Line Segments, Distance, and Midpoint


 Julie Sherman
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1 Geometry, Quarter 1, Unit 1.1 Line Segments, Distance, and Midpoint Overview Number of instructional days: 8 (1 day = minutes) Content to be learned Know the precise definition of line segment, based on the undefined notions of point, line, and distance along a line. Find distance and midpoint between two points on the coordinate plane and/or number line. Solve problems on the coordinate plane using distance and midpoint formulas. Apply the distance and midpoint formulas to solve word problems. Derive the distance formula using the Pythagorean Theorem. Construct and bisect a segment using a variety of tools (e.g., straight edge, protractor, compass, geometry software). Essential questions Under which circumstances would you use the distance and midpoint formulas? How is the Pythagorean Theorem related to the distance formula? Mathematical practices to be integrated Use appropriate tools strategically. Use various tools, including technology, to help solve problems. Use estimation to know if tools have worked correctly. Attend to precision. Use labels and units of measure correctly. Calculate and compute accurately. Where can the concepts of distance and midpoint be applied in the real world? How can you define line segment? 1
2 Geometry, Quarter 1, Unit 1.1 Line Segments, Distance, and Midpoint (8 days) Written Curriculum Common Core State Standards for Mathematical Content Congruence GCO Experiment with transformations in the plane GCO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Expressing Geometric Properties with Equations GGPE Use coordinates to prove simple geometric theorems algebraically [Include distance formula; relate to Pythagorean theorem] GGPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. GCO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Common Core Standards for Mathematical Practice 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 2
3 Geometry, Quarter 1, Unit 1.1 Line Segments, Distance, and Midpoint (8 days) 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Clarifying the Standards Prior Learning In grade 4, students learned to draw and identify points, lines, and line segments. In grade 8, students explained the proof of the Pythagorean Theorem and its converse, applied the Pythagorean Theorem to solve problems in two and three dimensions, and applied the Pythagorean Theorem to find the distance between two points in the coordinate system. Current Learning Students find distance and midpoint on the coordinate plane. They derive the distance formula using the Pythagorean Theorem. The distance and midpoint are also used to justify properties of polygons. Future Learning In algebra 2, students will use the distance formula to write the equation of a conic. In calculus, students will use the distance formula to minimize the distance in a curve. In geometry unit 4.1, students will use the distance formula to find the equation of a circle. Additional Findings According to Principles and Standards for School Mathematics, Students in grades 9 12 are to specify locations and describe spatial relationships using coordinate geometry and other representational systems (p. 308). According to Beyond Numeracy, Students at this level should be introduced to the coordinate system and its uses in analytic geometry (pp ). According to Principles and Standards for School Mathematics, In grades 9 12 students have to write an algebraic justification of the Pythagorean Theorem, and students use the Pythagorean Theorem as one of the many multiple approaches to solving geometric problems (p. 301). According to Benchmarks for Science Literacy, In grades 6 8 students should use the Pythagorean Theorem as a model for problem solving (p. 269). 3
4 Geometry, Quarter 1, Unit 1.1 Line Segments, Distance, and Midpoint (8 days) 4
5 Geometry, Quarter 1, Unit 1.2 Transformations Overview Number of instructional days: 13 (1 day = minutes) Content to be learned Explore different transformations and compare their properties. Use function notation to describe transformations. Develop definitions for different transformations. Predict the effect of a transformation on a figure. Mathematical practices to be integrated Make sense of problems and persevere in solving them. Use multistep transformations to persevere and find a result. Use prior knowledge of functions to demonstrate understanding of coordinate inputs and outputs. Use appropriate tools strategically. Use the available tools (i.e., transparencies and geometric software). Attend to precision. Specify a sequence of transformations that will carry a given figure onto another. Maintain integrity of shapes through transformations. Look for and make use of structure. Recognize which transformations maintain shape and attribute integrity. Describe functions that produce outputs for particular inputs in the coordinate plane. Essential questions What are the similarities and differences among a rotation, reflection, and translation? What are the operations that cause a figure to transform? What are the ways that a figure can change position? How does this change of position affect its shape? How would you describe a transformation as a function? 5
6 Geometry, Quarter 1, Unit 1.2 Transformations (13 days) Written Curriculum Common Core State Standards for Mathematical Content Congruence GCO Experiment with transformations in the plane GCO.2 GCO.3 GCO.4 GCO.5 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Understand congruence in terms of rigid motions [Build on rigid motions as a familiar starting point for development of concept of geometric proof] GCO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 6
