Resources:

Benchmarks for Science literacy provided by American Association for the Advancement of Science http://www.project2061.org/publications/bsl/online/bolintro.htm

National Science Education Standards provided by National Academy of Sciencehttp://www.nap.edu/readingroom/books/nses/6d.html

Texas Essential Knowledge and Skills (TEKS) for middle school Mathematics/Sciencehttp://www.tea.state.tx.us/teks/

National Standards provided by National Council for Teachers of Mathematics http://standards.nctm.org/document/chapter6/index.htm

Technology and science MSNBC web page.http://www.msnbc.msn.com/id/3077384

The earth and its seasons. (La tierra y sus movimientos)http://www.educabolivia.bo/Portal.Base/Web/VerContenido.aspx?ID=131098&GUID=f080978f-8e9c-4667-bb39-52d5c40f6ffd

National Library of Manipulativeshttp://nlvm.usu.edu/en/applets/controller/query/query.htm?qt=rotation&lang=enWikimedia.http://upload.wikimedia.org/wikipedia/commons/2/2c/Rotating_earth_%28large%29.gif

Math warehousehttp://www.mathwarehouse.com/transformations/index.php