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Lesson Plan -- Graphing Chapter Resources - Lesson 5-12 Ordered Pairs - Lesson 5-13 Plot Points in the Coordinate Plane - Lesson 5-15 Graph Linear Equations - Lesson 5-12 Ordered Pairs Answers - Lesson 5-13 Plot Points in the Coordinate Plane Answers - Lesson 5-15 Graph Linear Equations Answers 1 Name ——————————————————————— LESSON 5-12 7ZLMZML8IQZ[ California Standards Words to Remember Gr. 3 AF 1.0: Students select appropriate symbols, operations, and properties to represent, describe, simplify, and solve simple number relationships. Ordered pair: A pair of numbers represented by x and y and written in the form (x, y) Input: The values of x or of the first numbers in the ordered pairs Output: The values of y or of the second numbers in the ordered pairs Equation in two variables: A rule relating input and output values Gr. 5 SDAP 1.5: Know how to write ordered pairs correctly; for example (x, y). Also included: Gr. 3 AF 2.1, Gr. 3/4/5/6/7 MR 1.1 Date ———————————— Getting Started You can use a rule to describe how the output values are related to the input values. The ordered pairs from the rule can be displayed in an input-output table. E@)584- Making an Input-Output Table Complete the input-output table for the rule y 5 x 2 1. Substitute each input value for x in the rule. Then simplify to find the output values y. Input, x Output, y T:A <0 1; 23 21 23 2 1 5 24 21 2 1 5 22 0 5 0 2 1 5 21 52154 Make a table. 1. Complete the input-output table for the rule y 5 2x. Input, x 22 0 1 4 Output, y 2. Complete the input-output table for the rule y 5 x 1 5. Input, x Output, y 48 Math Intervention Book 5 Algebraic Thinking 23 21 2 4 Copyright © by McDougal Littell, a division of Houghton Mifflin Company. Solution Name ——————————————————————— E@)584- Date ———————————— Writing Ordered Pairs from a Table Write the values in the table as ordered pairs. 3FNFNCFS In an ordered pair, you write the x-values first and the y-values second. x 23 21 0 2 5 y 1 3 4 6 9 Solution (23, 1), (21, 3), (0, 4), (2, 6), (5, 9) T:A <0 1; Write ordered pairs. 3. Write the values in the table as ordered pairs. Copyright © by McDougal Littell, a division of Houghton Mifflin Company. -PPLGPSB 1BUUFSO How can the y-values be obtained from the corresponding x-values? x 215 25 0 10 y 23 21 0 2 E@)584- Writing a Rule from an Input-Output Table Write a rule that shows how y relates to x. x 26 23 0 5 y 22 1 4 9 Solution The y-values are 4 more than the corresponding x-values. Therefore, the rule is y 5 x 1 4. T:A <0 1; Write a rule. 4. Write a rule that shows how y relates to x. x 22 21 0 1 2 y 26 23 0 3 6 5. Write a rule that shows how y relates to x. x 24 22 0 2 4 y 27 25 23 21 1 Math Intervention Book 5 Algebraic Thinking 49 Name ——————————————————————— Date ———————————— Summarize Making an Input-Output Table Using a rule describing a relationship between two values x and y, substitute input values for x to find the corresponding output values for y. Writing Ordered Pairs from a Table The x-values or input values are the first number in each ordered pair. The y-values or output values are the second. Writing a Rule from an Input-Output Table Look for a relationship between the x-values and the y-values. Find the pattern that determines how each value of y can be obtained from the corresponding value of x. Practice Complete the input-output table using the given rule. 1. y 5 4x Input, x 1 2 5 7 2. The number of sailboats x is 20 less than the number of speed boats y. Sailboats, x 0 2 3 5 Speed boats, y Write the values in the table as ordered pairs. 