UNIT 6 CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Station Activities Set 1: Parallel Lines, Slopes, and Equations
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1 Station Activities Set 1: Parallel Lines, Slopes, and Equations Station 1 At this station, you will find rulers. Use these to help you determine whether or not the following lines are parallel. Look at the graph below. 1. Are these lines parallel? 2. Explain two ways you can tell lines are parallel. 3. What is the shortest distance between these two lines? 4. Draw a line that is parallel to the given line below. 5. How do you know your line is parallel? U6-105
2 Station Activities Set 1: Parallel Lines, Slopes, and Equations Station 2 At this station, you will find graph paper, a ruler, a red marker, and a blue marker. As a group, construct an x- and y-axis on your graph paper. 1. On your graph paper, construct a straight line that passes through ( 2, 5) and (7, 1). What is the slope of this line? Show how you found this slope by using the red and blue markers to represent rise and run on your graph. 2. Construct a second line that passes through (0, 4) and (18, 16). What is the slope of this line? Show how you found this slope by using the red and blue markers to represent rise and run on your graph. 3. Construct a third line that passes through (0, 0) and (10, 8). What is the slope of this line? Show how you found this slope by using the red and blue markers to represent rise and run on your graph. 4. Of the three straight lines you created, which two lines do you think are parallel? Explain your answer. continued U6-106
3 Station Activities Set 1: Parallel Lines, Slopes, and Equations For problems 5 8, determine whether the lines are parallel. Answer yes or no. Justify your answers. 5. y x 6. Are the lines parallel? y x Are the lines parallel? 7. One line has a slope of 2 3 Are the lines parallel? and another line has a slope of One line has a slope of 1 2 and another line has a slope of 1 2. Are the lines parallel? U6-107
4 Station Activities Set 1: Parallel Lines, Slopes, and Equations Station 3 At this station, you will find spaghetti noodles, graph paper, and a ruler. Work as a group to construct the lines and answer the questions. 1. On your graph paper, construct a straight line that passes through ( 2, 2) and (5, 10). Place a spaghetti noodle on this line. 2. On the same graph, construct a straight line that passes through ( 6, 4) and (1, 16). Place a spaghetti noodle on this line. By looking at the spaghetti noodles, the two lines may appear to be parallel, but are they? 3. How can you use the points on each line to determine if the two lines are parallel? Show your work. Are the lines parallel? Why or why not? 4. On the same graph, construct a straight line that passes through (5, 10) and (17, 3). Place a spaghetti noodle on this line. What is the slope of this line? Is this line perpendicular to the line you created in problem 1? Why or why not? Is this line also perpendicular to the line you created in problem 2? Why or why not? 5. Perpendicular lines create four angles that each measure. U6-108
5 Station Activities Set 1: Parallel Lines, Slopes, and Equations Station 4 At this station, you will find graph paper and a ruler. Work as a group to construct the graphs and answer the questions. 1. On your graph paper, graph the straight line that passes through the points ( 4, 5) and (4, 5). 2. What is the slope of this line? 3. Use the formula for point-slope form to find an equation for this line. Show your work and answer in the space below. Point-slope form: y y 1 = m(x x 1 ) Equation for the line: 4. If you were to construct a line parallel to the line in problem 1, what slope would this line have? Explain your answer. 5. How can you use your graph paper and the slope you found in problem 2 to construct a line parallel to the line in problem 1? Draw this line on your graph paper. continued U6-109
6 Station Activities Set 1: Parallel Lines, Slopes, and Equations 6. Use the formula for point-slope form to find an equation for this line. Show your work and answer in the space below. Point-slope form: y y 1 = m(x x 1 ) Equation for the line: 7. Construct another parallel line on your graph paper. 8. Use the formula for point-slope form to find an equation for this line. Show your work and answer in the space below. Point-slope form: y y 1 = m(x x 1 ) Equation for the line: U6-110
7 Station Activities Set 2: Perpendicular Lines Station 1 At this station, you will find graph paper and a ruler. As a group, construct an x- and y-axis on the graph paper. 1. Construct a line that passes through ( 2, 5) and (8, 5). What type of line have you created? 2. On the same graph, plot and label the point (3, 12). 3. How can you use perpendicular lines to find the distance between the line you created in problem 1 and the point you plotted in problem 2? 4. What is the distance between the line you created in problem 1 and the point you plotted in problem 2? Explain your answer. 5. On a new graph, construct a line that passes through (2, 10) and (2, 0). What type of line have you created? 6. On the same graph, plot and label the point (6, 3). continued U6-117
8 Station Activities Set 2: Perpendicular Lines 7. How can you use perpendicular lines to find the distance between the line you created in problem 5 and the point you plotted in problem 6? 8. What is the distance between the line you created in problem 5 and the point you plotted in problem 6? Explain your answer. U6-118
9 Station Activities Set 2: Perpendicular Lines Station 2 At this station, you will find a protractor. For problems 1 and 2, work as a group to determine whether or not the two lines are perpendicular using the coordinate graph. Do NOT use your protractor. 1. y x Are the lines perpendicular? Explain. 2. y x Are the lines perpendicular? Explain. continued U6-119
