Unit 3, Activity 3, Distance in the Plane Process Guide

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1 Unit 3, Activity 3, Distance in the Plane Process Guide Name: Date: Section A: Use the Pythagorean Theorem to find the missing side in each right triangle below. Give exact answers (in simplest radical form, if necessary). A. 8 x 15 B b C. d 7 7 Blackline Masters, Geometry Page 3-1

2 Unit 3, Activity 3, Distance in the Plane Process Guide Section B: Find the coordinates of the vertices on each right triangle. Find the lengths of the legs of each right triangle. Use the Pythagorean Theorem to find the length of the hypotenuse of each right triangle. Then answer the questions that follow. A D E C B F 1. How do the lengths of the horizontal legs of each triangle relate to distance on a number line? Which operation could you perform on the two x-coordinates that would result in the length of the horizontal legs? What number sentence would represent this operation? (Hint: Remember the distance on a number line is found by b a.). How do the lengths of the vertical legs of each triangle relate to distance on a number line? Which operation you could perform on the two y-coordinates that would result in the length of the vertical legs? What number sentence would represent this operation? (Hint: Remember the distance on a number line is found by b a.) Blackline Masters, Geometry Page 3-

3 Unit 3, Activity 3, Distance in the Plane Process Guide 3. For each triangle, substitute the number sentence you created in question 1 for a and the number sentence you created in question for b in the Pythagorean Theorem. Then solve for c without simplifying the expression you have for a +b. (Hint: ) Section C: Given the right triangle below, use the Pythagorean Theorem to find the hypotenuse of the given triangle using points and. Remember to find the legs of the right triangle just as you did in Section B. Solve the equation you create for c. Your final answer will have all variables and should resemble your final equation from number 3 above. Answer the questions that follow to help guide you through this process. M P N 4. To find the length of, which two coordinates would you need to use? What expression represents the distance from N to P? 5. To find the length of, which two coordinates would you need to use? What expression represents the distance from M to P? 6. To find the length of, use the Pythagorean Theorem and replace a with the expression from question 4 and replace b with the expression from question 5. Solve the equation for c. Remember, your result should have variables and should resemble. Blackline Masters, Geometry Page 3-3

4 Unit 3, Activity 3, Distance in the Plane Process Guide 7. The length of any segment is the between the endpoints. Based on that knowledge, how can the equation created in question 6 be useful? Section D: In Section C, you derived the Distance Formula. Typically, the formula is written as, where d represents the between two points (or the length of a segment that connects the two points). Use the formula you derived to find the length of the segments given on the grid below. Give the exact answer, in simplest radical form if necessary D 6 C 4 S F - G T Formula Investigation: Would the formula d ( x x ) ( y y ) = + also work? What about 1 1 ( ) ( ), ( ) ( ), or ( ) ( ) d = x x + y y d = x x + y y d = y y + x x? Why or why not? Explain your reasoning. Blackline Masters, Geometry Page 3-4

5 Unit 3, Activity 3, Distance in the Plane Process Guide Section E: Real-World Application Saints quarterback, Drew Brees, threw a pass from the Saints 10-yard line, 1 yards from the Saints sideline. The pass was caught by Marques Colston on the Saints 50-yard line, 40 yards from the same sideline. Brees gets credit for a 40-yard pass, but how much credit should Brees get for the pass? Imagine a grid laid over the representation of the field below with the horizontal axis on the bottom sideline and the vertical axis on left goal line, as shown. Goal Line SAINTS SAINTS Saints Sideline Remember, the intersection of the axes is (0,0) or the. What two ordered pairs would be used to represent the locations of Drew Brees and Marques Colston? (Hint: (yard line, yards from sideline)) How many yards did Brees throw the football? How did you find that distance? How many more yards should he get credit for on this pass? Blackline Masters, Geometry Page 3-5

6 Unit 3, Activity 3, Distance in the Plane Process Guide with Answers Name: Date: Section A: Use the Pythagorean Theorem to find the missing side in each right triangle below. Give exact answers (in simplest radical form, if necessary). A. 8 x 15 x = 17 B b C. d 7 7 Blackline Masters, Geometry Page 3-6

