CCSS IP Math I Scaffolded SWB Unit 6.indd
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CCSS IP Math I Scaffolded SWB Unit 6.indd
Student Workbook with Scaffolded Practice Unit 6 1 This book is licensed for a single student’s use only. The reproduction of any part, for any purpose, is strictly prohibited. © Common Core State Standards. Copyright 2010. National Governor’s Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. 1 2 3 4 5 6 7 8 9 10 ISBN 978-0-8251-7778-1 Copyright © 2014 J. Weston Walch, Publisher Portland, ME 04103 www.walch.com Printed in the United States of America WALCH EDUCATION 2 Table of Contents Program pages Introduction Workbook pages 5 Unit 6: Connecting Algebra and Geometry Through Coordinates Lesson 1: Slope and Distance Lesson 6.1.1: Using Coordinates to Prove Geometric Theorems with Slope and Distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-4–U6-31 Lesson 6.1.2: Working with Parallel and Perpendicular Lines. . . . . . . . . . . . . . . U6-32–U6-52 Lesson 2: Lines and Line Segments Lesson 6.2.1: Calculating Perimeter and Area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-60–U6-84 Station Activities Set 1: Parallel Lines, Slopes, and Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-105–U6-110 Set 2: Perpendicular Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-117–U6-122 Set 3: Coordinate Proof with Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-127–U6-132 Coordinate Planes Formulas Bilingual Glossary 7–16 17–26 27–38 39–44 45–50 51–56 57–62 63–66 67–90 iii © Walch Education 3 CCSS IP Math I Teacher Resource 4 Introduction The CCSS Mathematics I Student Workbook with Scaffolded Practice includes all of the student pages from the Teacher Resource necessary for your day-to-day classroom use. This includes: • Warm-Ups • Problem-Based Tasks • Practice Problems • Station Activity Worksheets In addition, it provides Scaffolded Guided Practice examples that parallel the examples in the TRB and SRB. This supports: • Taking notes during class • Working problems for preview or additional practice The workbook includes the first Guided Practice example with step-by-step prompts for solving, and the remaining Guided Practice examples without prompts. Sections for you to take notes are provided at the end of each sub-lesson. Additionally, blank coordinate planes are included at the end of the full unit, should you need to graph. The workbook is printed on perforated paper so you can submit your assignments and three-hole punched to let you store it in a binder. v © Walch Education 5 CCSS IP Math I Teacher Resource 6 Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 1: Slope and Distance Lesson 6.1.1: Using Coordinates to Prove Geometric Theorems with Slope and Distance Warm-Up 6.1.1 The graph below shows the average number of miles an airplane travels over the course of its flight. 4200 3900 3600 3300 Distance (miles) 3000 2700 2400 2100 1800 1500 1200 900 600 300 -60 0 60 120 180 240 300 360 420 Time (minutes) 1. What is the average rate of change per hour? 2. How many miles would the airplane travel in 9 hours? U6-4 CCSS IP Math I Teacher Resource 6.1.1 7 © Walch Education 8 Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 1: Slope and Distance Scaffolded Practice 6.1.1 Example 1 Calculate the distance between the points (4, 9) and (–2, 6) using both the Pythagorean Theorem and the distance formula. 1. To use the Pythagorean Theorem, first plot the points on a coordinate system. 10 9 8 7 6 5 4 3 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 2. Draw lines to form a right triangle, using each point as the end of the hypotenuse. 3. Calculate the length of the vertical side, a, of the right triangle. 4. Calculate the length of the horizontal side, b, of the right triangle. 5. Use the Pythagorean Theorem to calculate the length of the hypotenuse, c. 6. Now use the distance formula to calculate the distance between the same points, (4, 9) and (–2, 6). continued U6-9 © Walch Education 9 CCSS IP Math I Teacher Resource 6.1.1 Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 1: Slope and Distance Example 2 Determine if the line through the points (–8, 5) and (–5, 3) is parallel to the line through the points (1, 3) and (4, 1). Example 3 Determine if the line through the points (0, 8) and (4, 9) is perpendicular to the line through the points (–9, 10) and (–8, 6). Example 4 A right triangle is defined as a triangle with 2 sides that are perpendicular. Triangle ABC has vertices A (–4, 8), B (–1, 2), and C (7, 6). Determine if this triangle is a right triangle. When disproving a figure, you only need to show one condition is not met. Example 5 A square is a quadrilateral with two pairs of parallel opposite sides, consecutive sides that are perpendicular, and all sides congruent, meaning they have the same length. Quadrilateral ABCD has vertices A (–1, 2), B (1, 5), C (4, 3), and D (2, 0). Determine if this quadrilateral is a square. U6-10 CCSS IP Math I Teacher Resource 6.1.1 10 © Walch Education Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 1: Slope and Distance Problem-Based Task 6.1.1: Field of Dreams A square is a quadrilateral with two pairs of parallel opposite sides, consecutive sides that are perpendicular, and all sides congruent. The bases and home plate of a baseball diamond form a square. The local recreation department has created a map of its newest baseball field. First base is represented by the point (7, –3), second base is represented by (4, 6), third base is represented by (–5, 3), and home plate is represented by (–2, –6). Is this field actually a square? What is the distance from second base to home plate? What is the distance from third base to first base? Is this field actually a square? What is the distance from second base to home plate? What is the distance from third base to first base? U6-22 CCSS IP Math I Teacher Resource 6.1.1 11 © Walch Education 12 Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 1: Slope and Distance Practice 6.1.1: Using Coordinates to Prove Geometric Theorems with Slope and Distance Use the distance formula to calculate the distance between the points indicated. 1. (–1, 1) and (4, 11) 2. (–3, –2) and (6, –1) A right triangle has a 90˚ angle made up of two perpendicular sides. Graph each triangle and then determine if each one is a right triangle. Use the slope formula and/or distance formula to justify your answer. 3. A (–6, 2), B (–2, –2), and C (2, 1) 4. A (–3, 2), B (1, 4), and C (3, 0) A parallelogram is a quadrilateral with opposite sides parallel. Graph each quadrilateral and then determine if each one is a parallelogram. Use the slope formula and/or distance formula to justify your answer. 5. A (–2, –1), B (–1, 3), C (4, 3), and D (3, –1) 6. A (–2, 2), B (1, 8), C (4, 5), and D (3, –2) A rectangle is a parallelogram with opposite sides that are congruent and consecutive sides that are perpendicular. Graph each quadrilateral and then determine if each one is a rectangle. Use the slope formula and/or distance formula to justify your answer. 7. A (–2, –1), B (–3, 1), C (1, 3), and D (2, 1) 8. A (1, –3), B (0, 0), C (4, 4), and D (5, 1) A square is a parallelogram with four congruent sides and four right angles. Graph each quadrilateral and then determine if each one is a square. Use the slope formula and/or distance formula to justify your answer. 9. A (–2, 1), B (1, 3), C (4, 1), and D (1, –1) 10. A (–1, 4), B (0, 6), C (2, 5), and D (1, 3) U6-31 © Walch Education 13 CCSS IP Math I Teacher Resource 6.1.1 14 Name: Notes Date: 15 Name: Notes Date: 16 Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 1: Slope and Distance Lesson 6.1.2: Working with Parallel and Perpendicular Lines Warm-Up 6.1.2 Civil engineers are planning a new shopping center downtown. They would like the entrance to the 1 center to run perpendicular to the street represented by the equation y = x + 2 . 5 et Stre Water 10 9 8 7 6 5 4 3 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 1 2 3 4 5 6 7 8 9 10 1. W rite an equation that would represent the entrance to the shopping center. Explain your reasoning. 2. A second entrance will run parallel to the first entrance. Write an equation that would represent the second entrance to the shopping center. U6-32 CCSS IP Math I Teacher Resource 6.1.2 17 © Walch Education 18 Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 1: Slope and Distance Scaffolded Practice 6.1.2 Example 1 Write the slope-intercept form of an equation for the line that passes through the point (5, –2) and is parallel to the graph of 8x – 2y = 6. 1. Rewrite the given equation in slope-intercept form. 2. Identify the slope of the given line. 3. Substitute the slope of the given line for m in the point-slope form of a linear equation. 4. Substitute x and y from the given point into the equation for x1 and y1. 5. Simplify the equation. continued U6-37 © Walch Education 19 CCSS IP Math I Teacher Resource 6.1.2 Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 1: Slope and Distance Example 2 Write the slope-intercept form of an equation for the line that passes through the point (–1, 6) and is perpendicular to the graph of –10x + 5y = 20. Example 3 Find the point on the line y = 4x + 1 that is closest to the point (–2, 8). U6-38 CCSS IP Math I Teacher Resource 6.1.2 20 © Walch Education Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 1: Slope and Distance Problem-Based Task 6.1.2: Pave the Way The civil engineers are busy again. This time they are expanding one of the well-traveled roads in your 3 town. The center of the road is represented by the equation y = x + 5 . The law says a two-lane road 5 must be at least 24 feet wide. Engineers have mapped out the edges of the road and have one edge running through the point (16, 1) and the other edge running through the point (4, 21). What are the equations of the lines that represent the edges of the road? Will this satisfy the minimum width required by law? What are the equations of the lines that represent the edges of the road? Will this satisfy the minimum width required by law? U6-46 CCSS IP Math I Teacher Resource 6.1.2 21 © Walch Education 22 Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 1: Slope and Distance Practice 6.1.2: Working with Parallel and Perpendicular Lines Write the slope-intercept form of an equation for the line that passes through the given point and is parallel to the graph of the given equation. 