Mathematics Task Arcs


 Edwina Edwards
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1 Overview of Mathematics Task Arcs: Mathematics Task Arcs A task arc is a set of related lessons which consists of eight tasks and their associated lesson guides. The lessons are focused on a small number of standards within a domain of the Common Core State Standards for Mathematics. In some cases, a small number of related standards from more than one domain may be addressed. A unique aspect of the task arc is the identification of essential understandings of mathematics. An essential understanding is the underlying mathematical truth in the lesson. The essential understandings are critical later in the lesson guides, because of the solution paths and the discussion questions outlined in the share, discuss, and analyze phase of the lesson are driven by the essential understandings. The Lesson Progression Chart found in each task arc outlines the growing focus of content to be studied and the strategies and representations students may use. The lessons are sequenced in deliberate and intentional ways and are designed to be implemented in their entirety. It is possible for students to develop a deep understanding of concepts because a small number of standards are targeted. Lesson concepts remain the same as the lessons progress; however the context or representations change. Bias and sensitivity: Social, ethnic, racial, religious, and gender bias is best determined at the local level where educators have indepth knowledge of the culture and values of the community in which students live. The TDOE asks local districts to review these curricular units for social, ethnic, racial, religious, and gender bias before use in local schools. Copyright: These task arcs have been purchased and licensed indefinitely for the exclusive use of Tennessee educators.
2 mathematics Geometry Investigating Coordinate Geometry and Its Use in Solving Mathematical Problems A SET OF RELATED S UNIVERSITY OF PITTSBURGH
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4 Table of Contents 3 Table of Contents Introduction Overview... 7 Identified CCSSM and Essential Understandings... 8 Tasks CCSSM Alignment Lesson Progression Chart Tasks and Lesson Guides TASK 1: My Point is That There Are Many Points! Lesson Guide TASK 2: The Distance Between Us Lesson Guide TASK 3: Will That Work for ANY Two Points? Lesson Guide TASK 4: Building a New Playground Lesson Guide TASK 5: Go Fly a Kite Lesson Guide TASK 6: However You Want to Slice It! Lesson Guide TASK 7: How Should We Divide This? Lesson Guide TASK 8: Partitioning Lesson Guide... 62
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6 Tasks and Lesson Guides mathematics Ratios and Proportions Grade 6 Geometry <ADD ARTWORK FROM THE COVER> Introduction Investigating Coordinate Geometry and Its Use in Solving Mathematical Problems A SET OF RELATED S
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8 Introduction 7 Overview In this set of related tasks, students explore coordinate geometry as a mechanism for proving simple geometric theorems and solving problems. They examine relationships between and among line segments and use the Pythagorean Theorem to develop common formulas associated with coordinate geometry. They learn how to generalize on the coordinate plane using (x 1, y 1 ) notation, and develop techniques for finding distance and dividing a segment into an a:b ratio. The first two tasks allow students to explore midpoint and distance before generalizing formulas for finding them. Task 3 solidifies understanding of how to determine the midpoint of a segment and its length, while introducing notation that will allow students to generalize their results in later tasks. The remaining tasks continue to solidify understanding of midpoint and distance concepts by using them to develop geometric theorems. Task 4 introduces the perpendicular bisector of a line segment by asking students to determine points that are equidistant from two points in a contextual situation. Task 5 encourages students to examine the converse in a different context. In Task 6, students use their understanding of midpoint and distance to explore midsegments, then in Task 7, move forward to segments divided into ratios other than 1:1. Finally, in Task 8, students are asked to solidify understanding of midpoint, distance, and dividing a segment into an a:b ratio. The tasks are aligned to the G.GPE.B.4, G.GPE.B.5, G.GPE.B.6, and G.GPE.B.7 Content Standards of the CCSSM. The prerequisite knowledge necessary to enter these lessons includes an understanding of the meaning of triangle congruence, recognition of characteristics of parallel and perpendicular lines, and familiarity with plotting points, calculating slope, and using the Pythagorean Theorem. Through engaging in the lessons in this set of related tasks, students will: generalize midpoint and distance formulas using coordinate geometry; use coordinate geometry to explore patterns and make conjectures in the context of midpoint, distance, perpendicular bisectors, and midsegment; and use coordinate geometry to prove or disprove conjectures. By the end of these lessons, students will be able to answer the following overarching questions: How is coordinate geometry used to prove or disprove conjectures? How is coordinate geometry used to generalize mathematical formulas? The questions provided in the guide will make it possible for students to work in ways consistent with the Standards for Mathematical Practice. It is not our expectation that students will name the Standards for Mathematical Practice. Instead, the teacher can mark agreement and disagreement of mathematical reasoning or identify characteristics of a good explanation (MP3). The teacher can note and mark times when students independently provide an equation and then recontextualize the equation in the context of the situational problem (MP2). The teacher might also ask students to reflect on the benefit of using repeated reasoning, as this may help them understand the value of this mathematical practice in helping them see patterns and relationships (MP8). In study groups, topics such as these should be discussed regularly because the lesson guides have been designed with these ideas in mind. You and your colleagues may consider labeling the questions in the guide with the Standards for Mathematical Practice.
