Unit 1 Foundations of Geometry: Vocabulary, Reasoning and Tools

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1 Number of Days: 34 9/5/17-10/20/17 Unit Goals Stage 1 Unit Description: Using building blocks from Algebra 1, students will use a variety of tools and techniques to construct, understand, and prove geometric concepts and relationships. Unit 1 will set the foundation for this course in Geometry as the students learn to use the vocabulary to communicate precisely and to use the appropriate tools to construct and transform figures effectively. Unit 1, also, includes the beginnings of geometric reasoning. The students will use these new skills to justify key relationships among lines and angles. Finally, the student will investigate the relationship between intersecting lines and their resultant angles. Parallel and/or perpendicular lines will be identified using the coordinate plane. Throughout this course, students will use basic tools, straightedge and compass, to develop their understanding of geometric relationships. Proofs will be a capstone for many of these units. Throughout this course, strict attention is paid to precise use of mathematical language. Materials: Patty paper, straightedge and compass, poster-sized graph paper, markers, dynamic tools (Geogebra) Standards for Mathematical Practice SMP 1 Make sense of problems and persevere in solving them. SMP 2 Reason abstractly and quantitatively. SMP 3 Construct viable arguments and critique the reasoning of others. SMP 4 Model with mathematics. SMP 5 Use appropriate tools strategically. SMP 6 Attend to precision. SMP 7 Look for and make use of structure. SMP 8 Look for and express regularity in repeated reasoning. Standards for Mathematical Content Clusters Addressed [s] G-CO.A Experiment with transformations in the plane. [m] G-CO.C Prove geometric theorems. Transfer Goals Students will be able to independently use their learning to Make sense of never-before-seen problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Making Meaning UNDERSTANDINGS Students will understand that Accurate definitions of essential geometric terms lead to precise communication. Points, lines, and planes are the foundations of geometry. Valid inductive and deductive reasoning are used to develop and prove conjectures. Both straightedge and compass, and patty paper, are tools and methods to construct, explore, and prove the properties and lines and angles. There are parts of a triangle that are necessary to allow for the construction of a unique triangle. The attributes of the four concurrency points of a triangle develop based on the constructions of those points. The words sketch, draw, and construct each imply a different level of representational precision. ESSENTIAL QUESTIONS Students will keep considering Why are point, line, and plane the undefined terms of geometry? How are properties of geometric figures related to their measurable attributes? How are the foundations of logical reasoning used to develop and prove conjectures? How does the application of logical reasoning facilitate understanding geometric relationships? How do you prove that lines are parallel or perpendicular? What components of a triangle are needed to construct a triangle that is unique? Why is it important to develop visualization skills, critical thinking and cooperative behavior? LONG BEACH UNIFIED SCHOOL DISTRICT 1 Posted 10/6/17

2 [s] G-CO.D Make geometric constructions. [a] G-C.A Understand and apply theorems about circles. [m] G-GPE.B Use coordinates to prove simple geometric theorems algebraically. [a] G-GMD.B Visualize relationships between twodimensional and threedimensional objects. Unit Goals Stage 1 KNOWLEDGE Students will know Precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, and distance along a line. The special angle pairs formed by parallel lines and a transversal are either congruent or supplementary. Whether or not a unique triangle can be constructed given certain parts of that triangle. The properties and construction of the points of concurrency of a triangle. Acquisition SKILLS Students will be skilled at and/or be able to Define and classify geometric objects based on their special characteristics. Translate descriptions into diagrams. Measure angles and line segments to a required level of precision. Construct angles and line segments using a compass and straightedge, patty paper, and/or geometry software. Make conjectures based on observed patterns using inductive reasoning. Explore relationships between special angle pairs. Construct special angles from a 90 angle. Construct the lines that meet at the points of concurrency of a triangle. LONG BEACH UNIFIED SCHOOL DISTRICT 2 Posted 10/6/17

