TEKS 8.6 (A,B) & 8.7 (A,D)

Shapes and the Coordinate System

The Coordinate System

The coordinate system we use today is called a Cartesian plane after Rene Descartes, the man who invented it. The coordinate system looks like the one pictured on the next slide. On the slide there is a vertical dark line and a horizontal dark line, representing what are called the x-axis (horizontal) and y-axis (vertical). The x and y axes are labeled and are numbered from -5 to 5 on both axes. The axes may have higher numbers also. Notice that the x-axis and y-axis meet at the number 0.

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The Origin

The origin of the coordinate system is where the x- and y-axis meet or intersect. At the origin, the number on the x- and y-axis is equal to 0. This point is described as the origin because it is where every other point on the coordinate system is measured from.

Find the origin on the coordinate system.

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Y - AxisWhere is the origin?

Place a dot where the origin is.

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There’s the origin!

Dots are how we represent points on the coordinate system.

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Measuring Distances

When distance is measured from the origin it is measured by determining how far away something is away from the x-axis and the y-axis. Each minor line represents 1 unit away from the origin.

The first point is zero units or blocks away from the origin on the x-axis but two units away from the origin on the y-axis

The second point is three units away from the origin on the x-axis but zero units away from the origin on the y-axis.

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Ordered Pairs

Every point on the Cartesian plane can be described by a coordinate. An ordered pair of a point is its coordinate (or location) on the plane. An ordered pair looks like this: (3,2). The first number is the point’s position on the x-axis; the second number is the point’s position on the y-axis. By convention, ordered pairs are always written with the x-axis coordinate first, followed by a comma, and then the y-axis coordinate. So the position of point (3,2) would be over three units to the right on the x-axis and up two units on the y-axis starting from the origin.

(0,2) (3,2)

(3,0)

This graph shows three points and their ordered pair coordinates.

What would the coordinate for the origin be?

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(0,0)

What would the ordered pair for these points be?

Remember one or both of the numbers in the ordered pair can be negative.

Negative numbers (<0) for the x-axis coordinate means LEFT of the origin.

Negative numbers (<0) for the y-axis coordinate means BELOW the origin

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Shapes

Shapes can be drawn on the coordinate system as well. Instead of being represented by just one point, they are represented by lines that connect many points. We can locate and describe a shape based on where it is centered around (like a circle) or what points its corners are at (like a rectangle). Then we can also calculate how big the shape is.

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How would this circle be described?

Where is its center?

What is its radius?

What is its diameter?

To determine these either find the corresponding ordered pair or count the distance.

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How would you describe this rectangle?

Where are the corners?

What is the length?

What is the width?

What is the area?

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3-D Shapes

Unfortunately, we cannot draw 3-D shapes on the coordinate system. The coordinate system only works for 2-D shapes.

With these models being passed around to you, draw what the 3-D shapes look like. Try drawing the shapes from different perspectives.

Moving Shapes

There are 4 ways to transform a shape on a plane:

• Dilation• Reflection• Translation• Rotation

• We will see how these transformations work on the next slides.

Definitions:

• Dilation – The object is made bigger or smaller but kept centered around the same point.

• Reflection – A ‘mirror’ image is made of the object.

• Translation – The object is shifted on the plane without changing anything other than location.

• Rotation – The object is turned like a clock about a fixed point called the center of rotation.

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Here is a graph of a stick figure person who will show us the difference between dilation, reflection, translation, and rotation. X - Axis

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Dilation

Now our stick figure is exactly twice as big as he was the first time. Notice that even though he is bigger, he is still centered at the same point. He kept his orientation and position, but his size changed.

A dilation can be mean either an object getting bigger or smaller.

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Reflection

Now our stick figure friend has been reflected across the y-axis. Notice how his arms are opposite to the position they were previously in. That is because it is a mirror image. The stick figure kept his size, but his orientation and position changed.

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Translation

Our stick figure has now moved over 5 units to the right. On the x-axis. Note that he retained his size and orientation. Only his x-axis position has changed.

