Transformations on the Coordinate Plane Halftime Salute

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Cameron Carr
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Transformations on the Coordinate Plane SUGGESTED LEARNING STRATEGIES:

1 To boost school spirit and get students excited about geometry, the

features formations in the shape of special quadrilaterals.

Scott, asks band members who are also on the Math Team to help him plan

The marching band decides to use a Cartesian plane to represent the

The diagram below shows the MIU HS football field in the Cartesian

The band members locate the origin (0,0) at the intersection of the

Positions on the field are described relative to the middle of the

1 Transformations on the Coordinate Plane SUGGESTED LEARNING STRATEGIES: Questioning the Text, Shared Reading, Visualization ACTIVITY.1 To boost school spirit and get students excited about geometry, the marching band at MIU High School is planning a halftime show that features formations in the shape of special quadrilaterals. The band director, Mr. Scott, asks band members who are also on the Math Team to help him plan the show. The marching band decides to use a Cartesian plane to represent the football field. The diagram below shows the MIU HS football field in the Cartesian plane. The band members locate the origin (0,0) at the intersection of the 0-yard line and the center line on the football field. Each unit in the Cartesian plane is equal to 1 yard on the field. An actual football field is 0 yards long and 3.3 yards wide. Positions on the field are described relative to the middle of the field where the 0-yard line and the horizontal center line intersect one another. Positions to the left of the 0-yard line have negative coordinates and positions to the right of the 0-yard line have positive coordinates. The center line divides the football field into and sides Unit Coordinate Geometry and Transformations 31 T S

2 32 SpringBoard Mathematics with Meaning Geometry Knowledge, Close Reading, Interactive Word Wall, Vocabulary Organizer, Think/Pair/Share, Group Presentation, Create Representations, Quickwrite, Self/Peer Revision 1. Band member Zack Cefone starts the first song, Going the Distance, at the coordinates (, 0). Plot Zack s location on the diagram of the field on the previous page. How would you explain to Zack where he starts the halftime show in terms of his position on the football field? 2. The band director asks Sue to start the halftime show on the 0-yard line, yards above the center line. Plot Sue s location on the diagram of the field on the previous page. What are Sue s starting position coordinates? 3. Zack and Sue are at the opposite ends of a line of 11 band members. Draw the line segment between Zack and Sue on the diagram on the previous page. How far apart are Zack and Sue? Show work to justify your answer. 4. When the music begins, Zack and Sue s line is going to march to a new location on the field. Zack marches to ( 30, ). Sue marches to (, 0). Draw Zack and Sue s line in its new location on the diagram of the football field. a. How far did Zack and Sue travel to get to their new locations? b. Did the length of Zack and Sue s line change? Explain your reasoning.. The Math Team members tell Mr. Scott that the movement described in Item 4 is called a translation. Write a definition for the term translation in your own words. 6. The transformation notation for a translation is: (x, y) (x + a, y + b) where a represents the change in the x-coordinates and b represents the change in the y-coordinates. Use transformation notation to represent the translation in Item 4.

3 SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Vocabulary Organizer, Think/Pair/Share, Group Presentation, Create Representations, Look for a Pattern, Quickwrite, Self/Peer Revision 7. Translate Zack and Sue s line from Item 4(a) so that Sue ends up at the center of the football field. a. What are Zack and Sue s coordinates now? b. How far did Sue move? c. How far did Zack move? d. Did all of the band members on Zack and Sue s line move the same distance as Zack and Sue? Explain your reasoning. e. Use transformation notation to express this translation. 8. For the next formation, Sue marches in place at the origin and Zack moves parallel to the center line, stopping on the 30-yard line to the right of the 0-yard line. The rest of the band line also moves parallel to the center line, stopping on the same yard line on the opposite side of the field as their starting points. a. What are Zack s coordinates at his new location? b. Draw Zack and Sue s line in its new location. How far did Zack travel to get to his new location? Show work to justify your answer. c. Did all of the band members on Zack and Sue s line move the same distance as Zack and Sue? Explain your reasoning. 9. The type of movement described in Item 8 is called a reflection. The line was reflected over the 0-yard line (y-axis). Define the term reflection in your own words.. Consider the coordinates for each band member on the line in Item 8. a. How did the x-coordinate change for each member? b. How did the y-coordinate change for each member? c. Express this reflection in transformation notation. Unit Coordinate Geometry and Transformations 33