10 Station Activities Set 2: Perpendicular Lines 3. What strategy did you use to determine whether or not the lines are perpendicular? For problems 4 and 5, use your protractor to determine whether or not the lines are perpendicular. 4. Are the lines perpendicular? Explain. 5. Are the lines perpendicular? Explain. 6. How did you use your protractor to determine whether or not the lines were perpendicular? U6-120
11 Station Activities Set 2: Perpendicular Lines Station 3 At this station, you will find graph paper and a ruler. As a group, create an x- and y-axis on the graph paper. Work together to answer the questions. 1. On your graph paper, construct a line that passes through points ( 4, 0) and (0, 8). What is the slope of this line? 2. What is the slope of a line that is perpendicular to the line you created in problem 1? Explain your answer. 3. Use the point-slope form of an equation, y y 1 = m(x x 1 ), to find the equation of a line that is perpendicular to the line from problem 1, and that passes through the point ( 2, 4). Show your work and answer in the space below. Equation: 5 4. Write the equation for the line that is perpendicular to y = x + 10 and passes through (1, 2). Show your work and answer in the space below. 6 Equation: 5. On your graph paper, graph the two lines in problem 4 to justify that the lines are perpendicular. U6-121
12 Station Activities Set 2: Perpendicular Lines Station 4 At this station, you will find 18 index cards. Each index card has an equation written on it. Your job is to pair the equations into sets of parallel and perpendicular lines. 1. Work with your group members to find the parallel and perpendicular lines. Record your results below. Parallel lines Perpendicular lines 2. What was your strategy for working through this activity? U6-122
13 Station Activities Set 3: Coordinate Proof with Quadrilaterals Station 1 At this station, you will find a geoboard and rubber bands. 1. With your group, use the geoboard to construct a polygon with only one set of parallel sides. Draw this figure below. 2. Construct a polygon with two sets of parallel sides. Draw this figure below. 3. Construct a polygon with at least one set of perpendicular sides. Draw this figure below. 4. How did you apply what you know about parallel and perpendicular lines to successfully complete this activity? U6-127
14 Station Activities Set 3: Coordinate Proof with Quadrilaterals Station 2 At this station, you will find graph paper and a ruler. Work as a group to construct the quadrilaterals and answer the questions. 1. On your graph paper, construct a quadrilateral that has vertices (1, 1), (2, 4), (8, 4), and (9, 1). What type of quadrilateral did you create? 2. On the same graph, construct a quadrilateral that is congruent to the quadrilateral in problem 1. What are the vertices for this new quadrilateral? Explain why the quadrilaterals in problems 1 and 2 are congruent. 3. On the same graph, construct a quadrilateral that is NOT congruent to the quadrilateral in problem 1. What are the vertices of this new quadrilateral? Explain why the quadrilaterals in problems 1 and 3 are NOT congruent. 4. Graph a quadrilateral with vertices (0, 0), (2, 5), ( 2, 5), and (0, 7). What type of quadrilateral did you create? continued U6-128
15 Station Activities Set 3: Coordinate Proof with Quadrilaterals 5. Jacob claims that a quadrilateral with vertices ( 2, 2), (0, 6), ( 2, 8), and ( 4, 6) is congruent to the quadrilateral in problem 4. Is Jacob correct? Why or why not? Graph the quadrilaterals on your graph paper to justify your answer. If Jacob is incorrect, what vertices would make this quadrilateral congruent to the quadrilateral in problem 4? U6-129
16 Station Activities Set 3: Coordinate Proof with Quadrilaterals Station 3 At this station, you will find graph paper and a ruler. Work as a group to construct the quadrilaterals and answer the questions. 1. On your graph paper, graph a quadrilateral that has vertices (2, 4), (8, 4), (2, 10), and (8, 10). What are three names for this quadrilateral? Justify each name using properties of that type of quadrilateral. 2. On your graph paper, graph a quadrilateral that has vertices ( 1, 0), ( 4, 0), ( 1, 7), and ( 4, 7). What is the name of this quadrilateral? 3. Is the quadrilateral in problem 1 a regular quadrilateral? Why or why not? 4. Is the quadrilateral in problem 2 a regular quadrilateral? Why or why not? 5. What strategy did you use for problems 1 4? continued U6-130
17 Station Activities Set 3: Coordinate Proof with Quadrilaterals 6. What is the definition of a regular quadrilateral? 7. Are all rhombi and rectangles regular quadrilaterals? Why or why not? U6-131
18 Station Activities Set 3: Coordinate Proof with Quadrilaterals Station 4 At this station, you will find graph paper and a ruler. Work together to construct the quadrilaterals and answer the questions. 1. On your graph paper, construct a parallelogram with vertices (0, 0), ( 6, 0), ( 7, 4), and ( 1, 4). 2. On the same graph, construct a similar parallelogram. What are the vertices of the parallelogram? Why is this new parallelogram similar to the parallelogram in problem 1? 3. On a new graph, construct a trapezoid with vertices (1, 1), (4, 1), (2, 4), and (3, 4). Construct a second trapezoid with vertices (0, 0), (6, 0), (2, 6), and (4, 6). Are the two trapezoids congruent or similar? Explain your answer. 4. On a new graph, construct a kite. Construct a second kite that is similar, but not congruent to the first kite. What are the vertices of the first kite? What are the vertices of the second kite? Why are the kites similar? 5. When would you use similar quadrilaterals in the real world? U6-132
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