1. (3, 1) and 4x + 2y = 10 2. (–5, 6) and 12x + 9y = 3 Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of the given equation. 3. (–4, –3) and 8x + 2y = 14 4. (–6, 5) and 2x – 2y = 10 Calculate the shortest distance from the given point to the line indicated. 5. (2, 1) to x – y = –1 For questions 6–10, refer to the following map. Each unit represents 100 yards. 10 9 Bridge 8 Desiree's House Union S 7 6 treet 5 4 3 Park Train Station 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 Ma in Str ee t -9 -10 continued U6-51 © Walch Education 23 CCSS IP Math I Teacher Resource 6.1.2 Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 1: Slope and Distance 3 6. Main Street is given by the equation y = − x − 2 . The train station is located at the point (8, 1). 4 What is the equation of the line that represents the train tracks that run parallel to Main Street through the point (8, 1)? 7. Desiree’s house is located at the point (–8, 6). Her driveway is perpendicular to Union 1 Street, which is represented by the equation y = x + 6 . What is the equation of the line that 5 represents Desiree’s driveway? 8. L ittle River runs parallel to Union Street. There is a bridge located at the point (–5, 9). What is the equation of the line that represents Little River? 9. T he center of the park is located at the point (3, 2). A walking trail starting at this point runs perpendicular to Main Street. What is the equation of the line that represents the walking trail? 10. What is the shortest distance from the train station to Union Street? U6-52 CCSS IP Math I Teacher Resource 6.1.2 24 © Walch Education Name: Notes Date: 25 Name: Notes Date: 26 Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 2: Lines and Line Segments Lesson 6.2.1: Calculating Perimeter and Area Warm-Up 6.2.1 Mikko wants to make an open-top plywood box to fit on a certain shelf in his closet. He has drawn the sketch below of the box that he needs. h = 8 in. w = 11 in. l = 18 in. 1. How much plywood would Mikko need to create this box? 2. M ikko’s dad has brought him a piece of plywood that measures 2 feet by 3 feet. Is this enough plywood for the project? Explain why or why not. U6-60 CCSS IP Math I Teacher Resource 6.2.1 27 © Walch Education 28 Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 2: Lines and Line Segments Scaffolded Practice 6.2.1 Example 1 Triangle ABC has vertices A (–3, 1), B (1, 3), and C (2, –4). Calculate the perimeter of the triangle. 10 9 8 7 6 5 4 B (1, 3) 3 2 A (-3, 1) 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 C (2, -4) -6 -7 -8 -9 -10 1. Calculate the length of each side of the triangle. Use the distance formula. 2. Calculate the perimeter of triangle ABC. continued U6-66 CCSS IP Math I Teacher Resource 6.2.1 29 © Walch Education Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 2: Lines and Line Segments Example 2 Quadrilateral ABCD has vertices A (–3, 0), B (2, 4), C (3, 1), and D (–4, –3). Calculate the perimeter of the quadrilateral. 10 9 8 7 6 5 B (2, 4) 4 3 A (-3, 0) 2 C (3, 1) 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1 1 2 3 4 5 6 7 8 9 10 -2 D (-4, -3) -3 -4 -5 -6 -7 -8 -9 -10 Example 3 Triangle ABC has vertices A (1, –1), B (4, 3), and C (5, –3). Calculate the area of triangle ABC. 10 9 8 7 6 5 4 B (4, 3) 3 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1 -2 -3 1 2 A (1, -1) -4 3 4 5 6 7 8 9 10 C (5, -3) -5 -6 -7 -8 -9 continued -10 U6-67 © Walch Education 30 CCSS IP Math I Teacher Resource 6.2.1 Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 2: Lines and Line Segments Example 4 Rectangle ABCD has vertices A (–3, –4), B (–1, 2), C (2, 1), and D (0, –5). Calculate the area of the rectangle. 10 9 8 7 6 5 4 B (-1, 2) 3 2 C (2, 1) 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1 1 2 3 4 5 6 7 8 9 10 -2 -3 A (-3, -4) -4 -5 -6 D (0, -5) -7 -8 -9 -10 U6-68 CCSS IP Math I Teacher Resource 6.2.1 31 © Walch Education 32 Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 2: Lines and Line Segments Problem-Based Task 6.2.1: Building Fences The local recreation department has created a map of its newest baseball field. The department is planning to install a rectangular fence around the field. The corners of the field are represented on the map by the points A (–5, –10), B (13, –4), C (4, 23), and D (–11, 18). How many feet of fencing are needed for the baseball field? What is the area of the fenced-in field? Each unit on the map represents 10 feet. How many feet of fencing are needed for the baseball field? What is the area of the fencedin field? U6-79 © Walch Education 33 CCSS IP Math I Teacher Resource 6.2.1 34 Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Lesson 2: Lines and Line Segments Practice 6.2.1: Perimeter and Area Calculate the perimeter of each of the polygons below. 1. Triangle ABC has vertices A (–2, 1), B (–3, 5), and C (3, 6). 2. Trapezoid ABCD has vertices A (–3, –2), B (3, 4), C (3, –1), and D (1, –3). 3. Quadrilateral ABCD has vertices A (–4, 0), B (–2, 3), C (2, 3), and D (2, 0). 4. Square ABCD has vertices A (–2, 2), B (0, 4), C (2, 2), and D (0, 0). 5. Parallelogram ABCD has vertices A (–5, 4), B (–1, 6), C (5, 2), and D (1, 0). Calculate the area of each of the polygons below. 6. Rectangle ABCD has vertices A (–5, 2), B (–5, 4), C (4, 4), and D (4, 2). 7. Rectangle ABCD has vertices A (–4, –4), B (0, 2), C (9, –4), and D (5, –10). 8. Right triangle ABC has vertices A (–4, 2), B (–4, 6), and C (9, 6). 9. Triangle ABC has vertices A (–2, 5), B (3, 1), and C (3, 5). 10. Triangle ABC has vertices A (3, 5), B (7, 8), and C (5, –3). U6-84 CCSS IP Math I Teacher Resource 6.2.1 35 © Walch Education 36 Name: Notes Date: 37 Name: Notes Date: 38 Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES 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 © Walch Education 39 CCSS IP Math I Teacher Resource Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES 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? S how 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? S how 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? S how how you found this slope by using the red and blue markers to represent rise and run on your graph. 4. O f the three straight lines you created, which two lines do you think are parallel? Explain your answer. continued U6-106 CCSS IP Math I Teacher Resource 40 © Walch Education Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES 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. y 5. 10 9 8 7 6 5 4 3 2 1 0 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 1 2 3 4 5 6 7 8 9 10 x 1 2 3 4 5 6 7 8 9 10 x Are the lines parallel? y 6. 10 9 8 7 6 5 4 3 2 1 0 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 Are the lines parallel? 7. One line has a slope of Are the lines parallel? 16 2 and another line has a slope of . 3 24 8. One line has a slope of − Are the lines parallel? 1 1 and another line has a slope of . 2 2 U6-107 © Walch Education 41 CCSS IP Math I Teacher Resource Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES 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. O n your graph paper, construct a straight line that passes through (–2, –2) and (5, 10). Place a spaghetti noodle on this line. 2. O n 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. H ow 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. O n 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 CCSS IP Math I Teacher Resource 42 © Walch Education Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES 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. U se 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 – y1 = m(x – x1) Equation for the line: __________________ 4. I f you were to construct a line parallel to the line in problem 1, what slope would this line have? Explain your answer. 5. H ow 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 © Walch Education 43 CCSS IP Math I Teacher Resource Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Station Activities Set 1: Parallel Lines, Slopes, and Equations 6. U se 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 – y1 = m(x – x1) Equation for the line: __________________ 7. Construct another parallel line on your graph paper. 8. U se 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 – y1 = m(x – x1) Equation for the line: __________________ U6-110 CCSS IP Math I Teacher Resource 44 © Walch Education Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES 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. H ow 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. W hat 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 © Walch Education 45 CCSS IP Math I Teacher Resource Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Station Activities Set 2: Perpendicular Lines 7. H ow 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. W hat is the distance between the line you created in problem 5 and the point you plotted in problem 6? Explain your answer. U6-118 CCSS IP Math I Teacher Resource 46 © Walch Education Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES 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. y 10 9 8 7 6 5 4 3 2 1 1. 