3. x 26 22 1 4 y 28 24 21 2 ____________________________________________________________________________ 4. x 23 22 21 0 y 4 5 6 7 ____________________________________________________________________________ 50 Math Intervention Book 5 Algebraic Thinking Copyright © by McDougal Littell, a division of Houghton Mifflin Company. Output, y Name ——————————————————————— Date ———————————— Write a rule that shows how y relates to x. 5. x 21 0 1 2 y 4 0 24 28 _________________________________________________________________________________ 6. x 21 0 1 2 y 22 21 0 1 _________________________________________________________________________________ 7. x 23 0 6 12 y 21 0 2 4 _________________________________________________________________________________ 8. Explain how to identify the input values in the ordered pairs (22, 21), (7, 6), and (10, 9). _________________________________________________________________________________ _________________________________________________________________________________ 9. The distance d that Maria travels in h hours can be described by the rule d 5 50h. Write three ordered pairs that make the rule true. Copyright © by McDougal Littell, a division of Houghton Mifflin Company. _________________________________________________________________________________ ,1,A7=/ -<1<' 10. Fill in the missing words. In the ordered pair (x, y), the value of ________ is the input and the value of ________ is the output. 11. Describe a process. How would you write a rule that shows how y relates to x ? x 22 21 0 1 y 28 24 0 4 __________________________________________________________________________ __________________________________________________________________________ 12. Explain your reasoning. Can the relationship between the number of weeks that have passed and the number of years that have passed be described by a rule relating two variables w and y ? Explain. __________________________________________________________________________ __________________________________________________________________________ Math Intervention Book 5 Algebraic Thinking 51 Name ——————————————————————— LESSON 5-13 Date ———————————— 8TW\8WQV\[QV\PM +WWZLQVI\M8TIVM California Standards Words to Remember Gr. 4 MG 2.0: Students use two-dimensional coordinate grids to represent points and graph lines and simple figures. y 4 x-axis: The horizontal axis 3 y-axis 2 y-axis: The vertical axis Origin: The point (0, 0), where the horizontal and vertical axes intersect Gr. 5 AF 1.4: Identify and graph ordered pairs in the four quadrants of the coordinate plane. 1 x-axis 24 23 (0, 0) origin 1 2 3 4 x 22 23 24 Getting Started In Lesson 5-12 you learned how to write ordered pairs from an input-output table. In this lesson you will graph ordered pairs in a coordinate plane. Identifying Points in a Coordinate Plane Name the two points on the graph with the coordinates (24, 0) and (1, 23). y B 24 23 22 A point with y-coordinate 0 lies on the x-axis. A point with x-coordinate 0 lies on the y-axis. 3 2 A 1 C D 3FNFNCFS 4 O G 1 2 3 4 x 22 F 23 24 E Solution Starting at the origin, if you move left 4 units, then up 0 units, you get to C. Point C has coordinates (24, 0). Starting at the origin, if you move right 1 unit, then down 3 units, you get to F. Point F has coordinates (1, 23). 52 Math Intervention Book 5 Algebraic Thinking y 4 3 2 1 C (24, 0) 24 23 22 O 22 23 24 2 3 4 x F (1, 23) Copyright © by McDougal Littell, a division of Houghton Mifflin Company. E@)584- Name ——————————————————————— T:A <0 1; Date ———————————— Use the graph in Example 1 to name the point with the given coordinates. 1. (4, 0) If you move __________ 4 units, then up units you get to point __________. Point __________ has coordinates (4, 0). 2. (23, 22) units, then _________________ 2 units you get to If you move left point __________. Point __________ has coordinates (23, 22). E@)584- Graphing Points in a Coordinate Plane Graph the points A(25, 2), B(4, 1), and C(0, 23). Solution Copyright © by McDougal Littell, a division of Houghton Mifflin Company. 