The Cartesian Coordinate Plane

Transcription

1 The Cartesian Coordinate Plane Air traffic controllers use radar to track tens of thousands of commercial airline flights. Controllers use quadrants to identif the locations, altitudes, and speeds of man different flights Four Quadrants Etending the Coordinate Plane Geometr and Graphs Graphing Geometric Figures Water, Water Everwhere Solving Problems with Multiple Representations Ever Graph Tells a Stor! Interpreting Graphs

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3 Four Quadrants Etending the Coordinate Plane Learning Goals In this lesson, ou will: Etend the coordinate sstem to four quadrants. Name points on the plane. Graph ordered pairs on the Cartesian Coordinate Plane. Calculate the distance between points on the coordinate plane which are on the same vertical or horizontal line. Ke Terms -ais -ais origin quadrant ordered pair Cartesian Coordinate Plane You ve worked with coordinate planes before, but ou ma not know how the were invented. As the stor goes, René Descartes (pronounced da-kart), a French mathematician and philosopher, was having trouble falling asleep one night. While tring to fall asleep, he looked up at the tiled ceiling and spotted a fl. His mind began to wander and a question popped in his head: Could he describe the path of the fl without tracing the actual path? From that question came the revolutionar invention of the coordinate sstem an invention which made it possible to link algebra and geometr. Where have ou seen eamples of coordinate planes? How do coordinate grids help ou identif the locations of objects? 11.1 Etending the Coordinate Plane 719

4 Problem 1 Epanding the Coordinate Sstem In earlier lessons, ou graphed points on a coordinate plane where both the - and -coordinates were zero or positive numbers Now let s include negative numbers in the coordinate plane. 1. To begin this process, first draw a horizontal line segment across the width of the grid that splits the grid in half. Draw arrowheads at the ends of our line segment. Label the line. This horizontal line on the coordinate plane is called the -ais. Perpendicular means that two lines intersect each other at a right angle.. Net, draw a line segment perpendicular to our first line segment from the top of the grid that splits the grid in half so that the line segments intersect. Label this line. This vertical line on the coordinate plane is called the -ais. 3. Label the point of intersection with 0. This point where the -ais and -ais intersect on the coordinate plane is known as the origin. Then, using an interval of 1, label the grid lines to the right and above 0 with positive numbers in numerical order. Finall, label the grid lines to the left and below 0 with negative numbers in numerical order. 70 Chapter 11 The Cartesian Coordinate Plane

5 . How man regions are created when the coordinate plane is divided b the perpendicular lines? These regions on the coordinate plane are called quadrants. The are numbered with Roman numerals from one to four (I, II, III, IV) starting in the upper right-hand quadrant and moving counterclockwise. 5. Label each of the quadrants on our coordinate plane. You can plot points on the coordinate plane using an ordered pair. An ordered pair is a pair of numbers which can be represented as (, ) that indicate the position of a point on the coordinate plane. For eample, the ordered pair for the origin is (0,0).. Plot a point on the coordinate plane anwhere in the first quadrant, and label the point with its ordered pair. 7. Plot a point on the coordinate plane anwhere in the second quadrant, and label the point with its ordered pair.. Plot a point on the coordinate plane anwhere in the third quadrant, and label the point with its ordered pair. 9. Plot a point on the coordinate plane anwhere in the fourth quadrant, and label the point with its ordered pair. Remember, the -coordinate alwas comes first and the -coordinate alwas comes second whether the numbers are positive or negative. 10. Compare the ordered pair for the point ou plotted in the first quadrant with the ordered pairs our classmates plotted. What is similar about the points ou graphed? 11. Compare the ordered pair for the point ou plotted in the second quadrant with the ordered pairs our classmates plotted. What is similar about the points ou graphed? 11.1 Etending the Coordinate Plane 71

6 1. Compare the ordered pair for the point ou plotted in the third quadrant with the ordered pairs our classmates plotted. What is similar about the points ou graphed? 13. Compare the ordered pair for the point ou plotted in the fourth quadrant with the ordered pairs our classmates plotted. What is similar about the points ou graphed? This coordinate plane, called the Cartesian Coordinate Plane, is named after René Descartes. 1. Draw and label - and -aes on the coordinate plane shown. Then, plot and label each point. a. A (, 3) b. B (, 3) c. C (, 3) d. D (0, ) e. E (, 0) f. F ( 3, ) 7 Chapter 11 The Cartesian Coordinate Plane

7 Problem Identifing Points and the Distances between Points On the coordinate plane are points labeled from A to H. D F B A G E H C 1. Identif the ordered pairs associated with each point. A B C D E F G H Use the coordinate plane shown to answer Questions through 11. B A D E F C. Identif the ordered pairs associated with each point. A B C D E F 3. Use the coordinate plane and scale from Question to determine the distance from point A to point B Etending the Coordinate Plane 73

