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The Adventures of Bob: Analyzing and Graphing Linear Equations

The Adventures of Bob: Analyzing and Graphing Linear Equations. EDTEC 544—Cynthia Jackson and Sonya Rea. Help Bob through his challenges in medieval times, space, the rainforest and the financial jungle by graphing and analyzing linear equations.

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The Adventures of Bob: Analyzing and Graphing Linear Equations

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  1. The Adventures of Bob: Analyzing and Graphing Linear Equations

    EDTEC 544—Cynthia Jackson and Sonya Rea Help Bob through his challenges in medieval times, space, the rainforest and the financial jungle by graphing and analyzing linear equations.
  2. This is Bob. He is about to find a portal in time and have the adventure of a lifetime. He will need your help. Are you ready? Click to Open Bob’s Notebook Each time you master a skill you will see this space portal. As you go through each level of the game you will learn about each skill you will need to help Bob. If you need help, click on Bob’s notebook to look at the notes for that level.
  3. Level One—Medieval Kingdom Bob skates his way into medieval times. You can help him become king by collecting swords, shields, and slaying a dragon. Good luck!
  4. Bob’s Notebook Level One—Medieval Kingdom Information You Will Need to Help Bob in Level One Slope: The slope of a line measures the steepness of the line. Most of you are probably familiar with associating slope with "rise over run".  Rise means how many units you move up or down from point to point.  On the graph that would be a change in the y values.Run means how far left or right you move from point to point.  On the graph, that would mean a change of x values. Here is a visual to help you with this definition: Positive slope: Note that when a line has a positive slope it goes up left to right. Negative slope: Note that when a line has a negative slope it goes down left to right.
  5. Bob’s Notebook Level One—Medieval Kingdom More Information You Will Need to Help Bob in Level One y-intercept: The y-intercept of a line is the point at which a line crosses the y-axis. Here is a visual to help you with this concept: How To find the y-intercept: Step 1: Put your finger on the origin (0,0). Step 2: Now find your line. Is it above your finger or below your finger? If it is above your number will be positive, if it is below your number will be negative. Step 3: Now move your finger up or down until it touches the point where your line crosses the y-axis. Where did you end up? This is the y-intercept. I am ready to help Bob!
  6. “If the line is going down it is negative!” Bob needs your help. Identify whether the slope of the line is positive or negative. If you do it correctly Bob will skate up or down the ramp. y Click to Open Bob’s Notebook x Positive Negative Next Problem
  7. “If the line is going up it is positive!” Bob needs your help. Identify whether the slope of the line is positive or negative. If you do it correctly Bob will skate up the ramp. Click to Open Bob’s Notebook Positive Negative Next Problem
  8. Bob needs your help. Identify whether the slope of the line is positive or negative. If you do it correctly Bob will skate down the ramp. Click on the space portal after Bob skates. “Try again! Click to Open Bob’s Notebook Positive Negative
  9. You have helped Bob so well that he is skating better than ever! He has found a space portal. Click on Bob to help him skate into the time portal and back into medieval times.