0 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 1 2 3 4 5 6 7 8 9 10 x 9 10 x Are the lines perpendicular? Explain. y 2. 10 9 8 7 6 5 4 3 2 1 0 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 1 2 3 4 5 6 7 8 Are the lines perpendicular? Explain. continued U6-119 © Walch Education 47 CCSS IP Math I Teacher Resource Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES 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 CCSS IP Math I Teacher Resource 48 © Walch Education Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES 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. W hat is the slope of a line that is perpendicular to the line you created in problem 1? Explain your answer. 3. U se the point-slope form of an equation, y – y1 = m(x – x1), 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. W rite the equation for the line that is perpendicular to y = − x + 10 and passes through 6 (1, 2). Show your work and answer in the space below. Equation: __________________ 5. O n your graph paper, graph the two lines in problem 4 to justify that the lines are perpendicular. U6-121 © Walch Education 49 CCSS IP Math I Teacher Resource Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES 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. W ork 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 CCSS IP Math I Teacher Resource 50 © Walch Education Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Station Activities Set 3: Coordinate Proof with Quadrilaterals Station 1 At this station, you will find a geoboard and rubber bands. 1. W ith 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. H ow did you apply what you know about parallel and perpendicular lines to successfully complete this activity? U6-127 © Walch Education 51 CCSS IP Math I Teacher Resource Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES 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. O n 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 CCSS IP Math I Teacher Resource 52 © Walch Education Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Station Activities Set 3: Coordinate Proof with Quadrilaterals 5. J acob 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. I f Jacob is incorrect, what vertices would make this quadrilateral congruent to the quadrilateral in problem 4? U6-129 © Walch Education 53 CCSS IP Math I Teacher Resource Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES 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). hat are three names for this quadrilateral? Justify each name using properties of that type of W 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 CCSS IP Math I Teacher Resource 54 © Walch Education Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES 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 © Walch Education 55 CCSS IP Math I Teacher Resource Name: Date: UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES 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. O n 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. O n 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 CCSS IP Math I Teacher Resource 56 © Walch Education 57 58 59 60 61 62 Formulas ALGEBRA General Exponential Equations (x, y) Ordered pair y = abx (x, 0) x-intercept (0, y) y-intercept General form x Symbols y = ab t Exponential equation y = a(1 + r)t Exponential growth y = a(1 – r)t Exponential decay ≈ Approximately equal to ≠ Is not equal to r A= P1+ n a Absolute value of a Compounded… a nt Compounded interest formula n (number of times per year) Yearly/annually 1 Square root of a Semi-annually 2 Quarterly 4 Slope Monthly 12 ax + b = c One variable Weekly 52 y = mx + b Slope-intercept form Daily 365 ax + by = c General form Linear Equations m= y2 − y1 x2 − x1 Functions y – y1 = m(x – x1) Point-slope form f(x) Notation, “f of x” Arithmetic Sequences f(x) = mx + b Linear function an = a1 + (n – 1)d Explicit formula f(x) = bx + k Exponential function an = an –1 + d (f + g)(x) = f(x) + g(x) Addition Recursive formula (f – g)(x) = f(x) – g(x) Subtraction Geometric Sequences (f • g)(x) = f(x) • g(x) Multiplication an = a1 • r Explicit formula (f ÷ g)(x) = f(x) ÷ g(x) Division an = an – 1 • r Recursive formula n–1 F-1 Formulas 63 Formulas Properties of Equality Property In symbols Reflexive property of equality a=a Symmetric property of equality If a = b, then b = a. Transitive property of equality If a = b and b = c, then a = c. Addition property of equality If a = b, then a + c = b + c. Subtraction property of equality If a = b, then a – c = b – c. Multiplication property of equality If a = b and c ≠ 0, then a • c = b • c. Division property of equality If a = b and c ≠ 0, then a ÷ c = b ÷ c. Substitution property of equality If a = b, then b may be substituted for a in any expression containing a. Properties of Operations Property General rule Commutative property of addition a+b=b+a Associative property of addition (a + b) + c = a + (b + c) Commutative property of multiplication a•b=b•a Associative property of multiplication (a • b) • c = a • (b • c) Distributive property of multiplication over addition a • (b + c) = a • b + a • c Properties of Inequality Laws of Exponents Property Law General rule If a > b and b > c, then a > c. Multiplication of exponents Power of exponents bm • bn = bm + n If a > b, then b < a. If a > b, then –a < –b. If a > b, then a ± c > b ± c. If a > b and c > 0, then a • c > b • c. If a > b and c < 0, then a • c < b • c. If a > b and c > 0, then a ÷ c > b ÷ c. If a > b and c < 0, then a ÷ c < b ÷ c. (b ) m n = b mn ( bc )n = bnc n Division of exponents bm n Exponents of zero Negative exponents F-2 Formulas 64 = b m− n b b0 = 1 b− n = 1 b n and 1 b −n = bn Formulas DATA ANALYSIS IQR = Q 3 – Q 1 Interquartile range Q 1 – 1.5(IQR) Lower outlier formula Q 3 + 1.5(IQR) Upper outlier formula y – y0 Residual formula GEOMETRY Symbols ( ) Translations d ABC Arc length T(h, k) = (x + h, y + k) Translation ∠ Angle Reflections Circle rx-axis (x, y) = (x, –y) Through the x-axis ≅ PQ Congruent ry-axis (x, y) = (–x, y) Through the y-axis Line ry = x (x, y) = (y, x) PQ PQ Line Segment Rotations Ray R90 (x, y) = (–y, x) Parallel ⊥ Perpendicular • Point Triangle A′ Prime ° Degrees Through the line y = x Counterclockwise 90º about the origin R180 (x, y) = (–x, –y) Counterclockwise 180º about the origin R270 (x, y) = (y, –x) Counterclockwise 270º about the origin Congruent Triangle Statements Side-Side-Side (SSS) Side-Angle-Side (SAS) A C ABC ≅ XYZ G D B Z Angle-Side-Angle (ASA) F X T Y W J E V DEF ≅ TVW Q S H R GHJ ≅ QRS F-3 Formulas 65 Formulas Pythagorean Theorem Distance Formula a2 + b2 = c2 d = ( x2 − x1 ) 2 + ( y2 − y1 ) 2 Distance formula Area A = lw Rectangle 1 A = bh 2 Triangle MEASUREMENTS Length Volume and Capacity Metric Metric 1 kilometer (km) = 1000 meters (m) 1 liter (L) = 1000 milliliters (mL) Customary 1 meter (m) = 100 centimeters (cm) 1 centimeter (cm) = 10 millimeters (mm) Customary 1 mile (mi) = 1760 yards (yd) 1 gallon (gal) = 4 quarts (qt) 1 quart (qt) = 2 pints (pt) 1 pint (pt) = 2 cups (c) 1 cup (c) = 8 fluid ounces (fl oz) 1 mile (mi) = 5280 feet (ft) 1 yard (yd) = 3 feet (ft) 1 foot (ft) = 12 inches (in) Weight and Mass Metric 1 kilogram (kg) = 1000 grams (g) 1 gram (g) = 1000 milligrams (mg) 1 metric ton (MT) = 1000 kilograms (kg) Customary 1 ton (T) = 2000 pounds (lb) 1 pound (lb) = 16 ounces (oz) F-4 Formulas 66 PROGRAM OVERVIEW Glossary English Español acute angle an angle measuring less than 90˚ but greater than 0˚ algebraic expression a mathematical statement that includes numbers, operations, and variables to represent a number or quantity algebraic inequality an inequality that has one or more variables and contains at least one of the following symbols: <, >, ≤, ≥, or ≠ altitude the perpendicular line from a vertex of a figure to its opposite side; height angle two rays or line segments sharing a common endpoint; the symbol used is ∠ angle of rotation the measure of the angle created by the preimage vertex to the point of rotation to the image vertex. All of these angles are congruent when a figure is rotated. A U5-2 U1-3 U1-167 U5-91 U5-2 U5-91 U5-254 angle-side-angle (ASA) if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent arc length the distance between the endpoints of an arc; written as d ABC U5-312 area the amount of space inside the boundary of a two-dimensional figure U6-58 ( ) U5-2 ángulo agudo ángulo que mide menos de 90˚ pero más de 0˚ expresión algebraica declaración matemática que incluye números, operaciones y variables para representar un número o una cantidad desigualdad algebraica desigualdad que tiene una o más variables y contiene al menos uno de los siguientes símbolos: <, >, ≤, ≥, o ≠ altitud línea perpendicular desde un vértice de una figura hasta su lado opuesto; altura ángulo dos semirrectas o segmentos de línea que comparten un extremo común; el símbolo utilizado es ∠ ángulo de rotación medida del ángulo creada por el vértice del preimagen hasta el punto de rotación del vértice del imagen. Todos estos ángulos son congruentes cuando una figura está rotada. ángulo-lado-ángulo (ASA) si dos ángulos y el lado incluido de un triángulo son congruentes con los dos ángulos y el lado incluido de otro triángulo, entonces los dos triángulos son congruentes longitud de arco distancia entre los extremos de un arco; se expresa como d ABC ( ) área cantidad de espacio dentro del límite de una figura bidimensional G-1 © Walch Education 67 CCSS IP Math I Teacher Resource PROGRAM OVERVIEW Glossary English arithmetic sequence a linear function with a domain of positive consecutive integers in which the difference between any two consecutive terms is equal asymptote a line that a graph gets closer and closer to, but never crosses or touches base the factor being multiplied together in an exponential