8 . Describe how the ordered pairs for A and B are similar. 5. Use the -coordinates of points A and B to calculate How can an absolute value equation help ou calculate the distance from one point to another on the coordinate plane when the points are on the same horizontal line? 7. Use the grid and scale to calculate the distance from point A to point F.. Describe how the ordered pairs for A and F are similar. 9. Use the -coordinates of points A and F to calculate How can an absolute value equation help ou calculate the distance from one point to another on the coordinate plane when the points are on the same vertical line? 11. Write an absolute value equation and calculate each distance from: a. point B to point C b. point D to point E c. point E to point F Be prepared to share our solutions and methods. 7 Chapter 11 The Cartesian Coordinate Plane

9 Geometr and Graphs Graphing Geometric Figures Learning Goals In this lesson, ou will: Plot points to form geometric figures. Identif points on the coordinate plane to form geometric figures. Identif geometric figures plotted on the coordinate plane. B 5. Hit! You sank it! These words are classic phrases ou hear with the ver popular game Battleship. To begin the game, each plaer places their fleet of ships on a grid that has letters and numbers. Then each plaer takes turn guessing points on their opponent s grid, and that opponent announces if that guessed coordinate hit one of the ships, or if it was a miss. Both plaers record the point guessed on a grid. Originall, this game was invented in the 1900s, but did not get popular until the 1930s. Have ou ever plaed this game before? How is Battleship similar to the coordinate planes ou have been using? How is Battleship and the coordinate planes ou have been using different from each other? 11. Graphing Geometric Figures 75

10 Problem 1 Coordinate Geometr One advantage of the Cartesian Coordinate Plane is that it enables mathematicians to use coordinates to analze geometric figures. 1. The points A (, ) and B (, ) are plotted on the coordinate plane shown. Plot and label points C, D, E, and F so that squares ABCD and ABEF are formed. A (, ) B (, ). Compare our square with our classmates. a. Are all the squares the same? b. How do ou know that the other squares are drawn correctl? 3. On the coordinate plane, the line segment AB is graphed. Plot and label points C and D to form parallelogram ABCD with a height of units. A (, ) B (, ) Remember a parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. 7 Chapter 11 The Cartesian Coordinate Plane

11 . Compare our parallelogram with our classmates. a. Are all the parallelograms the same? b. How do ou know that the other parallelograms are drawn correctl? 5. Draw a trapezoid with a height of 5 units, and a base AB on the coordinate plane shown. A ( 5, ) B (, ) Remember, a trapezoid is a quadrilateral with eactl one pair of parallel sides.. Compare our trapezoid with our classmates. a. Are all the trapezoids the same? b. How do ou know that other trapezoids are drawn correctl? 11. Graphing Geometric Figures 77

12 Problem What Am I? 1. Graph the points on the coordinate plane. Connect the points with line segments, and then identif the geometric figure Chapter 11 The Cartesian Coordinate Plane

13 . Graph the points on the coordinate plane. Connect the points with line segments and identif the resulting figure Graphing Geometric Figures 79

14 Problem 3 Draw Me! 1. Points A and B are endpoints of one diagonal of a rectangle. Determine the locations of vertices C and D of the rectangle, plot them, and then draw the rectangle. A B. Plot and identif five points that will create the vertices of a pentagon. Name the pentagon PQRST. Be prepared to share our solutions and methods. 730 Chapter 11 The Cartesian Coordinate Plane

15 Water, Water Everwhere Solving Problems with Multiple Representations Learning Goal In this lesson, ou will: Analze and solve problems with multiple representations. Have ou ever heard of the saing sink or swim? Well, for some competitive swimmers, sinking is the name of the game. Actuall, the technical term is free diving. Free diving involves a person swimming into the ocean without the use of breathing devices. So, how deep do ou think the most successful free diver has swum? Currentl, William Trubridge of New Zealand holds the record for free diving. He has free dove almost feet into the ocean! And don t forget, that dive was without an breathing devices! So, how long do ou think it took him to reach that depth? How long do ou think he can hold his breath? How long can ou hold our breath? 11.3 Solving Problems with Multiple Representations 731

16 Problem 1 The Diver A free diver is a person who dives into the ocean without the use of an breathing device like scuba equipment. William Trubridge holds the record for free diving. Suppose Trubridge dives at the rate of 1.37 feet per second. 1. How deep will he be after: a. 10 seconds? b. 5 seconds? c. 1 minute? d. 1 1 minutes? When a diver dives, he or she is going below sea level.. What quantit or quantities are changing in this situation? 3. Define variables for the independent and the dependent quantities.. Write an equation for this problem situation. Since he is diving below the surface of the ocean, the distances he dives will be defined as negative numbers to represent below the surface. 73 Chapter 11 The Cartesian Coordinate Plane