  10. “Find the x- axis!” Bob needs your help. Identify the coordinates of the y-intercept of the given line. If you do it correctly Bob will collect swords and shields that will help him slay a dragon. “Find the origin!” “Find the y-axis!” y Click to Open Bob’s Notebook x (0, 0) (-1, -1) (-2, 2) (-3, -3) Next Problem
  11. “This is the x-intercept!” Bob needs your help. Identify the coordinates of the y-intercept of the given line. If you do it correctly Bob will collect swords and shields that will help him slay a dragon. “Look at where the line crosses the y axis!” “x is written before y!” Click to Open Bob’s Notebook (-2, 2) (2, 0) (-2, 0) (0, 2) Next Problem
  12. “This is the x-intercept!” Bob needs your help. Identify the coordinates of the y-intercept of the given line. If you do it correctly Bob will collect swords and shields that will help him slay a dragon. “Check my notebook!” “Check my Notebook!” Click to Open Bob’s Notebook (-2, 0) (0, -2) (-4, 0) (0,- 4) Next Problem
  13. “Look at where the line crosses the y-axis!” Good job! Now Bob needs a shield. Identify the coordinates of the y-intercept of the given line. Help Bob to collect a shield so he can slay the dragon. “x is always written first!” Click to Open Bob’s Notebook “This is the x-intercept!” (-3, 0) (2, -2) (2, 0) (0, -3 ) Next Problem
  14. Bob needs your help to slay the dragon and become the king. Find the equation of the line based on the y-intercept and the slope. Click on me y = - x + 2 y = x + 2 y = 2x + 2 y = - 1x + 3 Click to Open Bob’s Notebook Hint: the y-intercept is 2. Hint: the slope is negative. Hint: the slope is negative. Click here after you help Bob slay the dragon
  15. Level One Assessment Help Bob pass this test with 80% accuracy so he can move on to Level Two. In Coordinate Plane 1, is the line positive or negative? 1 Positive Negative 2. What are the coordinates of the y-intercept of line A ? A (-1, 0) (1, 0) (0, -2) (-2, 0) Next Problem
  16. What are the coordinates of the y-intercept of line A ? (0, 3) (0, 4) (4, 0) (3, 0) 4. Find the equation of the line based on the y-intercept and the slope. y= -3x - 3 y= 2x + 3 y= x 3 -3y= x - 3 Next Problem
  17. 5. Find the equation of the line based on the y-intercept and the slope. - y = x - 4 y = -2x + 2 y = x + 4 y = 4x - 4 Check Your Results
  18. You answered … 4 Out of five questions correctly. This gives you a percentage of… 80% Congratulations on Passing Level One!

  19. Congratulations! You have mastered Level One. Click on Bob to travel into the time portal.

  20. Bob’s Notebook Level Two—Space Now that Bob is King he wants to rule the universe. He needs your help to launch his spaceship and cross the evil empire of Yint. Information You Will Need to Help Bob in Level Two The x-coordinate is always the first number in the pair and the y-coordinate is always the second number in the pair. The point below shows the following ordered pair:(4, -2). The point is represented by a black dot. Therefore (x,y) = (4, -2). To identify the ordered pairs or to locate points, you start at the origin and count the units along each axis. This point show 4 to the right, and 2 down.
  21. Bob’s Notebook Level Two—Space Steps for Slope Formula Formula for Slope: In order to use the slope formula, you need to be given 2 points. Example, (4,3) and (5,-2) Steps: Label your points with the x and y labels Substitute your labeled values into the formula Simplify both the numerator and the denominator EX: Find the slope of a line with the points (4,3) and (5,-2) Step 1-Label your points with the x and y labels (4,3) and (5,-2) x1 y1 x2 y2 Step 2-Substitute your labeled values into the formula Slope= -5
  22. Bob’s Notebook Level Two—Space The slope intercept form of a line is y =mx + b. Where m is the slope and b is the y-intercept. The y-intercept is the point where the line crosses the y-axis. In the example to the right, the line crosses the y-axis at point (0,3). Therefore b = 3. The slope is 2. The slope intercept equation isy = 2x + 3.
  23. Bob’s Notebook Level Two—Space Steps for finding coordinates of y-intercept based on 2 points. Find the slope of the line based upon the two points Using the slope, count coordinates and fill in points on the graph Label the coordinate that crosses the Y-axis. That is your y-intercept Example:Points (2,-2) and (-2,2) lie on the same line. Find the coordinates of the y-intercept of that line. Step 1-Find the slope of the line Slope= Step 2-Using the slope, count the coordinates So, starting with point (2,-2), you would rise -1 and run +1 Your next coordinate would be (1,-1) and then (0,0). Step 3-Label the coordinates Because your line passes through point (0,0), your y-intercept is (0,0)
  24. Bob’s Notebook Level Two—Space Finding an equation of a line based on 2 points: Find the slope of the line with the 2 given points Using the slope-intercept form of y=mx+b, write your equation. Coordinate Pair (1,-2) and (5,-6) lie on the same line. Give the equation of that line. Step 1-Find the slope of the line Find the y-intercept of the line by graphing Step 3-Use the slope-intercept form y=mx+b y=-1x+(-1), so your equation is… y=-x-1 I am ready to help Bob!
