Coordinate graphing project transformations

|1Coordinate Graphing / Geometry Project

The purpose: The following activities allow students to demonstrate their understanding of the coordinate system, apply that knowledge to various geometric concepts, and create their own graphs utilizing the concepts outlined. This portfolio will enable the Algebra 1 students to apply, analyze, synthesize and evaluate their knowledge of middle school math and bring it into the next level, Geometry, through this geometric application. Students will reflect on the transformational concept as a metaphor for other changes that have been an integral part of our world.

Goals: This project addresses the following goals in the North Carolina Standard Course of Study for Math, for Middle Grades 6, 7 and 8 along with the higher level learning in Algebra 1 transitioning into High School Geometry.

6.3.03 Transform figures in the coordinate plane and describe the transformation6.3.04 Solve problems involving geometric figures in the coordinate plane7.3.03 Use scaling and proportional reasoning to solve problems related similar and congruent polygons7.3.02 Identify, define and describe similar and congruent polygons with respect to angle measures, length of sides, and proportionality of sides8.3.03 Identify, predict and describe dilations in the coordinate planeA3.02 Operate (addition, subtraction and scalar multiplication) with matrices to solve problemsG3.03 Describe the transformation (translation, reflection, rotation, dilation) of polygons in the coordinate plane in simple algebraic termsCommon Core Connection8.G.1a. Verify experimentally the properties of rotation, reflections, and translations8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations.8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two dimensional figures using coordinatesG-CO- Describe the effect of dilations, translation, rotations, and reflection on two dimensional fiures using coordinates and understand congruence in terms of rigid motions

Graphing Procedure: The student will complete all 9 activities. They must graph all activities on 1/4” graph paper and answer all questions connected with each activity. Answers must be complete sentences and in appropriate mathematical terms. Each graph must be drawn using a ruler or straight edge and must be colored.Essay Procedure: Chose a topic from the choice board provided and use the pre-writing graphic organizer. Write your essay with an introduction including a thesis statement, supporting paragraphs and a conclusion. Proper grammar and spelling are required

Grade: Students will receive two test grades, one on the graphing activities and one on the essay. This portfolio will count as 2 Test grades for the student. There will be a 5 percentage point deduction for each late school day. The portfolio will be accepted early.

Graphing component is due Thursday October 20, 2011Reflection essay is due November 1, 2011

Parent Signature: __________________________________Date __________________

Student Signature: _________________________________Date __________________

Name: ____________________________________Graphing Project Date ____________Key:Questions: 3 points – Answered all questions accurately.

2 points – Answered more than half of the questions1 point – Answered less than half of the questions or did not answer them at all.

Accuracy: 3 points – Points were graphed correctly. 2 points – Points were graphed partially accurate1 point – Points were graphed incorrectly.

Color: 3 points – Colored all geometric figures and used a straight edge.2 points – Only outlined all geometric figures and used a straight edge1 point – Only outlined all geometric figures and did not use a straight edge

Part 1Questions Accuracy Color Total

King Tut(Dilation)Cube(Dilation)Your Own(Dilation)Slides(Translation)Your Own(Translation)Trapezoid(Reflection)Your Own(Reflection)Arrow(Rotation)Your Own (Rotation)

Part 2 Subtotal

5 points turned project in on time 0 points did not turn project in on time

5 points overall presentation (in order) 0 overall presentation (not in order)

1st Test Grade=

Part 2Scale 0-5, 5 High Thesis/Argument Organization Grammar/Spelling

Written Reflection

5 points turned project in on time 0 points did not turn project in on time

5 points overall presentation (in order) 0 overall presentation (not in order)

Part 2-– Subtotal2nd Test Grade

Dilation – Activity 1: King Tut

1. Use the graph paper vertically. Put the origin in the center

2. Plot and label these points.A = ( 1, 5 ) B = ( 7, -2 ) C = ( 4, -3 ) D = ( -4, -3 ) E = (-1, -2 )

Rewrite as a matrix: x’s 1 7 4 -4 -1y’s 5 -2 -3 -3 -2

3. Make solid lines AB, AC, BC, CD and AD

4. Make dashed lines AE, DE and EB

5. Dilate each coordinate of A, B, C, D, E by a scale factor of 2 to get new points A’, B’, C’, D’ and E’. Remember ( x, y ) = ( 2x, 2y )

Show matrix multiplication:

Rewrite as points: A’ = ( , ) B’= ( , ) C’ = ( , ) D’ = ( , ) E’= ( , )