17 5. Complete the table using our answers from Question 1. Time (seconds) Depth (feet). Graph the equation using the values from the table. The convention for graphing relationships between variables is to graph the independent variable on the horizontal ais, or -ais, and the dependent variable on the vertical ais, or -ais Time (seconds) Remember what conventions are? Conventions are rules that are developed to maintain an order. Depth (feet) Solving Problems with Multiple Representations 733

18 7. At what depth was William Trubridge at 3 minutes and 30 seconds into his world-record breaking dive? a. Use our graph to determine his depth. b. Use our equation to determine his depth.. Compare our answers from our graph and from our equation. Which answer is more accurate? Wh? 9. Eplain wh the points should be connected in this graph. For what value of time should the graph begin and end? 73 Chapter 11 The Cartesian Coordinate Plane

19 Problem In the Pool 1. Create a table of values for the points on this graph. Suppose the -ais of this graph represents the water level of a pool (in inches) and the -ais represents time (in hours). The origin represents 3:00 pm, and the preferred water level.. Describe the meaning of each point: a. the first point on the left b. the second point on the left c. the last point on the right 11.3 Solving Problems with Multiple Representations 735

20 3. At what rate did the water go into the pool? Eplain our reasoning.. Describe a situation that could match the graph. 5. Define variables for the quantities that are changing, and write an equation for this situation.. Would it make sense to connect the points on the graph? If so, connect the points. 7. What would happen to the graph if the water did not go into the pool at the same rate throughout? 73 Chapter 11 The Cartesian Coordinate Plane

21 Problem 3 No Place Like Home Let s consider another graph. The -ais of this graph represents time in minutes from 1:00 pm, and the -ais represents our distance from home in blocks. A point at the origin represents ou being home at 1:00 pm. 1. Describe our distance from home over the time from 9 minutes before 1:00 pm to 1:00 pm.. Describe our distance from home over the time from 1:00 pm to minutes after 1:00 pm Solving Problems with Multiple Representations 737

22 Let s consider another graph comparing distance from home with time in minutes. Again, a point at the origin represents ou begin at home at 1:00 pm. 3. Describe our distance from home over the time from 9 minutes before 1:00 pm to 1:00 pm. Be prepared to share our solutions and methods. 73 Chapter 11 The Cartesian Coordinate Plane

23 Ever Graph Tells a Stor! Interpreting Graphs Learning Goals In this lesson, ou will: Interpret information about a situation from a graphical representation. Identif the graphs of situations. You can read about business, but ou also read graphs about financial markets. Graphs are a part of almost ever business section of newspapers, or financial tabs on news web sites. Graphs will almost alwas displa the Dow Jones, Nasdaq, and Standard and Poors 500 (also known as the S & P 500). These three names represent the financial markets that are publicall traded ever weekda in the United States. And as a part of ever weekda trading session, graphs are used to show the trends of trading during the da. News of the U.S econom, press releases from companies, and unemploment reports can quickl change the trend of trading on a dail basis and graphs can quickl capture these changes in a visual wa. What other things use graphs to depict trends? Can graphs be used to show trends in data for months and ears too? 11. Interpreting Graphs 739

24 Problem 1 It s Not a Tall Tale! 1. Write a few short sentences to describe each graph. a. The Water Level in the Bathtub b. Mone in Your Bank Account Time (minutes) Amount of Water (gallons) Amount of Mone (dollars) Time (weeks) 70 Chapter 11 The Cartesian Coordinate Plane

25 c. Running a Race d. Jogging for Eercise Time Speed Distance from Home Time 11. Interpreting Graphs 71

26 e. Create a graph. Be sure to label the aes, the intervals, and name our graph. Then, ask our partner to tell the relationship it shows. Did our partner tell the stor ou had in mind? Problem Hot Air Nadja is coordinating a student council banquet. She asks Matthew to blow up balloons for the event. The graphs shown represent his efforts. Graph 1 Graph Volume of Air in the Balloon (cubic feet) Time (seconds) Volume of Air in the Balloon (cubic feet) Time (seconds) 7 Chapter 11 The Cartesian Coordinate Plane

27 Graph 3 Graph Volume of Air in the Balloon (cubic feet) Volume of Air in the Balloon (cubic feet) Time (seconds) Time (seconds) 1. Analze each graph shown, and then answer each question. a. What quantit is represented on the -ais in each graph? b. What quantit is represented on the -ais in each graph? c. What quantit or quantities change in each graph? d. What quantit or quantities depends on the other quantit? 11. Interpreting Graphs 73