3 Assessed Grade Level Standards Standards for Mathematical Practice SMP 1 Make sense of problems and persevere in solving them. SMP 2 Reason abstractly and quantitatively. SMP 3 Construct viable arguments and critique the reasoning of others. SMP 4 Model with mathematics. SMP 5 Use appropriate tools strategically. SMP 6 Attend to precision. SMP 7 Look for and make use of structure. SMP 8 Look for and express regularity in repeated reasoning. Standards for Mathematical Content [s] G-CO.A Experiment with transformations in the plane. G.CO.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. G.CO.2 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). [m] G-CO.C Prove geometric theorems. [Focus on validity of underlying reasoning while using variety of ways of writing proofs.] G-CO.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. G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. [s] G-CO.D Make geometric constructions. G-CO.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. G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [a] G-C.A Understand and apply theorems about circles. G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [m] G-GPE.B Use coordinates to prove simple geometric theorems algebraically. [Include distance formula; relate to Pythagorean theorem.] G-GPE.5 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). [a] G-GMD.B Visualize relationships between two-dimensional and three-dimensional objects. G-GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Key: [m] = major clusters; [s] = supporting clusters, [a] = additional clusters Indicates a modeling standard linking mathematics to everyday life, work, and decision-making LONG BEACH UNIFIED SCHOOL DISTRICT 3 Posted 10/6/17

4 Assessment Evidence Evidence of Learning Stage 2 Unit Assessment Claim 1: Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency. Concepts and skills that may be assessed in Claim1: G-CO.A The students will define geometric terms precisely. The student will make precise use of vocabulary throughout their geometric arguments and error analysis. G-CO.C Students will use inductive reasoning to make conjectures about linear pairs, vertical angles, and angles formed when a transversal crosses parallel lines. Students will follow informal arguments (proofs) of some of these conjectures. Students will construct the points of concurrency for a triangle: centroid, incenter, circumcenter and orthocenter. G-CO.D Students will choose tools strategically to make formal geometric constructions. Students will construct a square, an equilateral triangle, and a regular hexagon inscribed in a circle. G-C.A Students will construct the inscribed and circumscribed circles of a triangle. G-GPE.B Students will prove the slope criteria for parallel and perpendicular lines. Students will use the slope criteria for parallel and perpendicular lines to solve geometric problems. G-GMD.B Students will identify the shapes of two-dimensional cross sections of three-dimensional objects. Students will identify three-dimensional objects generated by rotations of two-dimensional objects. Claim 2: Students can solve a range of wellposed problems in pure and applied mathematics, making productive use of knowledge and problem-solving strategies. Standard clusters that may be assessed in Claim 2: NONE Claim 3: The student can clearly and precisely construct viable arguments to support their own reasoning and critique the reasoning of others. Standard clusters that may be assessed in Claim 3: G-CO.A G-CO.C Claim 4: The student can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Standard clusters that may be assessed in Claim 4: NONE LONG BEACH UNIFIED SCHOOL DISTRICT 4 Posted 10/6/17

5 Other Evidence Evidence of Learning Stage 2 Formative Assessment Opportunities Informal teacher observations Checking for understanding using active participation strategies Exit slips/summaries Tasks Modeling Lessons (SMP 4) Formative Assessment Lessons (FAL) Quizzes/Chapter Tests SBAC Interim Assessment Blocks SBAC Interim Assessment Blocks Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website M Mathematics Curriculum Documents LONG BEACH UNIFIED SCHOOL DISTRICT 5 Posted 10/6/17

6 1 day I will explore writing a well-constructed and valid argument by participating in the Opening Task. OPENING TASK Dear Mom and Dad This Opening Task has the students brainstorming in small groups about requests they recently made of their parent(s). They are asked to construct an argument as to why the request should be granted, first in 2-column form, then in paragraph form. This letter will be revisited at the end of the unit to see if studying the structure of proof had any effect on the students ability to argue for their cause. This task is a gateway into the entire vocabulary and reasoning unit. Application: Dear Mom and Dad 7-9 I will communicate precisely and accurately by Knowing the basic terms of geometry: point, line, plane, segment, ray and collinear and coplanar points. (SMP 6) Knowing what is required of a precise definition. (SMP 6) Writing precise definitions that will be used throughout this course. (SMP 6) Answering questions such as o Why do some terms in geometry remained undefined? o How do you use points, lines, and planes as the basic elements of geometry? o How can a line have infinite length when it is made up of points that have no size? o What makes a good definition? o Do some math words mean the same thing in regular language? Give examples. Lesson 1-1 Lesson 1-2 Lesson 1-3 Lesson 1-4 Lesson 1-5 Lesson 1-6 Lesson 1-7 Lesson 1-8 Lesson 1-9 Which One Doesn t Belong Shapes MathOpenRef: Using a Protractor to Measure Angles Dynamic Tool MathOpenRef: Angles Dynamic Tool MathOpenRef: Points, Lines and Planes Dynamic Tool MathOpenRef: Quadrilaterals Dynamic Tool MathOpenRef: Circles Dynamic Tool Khan Academy: Intro to Euclidean Geometry Khan Academy: Angles Khan Academy: Polygons Khan Academy: Circle Basics Khan Academy: Constructing Regular LONG BEACH UNIFIED SCHOOL DISTRICT 6 Posted 10/6/17