9 8 Introduction Identified CCSSM and Essential Understandings CCSS for Mathematical Content: Expressing Geometric Properties with Equations. Essential Understandings Use coordinates to prove simple geometric theorems algebraically. GGPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). Coordinate geometry can be used to prove geometric theorems because it is possible to replace specific coordinates with variables to show that a relationship remains true, regardless of the coordinates. In some situations, both a theorem and its converse are true and can be shown to be so using coordinate geometry. When both statements, if a then b (a4 b) and if b then a (b4 a) are true, the notations a iff b and a 1 b are used to show this relationship. The use of coordinate geometry techniques for finding length and slope verify that a line segment that connects two sides of a triangle, each divided into an m:n ratio, is parallel to and m/n of the length of the third side.
10 Introduction 9 CCSS for Mathematical Content: Expressing Geometric Properties with Equations. Essential Understandings GGPE.B.5 GGPE.B.6 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Find the point on a directed line segment between two given points that partitions the segment in a given ratio. The set of points that are equidistant from two points A and B lie on the perpendicular bisector of line segment AB, because every point on the perpendicular bisector can be used to construct two triangles that are congruent by definition of triangle congruence, reflection and/or Side AngleSide; corresponding parts of congruent triangles are congruent. For any point C that lies on the perpendicular bisector of points A and B, C is equidistant from points A and B because the perpendicular bisector divides triangle ABC into two congruent right triangles. A midsegment connects the midpoints of two sides of a triangle and divided the side lengths into a 1:2 ratio. The use of coordinate geometry techniques for finding length and slope verify that a midsegment s length is half of the third side and it is also parallel to the third side. A midpoint of a line is a point that partitions the segment into two segments that have the same length. Each of these segments is in a 1:2 ratio with the whole segment. For a line segment with endpoints at A(x 1,y 1 ) and B(x 2,y 2 ), since the midpoint is in the middle of segment AB, it is located at, because the midpoint represents the average value for both the x and yvalues of the two coordinates of segment AB. On any line segment AB it is possible to locate a point C such that the ratio of AC:AB is m:n. If the endpoints of segment AB are located at (x 1,y 1 ) and (x 2,y 2 ). C is located at the point, because C is located m/n of the, distance of the segment from the endpoint in both the horizontal and vertical directions.
11 10 Introduction CCSS for Mathematical Content: Expressing Geometric Properties with Equations. Essential Understandings GGPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. The distance between two points on a coordinate plane is the length of the line segment that connects them and is the count of the number of units that form the line. The length of a line segment parallel to the xaxis (yaxis) can be determined by subtracting the values of the xcoordinates (ycoordinates) of the points forming the segment because subtraction counts the number of units in the segment. The length of a diagonal line segment can be determined by utilizing the line segment as the hypotenuse of a right triangle and computing its length using the Pythagorean Theorem. The Distance Formula can then be derived by replacing specific coordinates with variables in such a situation. The perimeter of the triangle formed by connecting points on two sides of a triangle whose side lengths are in an m:n ratio will also be in an m:n ratio because each side length in such a triangle is m/n of the lengths of the sides of the original triangle.