28 . Match the descriptions with the appropriate graph. a. Matthew blows air into a balloon at a stead rate, then ties it off when it is full. b. Matthew blows air into a balloon, and then the balloon pops! c. Matthew blows air into a balloon and then lets the air out. d. Matthew blows air into a balloon slowl. As the balloon stretches out, it becomes easier for him to blow into the balloon, and he can blow more air into the balloon. He then ties off the balloon when it is full. Problem 3 Match and Sort In this activit, ou will match a specific graph to a scenario. Follow the steps given. 1. Cut out each graph.. Tape each graph in the bo with the appropriate scenario. 3. Label the aes with the appropriate quantities.. Cut out the scenarios, and sort them into similar groups. 7 Chapter 11 The Cartesian Coordinate Plane

29 A. B. C. D. E. F. 11. Interpreting Graphs 75

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31 G. H. I. J. 11. Interpreting Graphs 77

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33 1. You bu T-shirts to sell for our school. There is a $5 design charge for each T-shirt. What is the total cost for different numbers of T-shirts?. A bus leaves school at the end of the da and stops to drops off its first passenger. 11. Interpreting Graphs 79

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35 3. You have Fig Newtons for our class part. How man Fig Newtons will each classmate receive (ou don t know how man classmates will show up)?. You are drinking our milk through a straw, and then the carton spills over. 11. Interpreting Graphs 751

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37 5. Your telephone calling card charges $0.0 for the first minute of calls and $0.0 for each additional minute of calls.. The video stores charges $3.00 for DVD rentals. How man DVDs can ou rent for different amounts of mone? 11. Interpreting Graphs 753

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39 7. You record the temperature for each hour on Februar, There is a record of our growth since ou were born. 11. Interpreting Graphs 755

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41 9. On Monda, the rain fell at a stead rate. Then, it let up for a few hours before a sudden downpour. Finall it let up. 10. You toss a basketball in the air. 11. Interpreting Graphs 757

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43 5. How did ou sort our graphs? Did our partner sort his or her graphs in the same wa? Be prepared to share our solutions and methods. 11. Interpreting Graphs 759

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45 Chapter 11 Summar Ke Terms - ais (11.1) - ais (11.1) origin (11.1) quadrant (11.1) ordered pair (11.1) Cartesian Coordinate Plane (11.1) Cartesian Coordinate Plane The Cartesian Coordinate Plane is formed b two perpendicular number lines that intersect at the zeros, or the origin. The intersecting number lines divide the plane into four regions, called quadrants. So, tai drivers in London have a larger than normal part of the brain that controls memor and spatial navigation, probabl because the are alwas learning new routes. I wonder if taking a different route to m net class will make m brain grow... Eample Point A at (, 3) has been plotted. It is in Quadrant II. A (, 3) Chapter 11 Summar 71

46 Calculating the Distance Between Points The distance between two points on a coordinate plane can be calculated b using the coordinates of the two points. Eample Points A and B are labeled on the coordinate plane. A B Identif the ordered pair associated with each point. Point A (, 5) Point B (3, 5) Write an absolute value equation to calculate the distance between Point A and Point B Chapter 11 The Cartesian Coordinate Plane

47 Graphing Geometric Figures One advantage of the Cartesian Coordinate Plane is that it enables mathematicians to use coordinates to analze geometric figures. Eample On the coordinate plane, the line segment AB is graphed. Plot and label points C and D to form parallelogram ABCD with a height of units. D C A B Chapter 11 Summar 73

48 Analzing and Solving Problems with Multiple Representations Problem situations can be represented in multiple was in order to analze the problem. Eample An airplane is taking off and climbing at a constant rate of 1000 feet per minute. Complete the table to show the plane s altitude, a, after t minutes. Time (minutes) Altitude (feet) Write an equation for this situation. a t Graph the equation using the values from the table. 10,000 Plane Altitude During Take-Off 9,000,000 Altitude (feet) 7,000,000 5,000,000 3,000,000 1, Time (minutes) 7 Chapter 11 The Cartesian Coordinate Plane

49 Describing a Graph in Your Own Words You can observe how a line or set of points in a graph move (up, down, horizontal, steep, gradual) to describe the relationships between the quantities represented on the - and - aes. You can describe the graph as increasing or decreasing quickl or slowl, or as remaining constant. Eamples Hours of Stud Das This graph represented the number of hours Clair studied in the das surrounding her mid-semester eam. Prior to the eam, there was a peak in the number of hours she studied. Then, after she took the eam, the amount of studing leveled off. Distance (miles) Time (hours) A man drove quickl to work because he was running late. He staed at work all da and then drove home at a stead pace. Chapter 11 Summar 75

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