6. Plot and label A’, B’, C’, D’ and E’ on the same graph.

7. Make solid lines A’B’, A’C’, B’C’, C’D’ and A’D’

8. Make dashed lines A’E’, D’E’ and E’B’

9. How does the two graphs compare? Discuss congruency.

10. What did the scale factor of 2 do to the original image?

Dilation – Activity 2: The Incredible Shrinking Cube

1. Use the graph paper horizontally. Put the origin the lower left-hand corner.2. Plot and label the following points. A= (12,12) B= (12,20) C= (20,20) D=( 20,12)

E=( 16,24) F= ( 24,24) G=( 24,16) H=(16,16)

3. Make solid lines AB, AD, AH, BE, EF, EH, DG, FG and GH

4. Make dashed lines BC, CF and CD

5. Dilate each coordinate of A, B, C, D, E, F, G and H by a scale factor of 1/2 to get new points A’, B’, C’, D’, E’, F’, G’ and H’. Remember ( x, y ) = (1/2x, 1/2y )Rewrite as a matrix and show multiplication:

Rewrite as points: A’ = ( , ) B’= ( , ) C’ = ( , ) D’= ( , ) E’= ( , ) F’= ( , ) G’=( , ) and H’= ( , )

6. Plot and label A’, B’, C’, D’, E’, F’, G’ and H’

7. Make solid lines A’B’, A’D’, A’H’, B’E’, E’F’, E’H’, D’G’, F’G’ and G’H’

8. Make dashed lines B’C’, C’F’ and C’D’

9. Using your new coordinates of A’, B’, C’, D’, E’, F’, G’ and H’ from #5 dilate each coordinate with a scale factor of ½ to get new points A”, B”, C”, D”, E”, F”, G” and H” Remember ( x, y ) = (1/2x, 1/2y ) Rewrite as a matrix and show multiplication:

A” = ( , ) B”= ( , ) C” = ( , ) D”= ( , ) E” = ( , ) F”= ( , ) G”= ( , ) and H” = ( , )

10. Make solid lines A”B”, A”D”, A”H”, B”E”, E”F”, E”H”, D”G”, F”G” and G”H”

11. Make dashed lines B”C”, C”F” and C”D”

12. Describe the size and location of the three cubes.

Activity 3: Create Your Own Dilation

1. Set up an x-axis and y-axis on your graph paper

1. Draw a design on your graph paper. (minimum 5 points)

2. Make a list of the ordered pairs necessary to create your design. Be sure to include directions that indicate where it is necessary to lift the pencil and where it is necessary to connect each point to the next one in the order that you have them listed.

3. Dilate your points with a reduction, locate and label (show your work). Your scale factor is ________

4. Dilate your points with an enlargement, locate and label (show your work). Your scale factor is _____

5. Color your design.

Activity 4: Translations: Sliding Trapezoids

1. Use the graph paper horizontally. Put the origin in the center. Locate these points.A = (-4, -2), B = (-2, 2), C = (1, 2), D = (5, -2)Connect ABCDA. The figure you his called a Trapezoid.

2. Add 10 to each x-coordinate and 5 to each y-coordinateMatrix

A = -4 -2 1 5 B = 10 10 10 10-2 2 2 -2 5 5 5 5

Add A + B =

Rewrite points A’ = ( , ), B’ = ( , ), C = ( , ) and D = ( , )

3. Locate A’B’C’D’ and connect to make a trapezoid

4. Draw a straight arrow from A to A’. How far over and how far up is it from A to A’?

5. Add 10 to each x-coordinate and subtract 5 from each right-hand coordinate in the original set of points.

Matrix A = -4 -2 1 5 B = 10 10 10 10

-2 2 2 -2 -5 - 5 - 5 - 5

Add A + B =

Rewrite points A” = ( , ), B” = ( , ), C” = ( , ) and D” = ( , )

6. Locate A”B”C”D” and connect to make a trapezoid

7. Draw an arrow from A to A”. How far over and down is tit from A to A”?

8. What type of motion will move the trapezoid ABCD onto A”B”C”D”

9. Suppose you wanted to move the original trapezoid eight units to the right and twelve units up. With out drawing it, give the coordinates of the vertices.

A’” = ( , ), B’” = ( , ), C’” = ( , ), D’” = ( , )

Activity 5: Create Your Own Translation

1. Set up an x-axis and y-axis on your graph paper



8. Translate your points to the right 5 units and down 3 units, locate and label (show your work). Matrices

9. Translate your points to the left 5 units and up 3 units, locate and label (show your work). Matrices

10.Color your design.

Activity 6: Reflection Trapezoid

1. Use the graph paper vertically. Put the origin in the center. Locate these points.

A= ( 3, 3 ), B= ( 5, 7 ), C= ( 8, 7) and D= (12, 3 )

Connect ABCDA to make a trapezoid.

2. Reflect over the y-axis by multiplying each x-coordinate by -1 to get A’, B’, C’, D’

A’= ( , ), B’= ( , ), C’= ( , ) and D’= ( , )

Locate these points and connect them to make a trapezoid.