12 Introduction 11 The CCSS for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. *Common Core State Standards, 2010, NGA Center/CCSSO
13 12 Introduction Tasks CCSSM Alignment Task G.GPE.B.4 G.GPE.B.4 G.GPE.B.6 G.GPE.B.7 Task 1 My Point is That There Are Many Points! Developing Understanding Task 2 The Distrance Between Us Developing Understanding Task 3 Will That Work for ANY Two Points? Solidifying Understanding Task 4 Building a New Playground Developing Understanding Task 5 Go Fly a Kite Developing Understanding Task 6 However You Want to Slice It! Solidifying Understanding Task 7 How Should We Divide This? Developing Understanding Task 8 Partitioning Solidifying Understanding
14 Introduction 13 Task MP 1 MP 2 MP 3 MP 4 MP 5 MP 6 MP 7 MP 8 Task 1 My Point is That There Are Many Points! Developing Understanding Task 2 The Distrance Between Us Developing Understanding Task 3 Will That Work for ANY Two Points? Solidifying Understanding Task 4 Building a New Playground Developing Understanding Task 5 Go Fly a Kite Developing Understanding Task 6 However You Want to Slice It! Solidifying Understanding Task 7 How Should We Divide This? Developing Understanding Task 8 Partitioning Solidifying Understanding
15 14 Introduction Lesson Progression Chart Overarching Questions How is coordinate geometry used to prove or disprove conjectures? How is coordinate geometry used to generalize mathematical formulas? TASK 1 My Point is That There Are Many Points! Developing Understanding TASK 2 The Distance Between Us Developing Understanding TASK 3 Will That Work for ANY Two Points? Solidifying Understanding TASK 4 Building a New Playground Developing Understanding Content Explores the concept of midpoint as a point that partitions a segment into two equal segments. Discusses the strategy of using coordinates to verify lengths. Deepens understanding of the use of coordinates to verify conjectures by examining lengths of segments on the coordinate plane. Solidifies understanding of the use of coordinate geometry to verify conjectures by using coordinate geometry to generalize the midpoint formula and the distance formula. Explores the use of coordinate geometry to make and verify conjectures about points equidistant from two points on a coordinate plane. Strategy Students use informal strategies to determine midpoints and endpoints, and then use coordinate geometry to support and justify the informal strategies. Students disprove incorrect conjectures using counterexamples and the reasoning of coordinate geometry. They make and prove conjectures. Students use coordinate geometry to generalize the midpoint formula and explore the use of the Pythagorean Theorem to generalize the distance formula. Students use informal strategies to determine the points along the perpendicular bisector. The midpoint and distance formulas, the Pythagorean Theorem, and congruent triangles may be used. Representations Students use graphs, coordinates, and equations. Students use graphs, coordinates, and equations. Students use graphs, coordinates, and equations. Students begin with a context and graph and move to using equations and graphs.
16 Introduction 15 TASK 5 Go Fly a Kite Developing Understanding TASK 6 However You Want to Slice It! Developing Understanding TASK 7 How Should We Divide This? Developing Understanding TASK 8 Partitioning Solidifying Understanding Content Uses coordinate geometry to examine the concept of converse in the case of the perpendicular bisector. Introduces iff and 1 notation. Uses coordinate geometry to explore partitions other than in a 1:2 ratio; builds off of midpoint s 1:2 ratio. Uses coordinate geometry to make and test conjectures about the midsegments of a triangle and to examine perimeter. Explores ratios other than 1:2. Solidifies understanding about the use of coordinate geometry in generalization. Solidifies understanding about partitioning and midsegments. Strategy Students use informal strategies and geometric tools such as patty paper, rulers, and protractors to examine a converse statement. They may also use coordinates. Students use informal strategies, coordinates, and geometric tools such as patty paper, rulers, and protractors to partition segments. Students use coordinate proofs, explanations, and counterexamples to verify conjectures about midsegments. Students use coordinate proofs. Representations Students begin with a context and use graphs, coordinates, and equations. Students use graphs, coordinates, and equations. Students use graphs, coordinates, and equations. Students use graphs, coordinates, and equations.
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18 Tasks and Lesson Guides mathematics Ratios and Proportions Grade 6 Geometry <ADD ARTWORK FROM THE COVER> Tasks and Lesson Guides Investigating Coordinate Geometry and Its Use in Solving Mathematical Problems A SET OF RELATED S
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20 Tasks and Lesson Guides 19 Name My Point is That There Are Many Points! TASK 1 1. Consider the 3 line segments through point A (4,2) on the graph below. Label the endpoints of each segment and identify their coordinates A Use coordinates to justify how you know A is the midpoint of each line segment. 2. Given a point B with coordinates (x,y). A. Draw two different line segments (AC) and (DE) with different slopes such that B is the midpoint of each line segment. B. Write the coordinates for points A, C, D, and E in terms of x and y. C. Use the coordinates of the endpoints to prove that B is the midpoint of each segment.
21 20 Tasks and Lesson Guides 1 My Point is That There Are Many Points! Rationale for Lesson: Explore the idea that a midpoint partitions a segment into two segments of equal length. Introduce coordinate geometry as a tool for justifying reasoning and making generalizations. Task: My Point Is That There Are Many Points! 1. Consider the 3 line segments through point A (4,2) on the graph below. Label the endpoints of each segment and identify their coordinates A Use coordinates to justify how you know A is the midpoint of each line segment. See student paper for complete task. Common Core Content Standards G.GPE.B.4 G.GPE.B.6 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Standards for Mathematical Practice Essential Understandings MP1 Make sense of problems and persevere in solving them. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP6 Attend to precision. MP8 Look for and express regularity in repeated reasoning. Coordinate geometry can be used to prove geometric theorems because it is possible to replace specific coordinates with variables to show that a relationship remains true regardless of the coordinates. A midpoint of a line is a point that partitions the segment into two segments that have the same length. Each of these segments is in a 1:2 ratio with the whole segment. For a line segment with endpoints at A(x 1,y 1 ) and B(x 2,y 2 ), since the midpoint is in the middle of segment AB, it is located at, because the midpoint represents the average value for both the x and yvalues of the two coordinates of AB.