7 Unit 3, Activity 3, Distance in the Plane Process Guide with Answers Section B: Find the coordinates of the vertices on each right triangle. Find the lengths of the legs of each right triangle. Use the Pythagorean Theorem to find the length of the hypotenuse of each right triangle. Then answer the questions that follow. A D E C B F 1. How do the lengths of the horizontal legs of each triangle relate to distance on a number line? Which operation could you perform on the two x-coordinates that would result in the length of the horizontal legs? What number sentence would represent this operation? (Hint: Remember the distance on a number line is found by b a.) Sample answer: The length of the horizontal legs is the distance from one x-coordinate to the next. This measure can be found by subtracting the x-values. The number sentences would be and.. How do the lengths of the vertical legs of each triangle relate to distance on a number line? Which operation you could perform on the two y-coordinates that would result in the length of the vertical legs? What number sentence would represent this operation? (Hint: Remember the distance on a number line is found by b a.) Sample answer: Even though the segments are vertical, the length of the vertical legs is the distance from one y-coordinate to the next. This measure can be found by subtracting the y- values. The number sentences would be and. Blackline Masters, Geometry Page 3-7

8 Unit 3, Activity 3, Distance in the Plane Process Guide with Answers 3. For each triangle, substitute the number sentence you created in question 1 for a and the number sentence you created in question for b in the Pythagorean Theorem. Then solve for c without simplifying the expression you have for a +b. (Hint: ) Triangle ABC: Triangle DEF: Note: students may write the equations without the absolute value which is acceptable. Discussions should be held to help students understand why the absolute value symbols are not needed once the expressions have been squared. Section C: Given the right triangle below, use the Pythagorean Theorem to find the hypotenuse of the given triangle using points and. Remember to find the legs of the right triangle just as you did in Section B. Solve the equation you create for c. Your final answer will have all variables and should resemble your final equation from number 3 above. Answer the questions that follow to help guide you through this process. M P N 4. To find the length of, which two coordinates would you need to use? What expression represents the distance from N to P? ; Note: is also acceptable. 5. To find the length of, which two coordinates would you need to use? What expression represents the distance from M to P? ; Note: is also acceptable. 6. To find the length of, use the Pythagorean Theorem and replace a with the expression from question 4 and replace b with the expression from question 5. Solve the equation for c. Remember, your result should have variables and should resemble. Blackline Masters, Geometry Page 3-8

9 Unit 3, Activity 3, Distance in the Plane Process Guide with Answers Note: students may write the equations with or without the absolute value which is acceptable. Discussions should be held to help students eventually understand why the absolute value symbols are not needed once the expressions have been squared. 7. The length of any segment is the distance between the endpoints. Based on that knowledge, how can the equation created in question 6 be useful? The equation in number 6 can be used to find the distance between any two endpoints or the length of segment MN in the triangle above. Section D: In Section C, you derived the Distance Formula. Typically, the formula is written as, where d represents the distance between two points (or the length of a segment that connects the two points). Use the formula you derived to find the length of the segments given on the grid below. Give the exact answer, in simplest radical form if necessary D 6 C 4 S F - G T 1 1 Formula Investigation: Would the formula d ( x x ) ( y y ) = + also work? What about ( ) ( ), ( ) ( ), or ( ) ( ) d = x x + y y d = x x + y y d = y y + x x? Why or why not? Explain your reasoning. Yes, all of these formulas work. Students should understand that because the differences are squared, in these cases they can change the order for the subtraction even though subtraction is not a commutative operation. Since addition is commutative, whether they subtract x-values or y- values first does not matter. Make sure students know they should subtract x-values from x- values and y-values from y-values. Blackline Masters, Geometry Page 3-9

10 Unit 3, Activity 3, Distance in the Plane Process Guide with Answers Section E: Real-World Application Saints quarterback, Drew Brees, threw a pass from the Saints 10-yard line, 1 yards from the Saints sideline. The pass was caught by Marques Colston on the Saints 50-yard line, 40 yards from the same sideline. Brees gets credit for a 40-yard pass, but how much credit should Brees get for the pass? Imagine a grid laid over the representation of the field below with the horizontal axis on the bottom sideline and the vertical axis on left goal line, as shown. Goal Line SAINTS SAINTS Saints Sideline Remember, the intersection of the axes is (0,0) or the origin. What two ordered pairs would be used to represent the locations of Drew Brees and Marques Colston? (Hint: (yard line, yards from sideline)) Drew Brees: (10, 1); Marques Colston: (50, 40) How many yards did Brees throw the football? How did you find that distance? How many more yards should he get credit for on this pass? The distance is 48.8 yards. The distance was found by using the formula developed in this activity and using the ordered pairs listed above. Brees should get 8.8 yards more credit. Blackline Masters, Geometry Page 3-10