  25. Check your sign. Use the Slope Formula to help Bob find the right slope to launch his spaceship. (4, 3) The y-coordinates are the numerator. rise run = Divide. Don’t subtract. Click to Open Bob’s Notebook x (5, -2) 5 1/3 Way to go! One more right answer and I can launch the spaceship. -5 4 y Next Problem
  26. Check your sign. Use the Point Slope Formula to help Bob find the right slope to launch his spaceship. Try again. (4, 1) Divide. Don’t subtract. -1 1/2 Click to Open Bob’s Notebook 0 1 (1, -2) Next Problem
  27. You need to help Bob cross the virtual space fence of Yint, the Evil Empire. (-2, 2) Points ( 2, -2) and ( -2, 2) lie on the same line. Find the coordinates of the y-intercept of that line. YINT x “ One more correct answer and we can cross the space fence of YINT!.” (2, -2) (-2, 2 Click to Open Bob’s Notebook (2, -2) (-1, 0) (0, 0) “Check Bob’s Notebook.” y “ This is one of the coordinates.” “This is one of the coordinates.” Next Problem
  28. You need to help Bob cross the virtual space fence of Yint, the Evil Empire. Points ( 1, -2) and ( -3, 2) lie on the same line. Find the coordinates of the y-intercept of that line. YINT (-4, 4) (0, -1) Click to Open Bob’s Notebook (-1, 0) (0, 1) “Check your sign!” “Keep working. Divide these numbers to find the slope!” “x always goes before y!” Next Problem
  29. You and Bob have mastered slope and y-intercept. Using the two coordinate pairs find the equation of the line so that Bob can become the ultimate Space Tyrant, ruler of all four quadrants of space. Coordinate pair (1, - 2) and (5,- 6) lie on the same line. Give the equation of the line. y = x + 1 y = -x - 1 Click to Open Bob’s Notebook y = x - 3 y = 2x -1 “The slope is not 2. Use the point intercept formula!” “The slope is negative!” II am the ultimate Space Tyrant, ruler of all four quadrants! “The y-intercept is negative!”
  30. Congratulations! You have helped Bob become the Ultimate Space Tyrant. Now you must complete the assessment to determine Bob’s preparedness for his next adventure.
  31. A given line contains the points (1,1) and (2,-1). What is the slope of that line? 2. A line contains the points (-2,-2) and (1,-5). What is the y-intercept of that line? 2 -3 (0,1) (0,-1) -2 1/2 (0,-4) (0, -3) 3. A line contains the points (3,0) and (-1,-2). What is the equation of that line? 4. A line connects points (-3,3) and (1,-5). What is the equation of the line? y=1/2 x – 3/2 y= 1/2x + 2 y= 2x-4 y=-2x-3 y= -1/2 x +2 y= -1/2x – 1/2 y= ½ x -3 y= 1/2x -4 Next Problem
  32. 5. The line connecting Bob’s planet with the nearest time portal contains points (-1,-1) and (0,1). Bob needs to know the equation of that line in order to navigate to the time portal. What is the equation of the line? y= 1/2x + 1 y=2x+1 y= -1/2x -1 y= -2x-1 Check Your Results
  33. You answered … 5 Out of five questions correctly. This gives you a percentage of… 100% Congratulations on Passing Level Two!
  34. Congratulations! You have mastered Level Two and helped Bob become the Ultimate Space Tyrant. Click on Bob to travel into the time portal.
  35. Bob’s Notebook Level Three—The Rainforest Now that Bob is the Ultimate Space Tyrant he wants to go back to Earth and find treasure that has been hidden by evil gorilla poachers. He needs your help to make his way through the jungle. Perpendicular Lines As you can see from the picture below: the slopeof these lines are negative reciprocals: (Ex: 2 and - ½ are negative reciprocals) these lines are perpendicular and intersect at 90 degrees Parallel Lines As you can see from the diagram below, these lines have the same slope and are never going to intersect:
  36. Bob’s Notebook Level Three—The Rainforest
  37. Bob’s Notebook Level Three—The Rainforest Steps for finding the equation of a line given a parallel line and a set of points 1. Look at the equations of your two lines: Steps for finding a linear equation in slope-intercept form: Use the formula: y-y1=m(x-x1) x1 and y1 is the point you were given Substitute the point into the formula Simplify by isolating and solving for y What is the equation of line A given the point (2,2) and is parallel to line B y=3x-1? Remember, parallel lines contain the same slope, so your slope is 3 Step 1 -Use the formula: y-y1=m(x-x1) Step 2 -Substitute the given point into the slope-intercept form y-2=3(x-2) Step 3 -Simplify by solving for y y-2=3(x-2) y-2=3x-6 +2 +2 y =3x-4 ANSWER: y=3x-4 I am ready to help Bob!