expression; in the expression ab, a is the base bisect to cut in half box plot a plot showing the minimum, maximum, first quartile, median, and third quartile of a data set; the middle 50% of the data is indicated by a box. Example: causation a relationship between two events where a change in one event is responsible for a change in the second event circle the set of points on a plane at a certain distance, or radius, from a single point, the center. The set of points forms a two-dimensional curve that measures 360˚. circular arc on a circle, the unshared set of points between the endpoints of two radii U2-457 U2-167 U2-246 B U1-3 U5-91 U4-3 C U4-175 U5-2 U5-175 U5-2 Español secuencia aritmética función lineal con dominio de enteros consecutivos positivos, en la que la diferencia entre dos términos consecutivos es equivalente asíntota línea a la que se acerca cada vez más un gráfico, pero sin cruzarlo ni tocarlo base factor que se multiplica en forma conjunta en una expresión exponencial; en la expresión ab, a es la base bisecar cortar por la mitad diagrama de caja diagrama que muestra el mínimo, máximo, primer cuartil, mediana y tercer cuartil de un conjunto de datos; se indica con una caja el 50% medio de los datos. Ejemplo: causalidad relación entre dos eventos en la que un cambio en un evento es responsable por un cambio en el segundo evento círculo conjunto de puntos en un plano a determinada distancia, o radio, de un único punto, el centro. El conjunto de puntos forma una curva bidimensional que mide 360˚. arco circular en un círculo, conjunto de puntos no compartidos entre los extremos de dos radios G-2 CCSS IP Math I Teacher Resource 68 © Walch Education PROGRAM OVERVIEW Glossary English clockwise rotating a figure in the direction that the hands on a clock move coefficient the number multiplied by a variable in an algebraic expression common difference the number added to each consecutive term in an arithmetic sequence compass an instrument for creating circles or transferring measurements that consists of two pointed branches joined at the top by a pivot compression a transformation in which a figure becomes smaller; compressions may be horizontal (affecting only horizontal lengths), vertical (affecting only vertical lengths), or both conditional relative frequency the percentage of a joint frequency as compared to the total number of respondents, total number of people with a given characteristic, or the total number of times a specific response was given congruency transformation a transformation in which a geometric figure moves but keeps the same size and shape congruent figures are congruent if they have the same shape, size, lines, and angles; the symbol for representing congruency between figures is ≅ congruent angles two angles that have the same measure U5-58 U5-254 U1-3 U2-457 U5-91 U5-254 U4-76 U5-254 U5-2 U5-91 U5-175 U5-254 U6-2 U5-312 Español sentido horario rotación de una figura en la dirección en que se mueven las agujas de un reloj coeficiente número multiplicado por una variable en una expresión algebraica diferencia común número sumado a cada término consecutivo en una secuencia aritmética compás instrumento utilizado para crear círculos o transferir medidas, que consiste en dos brazos terminados en punta y unidos en la parte superior por un pivote compresión transformación en la que una figura se hace más pequeña; las compresiones pueden ser horizontales (cuando afectan sólo la longitud horizontal), verticales (cuando afectan sólo la longitud vertical), o en ambos sentidos frecuencia condicional relativa porcentaje de una frecuencia conjunta en comparación con la cantidad total de respondedores, cantidad total de personas con una determinada característica, o cantidad total de veces que se dio una respuesta específica transformación de congruencia transformación en la que se mueve una figura geométrica pero se mantiene el mismo tamaño y la misma forma congruente las figuras son congruentes si tienen la misma forma, tamaño, rectas y ángulos; el símbolo para representar la congruencia entre figuras es ≅ ángulos congruentes dos ángulos con la misma medida G-3 © Walch Education 69 CCSS IP Math I Teacher Resource PROGRAM OVERVIEW Glossary English congruent sides two sides that have the same length congruent triangles triangles having the same angle measures and side lengths consistent a system of equations with at least one ordered pair that satisfies both equations constant a quantity that does not change constant ratio the number each consecutive term is multiplied by in a geometric sequence constraint a restriction or limitation on either the input or output values construct to create a precise geometric representation using a straightedge along with either patty paper (tracing paper), a compass, or a reflecting device construction a precise representation of a figure using a straightedge and a compass, patty paper and a straightedge, or a reflecting device and a straightedge continuous having no breaks coordinate plane a set of two number lines, called the axes, that intersect at right angles correlation a relationship between two events, where a change in one event is related to a change in the second event. A correlation between two events does not imply that the first event is responsible for the change in the second event; the correlation only shows how likely it is that a change also took place in the second event. U5-312 U5-312 U3-67 U1-3 U2-457 U1-167 U5-91 U5-91 U5-175 U2-167 U1-93 U4-175 Español lados congruentes dos lados con la misma longitud triángulos congruentes triángulos con las mismas medidas de ángulos y longitudes de lados consistente sistema de ecuaciones con al menos un par ordenado que satisface ambas ecuaciones constante cantidad que no cambia proporción constante el número que cada término esta multiplicado por en una secuencia geométrica limitación restricción o límite en los valores de entrada o salida construir crear una representación geométrica precisa mediante regla de borde recto y papel encerado (papel para calcar), compás o un dispositivo de reflexión construcción representación precisa de una figura mediante regla de borde recto y compás, papel encerado y una regla de borde recto, o un dispositivo de reflexión y una regla de borde recto continuo sin interrupciones plano de coordenadas conjunto de dos rectas numéricas, denominadas ejes, que se cortan en ángulos rectos correlación relación entre dos eventos en la que el cambio en un evento se relaciona con un cambio en el segundo evento. Una correlación entre dos eventos no implica que el primero sea responsable del cambio en el segundo; la correlación sólo demuestra cuán probable es que también se produzca un cambio en el segundo evento. G-4 CCSS IP Math I Teacher Resource 70 © Walch Education PROGRAM OVERVIEW Glossary English correlation coefficient a quantity that assesses the strength of a linear relationship between two variables, ranging from –1 to 1; a correlation coefficient of –1 indicates a strong negative correlation, a correlation coefficient of 1 indicates a strong positive correlation, and a correlation coefficient of 0 indicates a very weak or no linear correlation corresponding angles angles of two figures that lie in the same position relative to the figure. In transformations, the corresponding vertices are the preimage and image vertices, so ∠A and ∠A′ are corresponding vertices and so on. Corresponding Parts of Congruent Triangles are Congruent (CPCTC) if two or more triangles are proven congruent, then all of their corresponding parts are congruent as well corresponding sides sides of two figures that lie in the same position relative to the figure. In transformations, the corresponding sides are the preimage and image sides, so AB and A′B′ are corresponding sides and so on. counterclockwise rotating a figure in the opposite direction that the hands on a clock move curve the graphical representation of the solution set for y = f(x); in the special case of a linear equation, the curve will be a line U4-175 U5-254 U5-312 U5-312 U5-254 U5-312 U5-58 U5-254 U2-2 Español coeficiente de correlación cantidad que evalúa la fuerza de una relación lineal entre dos variables, que varía de –1 a 1; un coeficiente de correlación de –1 indica una fuerte correlación negativa, un coeficiente de correlación de 1 indica una fuerte correlación positiva, y un coeficiente de correlación de 0 indica una correlación muy débil o no lineal ángulos correspondientes ángulos de dos figuras que se ubican en la misma posición relativa a la figura. En las transformaciones, los vértices correspondientes son los vértices de preimagen e imagen, de manera que ∠A y ∠A′ son los vértices correspondientes, etc. Las partes correspondientes de triángulos congruentes son congruentes (CPCTC) si se comprueba que dos o más triángulos son congruentes, entonces todas sus partes correspondientes son también congruentes lados correspondientes lados de dos figuras que están en la misma posición relativa a la figura. En las transformaciones, los lados correspondientes son los de preimagen e imagen, entonces AB y A′B′ son los lados correspondientes, etc. en sentido antihorario rotación de una figura en la dirección opuesta a la que se mueven las agujas de un reloj curva representación gráfica del conjunto de soluciones para y = f(x); en el caso especial de una ecuación lineal, la curva será una recta G-5 © Walch Education 71 CCSS IP Math I Teacher Resource PROGRAM OVERVIEW Glossary English dependent a system of equations that has an infinite number of solutions; lines coincide when graphed dependent variable labeled on the y-axis; the quantity that is based on the input values of the independent variable; the output variable of a function diameter a straight line passing through the