11 Unit 3, Activity 4, Dividing Number Line Segments Name: Date: A B Point D lies on between points A and B and divides into a ratio of :3. What is the coordinate of point D? 1. What is the length of?. If is divided into a ratio of :3, how many equal parts would there be? 3. Find the coordinate of D. 4. Explain in your own words how you found the coordinate of D. Given the number line below, answer the following questions If M is located at -3 and N is located at 4, find P such that is divided into a 1:5 ratio. 6. If X is located at - and Y is located at -5, find Z such that is divided into a ratio of :1. 7. Given A and C as arbitrary points on the number line (they could be located at any value), develop a formula that could help you find the location of any point that divides the segment into the ratio. Blackline Masters, Geometry Page 3-11

12 Unit 3, Activity 4, Dividing Number Line Segments with Answers Name: Date: A B Point D lies on between points A and B and divides into a ratio of :3. What is the coordinate of point D? 1. What is the length of? 10 units. If is divided into a ratio of :3, how many equal parts would there be? 5 3. Find the coordinate of D. ( 10 ) = 4 5 D is located at coordinate ( 1) = 3 4. Explain in your own words how you found the coordinate of D. First, find the length of segment AB which is 10. Then, multiply 10 by /5 because we want D to be of the five equal parts (or /5 of the distance) way from A. /5 of 10 is 4. Then, add 4 to the coordinate for A which is 3. **Note some students may use the coordinate for B; if that happens they should understand that D is to be 3/5 of the length of segment AB from B and that result (6) should be subtracted from B to make sure it is between A and B. Students should be encouraged to always use the leftmost endpoint. Given the number line below, answer the following questions If M is located at -3 and N is located at 4, find P such that is divided into a 1:5 ratio. P should be located at. 6. If X is located at - and Y is located at -5, find Z such that is divided into a ratio of :1. Z should be located at Given A and C as arbitrary points on the number line (they could be located at any value), where A is the leftmost point, develop a formula that could help you find the location of any point that divides the segment into the ratio. k1 ( C A) + A k + k 1 Blackline Masters, Geometry Page 3-1

13 Unit 3, Activity 7, Parallel Line Facts Use the given diagram to complete the steps and answer the questions. Date Name 1. Draw a line through vertex B so that the line is parallel to AC. Locate one point to the left of B and label it L. Locate one point to the right of B and label it R.. Given the diagram above with the parallel line drawn through B, prove that m BAC + m ABC + m ACB = Using the same diagram above, extend AC so that it is a line. Draw two points on AC : one to the left of A labeled D and the other to the right of C and label it E. Using the new diagram, prove that m BAD = m ABC + m ACB. Blackline Masters, Geometry Page 3-13

14 Unit 3, Activity 7, Parallel Line Facts 4. Remember, the area of a triangle can be written as answer the following questions. 1 A = bh. Use the diagram below to a. Using D, draw triangle ADC. b. Choose a point anywhere on BD and label it E. Draw triangle AEC. c. What do you notice about the base of each triangle: ABC, AB ' C, ADC, and AEC? d. What do you notice about the height of each triangle? e. What conjecture can you make about the area of any triangle that would be drawn between these parallel lines if A and C are not moved to different positions? Explain your reasoning. f. Would your conjecture still be true if you were able to choose any three points on the two lines to draw your triangles? Explain your reasoning. Blackline Masters, Geometry Page 3-14