22 Tasks and Lesson Guides 21 Materials Needed Graph paper Straight edge 1 SETUP PHASE Please read the task aloud. In your own words, define midpoint. We will take three minutes of private think time to begin. Then, you will work with your group to complete the task. There is a container of rulers, protractors, graph paper, and any other supplies that you might need at each table. EXPLORE PHASE Possible Student Pathways Group can t get started. Counts spaces on the grid using only horizontal and/ or vertical lines. Draws slope triangles (counts up and over). Uses the Pythagorean Theorem to determine length. Assessing Questions What does it mean for a point to be the midpoint of a segment? Can you explain your method to me? How do you know A is the midpoint of your segments? Can you explain your method to me? How are you using them to determine length? How does this relate to midpoint? What does your work say about the lengths of the segments? How does it verify or refute that A is the midpoint of the segments? Advancing Questions What does it mean to be in the middle? How do you think the xcoordinate of the midpoint compares to the xcoordinate of the endpoints? What about the ycoordinate? What can you tell me about the length of the green segment from (0, 0) to point A? How can you decide how it compares to the length of the rest of the segment? Are the triangles congruent? How do you know? How does this tell you that the point is the midpoint? Can you always use this method to verify that a point is a midpoint? What do you expect might happen if you replace your endpoints with variables? Why?
23 22 Tasks and Lesson Guides 1 SHARE, DISCUSS, AND ANALYZE PHASE EU: A midpoint of a line is a point that partitions the segment into two segments that have the same length. Each of these segments is in a 1:2 ratio with the whole segment. Tell us about how you used a counting method to find the endpoints when given the midpoint. Who can restate what this group just shared about counting up and over? Who can show us how this counting method can be used to locate the endpoints in #2? (You can count up and over in each direction, making sure that the x and yvalues are the same distance from the midpoint). How do you know the segment is in a 1:2 ratio? I saw another group using triangles. Tell us about how you used the triangles to determine the endpoints. Who can restate what this group shared about congruent triangles? Who can come up here and show us how these triangles are congruent? How can we use congruent triangles to prove B is the midpoint? (If the triangles are congruent, we can lay one directly on top of the other. This means that the midpoint is directly in the middle because the distance on each side is the same). How are the counting method and the triangle method similar? How are they different? How do the segments on either side compare to the original segment? What is the ratio of the length of the smaller segments to the whole? How do you know? So, we explored two methods for constructing a segment with a given midpoint. Starting at the midpoint, we can count up and over in one direction and then count the same amount down and over in the other direction. Another strategy is to draw congruent slope triangles with one vertex at the midpoint extending in opposite directions. (Recapping) How can we write the coordinates in terms of x and y? You stated that the endpoints are (x + 2, y  1) and (x  2, y + 1) in this specific example. How do we know that (x,y) is the midpoint?
24 Tasks and Lesson Guides 23 EU: For a line segment with endpoints at A(x 1,y 1 ) and B(x 2,y 2 ), since the midpoint is in the middle of segment AB, it is located at, because the midpoint represents the average value for both the x and yvalues of the two coordinates of AB. EU: Coordinate geometry can be used to prove geometric theorems because it is possible to replace specific coordinates with variables to show that a relationship remains true regardless of the coordinates. What pattern did this group notice about the xcoordinates of the midpoints compared to the xcoordinates of the endpoints? The ycoordinates? (Student name) said the midpoint is the average of the xcoordinates and ycoordinates of the endpoints. Do you agree or disagree? Why? Who can come up and show us what they mean by average? Does this idea of an average always work? Will it work for horizontal and vertical line segments? (Challenging) Explain. (The average still works. For example, in the case of horizontal lines the xvalues are changing and y remains zero. This means you really only have to calculate the average for the xvalues.) This group came up with a formula. Tell us about your formula. (The formula is the sum of the xvalues divided by 2 and the sum of the yvalues divided by 2. We often use coordinates like (x 1, y 1 ) and (x 2, y 2 ) to say, the coordinates of the first point and the coordinates of the second point. How can we express this idea of average symbolically, if we use endpoints (x 1, y 1 ) and (x 2, y 2 )? How does this groups formula relate to the idea of finding the average of the coordinates? I heard you say that the midpoint formula is,, so let s include this on our formula sheet. (Marking) 1 Application Summary Quick Write Determine the midpoint of segment AB for A(2,4) and B(3,6). A midpoint is a point that divides a segment into two equal segments. How did we use coordinates to prove that a point divides a segment into two equal segments? Given any two points, explain in words how to determine the midpoint. Support for students who are English Learners (EL) 1. Take time during the SetUp phase and throughout the lesson to assess understanding of vocabulary terms and to discuss the meaning of terms such as midpoint, endpoints, segment, coordinates, and more. Point to each of these as the vocabulary is discussed. 2. Private think time during the Explore phase is provided so students have time to organize their thoughts and struggle with the material individually. Cooperative learning is beneficial for all students, but in particular it gives students who are English Learners (EL) the opportunity to work through ideas in a small group before sharing out to the class.