center of a circle connecting two points on the circle; twice the radius dilation a transformation in which a figure is either enlarged or reduced by a scale factor in relation to a center point Español D U3-67 U1-93 U2-2 U5-175 U5-254 dependiente sistema de ecuaciones con una cantidad infinita de soluciones; las rectas coinciden cuando se grafican variable dependiente designada en el eje y; cantidad que se basa en los valores de entrada de la variable independiente; variable de salida de una función diámetro línea recta que pasa por el centro de un círculo y conecta dos puntos en el círculo; dos veces el radio dilatación transformación en la que una figura se amplía o se reduce por un factor de escala en relación con un punto central discrete individually separate and distinct distance along a line the linear distance between two points on a given line; written as d ( PQ ) U2-139 U5-2 discreto individualmente aparte y distinto distancia a lo largo de una recta distancia lineal entre dos puntos de una determinada línea; se expresa como d ( PQ ) distance formula formula that states the U6-2 U6-58 fórmula de distancia fórmula que establece distance between points (x1, y1) and (x2, y2) is equal to ( x2 − x1 ) 2 + ( y2 − y1 ) 2 domain the set of all inputs of a function; the set of x-values that are valid for the function dot plot a frequency plot that shows the number of times a response occurred in a data set, where each data value is represented by a dot. Example: la distancia entre los puntos (x1, y1) y (x2, y2) equivale a ( x2 − x1 ) 2 + ( y2 − y1 ) 2 U2-2 U2-168 U4-3 dominio conjunto de todas las entradas de una función; conjunto de valores x que son válidos para la función diagrama de puntos diagrama de frecuencia que muestra la cantidad de veces que se produjo una respuesta en un conjunto de datos, en el que cada valor de dato está representado por un punto. Ejemplo: G-6 CCSS IP Math I Teacher Resource 72 © Walch Education PROGRAM OVERVIEW Glossary English drawing a precise representation of a figure, created with measurement tools such as a protractor and a ruler U5-92 elimination method adding or subtracting the equations in the system together so that one of the variables is eliminated; multiplication might be necessary before adding the equations together E U3-67 end behavior the behavior of the graph as x approaches positive infinity and as x approaches negative infinity endpoint either of two points that mark the ends of a line segment; a point that marks the end of a ray equation a mathematical sentence that uses an equal sign (=) to show that two quantities are equal equidistant the same distance from a reference point equilateral triangle a triangle with all three sides equal in length explicit equation an equation describing the nth term of a pattern U2-246 explicit formula a formula used to find the nth term of a sequence; the explicit formula for an arithmetic sequence is an = a1 + (n – 1)d; the explicit formula for a geometric sequence is an = a1 • r n – 1 U2-139 U2-457 exponent the number of times a factor is being multiplied together in an exponential expression; in the expression ab, b is the exponent U1-3 U5-92 U1-33 U2-371 U5-92 U5-255 U5-175 U2-371 Español dibujo representación precisa de una figura, creada con herramientas de medición tales como transportador y regla método de eliminación suma o sustracción conjunta de ecuaciones en el sistema de manera de eliminar una de las variables; podría requerirse multiplicación antes de la suma conjunta de las ecuaciones comportamiento final el comportamiento de la gráfica al aproximarse x a infinito positivo o a infinito negativo extremo uno de los dos puntos que marcan el final de un segmento de recta; punto que marca el final de una semirrecta ecuación declaración matemática que utiliza el signo igual (=) para demostrar que dos cantidades son equivalentes equidistante a la misma distancia de un punto de referencia triángulo equilátero triángulo con sus tres lados de la misma longitud ecuación explícita ecuación que describe el enésimo término de un patrón fórmula explícita fórmula utilizada para encontrar el enésimo término de una secuencia; la fórmula explícita para una secuencia aritmética es an = a1 + (n – 1)d; la fórmula explícita para una secuencia geométrica es an = a1 • r n – 1 exponente cantidad de veces que se multiplica un factor en forma conjunta en una expresión exponencial; en la expresión ab, b es el exponente G-7 © Walch Education 73 CCSS IP Math I Teacher Resource PROGRAM OVERVIEW Glossary English exponential decay an exponential equation with a base, b, that is between 0 and 1 (0 < b < 1); can be represented by the formula y = a(1 – r) t, where a is the initial value, (1 – r) is the decay rate, t is time, and y is the final value exponential equation an equation that has a variable in the exponent; the general form is y = a • b x, where a is the U1-33 U1-93 U1-33 U1-93 U2-371 Español decaimiento exponencial ecuación exponencial con una base, b, que está entre 0 y 1 (0 < b < 1); puede representarse con la fórmula y = a(1 – r) t, en la que a es el valor inicial, (1 – r) es la tasa de decaimiento, t es el tiempo, y y es el valor final ecuación exponencial ecuación con una variable en el exponente; la forma general es y = a • b x, en la que a es el valor inicial, b initial value, b is the base, x is the time, es la base, x es el tiempo, y y es el valor final and y is the final output value. Another de salida. Otra forma es y = ab t , en la que t form is y = ab t , where t is the time it es el tiempo que tarda la base en repetirse. x x takes for the base to repeat. exponential function a function that has a variable in the exponent: • the general form is f (x) = ab x, where a is the initial value, b is the growth or decay factor, x is the time, and f (x) is the final output value • can also be written in the form f(x) = b x + k, where b is a positive integer not equal to 1 and k can equal 0; the parameters are b and k. b is the growth factor and k is the vertical shift. exponential growth an exponential equation with a base, b, greater than 1 (b > 1); can be represented by the formula y = a(1 + r) t, where a is the initial value, (1 + r) is the growth rate, t is time, and y is the final value U2-246 U2-296 U2-484 U1-33 U1-93 función exponencial función con una variable en el exponente: • la forma general es f (x) = ab x, en la que a es el valor inicial, b es el factor de crecimiento o decaimiento, x es el tiempo, y f (x) es el valor de salida • también puede expresarse en la forma f(x) = b x + k, en donde b es un entero positivo diferente de 1 y k puede ser igual a 0; los parámetros son b y k. b es el factor de crecimiento y k es el desplazamiento vertical. crecimiento exponencial ecuación exponencial con una base, b, mayor que 1 (b > 1); puede representarse con la fórmula y = a(1 + r) t, en la que a es el valor inicial, (1 + r) es la tasa de crecimiento, t es el tiempo, y y es el valor final G-8 CCSS IP Math I Teacher Resource 74 © Walch Education PROGRAM OVERVIEW Glossary English expression a combination of variables, quantities, and mathematical operations; 4, 8x, and b + 102 are all expressions. extrema the minima and maxima of a function U2-371 U2-168 F U1-3 U2-296 factor one of two or more numbers or expressions that when multiplied produce a given product first quartile the value that identifies the lower 25% of the data; the median of the lower half of the data set; written as Q 1 U4-4 formula a literal equation that states a specific rule or relationship among quantities function a relation in which every element of the domain is paired with exactly one element of the range; that is, for every value of x, there is exactly one value of y. function notation a way to name a function using f(x) instead of y U1-187 U2-2 U2-371 U2-422 U4-76 U2-3 G U2-457 geometric sequence an exponential function that results in a sequence of numbers separated by a constant ratio graphing method solving a system by graphing equations on the same coordinate plane and finding the point of intersection U3-67 growth factor the multiple by which a quantity increases or decreases over time U2-296 Español expresión combinación de variables, cantidades y operaciones matemáticas; 4, 8x, y b + 102 son todas expresiones. extremos los mínimos y máximos de una función factor uno de dos o más números o expresiones que cuando se multiplican generan un producto determinado primer cuartil valor que identifica el 25% inferior de los datos; mediana de la mitad inferior del conjunto de datos; se expresa Q 1 fórmula ecuación literal que establece una regla específica o relación entre cantidades función relación en la que cada elemento de un dominio se combina con exactamente un elemento del rango; es decir, para cada valor de x, existe exactamente un valor de y. notación de función forma de nombrar una función con el uso de f(x) en lugar de y secuencia geométrica función exponencial que produce como resultado una secuencia de números separados por una proporción constante método de representación gráfica resolución de un sistema mediante graficación de ecuaciones en el mismo plano de coordenadas y hallazgo del punto de intersección factor de crecimiento múltiplo por el que aumenta o disminuye una cantidad con el tiempo G-9 © Walch Education 75 CCSS IP Math I Teacher Resource PROGRAM OVERVIEW Glossary English half plane a region containing all points that has one boundary, a straight line that continues in both directions infinitely histogram a frequency plot that shows the number of times a response or range of responses occurred in a data set. Example: image the new, resulting figure after a transformation included angle the angle between two sides included side the side between two angles of a triangle inclusive a graphed