15 Unit 3, Activity 7, Parallel Line Facts with Answers Use the given diagram to complete the steps and answer the questions. Date Name 1. Draw a line through vertex B so that the line is parallel to AC. Locate one point to the left of B and label it L. Locate one point to the right of B and label it R.. Given the diagram above with the parallel line drawn through B, prove that m BAC + m ABC + m ACB = 180. Given that BR AC, we know that alternate interior angles are congruent. So, BAC ABL and ACB CBR. By definition of congruence, m BAC = m ABL and m ACB = m CBR. Because ABL, ABC, and CBR are adjacent and form a line, m ABL + m ABC + m CBR = 180. Using the substitution property of equality, we now have m BAC + m ABC + m ACB = Using the same diagram above, extend AC so that it is a line. Draw two points on AC : one to the left of A labeled D and the other to the right of C and label it E. Using the new diagram, prove that m BAD = m ABC + m ACB. Given that BR AC, we know that alternate interior angles are congruent. So, BAD ABR and ACB CBR. By definition of congruence, we also know that m BAD = m ABR and m ACB = m CBR. Using the Angle Addition Postulate, we know m ABR = m ABC + m CBR. Next, using the Substitution Property of Equality, we find m ABR = m ABC + m ACB. Using the Substitution Property of Equality one more time, we get m BAD = m ABC + m ACB. Blackline Masters, Geometry Page 3-15

16 Unit 3, Activity 7, Parallel Line Facts with Answers 4. Remember, the area of a triangle can be written as answer the following questions. 1 A = bh. Use the diagram below to a. Using D, draw triangle ADC. b. Choose a point anywhere on BD and label it E. Draw triangle AEC. c. What do you notice about the base of each triangle: ABC, AB ' C, ADC, and AEC? The base of all four triangles is segment AC. The measure of the base doesn t change. d. What do you notice about the height of each triangle? Since the distance between parallel lines is equal everywhere, the height of all four triangles is the same. e. What conjecture can you make about the area of any triangle that would be drawn between these parallel lines if A and C are not moved to different positions? Explain your reasoning. Since both the base and height of these triangles are the same, they will have the same area. f. Would your conjecture still be true if you were able to choose any three points on the two lines to draw your triangles? Explain your reasoning. No, the conjecture would not necessarily work. If the measure of the base were changed each time, the area of each triangle would also change despite the fact that the height remained the same. Blackline Masters, Geometry Page 3-16

17 Unit 3, General Assessment, Scrapbook Rubric Parallel and Perpendicular Lines Scrapbook CATEGORY 4 points 3 points points 1 point 0 points Score Comments Quantity (4) Minimum of 3 photos per term (Parallel or Perpendicular) Only two photos/ pictures per term Only one photo/picture per term Only one picture to demonstrate both terms No photos or pictures x 6 Quality (4) Title Page (8) Reflection (1) Photos are of excellent quality; clear; description is written clearly Excellent quality; typed; includes project title, date, and class period Typed; tells what the student learned from project; grammatically correct Photos/ pictures are of good quality; description is clear but missing some elements Typed; missing date or class period Typed; 1- grammar errors; some evidence of learning Photos/pictures are grainy; term is not clearly depicted in picture; description is vague Handwritten with all information; or typed and missing date and class period Handwritten; 3-4 grammar errors; vague evidence of learning Photos/pictures do not depict term at all; description only gives definition Handwritten and missing date and class period; missing title (typed with all other info) Handwritten; 5-6 grammar errors; little to no evidence of learning No description x 6 given No title page No reflection x x 3 Neatness/ Creativity (8) Timeliness (8) Typed; clean; neatly bound pages; original project title; attractive; etc. Turned in on time Typed; project name not original; some pages loose Turned in one day late Handwritten; project is less than attractive; Turned in two days late Dirty, crumpled pages; if handwritten there are scratchouts or places with liquid paper Turned in three days late Pages are not bound Turned in more than three days late x x Blackline Masters, Geometry Page 3-17

18 Unit 3, Activity 1, Specific Assessment, What s My Line? What s My Line? On the attached page, you have been given a line and a point. Every line can be unique and can have its own unique equation. It is your job to find out as much about the line as possible. Listed below is the information that you must determine about the line. 1. Locate, draw, and label an x-axis and y-axis.. Find two points on your line. Label their (x,y) coordinates. Using the two points find the slope of your line. Show your calculations below. Write the slope next to the line as m=. 3. Determine the slope-intercept form of the equation for your line. Show your calculations below. Label your line with the equation. 4. Calculate the x and y-intercepts. Show your calculations below. On your graph, identify and label by giving their (x,y) coordinates. 5. Draw the line that is perpendicular to your line that passes through the point that was given on the page. Write the word perpendicular next to this line. Write the slopeintercept form of the equation for the perpendicular line and label the line with this equation. Show your calculations below. 6. Draw the line that is parallel to your line that passes through the point that was given on the page. Write the word parallel next to this line. Write the slope-intercept form of the equation for the parallel line and label the line with this equation. Show your calculations below. Blackline Masters, Geometry Page 3-18