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Rotation

Now our friend is rotated 35 degrees counterclockwise with the center of rotation at (-3,0) Notice how it looks like he spins around his mid-section to the left. Where would our friend end up if he were spun 360 degrees from his starting position?

Center of Rotation

Three Types of Changes

Three shape properties can change: size, position, and orientation. Each transformation changes at least one of them.

• What does dilation change?• What does reflection change?• What does translation change?• What does rotation change?

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We can do all four transformations together. In this graph our stick figured was reflected across the y-axis, then dilated to half its size, then translated 4 units up on the y-axis, and rotated 45 degrees clockwise.

Whew!

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Now it’s your turn!

Take this object and draw a dilation.

Next draw a reflection across the y-axis.

Then draw a translation.

Then rotate the object 45 degrees clockwise.

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Additional Resources

TEKS 8.6• LESSON: Explore the world of translations, reflections, and

rotations in the Cartesian coordinate system by transforming squares, triangles and parallelograms.

• The Transmographer applet allows the user to explore the world of transformations, reflections, and rotations. You can translate triangles, squares, and parallelograms on both the x and y-axes. You can also reflect the figure around x values, y values, and the line x = y. The applet can also rotate the figure any given number of degrees.

• http://www.shodor.org/interactivate/activities/transform/index.html• http://www.shodor.org/interactivate/activities/transform/what.html

Additional Resources

TEKS 8.6• LESSON: Shapes and Shape Relationships

• Benchmarks– Distinguish among shapes and differentiate between examples and non-

examples of shapes based on their properties; generalize about shapes of graphs and data distributions

– Generalize the characteristics of shapes and apply their generalizations to classes of shapes

– Derive generalizations about shapes and apply those generalizations to develop classifications of familiar shapes

• http://www.svsu.edu/mathsci-center/mshape.htm

TEKS 8.6• LESSON: Shapes and Shape Relationships

• Benchmarks– Translate, reflect, rotate, and dilatate geometric figures using mapping

notation in the coordinate plane. – Analyze a given transformation and describe it using mapping notation. – Recognize, and describe in mapping notation and image from a

combination of any two transformations. – Demonstrate whether congruence, similarity, and orientation are

maintained under translations, reflections, rotations, and dilatations

• http://plato.acadiau.ca/courses/educ/reid/up/Trans_geometry_unit_plan_final.htm

Additional Resources

Additional ResourcesTEKS 8.6• LESSON: Simple Transformations in Geometry

• Sixth grade students engage in an authentic learning experience as they identify the meaning of translations, reflections, rotations, and dilations of two-dimensional shapes. After solidifying their understanding of each type of symmetry, students will work together to create a movie that not only explains the meaning and characteristics, but also provides an example of a natural occurrence of each type of symmetry.   

• http://newali.apple.com/ali_sites/mili/exhibits/1000886/the_lesson.html

Additional ResourcesTEKS 8.7

• Benchmarks– Identify, describe, compare, and classify geometric figures– Identify, draw, and construct three-dimensional geometric figures

from nets– Identify congruent and similar figures– Explore transformations of geometric figures– Understand, apply, and analyze key concepts in transformational

geometry using concrete materials and drawings– Use mathematical language effectively to describe geometric

concepts, reasoning, and investigations

• http://www.gecdsb.on.ca/d&g/math/Math%20Menus/gr7gass.htmhttp://www.gecdsb.on.ca/d&g/math/Math%20Menus/gr8gass.htm

Additional Resources

TEKS 8.7• Overview

– In this lesson, students develop informal geometry and spatial thinking. They are given opportunities to create plans, build models, draw, sort, classify, and engage in geometric and mathematical creativity through problem solving.

• http://illumtest.nctm.org/lessonplans/6-8/geomiddlegrades/

Additional Resources

TEKS 8.7• LESSON: Geography Geometry Grades: 5-8

Benchmarks– Geography: Students should be able to specify locations and describe

spatial relationships using coordinate geometry and other representational systems

– Geometry: Students should be able to use visualization, spatial reasoning, and geometric modeling to solve problems.

• http://www.microsoft.com/Education/GeographyGeometry.aspx