4 34 SpringBoard Mathematics with Meaning Geometry Knowledge, Close Reading, Think/Pair/Share, Group Presentation, Identify a Subtask, Quickwrite, Self/Peer Revision 11. To get to the line s final formation, reflect Zack and Sue s line across the right side 30-yard line. a. Write instructions to tell the band line members how they should move to get to their new positions. b. What are Zack and Sue s coordinates now? c. How far did Zack move? d. How far did Sue move? 12. Claire Anett is the band member in the middle of Zack and Sue s line. Recall that Zack started at (-, 0) and Sue started at (0, ). a. What were Claire s coordinates when Zack and Sue s line started the song? Explain your answer. b. What are Claire s coordinates once the band line is in its final location? Show work to justify your answer. c. How many total yards did Claire travel during this song? Show work to justify your answer.

5 Knowledge, Marking the Text, Think/Pair/Share, Group Presentation, Identify a Subtask, Quickwrite, Self/Peer Revision At the beginning of the band s second song, The Quadrilateral Quickstep, Zack and Sue s line is part of a quadrilateral formation with other band members. Zack s position is (, -) and Sue s position is at (40, 0). The other vertices of this new quadrilateral are Tom Pratt at (40, ) and Patty Drum at (0, -) Plot the coordinates of the vertices of the quadrilateral and draw the band formation on the diagram above. Label the vertices using the first letter of each player s name. a. Describe Zack, Sue, Tom, and Patty s positions on the football field. b. What is the best name for Quad ZSTP? Show the work that supports your answer. T S Unit Coordinate Geometry and Transformations 3

6 MATH TERMS Recall that the median of a trapezoid is the segment that joins the midpoints of the legs of the trapezoid. 36 SpringBoard Mathematics with Meaning Geometry Knowledge, Think/Pair/Share, Identify a Subtask, Self/ Peer Revision P 14. For the next formation, the bases of the trapezoid ( PT and ZS ) march toward each other and form the median of Quad PTSZ. a. What are the coordinates of the midpoints of the legs of this t r ap e z oi d? b. Plot the midpoints and draw the new formation made by the band. How long is the median made by the band members? c. Show that the length of the median is half the sum of the length of the bases. Z T S

7 Knowledge, Marking the Text, Think/Pair/Share, Identify a Subtask, Self/Peer Revision S The band is going to form a parallelogram at the end of The Quadrilateral Quickstep. Zack, Sue, Tom, and Patty are again at the vertices as shown above. They march to new positions Z(, ), S(0, 0), T(30, ) and P(0, ) to form the parallelogram. Plot and label the vertices of the parallelogram on the diagram above. a. Verify that Quad ZSTP is a parallelogram. Show your work. b. The band members located along ZP moved to their new formation by rotating about the midpoint of the line. Compare the new position of ZP to its old position. Through how many degrees about its midpoint did ZP rotate as the band moved into formation? P Z T ACADEMIC VOCABULARY A rotation is a transformation in which each point of the pre-image travels clockwise or counterclockwise around a fixed point a certain number of degrees. Unit Coordinate Geometry and Transformations 37

8 Knowledge, Think/Pair/Share, Group Presentation, Create Representations, Quickwrite, Self/Peer Revision T S P Z MATH TERMS The center of a quadrilateral is the point of intersection of its diagonals. 16. During this song, the drum major moves to the center of the parallelogram. a. Draw in the diagonals of the parallelogram above and find the coordinates of the midpoint of each diagonal. b. Which property of a parallelogram does this illustrate? Explain your reasoning. c. Describe the drum major s position on the football field in the diagram above. 17. Mr. Scott notices that the final formation looks like a rectangle, but the Math Team disagrees. Who is correct? Show work to justify your answer. 38 SpringBoard Mathematics with Meaning Geometry