line or boundary is part of an inequality’s solution inconsistent a system of equations with no solutions; lines are parallel when graphed independent a system of equations with exactly one solution Español H U2-83 U4-4 I U5-3 U5-255 U5-312 U5-312 U2-83 U3-67 U3-67 semiplano región que contiene todos los puntos y que tiene un límite, una línea recta que continúa en ambas direcciones de manera infinita histograma diagrama de frecuencia que muestra la cantidad de veces que se produce una respuesta o rango de respuestas en un conjunto de datos. Ejemplo: imagen nueva figura resultante después de una transformación ángulo incluido ángulo entre dos lados lado incluido lado entre dos ángulos de un triángulo inclusivo línea graficada o límite que forma parte de una solución de desigualdad inconsistente sistema de ecuaciones sin soluciones; las líneas son paralelas cuando se las grafica independiente sistema de ecuaciones con una solución exacta G-10 CCSS IP Math I Teacher Resource 76 © Walch Education PROGRAM OVERVIEW Glossary English independent variable labeled on the x-axis; the quantity that changes based on values chosen; the input variable of a function U1-93 U2-3 inequality a mathematical sentence that shows the relationship between quantities that are not equivalent inscribe to draw one figure within another figure so that every vertex of the enclosed figure touches the outside figure integer a number that is not a fraction or a decimal intercept the point at which a line intersects the x- or y-axis interquartile range the difference between the third and first quartiles; 50% of the data is contained within this range interval a continuous series of values inverse a number that when multiplied by the original number has a product of 1 irrational numbers numbers that cannot a be written as , where a and b are b integers and b ≠ 0; any number that U1-33 U1-167 U5-175 U2-168 U2-83 U2-168 U4-4 U2-168 U2-296 U1-187 U2-168 cannot be written as a decimal that ends Español variable independiente designada en el eje x; cantidad que cambia según valores seleccionados; variable de entrada de una función desigualdad declaración matemática que demuestra la relación entre cantidades que no son equivalentes inscribir dibujar una figura dentro de otra de manera que cada vértice de la figura interior toque la exterior entero número que no es una fracción ni un decimal intersección punto en el que una recta corta el eje x o y rango intercuartílico diferencia entre el tercer y primer cuartil; el 50% de los datos está contenido dentro de este rango intervalo serie continua de valores inverso número que cuando se lo multiplica por el número original tiene un producto de 1 números irracionales números que no a pueden expresarse como , en los que a b y b son enteros y b ≠ 0; cualquier número que no puede expresarse como decimal or repeats isometry a transformation in which the preimage and image are congruent U5-3 U5-255 J U4-76 joint frequency the number of times a specific response is given by people with a given characteristic; the cell values in a two-way frequency table finito o periódico isometría transformación en la que la preimagen y la imagen son congruentes frecuencia conjunta cantidad de veces que personas con una determinada característica brindan una respuesta específica; valores de celdas en una tabla de frecuencia de doble entrada G-11 © Walch Education 77 CCSS IP Math I Teacher Resource PROGRAM OVERVIEW Glossary English laws of exponents rules that must be followed when working with exponents Español L U3-2 like terms terms that contain the same variables raised to the same power U1-3 line the set of points between two points P and Q in a plane and the infinite number of points that continue beyond those points; written as PQ U5-3 U5-92 line of reflection the perpendicular bisector of the segments that connect the corresponding vertices of the preimage and the image line of symmetry a line separating a figure into two halves that are mirror images; written as l line segment a line with two endpoints; written as PQ U5-255 line symmetry exists for a figure if for every point on one side of the line of symmetry, there is a corresponding point the same distance from the line linear equation an equation that can be written in the form ax + by = c, where a, b, and c are rational numbers; can also be written as y = mx + b, in which m is the slope, b is the y-intercept, and the graph is a straight line. The solutions to the linear equation are the infinite set of points on the line. linear fit (or linear model) an approximation of data using a linear function U5-3 U5-3 U5-3 U1-33 U1-93 U2-3 U2-371 U4-175 leyes de los exponentes normas que deben cumplirse cuando se trabaja con exponentes términos semejantes términos que contienen las mismas variables elevadas a la misma potencia línea recta conjunto de puntos entre dos puntos P y Q en un plano y cantidad infinita de puntos que continúan más allá de esos puntos; se expresa como PQ línea de reflexión bisectriz perpendicular de los segmentos que conectan los vértices correspondientes de la preimagen y la imagen línea de simetría línea que separa una figura en dos mitades que son imágenes en espejo; se expresa como l segmento de recta recta con dos extremos; se expresa como PQ simetría lineal la que existe en una figura si para cada punto a un lado de la línea de simetría, hay un punto correspondiente a la misma distancia de la línea ecuación lineal ecuación que puede expresarse en la forma ax + by = c, en la que a, b, y c son números racionales; también puede escribirse como y = mx + b, en donde m es la pendiente, b es el intercepto de y, y la gráfica es una línea recta. Las soluciones de la ecuación lineal son el conjunto infinito de puntos en la recta. ajuste lineal (o modelo lineal) aproximación de datos con el uso de una función lineal G-12 CCSS IP Math I Teacher Resource 78 © Walch Education PROGRAM OVERVIEW Glossary English linear function a function that can be written in the form f(x) = mx + b, in which m is the slope, b is the y-intercept, and the graph is a straight line literal equation an equation that involves two or more variables marginal frequency the total number of times a specific response is given, or the total number of people with a given characteristic mean the average value of a data set, found by summing all values and dividing by the number of data points U2-246 U2-296 U2-484 U1-187 M U4-76 U4-4 mean absolute deviation the average absolute value of the difference between each data point and the mean; found by summing the absolute value of the difference between each data point and the mean, then dividing this sum by the total number of data points U4-4 measures of center values that describe expected and repeated data values in a data set; the mean and median are two measures of center measures of spread a measure that describes the variance of data values, and identifies the diversity of values in a data set median 1. the middle-most value of a data set; 50% of the data is less than this value, and 50% is greater than it 2. the segment joining the vertex to the midpoint of the opposite side U4-4 U4-4 U4-4 U5-92 Español función lineal función que puede expresarse en la forma f(x) = mx + b, en la que m es la pendiente, b es el intercepto de y, y la gráfica es una línea recta ecuación literal ecuación que incluye dos o más variables frecuencia marginal cantidad total de veces que se da una respuesta específica, o cantidad total de personas con una determinada característica media valor promedio de un conjunto de datos, que se determina al sumar todos los valores y dividirlos por la cantidad de puntos de datos desviación media absoluta valor promedio absoluto de la diferencia entre cada punto de datos y la media; se determina mediante la suma del valor absoluto de la diferencia entre cada punto de datos y la media, y luego se divide esta suma por la cantidad total de puntos de datos medidas de centro valores que describen los valores de datos esperados y repetidos de un conjunto de datos; la media y la mediana son dos medidas de centro medidas de dispersión medidas que describen la varianza de los valores de datos e identifican la diversidad de valores en un conjunto de datos mediana 1. valor medio exacto de un conjunto de datos; el 50% de los datos es menor que ese valor, y el otro 50% es mayor 2. segmento que une el vértice con el punto medio del lado opuesto G-13 © Walch Education 79 CCSS IP Math I Teacher Resource PROGRAM OVERVIEW Glossary English midpoint a point on a line segment that divides the segment into two equal parts midsegment a line segment joining the midpoints of two sides of a figure natural numbers the set of positive integers {1, 2, 3, …, n} negative function a portion of a function where the y-values are less than 0 for all x-values non-inclusive a graphed line or boundary is not part of an inequality’s solution non-rigid motion a transformation done to a figure that changes the figure’s shape and/or size obtuse angle an angle measuring greater than 90° but less than 180° one-to-one a relationship wherein each point in a set of points is mapped to exactly one other point order of operations the order in which expressions are evaluated from left to right (grouping symbols, evaluating exponents, completing multiplication and division, completing addition and subtraction) ordered pair a pair of values (x, y) where the order is significant outlier a data value that is much greater than or much less than the rest of the data in a data set; mathematically, any data less than Q 1 – 1.5(IQR) or greater than Q 3 + 1.5(IQR) is an outlier U5-92 U5-92 N U2-139 U2-168 U2-168 U2-83 U5-255 O U5-3 U5-3 U1-3 U2-3 U4-4 Español punto medio punto en un segmento de recta que lo divide en dos partes iguales