UNIT 6. Coordinate Plane CCM6+ Name: Math Teacher:

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1 UNIT Coordinate Plane CCM6+ Name: Math Teacher: Main Concept 1 P a g e Page(s) Descartes and the Fly / Unit 6 Vocabulary 2 Daily Warm-Ups 3 5 Graphing on the Coordinate Plane 6 11 Reflections (x & y axis) Distance Between Points on the Coordinate Plane Unit 6 Study Guide Projected Test Date: Some work is derived from Eureka Math and licensed by Great Minds Great Minds. eureka-math.org This file derived from G6-M2-TE

2 Descartes and the Fly Who invented the coordinate plane? Sam knows a math legend that answers this question: René Descartes was a French man who lived in the 1600s. When he was a child, he was often sick, so the teachers at his boarding school let him stay in bed until noon. He went on staying in bed until noon for almost all his life. While in bed, Descartes thought about math and philosophy. One day, Descartes noticed a fly crawling around on the ceiling. He watched the fly for a long time. He wanted to know how to tell someone else where the fly was. Finally he realized that he could describe the position of the fly by its distance from the walls of the room. When he got out of bed, Descartes wrote down what he had discovered. Then he tried describing the positions of points, the same way he described the position of the fly. Descartes had invented the coordinate plane! In fact, the coordinate plane is sometimes called the Cartesian plane, in his honor. Sam likes this story, because it is about flies. Sam spends lots of time trying to find flies, just like Descartes does in the story. But is the story true? Or is it just a legend? The story of the coordinate plane turns out to be a long story, with many parts. It starts long before Descartes, in Ancient Greece. Unit 6: Coordinate Plane Vocabulary coordinate plane x-axis y-axis quadrants origin ordered pairs x-coordinate y-coordinate reflection integers opposites absolute value A plane formed by the intersection of the x-axis and the y-axis. The horizontal number line The vertical number line The x- and y-axes divide the coordinate plane into four regions. Each region is called a quadrant. The point where the x-axis and y-axis intersect on the coordinate plane. A pair of numbers that can be used to locate a point on a coordinate plane. The first number in an ordered pair; it tells the distance to move right or left from the origin. The second number in an ordered pair; it tells the distance to move up or down from the origin. a transformation of a figure that flips the figure across a line The set of whole numbers and their opposites. Two numbers that are equal distance from zero on the number line. The distance of a number from zero on a number line; shown by the symbol: 2 P a g e

3 Unit 6 - Coordinate Plane Daily Warm-Ups 3 P a g e

4 Label the axes. 2. A coordinate is written in the form: (, ) 3. Graph the following coordinates on the plane above: A. (-6, 2) B. (0, 8) C. (-10, 5) D. (3, -7) E. (9, 3) F. (4, 0) G. (9, -9) H. (-2, -8) 4. Label the quadrants. 5. Label the origin - the ordered pair is (, ) 4 P a g e

5 What are the new coordinates when these points are reflected over the x-axis? (-4, 3) and (-5, 3) (2, 9) and (2, 5) What are the new coordinates when these points are reflected over the y-axis? (-4, 3) and (-5, 3) (2, 9) and (2, 5) Describe why we use absolute value to find the distance between two points. (Focus on the concept of distance not being negative) What is the distance between (3, ) and (3, )? 5 P a g e

6 Components of the Coordinate Plane To describe locations of points in the coordinate plane, we use of numbers. The first number of an ordered pair is called the. The second number of an ordered pair is called the. Order is important, so on the coordinate plane, we use the form ( ). The first coordinate represents the point s location from zero on the The second coordinate represents the point s location from zero on the -axis. -axis. All points on the coordinate plane are described with reference to the origin. What is the origin, and what are its coordinates? Exercises Use the coordinate plane to answer parts (a) (c). a. Graph three points on the x-axis, and label their coordinates. b. What do the coordinates of your points have in common? c. What must be true about any point that lies on the x-axis? Explain. 6 P a g e