25 24 Tasks and Lesson Guides TASK 2 Name The Distance Between Us 1. After studying the coordinate plane below, Louis claims that because B, C, D, and E are 2 units apart from each other, then AC is two units greater than AB, AD is 2 units greater than AC, and AE is 2 units greater than AD. Do you agree or disagree with Louis? Explain your reasoning. Use diagrams, words, and/or equations in your explanation.
26 Tasks and Lesson Guides Draw line segments AB and CD on the coordinate plane below. TASK 2 Donna claims: To get from A to B, I can move over 3 and up 4. To get from C to D, I can move over 2 and up 5. Since both of these paths are 7 units long, AB = CD. Explain to Donna why her reasoning is incorrect. Use diagrams, words, and/or equations in your explanation.
27 26 Tasks and Lesson Guides 2 The Distance Between Us Rationale for Lesson: Students build on their understanding of midpoint and informal distance calculations in the previous task to analyze thinking about the distance between any two points on the coordinate plane. They approach the problem by counting up and over as they did in the previous task to construct right triangles and use the Pythagorean theorem. Task: The Distance Between Us 1. After studying the coordinate plane below, Louis claims that because B, C, D, and E are 2 units apart from each other, then AC is two units greater than AB, AD is 2 units greater than AC, and AE is 2 units greater than AD. Do you agree or disagree with Louis? Explain your reasoning. Use diagrams, words, and/or equations in your explanation. See student paper for complete task. Common Core Content Standards GGPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Standards for Mathematical Practice Essential Understandings Materials Needed MP1 Make sense of problems and persevere in solving them. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP6 Attend to precision. MP8 Look for and express regularity in repeated reasoning. The distance between two points on a coordinate plane is the length of the line segment that connects them and is the count of the number of units that form the line. The length of a diagonal line segment can be determined by utilizing the line segment as the hypotenuse of a right triangle and computing its length using the Pythagorean Theorem. The Distance Formula can then be derived by replacing specific coordinates with variables in such a situation. Graph paper Straight edge Calculator
28 Tasks and Lesson Guides 27 2 SETUP PHASE I need a volunteer to read #1. Who can explain Louis argument in their own words, without giving your opinion of whether or not it is correct? Please read #2. Who can restate what Donna s argument is without critiquing it? We will take five minutes of private think time to individually respond to Louis and Donna s arguments. I will let you know when it is time to work with your small group. Be prepared to share your reasoning with your group members, and then with the rest of the class. 2 EXPLORE PHASE Possible Student Pathways Group can t get started. Uses a counting strategy to estimate the distance. Uses a ruler. Creates right triangles and uses the Pythagorean Theorem. Assessing Questions What is the problem asking you to do? Draw segment AB. How can you determine the length of segment AB? Which segments lengths can be determined by counting? How did you estimate the length of diagonal segments? What did you find when you measured the segments? Are your measurements precise? Why or why not? How long are the legs of the right triangles? How did you use that information to determine the length of the hypotenuse of each triangle? Advancing Questions How can we compare the length of segment AC to segment AB? What shapes do you see? How can we use these right triangles to determine the exact distance? How can we use the coordinates to find the exact length of the segments? If the right angle were not located at (0,0), would this method still work? Use coordinates to convince me that it will work, no matter where the line segment is located.