7 Geometry, Quarter 1, Unit 1.2 Transformations (13 days) 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Clarifying the Standards Prior Learning In grade 7, students created scale drawings of geometric figures, including lengths and area. In grade 8, students used physical models, transparencies, or geometric software to understand similarity. Current Learning Students represent transformations in the plane, describe transformations as functions that take points in the plane as inputs and give other points as outputs, and compare transformations that preserve distance and angle to those that do not (translation versus horizontal stretch). Students develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Given a geometric figure and a rotation, reflection, or translation, students draw the transformed figure (using, for example, graph paper, tracing paper, or geometry software). Students specify a sequence of transformations that will carry a given figure onto another. They use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure. Given two figures, students use the definition of congruence in terms of rigid motions to decide if they are congruent. Students prove theorems about lines and angles. (These theorems include, Points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. ) 7
8 Geometry, Quarter 1, Unit 1.2 Transformations (13 days) Future Learning In fourthyear mathematics courses, students will use matrices to represent reflections, translations, and rotations. Additional Findings According to Benchmarks of Science Literacy, by the end of grade 5, students should know that symmetry can be found by reflection, turns, or slides (p. 227). According to Principles and Standards for School Mathematics, high school students should conduct increasingly independent explorations, which will allow them to develop a deeper understanding of important geometric ideas such as transformation and symmetry (p. 309). 8
9 Geometry, Quarter 1, Unit 1.3 Segment and Angle Relationships Overview Number of instructional days: 16 (1 day = minutes) Content to be learned Make and defend conjectures about parallel and perpendicular lines off the coordinate plane. Use angles formed by parallel lines cut by transversals to solve realworld problems. Make and defend conjectures about angle and segment addition postulates. Make and defend conjectures about angle pairs (adjacent, linear, vertical, supplementary, complementary) and angle and segment bisectors. Use a variety of tools to copy and bisect an angle (e.g., straight edge, protractor, compass, geometry software). Mathematical practices to be integrated Reason abstractly and quantitatively. Use abstract reasoning; check answers to ensure they are quantitatively sound. Rewrite the problem in simpler terms. Construct viable arguments and critique the reasoning of others. Make conjectures and prove them (including using counterexamples). Use proofs to make logical arguments and conclusions. Use appropriate tools strategically. Use tools and technology to explore and deepen understanding of mathematics and be able to detect errors. Look for and make use of structure. See complicated problems as a group of smaller, easier problems. Essential questions What are the differences and similarities among vertical, linear, adjacent, supplementary, and complementary angle pairs? How do you know if an angle or segment has been bisected? How can vertical, linear, adjacent, supplementary, and complementary angle pairs be used to solve problems? What is different about the angle pairs formed when a transversal intersects parallel lines versus the angle pairs formed when a transversal intersects nonparallel lines? How do you use special angle pairs to determine if two lines cut by a transversal are parallel? How do you know if an angle or segment has been bisected, and what evidence would you need to support it? 9
10 Geometry, Quarter 1, Unit 1.3 Segment and Angle Relationships (16 days) Written Curriculum Common Core State Standards for Mathematical Content Congruence GCO Prove geometric theorems [Focus on validity of underlying reasoning while using variety of ways of writing proofs] GCO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. GCO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Common Core Standards for Mathematical Practice 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 10
11 Geometry, Quarter 1, Unit 1.3 Segment and Angle Relationships (16 days) 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning In grade 7, students used facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure. In grade 8, students used informal arguments to establish facts about angles created when parallel lines are cut by a transversal. Current Learning In this unit, students prove theorems about lines and angles. Theorems include: Vertical angles are congruent. and When a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent. Future Learning In a later geometry unit, students will apply theorems involving vertical, supplementary, complementary, and angles formed by parallel lines cut by a transversal to prove similar polygons and properties of polygons. 11
12 Geometry, Quarter 1, Unit 1.3 Segment and Angle Relationships (16 days) Additional Findings According to Benchmarks for Science Literacy, by the end of grade 8, students should know that lines can be parallel, perpendicular, or oblique (p. 224). Also, as shown Principles and Standards for School Mathematics, in grades 9 12, students should be able to apply the angle relationships formed by two parallel lines and a transversal to solve problems (p. 310). According to A Research Companion to Principles and Standards for School Mathematics, by grade 8, students should know that parallel lines should not intersect and that they are equidistant (p. 164). 12
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