7 2. Use the coordinate plane to answer parts (a) (c). a. Graph three points on the y-axis, and label their coordinates. b. What do the coordinates of your points have in common? c. What must be true about any point that lies on the y-axis? Explain. 3. If the origin is the only point with 0 for both coordinates, what must be true about the origin? 7 P a g e

8 Quadrants of the Coordinate Plane Exercises Locate and label each point described by the ordered pairs below. Indicate which of the quadrants the points lie in. A(7, 2) B(3, 4) C(1, 5) D( 3, 8) E( 2, 1) 8 P a g e

9 5. Write the coordinates of another point in each of the four quadrants. a. Quadrant I b. Quadrant II c. Quadrant III d. Quadrant IV 6. Do you see any similarities in the points within each quadrant? Explain your reasoning. 9 P a g e

10 1. Name the quadrant in which each of the points lies. If the point does not lie in a quadrant, specify which axis the point lies on. a. ( 2, 5) b. (8, 4) c. ( 1, 8) d. (9.2, 7) e. (0, 4) 2. Jackie claims that points with the same x- and y-coordinates must lie in Quadrant I or Quadrant III. Do you agree or disagree? Explain your answer. 3. Locate and label each set of points on the coordinate plane. Describe similarities of the ordered pairs in each set, and describe the points on the plane. a. {( 2, 5), ( 2, 2), ( 2, 7), ( 2, 3), ( 2, 0.8)} b. {( 9, 9), ( 4, 4), ( 2, 2), (1, 1), (3, 3), (0, 0)} c. {( 7, 8), (5, 8), (0, 8), (10, 8), ( 3, 8)} 10 P a g e

11 1. Label the second quadrant on the coordinate plane, and then answer the following questions: a. Write the coordinates of one point that lies in the second quadrant of the coordinate plane. b. What must be true about the coordinates of any point that lies in the second quadrant? 2. Label the third quadrant on the coordinate plane, and then answer the following questions: a. Write the coordinates of one point that lies in the third quadrant of the coordinate plane. b. What must be true about the coordinates of any point that lies in the third quadrant? 3. An ordered pair has coordinates that have the same sign. In which quadrant(s) could the point lie? Explain. 4. Another ordered pair has coordinates that are opposites. In which quadrant(s) could the point lie? Explain. 11 P a g e

12 Reflecting a Point over x- or y-axis Give an example of two opposite numbers, and describe where the numbers lie on the number line. How are opposite numbers similar, and how are they different? Extending Opposite Numbers to the Coordinates of Points on the Coordinate Plane Locate and label your points on the coordinate plane to the right. For each given pair of points in the table below, record your observations and conjectures in the appropriate cell. Pay attention to the absolute values of the coordinates and where the points lie in reference to each axis. (3, 4) and ( 3, 4) (3, 4) and (3, 4) (3, 4) and ( 3, 4) Similarities of Coordinates Differences of Coordinates Similarities in Location Differences in Location Relationship Between Coordinates and Location on the Plane 12 P a g e

13 Exercises In each column, write the coordinates of the points that are related to the given point by the criteria listed in the first column of the table. Point S(5, 3) has been reflected over the x- and yaxes for you as a guide, and its images are shown on the coordinate plane. Use the coordinate grid to help you locate each point and its corresponding coordinates. Given Point: S(5, 3) ( 2, 4) (3, 2) ( 1, 5) The given point is reflected across the x-axis. The given point is reflected across the y-axis. A y S x The given point is reflected first across the x-axis and then across the y-axis. x M The given point is reflected first across the y-axis and then across the x-axis. 1. When the coordinates of two points are (x, y) and ( x, y), which axis did you reflect over? Explain. 2. When the coordinates of two points are (x, y) and (x, y), which axis did you reflect over? Explain. 13 P a g e