29 28 Tasks and Lesson Guides 2 SHARE, DISCUSS, AND ANALYZE PHASE EU: The distance between two points on a coordinate plane is the length of the line segment that connects them and is the count of the number of units that form the line. What does distance mean on the coordinate plane? How is segment AC related to the distance between A and C? (The length of the segment is the distance between the two points). Who can restate what (student name) just said? So, the length of the straight line connecting the points is the distance between the two points. (Marking) How did different groups calculate distance in the first problem? How does this understanding of distance help us respond to Donna s conjecture in part 2? What is Donna s misconception about how distance is measured? Will measurement tools help us find the exact difference in problem 2? Why or why not? EU: The length of a diagonal line segment can be determined by utilizing the line segment as the hypotenuse of a right triangle and computing its length using the Pythagorean Theorem. The Distance Formula can then be derived by replacing specific coordinates with variables in such a situation. How did you use triangles to determine the distance between two points on the coordinate plane? Who can restate this group s strategy? (They created right triangles by drawing horizontal lines and vertical lines). Do you agree or disagree that the Pythagorean Theorem can be used to determine the length of diagonal segments on the coordinate plane? Explain your thinking. Who can tell us how the Pythagorean Theorem can be used to disprove Donna s conjecture? Will the Pythagorean Theorem always work? What if the scale is not convenient for seeing the exact length of the legs of the triangle? How can we determine the length of the horizontal and vertical legs? (If you know the coordinates, you can subtract to determine the length of the horizontal and vertical segments). Who can restate what (student name) said? How can we use subtraction to calculate the length of the legs? What are we subtracting? Once we know the length of the legs of the triangle, how does that help us calculate the distance between the two points? Let s summarize what has been shared. While I m explaining, I d like somebody to indicate on the diagram what is being described. I am hearing that for any diagonal segment, we can think of that segment as the hypotenuse of a right triangle. We can draw in the horizontal and vertical legs and find their lengths by subtracting the xvalues of the endpoints for the horizontal leg and subtracting the ycoordinates for the vertical leg. Once we know these leg lengths, we can substitute them for a and b in the Pythagorean Formula, a 2 + b 2 = c 2, and solve for c, which is the length of the diagonal line segment. Is that what we figured out? (Recapping)
30 Tasks and Lesson Guides 29 Application Summary Quick Write Calculate the distance between the points (5, 5) and (2, 9). Show and explain your work. How do we define distance on the coordinate plane? How do we use what we know about right triangles to calculate distance on the coordinate plane? Explain how the distance between two points in the coordinate plane can be calculated using the Pythagorean Theorem. 2 Support for students who are English Learners (EL) 1. Take time during the SetUp phase and throughout the lesson to assess understanding of vocabulary terms and to discuss the meaning of terms such as distance, segment, coordinates, and more. 2. Use a document projector, overhead projector, poster paper, or other means of displaying student work to the class. Ask students who are English Learners to show where the relationships being discussed appear in the student work.
31 30 Tasks and Lesson Guides TASK 3 Name Will That Work for ANY Two Points? 1. Determine the midpoint of the line segment drawn between each of the end points. A. A (5,1) and B (5,1). B. C (0,0) and D (5,8). C. E (3,6) and F (6,4). D. Summarize your strategy for determining the midpoint of a line segment, and use this strategy to write a formula for the midpoint of any two points (x 1, y 1 ) and (x 2, y 2 ). 2. Use the Pythagorean Theorem to determine the distance between each pair of points. A. A (5,1) and B (5,1). B. C (0,0) and D (5,8) C. E (3,6) and F (6,4) D. Describe your strategy for determining the distance between any two points.
32 Tasks and Lesson Guides Consider the two points B and C on the coordinate plane below. Use their coordinates to write an equation that can be used to calculate the distance between any two points on the coordinate plane. TASK 3
33 32 Tasks and Lesson Guides 3 Will That Work for ANY Two Points? Rationale for Lesson: Solidify an understanding of midpoints and distance on the coordinate plane by using repeated reasoning to generalize midpoint and distance formulas. Task: Will That Work for ANY Two Points? 1. Determine the midpoint of the line segment drawn between each of the end points. A. A (5,1) and B (5,1). B. C (0,0) and D (5,8). C. E (3,6) and F (6,4). D. Summarize your strategy for determining the midpoint of a line segment and use this strategy to write a formula for the midpoint of any two points (x 1, y 1 ) and (x 2, y 2 ). 2. Use the Pythagorean Theorem to determine the distance between each pair of points. See student paper for complete task. Common Core Content Standards G.GPE.B.4 G.GPE.B.6 G.GPE.B.7 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Standards for Mathematical Practice Essential Understandings Materials Needed MP1 Make sense of problems and persevere in solving them. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP6 Attend to precision. MP8 Look for and express regularity in repeated reasoning. Coordinate geometry can be used to prove geometric theorems because it is possible to replace specific coordinates with variables to show that a relationship remains true regardless of the coordinates. For a line segment with endpoints at A(x 1,y 1 ) and B(x 2,y 2 ), since the midpoint is in the middle of segment AB, it is located at, because the midpoint represents the average value for both the x and yvalues of the two coordinates of AB. The length of a diagonal line segment can be determined by utilizing the line segment as the hypotenuse of a right triangle and computing its length using the Pythagorean Theorem. The Distance Formula can then be derived by replacing specific coordinates with variables in such a situation. Graph paper Straight edge Patty paper Calculator
34 Tasks and Lesson Guides 33 SETUP PHASE Read along silently while (student name) reads the task aloud. In this task, we will revisit the concept of midpoint that we explored in Task 1 and continue to look at how to determine exact distances on the coordinate plane. Graph paper, rulers, and other tools are located on each table for you to use as needed. We will take five minutes of private think time to enter into the task before working in small groups. 3 EXPLORE PHASE Possible Student Pathways Uses counting methods to determine midpoint and horizontal and vertical distance. Determines midpoint and distance for specific examples, but does not write formulas. Uses the midpoint and distance formulas. Group finishes early. Assessing Questions Show me how you count from the endpoints to calculate the midpoint. What calculations did you perform when the coordinates were given? Can you explain your reasoning? What is represented by each of the variables? Why do these formulas work? How did you determine the formula for midpoint and distance? What s the relationship between the distance formula and the Pythagorean Theorem? Advancing Questions How is the xcoordinate of the midpoint related to the xcoordinates of the endpoints? What operations can we use to determine a value that is exactly in the middle of two other values? What are the x 1 and y 1 values in part a? x 2 and y 2? How can you use the operations you performed on those coordinates to write a general formula? What is the relationship between the Pythagorean Theorem and the distance formula? How can you be sure your formulas work for any coordinates?
