 # TEKS 8.6 (A,B) & 8.7 (A,D) This slide is meant to be a title page for the whole presentation and not an actual slide. 8.6 (A) Generate similar shapes using.

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TEKS 8.6 (A,B) & 8.7 (A,D) This slide is meant to be a title page for the whole presentation and not an actual slide. 8.6 (A) Generate similar shapes using dilations 8.6 (B) Graph dilations, reflections, and translations on a coordinate plane 8.7 (A) Draw solids from different perspectives 8.7 (D) Locate and name points on a coordinate plane using ordered pairs 8.6 (A) The students would need to learn what similar shapes are. After drawing a few shapes on the board with dimensions (say a rectangle with sides 5 x 8), have each student draw their own similar shape with dimensions. Ask a few students to share what sizes their shapes were to point out that there is more than just one similar shape. In the example before rectangles measuring 10 x 16, 2.5 x 4, and 15 x 24 could all be used as a similar shapes. 8.6 (B) This TEKS can be taught along with 8.7 (D). When the students have gotten used to the coordinate system draw a shape on the coordinate system (it could even be a picture like a stick figure). Then show the dilation, reflection, and translation of that figure. Show the students how to do this using the coordinate system. 8.7 (A) For this pass around different shapes to the students and ask them to draw them. 8.7 (D) To teach the coordinate plane you could have the students use a map to find locations by the grid on the map. Maps usually have locations listed by the coordinates where they are located. An example would be a map of Texas having College Station located in E7 of the map. Have the students find E7 to find College Station. Once they are accustomed to the idea of finding places on the map based on their given coordinates you can take the same map and lay the Cartesian coordinate system on top of it. Then give them the new coordinates of College Station and have them find it again. Another option is to make a Cartesian coordinate system of the room. Label a center, how far away each unit on the coordinate system is, then have the students label where they are and maybe three other people in the classroom based on the Cartesian coordinate plane in their classroom.

Shapes and the Coordinate System
This is the first slide of the presentation

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.

5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis X - Axis

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.

Place a dot where the origin is.
5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis Where is the origin? Place a dot where the origin is. X - Axis

Dots are how we represent points on the coordinate system.
5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis There’s the origin! Dots are how we represent points on the coordinate system. X - Axis

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.

5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis 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. X - Axis

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.

This graph shows three points and their ordered pair coordinates.
5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis This graph shows three points and their ordered pair coordinates. What would the coordinate for the origin be? (0,2) (3,2) (0,0) (3,0) X - Axis The ordered pair for the origin is (0,0)

What would the ordered pair for these points be?
5 4 3 2 1 -1 -2 -3 -4 -5 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 Y - Axis B A E X - Axis C The points appear after each click so if more time needs to be spent on any particular point the other points can wait. A = (-3, 3) B = (4, 4) C = (3, -1) D = (-3, -3) E = (1, 2) D

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.

How would this circle be described?
5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis 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. X - Axis Center = (0,0), the origin Radius = 2 = abs (0-2) Diameter = 4 = (2 – (-2)) on either axis

How would you describe this rectangle?
5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis How would you describe this rectangle? Where are the corners? What is the length? What is the width? What is the area? X - Axis Corners = (1, 1), (1, 3), (4, 3), (4, 1) Length = 3 Width = 2 Area = 6 square units

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. Pass around 3-D shapes such as a sphere, cube, prisms, etc. Let the students draw them for a while and see if they can draw them from different perspectives. Ask why 3-D shapes cannot be represented on the coordinate system? What new axis would you need?

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.

5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis Here is a graph of a stick figure person who will show us the difference between dilation, reflection, translation, and rotation. X - Axis

5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis 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. X - Axis

5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis 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. X - Axis

5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis 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. X - Axis

5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis 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 X - Axis

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? Dilation changes size only. Reflection changes orientation and position but not size. Translation changes position but not size nor orientation. Rotation changes orientation and position with respect to the center of rotation but not size.

5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis 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! X - Axis

Now it’s your turn! Take this object and draw a dilation.
5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis 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. X - Axis Check the students to see if they did it correctly.

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.

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

Additional Resources 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

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.    APPLE software facilitates portions of the project.

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