  38. Bob needs to build equivalent poles in order to make his way over the snake pit. Line A is y = 2x + 4. Line B is 3y = 6x + 12. What do you know about these lines? The lines are dependent The lines intersect The lines are perpendicular Click to Open Bob’s Notebook The lines have different slopes “These lines can’t intersect because they have the same slope.” “Try dividing both sides of line B by 3.” “Perpendicular lines have a reciprocal slope and it differs in sign.” Next Problem
  39. Line A is -2y = 6x - 4. Line B is y = -3x + 2. What do you know about these lines? The lines intersect at (0,0) The lines have a positive slope The lines are dependent The lines are perpendicular Click to Open Bob’s Notebook “These lines can’t intersect because they have the same slope.” Thanks for helping me! Click on me so I can make my way across the snake pit.” “These lines have a negative slope.” “Perpendicular lines have a reciprocal slope and it differs in sign.” Next Problem
  40. “Great job! One more right answer and we can unlock the code and cross the bridge. Click Next Problem.” Bob needs to unlock the code to the hidden bridge so he can cross the gorge. Can you help him? Click to Open Bob’s Notebook origins y-intercepts slopes x-intercepts “Only one of these lines goes through the origin.” “Look at where each line on the graph crosses the y-axis.” These two lines are parallel. This means that their ________ are equal. “Look at where each line on the graph crosses the x-axis.” Next Problem
  41. Bob needs to unlock the code to the hidden bridge so he can cross the gorge. Can you help him? Line A is y = 2x – 3. Line B is y = 2x + 5. What do you know about these lines? These lines are perpendicular They will intersect Click to Open Bob’s Notebook The lines are parallel They are the same line “Perpendicular lines have negative reciprocals.” When the bridge appears click on Bob so he can cross over. “These lines will never intersect.” “If they were the same line, besides having the same slope they would have the same y-intercept.” Next Problem
  42. “Be careful! They’re shooting at us. Check Bob’s notebook.” Help Bob rescue the hidden treasure. It is being guarded by endangered gorillas. Bob needs to be careful not to hurt them. Answer carefully. Don’t alert the poachers to Bob! Line A contains the point (2, 2) and is parallel to line B y = 3x - 1. What is the equation of line A? “Be careful! They’re shooting at us. The line you chose has the same y-intercept not the same slope.” y= x - 2 Click to Open Bob’s Notebook 3y= 3x + 4 y=-3x + 1 y = 3x + 4 “Be careful! They’re shooting at us. The line you chose has a negative slope. “ Click on Bob when if he is able to retrieve the treasure
  43. Level Three Assessment Help Bob pass this test with 80% accuracy so he can move on to Level Four. The lines intersect at (0,0) The lines are parallel Line A is y = -x + 3 Line B is -4y = 4x – 12. What can you say about these lines? These are dependent lines They are perpendicular lines Next Problem
  44. 2. These two lines are parallel. This means that they have ______ that are equal. y-intercepts slopes origins x-intercepts Next Problem
  45. 3. Line A is y = -2x + 3 Line B is y = -2x – 5. What can you say about these lines? These are perpendicular lines They are the same line These are parallel lines The lines will intersect at the y-intersect 4. Line A contains the point (1, 2) and is parallel to line B y = 4x + 3. What is the equation of line A? y = 4x - 2 y = 2x + 1 y = x + 2 y = 4x Next Problem
  46. 5. Line A contains the point (1, 6) and is parallel to line B y = x - 2. What is the equation of line A? y = x + 6 y = 6x + 1 y = x + 2 y = x + 5 Check Your Results
  47. You answered … 5 Out of five questions correctly. This gives you a percentage of… 100% Congratulations on Passing Level Three!