35 34 Tasks and Lesson Guides 3 SHARE, DISCUSS, AND ANALYZE PHASE EU: For a line segment with endpoints at A(x 1,y 1 ) and B(x 2,y 2 ), since the midpoint is in the middle of segment AB, it is located at, because the midpoint represents the average value for both the x and yvalues of the two coordinates of AB. Tell us about how you counted to calculate the midpoint. Who can explain what this group s method is? Why does it work? Will counting always work? Give us an example of when counting might not work. (It will always work, but it might be difficult with large numbers). I m hearing that we need another way of calculating midpoint a formula that will always work. (Marking) While circulating, I heard somebody say that this is like calculating an average. Somebody explain this to me how is this like calculating an average? What arithmetic operations are we performing when calculating an average? This group wrote the formula, to represent this process. What do the variables represent? Who can show us how this formula uses averages to find the midpoint? EU: The length of a diagonal line segment can be determined by utilizing the line segment as the hypotenuse of a right triangle and computing its length using the Pythagorean Theorem. The Distance Formula can then be derived by replacing specific coordinates with variables in such a situation. Tell us about how you used right triangles to calculate the distance between points when you know their coordinates. Who can restate this strategy? Who can point to the a, b, and c values on the graph? What operations are we using to determine the lengths of the legs? How do we use the lengths of the legs to determine the hypotenuse? (We can subtract to determine the length of the legs and then use the Pythagorean Theorem to calculate the hypotenuse). So I m hearing that if you create a right triangle, calculating the distance between two points is the same as calculating the length of the hypotenuse. (Marking) Tell us about how you used right triangles for the generic coordinates in #3. Where is the 3rd vertex of that triangle located? How did they determine the coordinates of the 3rd vertex? Another group drew a different right triangle. Come up and show us what triangle you drew. Who can say how these two right triangles are related? Do both give us the same lengths? We used subtraction to calculate the length of the legs when we knew the coordinates. Does subtraction still work for the generic coordinates? Why or why not? What does it look like when we use these expressions for the length of each leg in the Pythagorean Theorem?
36 Tasks and Lesson Guides 35 EU: Coordinate geometry can be used to prove geometric theorems because it is possible to replace specific coordinates with variables to show that a relationship remains true regardless of the coordinates. How did completing several examples help us generate a formula? Who can restate what (student name) just said about patterns? How do patterns help us write formulas? (You can see the operation that is used each time and how it works. You can look across many problems and see a pattern). By looking at multiple examples, we can recognize the operations that are used every time and then we can write a rule using variables instead of specific coordinates. (Marking) What are the advantages of creating a general rule/formula? 3 Application Summary Quick Write Determine the midpoint and the length of the segment connecting the points (3,7) and (2,8.5). What formulas did we agree upon today? What are they used for? Why was it helpful to do several examples before deriving the formulas? No Quick Write for students. Support for students who are English Learners (EL) 1. Create a running list of the problems so that the patterns in the numbers become more evident to students who are English Language learners.
37 36 Tasks and Lesson Guides TASK 4 Name Building a New Playground The City Planning Commission is considering building a new playground. They would like the playground to be equidistant from the two elementary schools, represented by points A and B in the coordinate grid that is shown. PART A 1. Determine at least three possible locations for the park that are equidistant from points A and B. Explain how you know that all three possible locations are equidistant from the elementary schools. 2. Make a conjecture about the location of all points that are equidistant from A and B. Prove this conjecture.
38 Tasks and Lesson Guides 37 Part B: (Extension) 1. The City Planning Commission is planning to build a third elementary school located at (8,6) on the coordinate grid. Determine a location for the park that is equidistant from all three schools. Explain how you know that all three schools are equidistant from the park. TASK 4 2. Describe a strategy for determining a point equidistant from any three points.