7 Polygons Inscribed in Circles Khan Academy: Geometric Solids (3D Shapes) Application: Pool Geometry Dynamic Tool 1 day I will identify crosssectional shapes and 3D rotations by Representing three-dimensional figures in twodimensional space as nets. (SMP 6) Folding nets to recreate three-dimensional figures. Drawing diagrams to help with visualizations. Using physical models to demonstrate a situation. Answering questions such as o What does the word dimension mean? o How is an isometric drawing different from a perspective drawing? o What is a locus of points? o What are some of the techniques that helped you translate descriptions into solutions in Section 1-9? P. 79 #18 and #19 P. 86 #23 and #24 Illuminations: Cubes (Nets) Illuminations: Cubes Nets Illuminations: Geometric Solids Interactives: Geometry 3-D Shapes Khan Academy: Geometric Solids (3D Shapes) 4-5 I will show my use of methods of geometric reasoning by Finding the next terms in sequences. Recognizing patterns. Formulate mathematical models to represent situations. (SMP 4) Differentiating between inductive and deductive reasoning. Explaining their process used when finding the patterns and next terms. (SMP 3) Answering questions such as o Are conjectures always true? o How would you convince a friend that a conjecture is true or false? o Summarize the process you used to find the function rules. Explain the role each parameter played in the rule. Lesson 2-1 Lesson 2-2 Lesson 2-3 Lesson 2-4 Patterns and Function Connections STEM Performance Task: Reasoning at the Zoo Better Lesson: Definitely, Maybe Week of Inspirational Math: Oh Hail the Elephant Conjectures Inductive vs. Deductive Reasoning Interactive Quiz o What is a mathematical model? What is its purpose? Digit Place Game LONG BEACH UNIFIED SCHOOL DISTRICT 7 Posted 10/6/17

8 o Compare and contrast inductive and deductive reasoning. Khan Academy: Deductive and Inductive Reasoning Khan Academy: Interpreting Function Notation Khan Academy: Introduction to Arithmetic Sequences 2-3 I will use inductive reasoning by Making conjectures about linear pairs, vertical angles, and angles formed when a transversal crosses parallel lines. (SMP 3) Informally arguing for the validity of conjectures. Answering questions such as o What kind of reasoning led to the conjectures that arose in Section 2-5? o Can you prove a conjecture using Patty Paper? o Explain how we drew the lines to create alternate interior/alternate exterior angles. o When may a conjecture be accepted as a theorem? Lesson 2-5 Lesson 2-6 Solve for Unknown Angles-Transversals Task Angle Chase and Justification Parallel Lines Investigation Activity MathOpenRef: Angle Relationships Dynamic Tool MathOpenRef: Angles Around a Transversal Dynamic Tool Khan Academy: Vertical, Complementary, and Supplementary Angles Khan Academy: Angles 1 day I will solve for the slope of parallel and perpendicular lines by Plotting given points, connecting them with a line, and drawing in a right triangle to illustrate the change in altitude and change in horizontal distance from point to point. Deriving the slope formula. Deriving the relationship between the slopes of parallel lines and perpendicular lines. Using Your Algebra Skills 2 P. 135 Using Your Algebra Skills 3 P. 167 Illuminations: Rise-Run Triangles Which One Doesn t Belong: Lines LONG BEACH UNIFIED SCHOOL DISTRICT 8 Posted 10/6/17