segmento medio segmento de recta que une los puntos medios de dos lados de una figura números naturales conjunto de enteros positivos {1, 2, 3, …, n} función negativa porción de una función en la que los valores y son menores que 0 para todos los valores x no inclusivo línea graficada o límite que no forma parte de una solución de desigualdad movimiento no rígido transformación hecha a una figura que cambia su forma o tamaño ángulo obtuso ángulo que mide más de 90˚ pero menos de 180˚ unívoca relación en la que cada punto de un conjunto de puntos se corresponde con otro con exactitud orden de las operaciones orden en el que se evalúan las expresiones de izquierda a derecha (con agrupación de símbolos, evaluación de exponentes, realización de multiplicaciones y divisiones, sumas y sustracciones) par ordenado par de valores (x, y), en los que el orden es significativo valor atípico valor de datos que es mucho mayor o mucho menor que el resto de los datos de un conjunto de datos; en matemática, cualquier dato menor que Q 1 – 1,5(IQR) o mayor que Q 3 + 1,5(IQR) es un valor atípico G-14 CCSS IP Math I Teacher Resource 80 © Walch Education PROGRAM OVERVIEW Glossary English Español parallel lines that never intersect and have equal slope P U6-2 parallel lines lines in a plane that either do not share any points and never intersect, or share all points; written as AB PQ U5-3 U5-92 parallelogram a quadrilateral with opposite sides parallel parameter a term in a function that determines a specific form of a function but not the nature of the function perimeter the distance around a twodimensional figure perpendicular lines that intersect at a right angle (90˚); their slopes are opposite reciprocals U6-2 U2-484 U6-58 U6-2 perpendicular bisector a line constructed through the midpoint of a segment U5-92 perpendicular lines two lines that intersect at a right angle (90°); written as AB ⊥ PQ U5-3 U5-92 point an exact position or location in a given plane point of intersection the point at which two lines cross or meet point of rotation the fixed location that an object is turned around; the point can lie on, inside, or outside the figure polygon two-dimensional figure with at least three sides positive function a portion of a function where the y-values are greater than 0 for all x-values U5-3 U3-67 U5-255 U6-58 U2-168 paralelas líneas que nunca llegan a cortarse y tienen la misma pendiente líneas paralelas líneas en un plano que no comparten ningún punto y nunca se cortan, o que compartentodos los puntos; se expresan como AB PQ paralelogramo cuadrilátero con lados opuestos paralelos parámetro término en una función que determina una forma específica de una función pero no su naturaleza perímetro distancia alrededor de una figura bidimensional perpendiculares líneas que se cortan en ángulo recto (90˚); sus pendientes son recíprocas opuestas bisectriz perpendicular línea que se construye a través del punto medio de un segmento líneas perpendiculares dos líneas que se cortan en ángulo recto (90˚); se expresan como AB ⊥ PQ punto posición o ubicación exacta en un plano determinado punto de intersección punto en que se cruzan o encuentran dos líneas punto de rotación ubicación fija en torno a la que gira un objeto; el punto puede estar encima, dentro o fuera de la figura polígono figura bidimensional con al menos tres lados función positiva porción de una función en la que los valores y son mayores que 0 para todos los valores x G-15 © Walch Education 81 CCSS IP Math I Teacher Resource PROGRAM OVERVIEW Glossary English postulate a true statement that does not require a proof preimage the original figure before undergoing a transformation properties of equality rules that allow you to balance, manipulate, and solve equations properties of inequality rules that allow you to balance, manipulate, and solve inequalities quadrant the coordinate plane is separated into four sections: • In Quadrant I, x and y are positive. • In Quadrant II, x is negative and y is positive. • In Quadrant III, x and y are negative. • In Quadrant IV, x is positive and y is negative. quadrilateral a polygon with four sides quantity something that can be compared by assigning a numerical value radius a line segment that extends from the center of a circle to a point on the circle. Its length is half the diameter. range the set of all outputs of a function; the set of y-values that are valid for the function rate a ratio that compares different kinds of units rate of change a ratio that describes how much one quantity changes with respect to the change in another quantity; also known as the slope of a line U5-312 U5-3 U5-255 U3-2 U3-2 Q U5-58 U6-2 U1-34 R U5-175 U2-3 U1-34 U2-168 U2-296 Español postulado afirmación verdadera que no requiere prueba preimagen figura original antes de sufrir una transformación propiedades de igualdad normas que permiten equilibrar, manipular y resolver ecuaciones propiedades de desigualdad normas que permiten equilibrar, manipular y resolver desigualdades cuadrante plano de coordenadas que se divide en cuatro secciones: • En el cuadrante I, x y y son positivos. • En el cuadrante II, x es negativo y y es positivo. • En el cuadrante III, x y y son negativos. • En el cuadrante IV, x es positivo y y es negativo. cuadrilátero polígono con cuatro lados cantidad algo que puede compararse al asignarle un valor numérico radio segmento de línea que se extiende desde el centro de un círculo hasta un punto de la circunferencia del círculo. Su longitud es la mitad del diámetro. rango conjunto de todas las salidas de una función; conjunto de valores y válidos para la función tasa proporción en que se comparan distintos tipos de unidades tasa de cambio proporción que describe cuánto cambia una cantidad con respecto al cambio de otra cantidad; también se la conoce como pendiente de una recta G-16 CCSS IP Math I Teacher Resource 82 © Walch Education PROGRAM OVERVIEW Glossary English ratio the relation between two quantities; can be expressed in words, fractions, decimals, or as a percent rational number a number that can be a written as , where a and b are integers b and b ≠ 0; any number that can be U2-168 U2-168 Español proporción relación entre dos cantidades; puede expresarse en palabras, fracciones, decimales o como porcentaje número racional número que puede a expresarse como , en los que a y b son b enteros y b ≠ 0; cualquier número que puede escribirse como decimal finito o written as a decimal that ends or repeats ray a line with only one endpoint; written as PQ U5-3 U5-92 real numbers the set of all rational and irrational numbers reciprocal a number that when multiplied by the original number has a product of 1 U2-168 rectangle a parallelogram with opposite sides that are congruent and consecutive sides that are perpendicular recursive formula a formula used to find the next term of a sequence when the previous term or terms are known; the recursive formula for an arithmetic sequence is an = an – 1 + d; the recursive formula for a geometric sequence is an = an – 1 • r U6-2 U1-187 U2-139 U2-457 reflection a transformation where a mirror image is created; also called a flip; an isometry in which a figure is moved along a line perpendicular to a given line called the line of reflection regular hexagon a six-sided polygon with all sides equal and all angles measuring 120˚ U5-3 U5-58 U5-175 periódico semirrecta línea con un solo extremo; se expresa como PQ números reales conjunto de todos los números racionales e irracionales recíproco número que cuando se lo multiplica por el número original tiene un producto de 1 rectángulo paralelogramo con lados opuestos congruentes y lados consecutivos que son perpendiculares fórmula recursiva fórmula que se utiliza para encontrar el término siguiente de una secuencia cuando se conoce el o los términos anteriores; la fórmula recursiva de una secuencia aritmética es an = an – 1 + d; la fórmula recursiva para una secuencia geométrica es an = an – 1 • r reflexión transformación por la cual se crea una imagen en espejo; isometría en la que se mueve una figura a lo largo de una línea perpendicular hacia una recta determinada llamada línea de reflexión hexágono regular polígono de seis lados con todos los lados iguales y en el que todos los ángulos miden 120˚ G-17 © Walch Education 83 CCSS IP Math I Teacher Resource PROGRAM OVERVIEW Glossary English regular polygon a two-dimensional figure with all sides and all angles congruent relation a relationship between two sets of elements relative maximum the greatest value of a function for a particular interval of the function relative minimum the least value of a function for a particular interval of the function residual the vertical distance between an observed data value and an estimated data value on a line of best fit residual plot provides a visual representation of the residuals for a set of data; contains the points (x, residual for x) rhombus a parallelogram with four congruent sides right angle an angle measuring 90˚ rigid motion a transformation done to a figure that maintains the figure’s shape and size or its segment lengths and angle measures rotation a transformation that turns a figure around a point; also called a turn; an isometry where all points in the preimage are moved along circular arcs determined by the center of rotation and the angle of rotation U5-175 U5-3 U2-3 U2-168 U2-168 U4-77 U4-77 U6-3 U5-3 U5-255 U5-313 U5-3 U5-58 Español polígono regular figura bidimensional con todos los lados y todos los ángulos congruentes relación conexión entre dos conjuntos de elementos máximo relativo el mayor valor de una función para un intervalo particular de la función mínimo relativo el menor valor de una función para un intervalo particular de la función residual distancia vertical entre un valor de datos