  48. Congratulations! You have helped Bob find the treasure and master Level Two. Click on him so he can run through the time portal.
  49. Bob’s Notebook Level Three—Financial World . Perpendicular Lines: Two lines are considered perpendicular to each other if they intersect to form right angles. Equation of line A is Line B is perpendicular to line A. What is the slope? Remember, perpendicular lines have slopes that are negative reciprocals to each other. Therefore, the slope of Line B would be -2, because the slope of line A is .
  50. - Bob’s Notebook Level Three—The Financial World Steps to finding an equation of a perpendicular line given a point . Find the slope of the given line 2. Find the reciprocal of the slope of the given line 3. Use the slope-intercept form equation y-y1=m(x-x1) to find the new line Example Line A contains (-2,4) and is perpendicular to line B y=2x+1. What is the equation of line A? Step 1 - Find the slope of the given line. y=2x+1, so the slope of this line is 2. Step 2-Find the reciprocal of the slope of the given line. If the slope of our given line is 2, then our reciprocal is - Step 3-Use the slope-intercept form equation. y-y1=m(x-x1) y-4=-1/2(x-(-2)) +4 +4 ______________ ANSWER:
  51. Bob needs money to compete in the financial world. Help him answer correctly and add money to his bank account. “Perpendicular lines form right angles.” These two lines are perpendicular. That means that they intersect to form ________ angles. “Click on my notebook for help.” Click to Open Bob’s Notebook “Click on my notebook for help.” negative right positive obtuse Next Problem
  52. Now that Bob has money, he plans to invest it in real estate. Help Bob acquire property by finding the slope of the line which is perpendicular to the given line. If you select the correct answer, he makes a wise purchase. If not,… Bob needs to know the slope of a line which is perpendicular to y=-1/3x + 1. The slope of that line would be.. Click to Open Bob’s Notebook -1/3 1/3 3 -3
  53. With your help, Bob can conquer the real estate world. You will be given a point and the equation of a line perpendicular to the one Bob needs. Give the equation of the line Bob needs, so he can get from home to work and make the deal of a lifetime! “”Look in my notebook for help.” The line from home to work contains point (2,1) and is perpendicular to y=-2x+3. What is the equation of the line? Click to Open Bob’s Notebook “”Look in my notebook for help.” “”Check your sign.” y=-2x+0 y=-1/2x+3 y=1/2x+0 y=2x+3
  54. Level Four Assessment Help Bob pass this test with 80% accuracy so he can go to his next adventure in Algebra. 1. Perpendicular lines create what angle? A right angle A straight angle A left angle A 45 degree angle 2. If a given line has a slope of 2, the line perpendicular to that one will have a slope of 2 -2 1/2 -1/2
  55. You are almost finished! Just a few more questions remain. 3. If a given line contains the point (3,1) and is perpendicular to line y=4x-2, what is the equation of the given line? y=1/4x - 1/4 y=-4x - 7/4 y=-1/4x + 7/4 y= -1/4x – 7/4 4. Bob’s line is perpendicular to Hank’s line. Bob’s line has the equation y=-4x+2. Hank’s line contains the point (0,0). What is the equation of Hank’s line? y = ¼ x y = ¼ x + 2 y= -1/4 x y= -1/4 x + 2
  56. 5. Bob and Sally are planning to have a road built to connect Bob’s street with Sally’s home. Because Bob likes right angles, the new road will, of course, be perpendicular to Bob’s street. Bob’s road follows the equation y=-1x +1. Sally’s home is at point (5,5). What will be the equation of the line for the new road? y=x y=1/4x y=x - 1 y= -x - 4 Check Your Results
  57. You answered … 4 Out of five questions correctly. This gives you a percentage of… 80% Congratulations on Passing Level Four!
  58. Congratulations! You and Bob have mastered Graphing and Analyzing Line Equations! Join Bob on his next adventure in Algebra: Polynomials. Click on Bob to send him and Sally through the Time Portal.
  59. You answered … 3 Out of five questions correctly. This gives you a percentage of… 60% Please return to Bob’s Notebook and review Bob’s notes
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