9 Answering questions such as o What are the various methods you can use to find the slope of a line? o What must you have to find for the slope of a line? o Compare and contrast parallel slopes and perpendicular slopes. Slopes of Parallel and Perpendicular Interactive Quiz Quizlet: Identifying Parallel and Perpendicular Lines 1 Quizlet: Identifying Parallel and Perpendicular Lines 2 Recognizing Parallel and Perpendicular Lines Game Khan Academy: Slope Khan Academy: Horizontal and Vertical Lines Khan Academy: Parallel and Perpendicular 2-3 I will check my understanding of recognizing patterns and mathematically modeling those patterns by participating in the FAL. FORMATIVE ASSESSMENT LESSON Dan Meyer: [Makeover] Checkerboard Border Dan Meyer: [Makeover] Checkerboard Border Task 6-8 I will make formal constructions of lines and angles by Strategically choosing the appropriate tools for the following constructions: 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. (SMP 5) Using a variety of construction tools and methods such as: straightedge and compass, patty paper or geometry software. Making conjectures based on the constructions. (SMP 3) Lesson 3-1 Lesson 3-2 Lesson 3-3 Lesson 3-4 Lesson 3-5 Lesson 3-6 Illustrative Mathematics: Bisecting an Angle Illustrative Mathematics: Construction of Perpendicular Bisector True or False: How Do You Know? LONG BEACH UNIFIED SCHOOL DISTRICT 9 Posted 10/6/17

10 Identifying properties of figures based on the constructions. Answering questions such as o What is the relationship between an arc and a circle? o Which do you prefer: patty paper or straightedge and compass? Why? o Why does the compass and straightedge technique that we used to draw a perpendicular bisector work? o How can we use our work in lesson 3-2 to find the midpoint of a given line segment? o How many altitudes can a triangle have? Outside altitudes? Give examples. o If you construct a right angle, what other angles can you now construct? o How did you use your knowledge of perpendicular bisector construction from lesson 3-3 to construct an angle bisector? o How do you know when two lines are parallel? Explain using conjectures. o In space, can lines not be parallel and not intersect? o What is the minimum amount of information you need to construct a unique triangle? Constructing a Perpendicular Line Using a Compass and Straightedge Video Constructing an Angle Bisector Video Khan Academy: Constructing Bisectors of Lines and Angles Khan Academy: Parallel and Perpendicular Application: Construction Castle Project 1-2 I will construct inscribed and circumscribed angles of a triangle by Drawing the angle bisectors, perpendicular bisectors, and altitudes of a triangle. (SMP 5) Constructing the incenter, circumcenter and orthocenter of a triangle and applying the properties of these points of concurrency to solve real-world applications. Asking questions such as o How can we remember the difference between the circumcenter, the incenter, and the orthocenter? o Is there a triangle in which the three points of concurrency will be the same point? Lesson 3-7 Copy and Bisect an Angle Task Construct a Perpendicular Bisector Task Points of Concurrencies Task Points of Concurrency Project LONG BEACH UNIFIED SCHOOL DISTRICT 10 Posted 10/6/17

11 MathOpenRef: Triangle Centers Overview Dynamic Tool Khan Academy: Perpendicular Bisectors Khan Academy: Altitudes Application: Illuminations: Hospital Locator 1 day I will construct a centroid by Drawing the medians of a triangle. (SMP 5) Constructing the centroid of a triangle and applying its properties to solve real-world applications. Asking questions such as o o What are the properties of the centroid? Compare and contrast the properties of the four points of concurrency. Lesson 3-8 MathVisionProject: Centers of a Triangle MathOpenRef: Centroid of a Triangle Dynamic Tool MathOpenRef: Medians of a Triangle Quizlet: Concurrency Khan Academy: Medians and Centroids Khan Academy: Altitudes 1 day I will apply my knowledge of concurrent points by Completing the exploration of the Euler Line. Asking questions such as o What happens as the shape of the triangle changes? o Why do these three points of currency lie on a straight line? o What is the relationship between the pieces into which the Euler segment is divided? Exploration: The Euler Line P. 191 Geogebra: Special Points that Construct Euler s Line Dynamic Tool MathOpenRef: Euler Line Khan Academy: Bringing It All Together LONG BEACH UNIFIED SCHOOL DISTRICT 11 Posted 10/6/17

12 I will prepare for Incorporating the Standards for Mathematical Practice the unit (SMPs) along with the content standards to review the 2 assessment by unit. 1-2 Unit Assessment Students will take the Synergy Online Unit Assessment. Unit Assessment Resources (Word or PDF) can be used throughout the unit. LONG BEACH UNIFIED SCHOOL DISTRICT 12 Posted 10/6/17