39 38 Tasks and Lesson Guides 4 Building a New Playground Rationale for Lesson: Students have worked with midpoint and distance. In this lesson, students explore a perpendicular bisector as a tool for solving problems. In a realworld context, students will determine that all of the points that are equidistant between two points lie on the perpendicular bisector. Task: Building a Playground The City Planning Commission is considering building a new playground. They would like the playground to be equidistant from the two elementary schools, represented by points A and B in the coordinate grid that is shown. PART A 1. Determine at least three possible locations for the park that are equidistant from points A and B. Explain how you know that all three possible locations are equidistant from the elementary schools. See student paper for complete task. Common Core Content Standards G.GPE.B.4 G.GPE.B.5 G.GPE.B.6 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
40 Tasks and Lesson Guides 39 Standards for Mathematical Practice Essential Understandings Materials Needed MP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP5 Use appropriate tools strategically MP6 Attend to precision. MP7 Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning. Coordinate Geometry can be used to prove geometric theorems by replacing specific coordinates with variables, thereby showing that a relationship remains true regardless of the coordinates. The set of points that are equidistant from two points A and B lie on the perpendicular bisector of line segment AB, because every point on the perpendicular bisector can be used to construct two triangles that are congruent by definition of triangle congruence, reflection, and/or Side AngleSide; corresponding parts of congruent triangles are congruent. Graph paper Patty paper Straight edge Compass Calculator 4
41 40 Tasks and Lesson Guides 4 SETUP PHASE Read along silently as (student name) reads the task aloud. What is a City Planning Commission? What other kinds of projects might this group manage? Who can summarize what we are being asked to do without suggesting how to do it? Work for five minutes independently to develop a strategy for finding several locations for the park. I will let you know when it is time to work in your groups. Your geometric tools, including patty paper, are available at each table. Use them as needed. EXPLORE PHASE Possible Student Pathways Can t get started. Determines only the midpoint. Intuitively determines points along the perpendicular bisector w/o recognizing the relationship between the points and A & B. Determines that the points fall along the perpendicular bisector. Assessing Questions Choose a point that you think is the same distance from both schools. What are the coordinates of that point? How did you determine this location? How do you know it fits the criteria? How did you determine the location of the points? How would you count up/ over from this last point to determine an additional point that is equidistant? How did you know that the points would be along the perpendicular bisector? What led you to make this conjecture? Advancing Questions How can you determine if the point you chose is equidistant from A and B? What tools will help you? Can you use the coordinates? How can there be more than one point? If somebody is the same distance from both of us, does s/he have to be in the middle of us? What other location could s/he have? What pattern do you notice? Do the points lie on a line? Can you make a conjecture about the relationship between the segment connecting A and B and the segment connecting the equidistant points? How can you test your conjecture? Will this perpendicular bisector relationship hold for any two points? How can you test this possibility?
42 Tasks and Lesson Guides 41 SHARE, DISCUSS, AND ANALYZE PHASE EU: The set of points that are equidistant from two points A and B lie on the perpendicular bisector of line segment AB, because every point on the perpendicular bisector can be used to construct two triangles that are congruent by definition of triangle congruence, reflection and/or SideAngleSide; corresponding parts of congruent triangles are congruent. EU: Coordinate geometry can be used to prove geometric theorems because it is possible to replace specific coordinates with variables to show that a relationship remains true regardless of the coordinates. Tell us about your strategy. How did you choose these points? I d like another group to summarize their method. I heard them mention the midpoint, but what pattern did they use to determine other points? How did your group use the Midpoint Formula? How did you determine the other points from there? Do all of the equidistant points lie on a line? How can we show that? What is the relationship between the segment connecting the two original points and the line containing all of the equidistant points? Somebody else tell us how this group knew the line was perpendicular. How did they use slopes and the Midpoint Formula here? (We used the midpoint formula to calculate one point, but then we counted up and over using the pattern. Then we drew the line and noticed that the slope of the lines is the opposite reciprocal.) I m hearing that you used the Midpoint Formula and slope to determine the points (Marking), but how did your group show that the points are equidistant? I d like another group to summarize how they calculated the distances, and then I d like some other groups that used a different method to volunteer. How is this related to using the Pythagorean Theorem? When they used the Distance Formula, does that prove that all points along this line will be equidistant? Why or why not? Can you prove that this will always be the case? I noticed some interesting work while circulating. (Student name) tell us about how your group used congruent triangles to prove that the points are equidistant. Can you tell us about the sides and the angles of these triangles? Which corresponding sides and angles are congruent? Identify the points and side lengths as you describe them. (We traced one triangle and laid it over the other. They exactly fall on top of one another. This kept happening, no matter what point we chose. We know the line is perpendicular because of the slopes, so this is a right angle. This means the triangles are congruent so all of their corresponding parts are congruent.) Somebody else summarize how this group used congruent triangles to prove that the points will always be on the perpendicular bisector. Are all three pairs of corresponding sides necessarily congruent? Why or why not? What I heard in this discussion is that if you prove the triangles congruent, then you prove that corresponding parts must be congruent. In other words, when they showed that the triangles must be congruent, they showed that the side lengths, or distance to the playground, must be equal because these parts are corresponding parts of congruent triangles. (Recapping) How is this the same as noticing that the distance represents a hypotenuse and that this hypotenuse will be the same for each triangle? 4
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