14 1. Locate a point in Quadrant IV of the coordinate plane. Label the point A, and write its ordered pair next to it. a. Reflect point A over an axis so that its image is in Quadrant III. Label the image B, and write its ordered pair next to it. Which axis did you reflect over? What is the only difference in the ordered pairs of points A and B? b. Reflect point B over an axis so that its image is in Quadrant II. Label the image C, and write its ordered pair next to it. Which axis did you reflect over? What is the only difference in the ordered pairs of points B and C? How does the ordered pair of point C relate to the ordered pair of point A? c. Reflect point C over an axis so that its image is in Quadrant I. Label the image D, and write its ordered pair next to it. Which axis did you reflect over? How does the ordered pair for point D compare to the ordered pair for point C? How does the ordered pair for point D compare to points A and B? 2. If (8, -5) was reflected over both axes, what is the new ordered pair? 3. What is the rule for crossing over both axes? 4. Graph the following coordinates and connect each point: A (-3, 2) B (-6, 2) C (-6,-2) D (-3,-2) Describe the figure shown. 6. Reflect the object across the y-axis P a g e

15 Distance between Points Consider the points ( 4, 0) and (5, 0). What do the ordered pairs have in common, and what does that mean about their location in the coordinate plane? How did we find the distance between two numbers on the number line? Use the same method to find the distance between ( 4, 0) and (5, 0). Consider the line segment with end points (0, 6) and (0, 11). What do the ordered pairs of the end points have in common, and what does that mean about their location in the coordinate plane? Find the distance between between (0, 6) and (0, 11). 15 P a g e

16 (5, 3) and (5, 8) Find the Distance Between the Points (-2, 3) and (-8,3) (-4, -6) and (-1, -6) (5, -10) and (5, -2) POINTS IN SAME QUADRANT: 16 P a g e

17 (8, -6) and (8, 6) Find the Distance Between the (-2, 10) and (5, 10) Points (-3, 8) and (-3, -3) (4, -5) and (-3, -5) POINTS IN DIFFERENT QUADRANTS: 17 P a g e

18 (5, -2) and (-6, -2) Find the Distance Between the Points (-3, 4) and (-3, 8) (9, -4) and (3, -4) (2, -7) and (-2, -7) Think same or different quadrants so what should I do? 18 P a g e

19 Distance Between Points Use the graph below to help solve the following problems. Find the distance between the following points: 1. (4, 5) and (4, -8) 2. (10, -7) and (10, 3) 3. (-9, 6) and (4, 6) 4. (-2, 5) and (-3, 5) Find the distance without using the graph. 1. (9, 5) and (9, -2) 2. (-6, 3) and (-7, 3) 3. (8, ) and (8, 31 2 ) 4. (8 2 3, 4) and (-61 4, 4) 5. Tammy started at home at (4, 5) and then went to the store at (4, 2). She decided to then stop for gas at (4, -3) and then to pick up her printed photos at (4, -5). She then went home. What was Tammy s total distance? 19 P a g e

20 Unit 6 Study Guide Use the coordinate plane on the left for part I and II. For part I, name each graphed point and tell what quadrant the point is in. 1. A (, ) is in Quadrant 2. B (, ) is in Quadrant 3. C (, ) is in Quadrant 4. D (, ) is in Quadrant 5. E (, ) is in Quadrant For part II, answer each question about coordinate graphing. 6. What is the origin? 7. On the graph above right, label the x-axis and the y-axis. 8. For the star, reflect it across the x-axis. Where is it now? (, ) 9. For the lightning bolt, reflect it across the y-axis. Where is it now? (, ) 10. Find the distance between points E and F. 11. Find the distance between points A and J. 12. How can you remember the order of the quadrants? 20 P a g e

21 For part III, answer each question about coordinate graphing without a plane. 13. Starting at (0,0) if you were to go 5 units left and 10 units down what coordinates would you end up at? What quadrant would you be in? 14. Starting at (0,0) if you were to go 3 units right and 1 unit up what coordinates would you end up at? What quadrant would you be in? 15. If two points are in a line and are in the same quadrant, just the absolute values of the coordinates that are not alike. 16. If two points are in a line and are in different quadrants, just the absolute values of the coordinates that are not alike. 17. Find the distance between points (7, -5) and (7, 4). 18. Find the distance between points (1, 10) and (8, 10). 21 P a g e