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. 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 -5 -4 -3 -2 -1 0 1 2 3 4 5 X - Axis

The Origin The origin of the coordinate system is where the x- and y-axis meet. 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. -5 -4 -3 -2 -1 0 1 2 3 4 5 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. -5 -4 -3 -2 -1 0 1 2 3 4 5 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 smaller line represents 1 unit away from the origin.

5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis The top left point is moved two positions up from the origin on the y-axis but no positions away from the origin on the x-axis. The bottom right point is moved three positions away from the origin on the x-axis but no positions away from the origin on the y-axis. -5 -4 -3 -2 -1 0 1 2 3 4 5 X - Axis

Ordered Pairs To represent where a point is it is given what is called an ordered pair. An ordered pair looks like this (3,2). The number to the left of the comma is the point’s position on the x-axis; the number to the right of the comma is the point’s position on the y-axis. So the point (3,2) would be up two and over three to the right.

This graph shows three points with their ordered pairs. 5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis This graph shows three points with their ordered pairs. What would the ordered pair for the origin be? (0,2) (3,2) (3,0) X - Axis -5 -4 -3 -2 -1 0 1 2 3 4 5 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 Y - Axis B A E What would the ordered pair for these points be? Remember one or both of the numbers in the ordered pair can be negative. -5 -4 -3 -2 -1 0 1 2 3 4 5 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 go through 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 give 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. -5 -4 -3 -2 -1 0 1 2 3 4 5 X - Axis Center = (0,0), the origin Radius = 2 Diameter = 4

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? -5 -4 -3 -2 -1 0 1 2 3 4 5 X - Axis Corners = (1, 1), (1, 3), (4, 3), (4, 1) Length = 3 Width = 2

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 to you. Try drawing the shapes in 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.

Moving Shapes There are 3 ways to move a shape on a graph: Dilation Reflection Translation Each of the three takes a shape from its original position to a new position

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 moved on the coordinate plane but still retains all the properties it had before.

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, and translation. -5 -4 -3 -2 -1 0 1 2 3 4 5 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 around the same point. He kept his orientation and position, but his size changed. A dilation can be both an object getting bigger or and object getting smaller. -5 -4 -3 -2 -1 0 1 2 3 4 5 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. -5 -4 -3 -2 -1 0 1 2 3 4 5 X - Axis

5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis Translation Our stick figure has now moved over 5 positions. Notice that he retained his size and orientation, only his position changed. -5 -4 -3 -2 -1 0 1 2 3 4 5 X - Axis

Three Changes Three things can change: size, position, and orientation. Each transition changes at least one of them. What does dilation change? What does reflection change? What does translation change? Dilation changes size Reflection changes orientation and position Translation changes position.

5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis We can do all three together. In this graph our stick figured was reflected across the y-axis, then dilated to half his original size, the translated up 4 positions -5 -4 -3 -2 -1 0 1 2 3 4 5 X - Axis

Now it is your turn Take this object and draw a dilation. 5 4 3 2 1 -1 -2 -3 -4 -5 Y - Axis Now it is your turn Take this object and draw a dilation. Next draw a reflection across the y-axis. Then draw a translation. -5 -4 -3 -2 -1 0 1 2 3 4 5 X - Axis Check the students to see if they did it correctly.