observado y un valor de datos estimado sobre una línea de ajuste óptimo diagrama residual brinda una representación visual de los residuales para un conjunto de datos; contiene los puntos (x, residual de x) rombo paralelograma con cuatro lados congruentes ángulo recto ángulo que mide 90˚ movimiento rígido transformación que se realiza a una figura que mantiene su forma y tamaño o las longitudes de sus segmentos y las medidas de ángulos rotación transformación que hace girar una figura alrededor de un punto; isometría en la que todos los puntos de la preimagen se mueven a lo largo de arcos circulares determinados por el centro de rotación y el ángulo de rotación G-18 CCSS IP Math I Teacher Resource 84 © Walch Education PROGRAM OVERVIEW Glossary English Español S U5-255 scale factor a multiple of the lengths of the sides from one figure to the transformed figure. If the scale factor is larger than 1, then the figure is enlarged. If the scale factor is between 0 and 1, then the figure is reduced. scatter plot a graph of data in two variables on a coordinate plane, where each data pair is represented by a point U4-77 segment a part of a line that is noted by two endpoints sequence an ordered list of numbers side-angle-side (SAS) if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent side-side-side (SSS) if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent sketch a quickly done representation of a figure; a rough approximation of a figure skewed to the left data concentrated on the higher values in the data set, which has a tail to the left. Example: 20 24 28 32 36 U5-92 U2-139 U5-313 U5-313 U5-92 U4-5 40 factor de escala múltiplo de las longitudes de los lados de una figura a la figura transformada. Si el factor de escala es mayor que 1, entonces la figura se agranda. Si el factor de escala se encuentra entre 0 y 1, entonces la figura se reduce. diagrama de dispersión gráfica de datos en dos variables en un plano de coordenadas, en la que cada par de datos está representado por un punto segmento parte de una recta comprendida entre dos extremos secuencia lista ordenada de números lado-ángulo-lado (SAS) si dos lados y el ángulo incluido de un triángulo son congruentes con dos lados y el ángulo incluido de otro triángulo, entonces los dos triángulos son congruentes lado-lado-lado (SSS) si los tres lados de un triángulo son congruentes con los tres lados de otro triángulo, entonces los dos triángulos son congruentes bosquejo representación de una figura realizada con rapidez; aproximación imprecisa de una figura desviados hacia la izquierda datos concentrados en los valores más altos del conjunto de datos, que tiene una cola hacia la izquierda. Ejemplo: 20 24 28 32 36 40 G-19 © Walch Education 85 CCSS IP Math I Teacher Resource PROGRAM OVERVIEW Glossary English skewed to the right data concentrated on the lower values in the data set, which has a tail to the right. Example: 20 24 28 32 36 U4-5 20 40 slope the measure of the rate of change of one variable with respect to another yy22 −− yy11 yy rise rise variable; slope = ; the == == xx22 −−xx11 run xx run slope in the equation y = mx + b is m. slope-intercept method the method used to graph a linear equation; with this method, draw a line using only two points on the coordinate plane solution a value that makes the equation true solution set the value or values that make a sentence or statement true; the set of ordered pairs that represent all of the solutions to an equation or a system of equations solution to a system of linear inequalities the intersection of the half planes of the inequalities; the solution is the set of all points that make all the inequalities in the system true square a parallelogram with four congruent sides and four right angles straightedge a bar or strip of wood, plastic, or metal having at least one long edge of reliable straightness Español desviados hacia la derecha datos concentrados en los valores más bajos del conjunto de datos, que tiene una cola hacia la derecha. Ejemplo: U1-94 U2-168 U2-296 U2-372 U4-175 U6-3 U2-168 U1-34 U1-34 U1-167 U2-3 U2-83 U5-175 U6-3 U5-92 24 28 32 36 40 pendiente medida de la tasa de cambio de una variable con respecto a otra; yy22 −− yy11 yy rise rise pendiente = == =;=la pendiente xx22 −−xx11 xx run run en la ecuación y = mx + b es m método pendiente-intercepto método utilizado para graficar una ecuación lineal; con este método, se dibuja una línea con sólo dos puntos en un plano de coordenadas solución valor que hace verdadera la ecuación conjunto de soluciones valor o valores que hacen verdadera una afirmación o declaración; conjunto de pares ordenados que representa todas las soluciones para una ecuación o sistema de ecuaciones solución a un sistema de desigualdades lineales intersección de los medios planos de las desigualdades; la solución es el conjunto de todos los puntos que hacen verdaderas todas las desigualdades de un sistema cuadrado paralelograma con cuatro lados congruentes y cuatro ángulos rectos regla de borde recto barra o franja de madera, plástico o metal que tiene, al menos, un borde largo de rectitud confiable G-20 CCSS IP Math I Teacher Resource 86 © Walch Education PROGRAM OVERVIEW Glossary English substitution method solving one of a pair of equations for one of the variables and substituting that into the other equation symmetric situation in which data is concentrated toward the middle of the range of data; data values are distributed in the same way above and below the middle of the sample. Example: 20 24 28 32 36 U3-67 U4-5 Español método de sustitución solución de un par de ecuaciones para una de las variables y sustitución de eso en la otra ecuación simétrico situación en la que los datos se concentran hacia el medio del rango de datos; los valores de datos se distribuyen de la misma manera por encima y por debajo del medio de la muestra. Ejemplo: 20 40 system a set of more than one equation system of equations a set of equations with the same unknowns system of inequalities two or more inequalities in the same variables that work together U2-3 U1-167 U3-67 U1-167 U2-83 term a number, a variable, or the product of a number and variable(s) third quartile value that identifies the upper 25% of the data; the median of the upper half of the data set; 75% of all data is less than this value; written as Q 3 transformation a change in a geometric figure’s position, shape, or size translation moving a graph either vertically, horizontally, or both, without changing its shape; a slide; an isometry where all points in the preimage are moved parallel to a given line T U1-3 U4-5 U2-422 U5-3 U2-422 U5-3 U5-58 24 28 32 36 40 sistema conjunto de más de una ecuación sistema de ecuaciones conjunto de ecuaciones con las mismas incógnitas sistema de desigualdades dos o más desigualdades en las mismas variables que operan juntas término número, variable o producto de un número y una o más variables tercer cuartil valor que identifica el 25% superior de los datos; mediana de la mitad superior del conjunto de datos; el 75% de los datos es menor que este valor; se expresa como Q 3 transformación cambio en la posición, la forma o el tamaño de una figura geométrica traslación movimiento de un gráfico en sentido vertical, horizontal, o en ambos, sin modificar su forma; deslizamiento; isometría en la que todos los puntos de la preimagen se mueven en paralelo a una línea determinada G-21 © Walch Education 87 CCSS IP Math I Teacher Resource PROGRAM OVERVIEW Glossary English trend a pattern of behavior, usually observed over time or over multiple iterations triangle a three-sided polygon with three angles two-way frequency table a table that divides responses into categories, showing both a characteristic in the table rows and a characteristic in the table columns; values in cells are a count of the number of times each response was given by a respondent with a certain characteristic undefined slope the slope of a vertical line unit rate a rate per one given unit variable a letter used to represent a value or unknown quantity that can change or vary vertical shift number of units the graph of the function is moved up or down; a translation whole numbers the set of natural numbers that also includes 0: {0, 1, 2, 3, ...} x-intercept the point at which the line intersects the x-axis at (x, 0) U4-77 U5-175 U4-77 U U2-168 U1-34 V U1-3 U1-34 U2-372 U2-422 U2-484 W U2-168 X U1-94 U2-83 U2-168 U2-296 Español tendencia patrón de comportamiento, que se observa por lo general en el tiempo o en múltiples repeticiones triángulo polígono de tres lados con tres ángulos tabla de frecuencia de doble entrada tabla que divide las respuestas en dos categorías, y muestra una característica en las filas y una en las columnas; los valores de las celdas son un conteo de la cantidad de veces que un respondedor da una respuesta con una determinada característica pendiente indefinida pendiente de una línea vertical tasa unitaria tasa de una unidad determinada variable letra utilizada para representar un valor o una cantidad desconocida que puede cambiar o variar desplazamiento vertical cantidad de unidades que el gráfico de la función se desplaza hacia arriba o hacia abajo; traslación números enteros conjunto de números naturales que incluye el 0: {0, 1, 2, 3, ...} intercepto de x punto en el que una recta corta el eje x en (x, 0) G-22 CCSS IP Math I Teacher Resource 88 © Walch Education PROGRAM OVERVIEW Glossary English Español y-intercept the point at which a line or curve intersects the y-axis at (0, y); the y-intercept in the equation y = mx + b is b. Y U1-94 U2-83 U2-246 U2-296 U2-372 U4-175 intercepto de y punto en el que una recta o curva corta el eje y en (0, y); el intercepto de y en la ecuación y = mx + b es b. G-23 © Walch Education 89 CCSS IP Math I Teacher Resource 90