19 Unit 3, Activity 1, Specific Assessment, What s My Line? with Answers What s My Line? On the attached page, you have been given a line and a point. Every line can be unique and can have its own unique equation. It is your job to find out as much about the line as possible. Listed below is the information that you must determine about the line. 1. Locate, draw, and label an x-axis and y-axis. Will vary by student. Find two points on your line. Label their (x,y) coordinates. Using the two points find the slope of your line. Show your calculations below. Write the slope next to the line as m=. 7 Graph A: m = ; Graph B: m = ; Graph C: m = ; Graph D: m = ; Graph E: m = 3 Other answers cannot be given since there are no axes on the graphs (students must draw these in on their own wherever they choose). The placement of the axes determines other answers. 3. Determine the slope-intercept form of the equation for your line. Show your calculations below. Label your line with the equation. Answers will depend on where x/y axes are drawn by each student. 4. Calculate the x and y-intercepts. Show your calculations below. On your graph, identify and label by giving their (x,y) coordinates. Answers will depend on where x/y axes are drawn by each student. 5. Draw the line that is perpendicular to your line that passes through the point that was given on the page. Write the word perpendicular next to this line. Write the slopeintercept form of the equation for the perpendicular line and label the line with this equation. Show your calculations below. 3 Graph A: m = ; Graph B: Graph D: m = ; Graph E: m = 3 5 m = ; Graph C: 7 5 m = ; 6. Draw the line that is parallel to your line that passes through the point that was given on the page. Write the word parallel next to this line. Write the slope-intercept form of the equation for the parallel line and label the line with this equation. Show your calculations below. Graph A: m = ; Graph B: 3 5 Graph D: m = ; Graph E: 7 m = ; Graph C: 1 m = 3 m = ; 5 Blackline Masters, Geometry Page 3-19

20 Unit 3, Activity 1, Specific Assessment, What s My Line? Graph A Blackline Masters, Geometry Page 3-0

21 Unit 3, Activity 1, Specific Assessment, What s My Line? Graph B Blackline Masters, Geometry Page 3-1

22 Unit 3, Activity 1, Specific Assessment, What s My Line? Graph C Blackline Masters, Geometry Page 3-

23 Unit 3, Activity 1, Specific Assessment, What s My Line? Graph D Blackline Masters, Geometry Page 3-3

24 Unit 3, Activity 1, Specific Assessment, What s My Line? Graph E Blackline Masters, Geometry Page 3-4

25 Unit 3, Activity 1, Specific Assessment, What s My Line? Rubric What s My Line? Rubric Description Points Possible Points Earned I. x-axis and y-axis drawn and labeled 5 II. III. IV. Slope of the line A. points labeled with (x,y) coordinates B. Slope calculated correctly 6 C. Labeled line with slope 1 Equation of the line A. Calculated correctly 6 B. Labeled on graph 1 x and y-intercepts A. Calculated correctly 6 B. Labeled on graph V. Perpendicular Line A. Line drawn correctly 3 B. Slope of line correctly identified C. y-intercept calculated correctly D. Equation written correctly based on calculations shown C. Line labeled with perpendicular and equation VI. VII. Parallel Line A. Line drawn correctly 3 B. Slope of line correctly identified C. y-intercept calculated correctly D. Equation written correctly based on calculations shown C. Line labeled with parallel and equation Following directions A. Cover sheet 3 B. Stapled 3 C. Submitted on or before due date 3 Total 60 Blackline Masters, Geometry Page 3-5

Unit 1, Activity 1, Extending Number and Picture Patterns Geometry

Unit 1, Activity 1, Extending Number and Picture Patterns Geometry Blackline Masters, Geometry Page 1-1 Unit 1, Activity 1, Extending Number and Picture Patterns Date Name Extending Patterns and Sequences