3FNFNCFS When you graph a point, always start at the origin. Move right or up for positive coordinates. Move left or down for negative coordinates. First move along the horizontal axis. Then move up or down. To graph A move left 5 units. Then move up 2 units. y 4 3 2 A B 1 25 24 23 22 O 1 2 3 x 22 C 23 24 To graph B move right 4 units. Then move up 1 unit. To graph C do not move right or left. Just move down 3 units. T:A <0 1; 3. Graph the points X(22, 24), Y(21, 3), and Z(2, 0). To graph X move left units. Then move ______________ 4 units. y To graph Y move ______________ 1 unit. O x Then move up units. To graph Z just move ______________ units. Do not move ______________ or ______________. Math Intervention Book 5 Algebraic Thinking 53 Name ——————————————————————— Date ———————————— Summarize Identifying Points in a Coordinate Plane (1) To name the coordinates of a point on a coordinate graph, first find the x-coordinate by counting how many units the point is to the left or right of the origin. (2) Then find the y-coordinate by counting how many units the point is above or below the origin. (3) Write the coordinates as an ordered pair with the x-coordinate first and the y-coordinate second. Plotting a Point in a Coordinate Plane (1) To plot a point, start at the origin and move right or left the number of units corresponding to the x-coordinate. Move right if the x-coordinate is positive, move left if it is negative. (2) Then move up or down the number of units corresponding to the y-coordinate. Move up if the y-coordinate is positive, move down if it is negative. (3) Make a dot at this location. 8ZIK\QKM In Exercises 1–7, write the coordinates of the points. 1. Point A is 1 unit right and 6 units down from the origin, so the coordinates of A are ___________ . C B G 2. Point B is 4 units right and 1 unit up from the origin, so the coordinates of B are ___________ . 3. Point C is D units right and 3 units ___________ from the origin, so the coordinates of C are ___________ . 4. Point D is units ___________ and units ___________ from the origin, so the coordinates of D are ___________. 5. Point E is units ___________ from the origin, so the coordinates of D are ___________ . 6. The coordinates of F are ___________ . 7. The coordinates of G are ___________ . 54 Math Intervention Book 5 Algebraic Thinking 28 26 24 O 24 26 28 2 4 E A 6 8 x Copyright © by McDougal Littell, a division of Houghton Mifflin Company. y 8 F 6 4 2 Name ——————————————————————— Date ———————————— Plot the point. y 8. M(26, 2) 9. N(0, 1) 10. P(23, 24) 11. Q(8, 21) 8 6 4 2 28 26 24 12. R(5, 6) 13. S(22, 0) 14. O(0, 0) 15. U(0, 27) 16. V(21, 8) 17. W(2, 26) 18. X(0, 22) 19. Y(3, 1) O 2 4 6 8 x 24 26 28 20. Describe the coordinates of a point that lies on either the x-axis or the y-axis. _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ Copyright © by McDougal Littell, a division of Houghton Mifflin Company. ,1,A7=/ -<1<' 21. Fill in the missing words. The horizontal axis in a coordinate plane is called the ____________ and the vertical axis is called the ____________. 22. Describe a process. How would you explain to your friend how to graph the point (6, 21)? __________________________________________________________________________ __________________________________________________________________________ 23. Explain your reasoning. Fred says the coordinates of point M are (4, 22). Vicki says the coordinates are (22, 4). Who is correct? Explain. M y 4 3 2 1 24 23 22 O 1 2 3 4 x 22 23 24 __________________________________________________________________________ __________________________________________________________________________ Math Intervention Book 5 Algebraic Thinking 55 Name ——————————————————————— LESSON 5-15 /ZIXP4QVMIZ-Y]I\QWV[ California Standards Words to Remember Gr. 4 AF 1.5: Understand that an equation such as y 5 3x 1 5 is a prescription for determining a second number when a first number is given. Gr. 5 AF 1.0: Students use variables in simple expressions, compute the value of the expression for specific values of the variable, and plot and interpret the results. Also included: Gr. 4 Date ———————————— y Graph of a linear equation: All the points in the coordinate plane that are solutions of the equation 2 1 23 The graph is a horizontal, vertical, or diagonal line. O 2 3 x 22 23 Getting Started In Lesson 5-13 you learned how to plot points in a coordinate plane. In this lesson you will learn how you can plot points to graph a line. MG 2.0, Gr. 4 MG 2.1 E@)584- Graphing Linear Equations Using a Table of Values Solution ;\MX Complete the table. 3FNFNCFS x Find y by substituting each value of x in the equation y 5 x 1 3. y –2 y5x13 y 5 –2 1 3 51 (x, y ) (–2, 1) –1 y5x13 y 5 –1 1 3 52 (–1, 2) 0 1 y5x13 y5013 53 y5x13 y5113 54 (0, 3) (1, 4) ;\MX Plot the points. y 4 ;\MX Draw a line through the points. 2 1 24 22 O 1 2 3 4 x 22 23 24 ;\MX Interpret the graph. Every ordered pair that makes y 5 x 1 3 true is a solution of the equation and lies on its graph. Since 13 5 10 1 3, (10, 3) is a solution of y 5 x 1 3 and lies on its graph. 58 Math Intervention Book 5 Algebraic Thinking Copyright © by McDougal Littell, a division of Houghton Mifflin Company. Use the table of values to graph y 5 x 1 3. Interpret the graph by explaining how you know that the point (10, 13) also lies on the graph of the line. Name ——————————————————————— T:A <0 1; Date ———————————— Use the equation y 5 2x 1 1. 1. Fill in the missing information to graph y 5 2x 1 1. ;\MX Complete the table. x –2 0 y 5 2x 1 1 y 5 2x 1 1 11 y52• y ( , ) , ), and ( , y5 5 ;\MX Plot the points ( ( y5 1 y52• 5 (x, y) 1 5 ( , ) ( ), , Copyright © by McDougal Littell, a division of Houghton Mifflin Company. E@)584- ) 1 2 3 y ). 3 2. The point (–8, –15) also lies on the graph of 52+ , 4 2 1 Then draw a line through the points. y 5 2x 1 1 because 1 • 1 24 23 22 O 4 x 22 23 24 . Graphing Horizontal and Vertical Lines Use three ordered pairs to determine whether the graphs of y 5 – 4 and x 5 3 are horizontal or vertical. Solution Choose 3 values for x. y is always 24. Choose 3 values for y. x is always 3. y 5 24 x53 (–2, –4) (0, –4) (1, –4) (3, –4) (3, 0) (3, 3) ANSWER The graph of y 5 24 is a horizontal line. The graph of x 5 3 is a vertical line. T:A <0 1; Use three ordered pairs to determine whether the graph is horizontal or vertical. 3. x 5 22 __________ 4. y 5 1 __________ Math Intervention Book 5 Algebraic Thinking 59 Name ——————————————————————— Date ———————————— Summarize Graphing a Linear Equation To graph a linear equation, make a table of values and plot the points. Graph at least three points to determine the line. Graphing a Vertical or Horizontal Line For a vertical line, x will have the same value for all values of y. For a horizontal line, y will have the same value for all values of x. Vertical line Horizontal line Diagonal line x52 x 5 –0.5 y 5 –10 y 5 12 y5x–5 y 5 –4x 1 1 8ZIK\QKM Complete the table of values to draw the graph. 1. y 5 x 2 1 y –2 y y5x–1 y5 21 –1 ( ) 22 4. y 5 4x 1 2 0 60 2 y 3 4 x y x O y x O ) 3 5 1 Math Intervention Book 5 Algebraic Thinking , 5. y 5 2} x 2 1 y y ( x x O 21 ) 3. y 5 2x 2 5 2 y x , y 0 2 5 ( 2. y 5 x 1 7 x y5 y5 5 , x O y5x–1 y5 2 5 (x, y) 0 25 0 y 5 O x Copyright © by McDougal Littell, a division of Houghton Mifflin Company. x Name ——————————————————————— Date ———————————— Complete the table of values to draw the graph. 6. x 1 y 5 4 x y 21 0 7. 3x 1 y 5 6 x 1 x O y 8. y 5 1.5 1.5 1.5 x x O 22 x 1.5 3 9. x 5 22 y O 2 y x y 1 y y 22 22 O y x Tell whether the graph of the equation is a vertical, horizontal, or diagonal line. Explain your reasoning. 