Shapes and the Coordinate System TEKS 8.6 (A,B) & 8.7 (A,D)
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Shapes and the Coordinate System TEKS 8.6 (A,B) & 8.7 (A,D)
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 horizontal dark line and a vertical 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.
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.
X - Axis Y - Axis Where is the origin? Place a dot where the origin is. -5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -2 -3 -4 -5 There’s the origin! Dots are how we represent points on the coordinate system.
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.
The point on the y- axis is moved two positions up from the origin on the y- axis but no positions away from the origin on the x-axis. The point on the x- axis is moved three positions away from the origin on the x- axis but no positions away from the origin on the y-axis. The third point in the upper right corner is 3 positions away on the x-axis and 2 positions away on the y-axis -5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -2 -3 -4 -5 X - Axis Y - 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.
(0,2)(3,2) (3,0) This graph shows three points with their ordered pairs. What would the ordered pair for the origin be? -5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -2 -3 -4 -5 X - Axis Y - Axis
What would the ordered pairs for these points be? Remember one or both of the numbers in the ordered pair can be negative. A E D B C -5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -2 -3 -4 -5 X - Axis Y - Axis
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.
-5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -2 -3 -4 -5 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 Y - Axis
-5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -2 -3 -4 -5 How would you describe this rectangle? Where are the corners? What is the length? What is the width? X - Axis Y - Axis
Moving Shapes There are 3 ways to move a shape on a graph: Rotations Translation Reflection Dilation Each of the three takes a shape from its original position to a new position
Definitions Rotation – The object moves in the plane so that one point is held fixed. Thus the object can only turn. The fixed point is called the center of rotation. Translations – The object slides in the plane without turning. Thus there are no fixed points and any edge remains parallel to the original edge.
Definitions Reflections – The object flips across a line becoming the mirror image of the original. The line is called the line of reflection. Note you will have to lift the shape out of the plane to perform the reflection whereas it only slides for a rotation or translation. Dilations – The object stretches or shrinks away or toward one fixed point. The fixed point is called the center of dilation.
-5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -2 -3 -4 -5 Here is a graph of a stick figure person who will show us the difference between dilation, reflection, and translation. X - Axis Y - Axis
-5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -2 -3 -4 -5 Rotation Our stick figure has now rotated about the point (-3, 0). He rotated 180 o (half a whole turn). X - Axis Y - Axis
-5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -2 -3 -4 -5 Translation Our stick figure has now moved over 5 positions. Notice that he retained his size and orientation, only his position changed. X - Axis Y - Axis
-5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -2 -3 -4 -5 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 Y - Axis
-5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -2 -3 -4 -5 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. X - Axis Y - 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?
-5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -2 -3 -4 -5 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 X - Axis Y - Axis
-5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -2 -3 -4 -5 Now it is your turn Take this object and draw a dilation to half its original size at its right hand corner. Next draw a reflection across the y-axis. Then draw a translation of 4 spaces down. Then rotate the triangle 180 o (half a whole turn) around its right angle corner. X - Axis Y - Axis
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 dilate 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_p lan_final.htmhttp://plato.acadiau.ca/courses/educ/reid/up/Trans_geometry_unit_p lan_final.htm Additional Resources
TEKS 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.ht mlhttp://newali.apple.com/ali_sites/mili/exhibits/1000886/the_lesson.ht ml
Additional Resources TEKS 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 analyse 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.htm http://www.gecdsb.on.ca/d&g/math/Math%20Menus/gr8gass.htmhttp://www.gecdsb.on.ca/d&g/math/Math%20Menus/gr7gass.htm http://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