9 Knowledge, Summarize/Paraphrase/Retell, Visualization, Think/Pair/Share, Create Representations, Identify a Subtask, Quickwrite, Self/Peer Revision 18. For the third song in the halftime show, the band is marching to Rhombi Are Forever. As the song begins, the band collapses their parallelogram to form a line along the center line to the right side of the 0-yard line, as shown by the dotted line below a. Patty is on the left end at (0, 0) and Sue is on the right end of the line. What are Sue s coordinates? b. Tom and Zack are standing back-to-back at the midpoint of this line. What are their coordinates? c. The band moves into formation as the song begins. Zack marches yards toward the side along the -yard line and Tom marches yards toward the side along the -yard line. The ends of the band line march in place. The rest of the band line splits with every other band member, either marching toward the side with Zack or toward the side with Tom. The band forms a new quadrilateral. Draw the band s new formation on the diagram above. What is the best name for this quadrilateral? Explain your answer. Unit Coordinate Geometry and Transformations 39

10 Knowledge, Think/Pair/Share, Group Presentation, Create Representations, Quickwrite, Self/Peer Revision 19. Mr. Scott wants Tom and Zack to continue leading their band lines above and below the center line to create a larger quadrilateral. He wants Sue and Patty to continue marching in place a. Use the diagram above to draw the rhombus formed by the band when Tom is yards above and Zack is yards below the center line. What happens if the entire formation is reflected over the x-axis? b. Where should Tom and Zack stop so their quadrilateral is a square? Draw your solution on the diagram above and explain your reasoning. 360 SpringBoard Mathematics with Meaning Geometry

11 SUGGESTED LEARNING STRATEGIES: Marking the Text, Visualization, Think/Pair/Share, Group Presentation, Create Representations, Quickwrite, Self/Peer Revision. For the final song, Regular and Righteous, Zack, Sue, Tom, and Patty move from their previous positions to the vertices of one final quadrilateral with the rest of the band along the figure s sides. Their final coordinates are: Z(4, ), S(40, ), T (, ) and P(, ) a. Sketch the square that you drew on the previous page and draw the final quadrilateral. What transformation(s) did the band use to move into their final formation: reflection, rotation or translation? Explain your reasoning. b. Mr. Scott wants the final formation to be centered on the field. Which transformation(s) will accomplish this? Where should the vertices be located? c. Consider the transformations for Quad TPZS in Parts a and b. What happens to the formation if the order is switched? Where would the vertices be located? 21. How will a geometric figure, called the pre-image, compare to its image, once it is translated, reflected, or rotated? Explain your answer. T S CONNECT TO ALGEBRA A transformation is a function whose domain is a set of ordered pairs (the pre-image) and whose range (the image) is the set of ordered pairs that results when operations are performed on the coordinates of the ordered pairs in the domain. Unit Coordinate Geometry and Transformations 361

12 Two triangles are congruent if and only if all their corresponding sides and angles are congruent. Translations, reflections, and rotations are referred to as rigid motions because the size and shape of a figure are not affected in the transformation. 22. How can you apply the concept of rigid motions to show that two triangles are congruent? CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work. 1. Given Quad TRAN with vertices T(1, 1), R(2, 4), A(6, 4) and N(4, 1). Perform each of the following transformations on Quad TRAN and list the coordinates of the vertices of the image. What is the best name for the transformation? a. (x, y) (x - 4, y) b. (x, y) (x, y) c. (x, y) (1 x, y) 2. Let AB be the pre-image with coordinates A(2, 3) and B( 2, 0). a. Find the slope and length of AB. b. Perform the transformation (x, y) (x 3, y + 1) on AB. How do the slope and length of the image compare to the slope and length of the pre-image? c. Perform the transformation (x, y) ( x, y) on AB. How do the slope and length of the image compare to the slope and length of the pre-image? 3. Given the line with equation y = 3 2 x + 1. Write the equation of the image of the line under the given transformation. a. (x, y) (x, y 3) b. (x, y) (x 3, y) c. (x, y) ( x, y) 4. Given the pre-image of PQR below. Sketch the image of PQR after each of the following transformations are performed. a. reflection over a vertical axis. b. reflection over a horizontal axis. c. 90 clockwise rotation. d. 180 rotation. P. Given the line with the equation y = x - 3. Write the equation of the image of the line under the given transformation. a. (x, y) (x + 1, y + 3) b. (x, y) (x, y) c. 90 clockwise rotation about its y-intercept d. 4 clockwise rotation about its y-intercept 6. MATHEMATICAL REFLECTION Q R How does an image compare to its pre-image once the pre-image is reflected over an axis of symmetry? 362 SpringBoard Mathematics with Meaning Geometry

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