10. y 5 3 _________________________________________________________________________________ 11. x 5 4 _________________________________________________________________________________ Copyright © by McDougal Littell, a division of Houghton Mifflin Company. 12. y 5 2x 2 6 _________________________________________________________________________________ ,1,A7=/ -<1<' 13. Fill in the missing words. The graph of a linear equation is a ___________ , ___________ , or ___________ line. 1 2 14. Describe a process. How would you graph y 5 } x 1 3? __________________________________________________________________________ __________________________________________________________________________ 15. Explain your reasoning. Your friend said that the graph of x 5 2 is a horizontal line. Is she correct? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ Math Intervention Book 5 Algebraic Thinking 61 Answer Key Lesson 5-12, pp. 48–51 Try this: 1. 24, 0, 2, 8 2. 2, 4, 7, 9 3. (215, 23), (25, 21), (0, 0), (10, 2) 4. y 5 3x 5. y 5 x – 3 Practice: 1. 4, 8, 20, 28 2. 20, 22, 23, 25 3. (26, 28), (22, 24), (1, 21), (4, 2) 4. (23, 4), (22, 5), (21, 6), (0, 7) 5. y 5 24x 6. y 5 x 2 1 x 7. y 5 } 3 8. Sample answer: The input values are the first values in each ordered pair. The input values are 22, 7, and 10. 9. Sample answer: (1, 50), (2, 100), (3, 150) 10. x; y 11. Sample answer: Look for a relationship between the x-values and the y-values. In the table each y-value is 4 times the corresponding x-value, so the rule is y 5 4x. 12. yes; 52 weeks in 1 year, so w 5 52y Answer Key Lesson 5-13, pp. 52–55 Try this: 1. right, 0, G, G 2. 3, down, D, D 3. 4 ; y Y 3 2 1 1 2 O 24 23 22 Z 3 4 x 22 23 X 24 (continued ) 2; down; left; 3; right; 2; up; down Practice: 1. (1, 26) 2. (4, 1) 3. 3; up; (3, 3) 4. 5; left; 2; down; (25, 22) 5. 4; down; (0, 24) 6. (23, 5) 7. (23, 0) y 8–19. V R 6 4 2 N M S Y O O2 4 28 26 24 P X 6 Q x 24 W 26 28 U 20. Sample answer: A point that lies on the x-axis has y-coordinate 0. A point that lies on the y-axis has x-coordinate 0. 21. x-axis; y-axis 22. Sample answer: Start at the origin and move 6 units right and then 1 unit down. 23. Vicki is correct. Sample answer: To get to M you move left 2 units and then up 4 units. Answer Key Lesson 5-15, pp. 58–61 Try this: 1. (continued on next page also) x y (x, y ) –2 y 5 2x 1 1 0 y 5 2x 1 1 1 y 5 2x 1 1 y 5 2 p 22 1 1 y 5 2 p 0 1 1 y 5 2 p 1 1 1 5 23 (22, 23) 5 1 (0, 1) 5 3 (1, 3) –2; 23; 0; 1; 1; 3; 4 3 y 2 1 24 23 22 O 1 2 3 4 x 22 23 24 2. 215; –8; 1 3. vertical 4. horizontal Practice: 1. x y –2 y5x21 –1 y5x21 0 y5x21 y 5 22 2 1 y 5 21 2 1 y5021 5 23 (x, y ) (22, 23) 2 5 22 (21, 22) y 1 22 O 2. 5, 7, 9; 1 2 x 10 O 210 y 5 10 x 5 21 (0, 21) Answer Key 3. 21, 1, 3; 10 5 y 5 10 x O 210 4. 22, 2, 6; y 4 2 2 4 x 24 22 5. 2, 21, 24; 4 2 y 2 4 x 24 22 24 6. 5, 4, 3; 4 2 y 2 4 O 24 x 24 7. 3, 0, 23; 2 1 22 O y 1 x 22 8. Sample answer: 1, 2, 3; 2 y 1 22 1 2 x O 22 9. Sample answer: 1, 2, 3; 2 y 1 O 1 2 x 22 10. The y-values are the same for each x-value, so the line is horizontal. Answer Key 11. The x-values are the same for each y-value, so the line is vertical. 12. The x-values are different for each different y-value, so the line is diagonal. 13. horizontal; vertical; diagonal 14. Sample answer: Make a table of values choosing at least 3 numbers for x. To make the arithmetic easier, use even numbers for x. Substitute them in the equation for x to find the y-values. Then graph the ordered pairs and draw a line through them. 15. no; Sample answer: For every ordered pair on the graph of x 5 2, the value of x is 2. This means the graph goes through the points (2, 21), (2, 0), and (2, 1), for example. This describes a vertical line 2 units to the right of the y-axis.