How is this trapezoid related to the one you made in part 1?

3. Reflect over the x-axis by multiplying each y-coordinate in A, B, C, D by -1 to get new points

A”= ( , ), B”= ( , ), C”= ( , ) and D”= ( , )


How does this trapezoid related to the one you made in part one?

4. Reflect over the origin by multiplying both the x- coordinate and y-coordinate in part 1 by -1 to get new points:

A’”= ( , ), B’”= ( , ), C’”= ( , ) and D’”= ( , )


How is this trapezoid related to the one you made in part 2?

5. Start a new picture on another vertical piece of graph paper. Put the origin in the center of the page.

Activity 7 –Create your own Reflection

1. Set up an x-axis and y-axis on your graph paper.



4. Reflect your points over the y-axis, locate and label (Show your work).

5. Reflect your points over the x-axis, locate and label. (Show your work).

6. Reflect your points over the origin, locate and label. (Show your work).


Activity 8: Rotations- Arrow

1. Use the graph paper vertically. Put the origin in the center of the paper.

2. Locate these points: A = ( 0, 0 ), B = ( 5 , 10 ), C = ( 5 , 4 ), D = ( 4, 6 ) and E = ( 1, 0 )

Connect ABCDE to make an arrow.

3. Rotate 900 by switching your x-coordinate with your y-coordinate and multiplying your new x-coordinate by a negative one. Notation ( x, y) → ( -y, x )

A’ = ( , ), B’ = ( , ), C’ = ( , ), D’ = ( , ) and E’ = ( , )

4. How is this one related to the original?

5. Rotate 1800 by switching your x-coordinate with your y-coordinate and multiplying your new x-coordinate by a negative one. Notation ( x, y) → ( -x, -y )

A” = ( , ), B” = ( , ), C” = ( , ), D” = ( , ) and E” = ( , )


7. How would you rotate the figure 2700 ? ( Try to graph it and analyze the two sets of points Notation ( x, y) → ( , )

A’” = ( , ), B’” = ( , ), C’” = ( , ), D’” = ( , ) and E’”= ( , )


Activity 9 –Create your own Rotation

1. Set up an x-axis and y-axis on your graph paper.



4. Rotate your points 900, locate and label (Show your work).

5. Rotate your points 1800 , locate and label. (Show your work).

6. Rotate your points 2700 , locate and label. (Show your work).


Activity 10-Essay

Transformation in the World Around Us

Throughout history, our geography has been gradually changing: Continents have shifted and landscapes have changed as continental drift and erosion have created new features in the world around us. These changes can be viewed as a metaphor for other changes that happen in the world around us. For example, American culture has changed over the decades. Attitudes towards race and ethnicity, gender, music, and fashion have all shifted over time. Likewise, America has seen shifts in its national borders: from the thirteen colonies to the fifty states that we have today. Your task is to examine one area of transformation in the world around us and fully explore that topic.

Your reflection must include the following:

1. A thesis statement that gives a strong opinion about how the world around us has changed. (Your thesis statement may answer one of the guiding questions in the chart below.)

2. An introduction and conclusion that explain why the change you identify is important for the reader to recognize and understand.

3. Supporting paragraphs that give clear examples and support your thesis.4. This reflection will also be used to assess grammar and spelling.

Getting Started: Areas of transformation that you might want to explore…How have attitudes towards race shifted from 1776 until today?

How have attitudes towards inter-racial marriage shifted over time?

As the United States has expanded its borders and settled new land, what has been the impact on Native American tribes?

How have attitudes towards Native Americans shifted from the French and Indian war until today?

How did the poetry, art, and music of the Harlem Renaissance transform attitudes about African-Americans?

How did the events of September 11, 2001 transform our nation?

How has environmental pollution changed the world around us? How has our response to it shaped our culture?

Science Fiction: What new technologies or inventions could have the most impact on our nation? (Robots, Artificial Intelligence, Cold fusion, etc.)

How has fashion changed from 1776 until today? Why has it changed and what does it say about our culture?

How has music changed from the Harlem Renaissance until today? What impact did these changes have on our culture?

How did the Civil Rights movement transform our nation?

Take Charge:If you were president for a day, what one thing would you do to transform our nation for the better? How would it impact our world?

Pre-Writing Graphic OrganizerThesis:______________________________________________________________________________________________________________________________________________________

Why is my thesis important?______________________________________________________________________________________________________________________________________________________

Splash: Arguments and Evidence that support my thesis:

Pe

Peer Review:Three things that I love about your thesis, arguments, and evidence:1.

2.

3.

Three things that you could improve about your thesis, arguments, and evidence:1.

2.

3.

Education

Coordinate graphing project transformations