Unit 2 Triangle and Polygon Properties

Transcription

1 Number of Days: 34 10/23/17 12/15/17 Unit Goals Stage 1 Unit Description: Building on the algebra, reasoning, and constructions of Unit 1, Unit 2 has the students discover and explore the properties of triangles and polygons. As the students make conjectures and construct interior angles, exterior angles and congruencies of triangles, their new knowledge leads naturally to the study of polygons. As the students progress through their study of polygons, they will see that the properties of a polygon of one category hold for all the subcategories of that polygon. Using paragraph, flowchart, and two-column proofs, students compose logical arguments to support their triangle and quadrilateral conjectures. Materials: Patty paper, isometric paper, straightedge and compass, poster-sized graph paper, markers 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 [m] G-CO.B Understand congruence in terms of rigid motions. [m] G-CO.C Prove geometric theorems. [a] G-GMD.B Visualize relationships between twodimensional and threedimensional objects. 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 The triangle congruence theorems are viewed as shortcuts to proving the congruence of two triangles. Determining the physical characteristics of a geometric figure is tied to logical determination, or implication, which is essential to deductive reasoning. Proving theorems within a deductive system ensures the avoidance of circular reasoning. The triangle congruence theorems allow us to conclude that two triangles are congruent after we have shown the congruence of three of the six pairs of corresponding angles and sides. There are relationships between the angles, sides and diagonals of convex polygons. The midsegment properties of a triangle are derived from the midsegment properties of a trapezoid. The properties of a category of polygons are inherited by all the subcategories of that polygon. ESSENTIAL QUESTIONS Students will keep considering What parts of triangles are needed to determine a unique triangle and, from this, what information is needed to determine the congruence of two triangles? Why are proofs considered to be such an large part of geometry and where, outside of geometry, is this an important skill? What are the advantages and disadvantages of the various proof formats: paragraph, flow, and two-column? Why do some congruence conjectures prove triangles to be congruent and others do not? Why can SSA and AAA not be used to prove congruence? Do a statement and its converse always have the same truth value? Does equilateral imply equiangular for polygons other than triangles? How are polygons related to each other? LONG BEACH UNIFIED SCHOOL DISTRICT 1 Posted 10/16/17

2 [m] G-SRT.B Prove theorems involving similarity. KNOWLEDGE Students will know The Triangle Sum Conjecture The base angles of an isosceles triangle are congruent. The converse is also true. Equal angles are opposite equal sides. The smallest angle of a triangle is opposite its shortest side and the largest angle is opposite the longest side. The Side-Angle Inequality Conjecture The Exterior Angle Conjecture That SSS, SAS, ASA and SAA are shortcuts for proving that triangles are congruent. SSA and AAA are not. Two polygons are congruent if their corresponding sides and angles are congruent. (CPCTC) A proof can be constructed using several methods: flowchart, paragraph, or two-column. Triangles can be used to form other polygons. That the angle bisector of the vertex angle and the median and altitude from that angle are the same line for isosceles triangles and, thus, equilateral triangles as well. Midsegment properties of triangles and trapezoids. The properties of kites, trapezoids, parallelograms, rhombuses, rectangles and squares. Proving theorems within a deductive system ensures the avoidance of circular reasoning. Acquisition SKILLS Students will be skilled at and/or be able to Explain the sum of the measures of the angles of a triangle. Prove the Triangle Sum Conjecture. Verify the Isosceles Triangle Conjecture and its converse. Given a statement, construct its converse or biconditional. Given two sides of a triangle, solve for possible lengths of the third side. Find the measure of an exterior angle in a triangle. Use congruence shortcuts (SSS, SAS, ASA, and SAA) to prove that triangles are congruent. Show that pairs of angles or pairs of sides are congruent by identifying related triangles and proving them congruent, then applying CPCTC. State conjectures as conditional statements. Find the length of the midsegment of a trapezoid or a triangle. Calculate the measures of missing sides and angles. Precisely identify a given polygon. Find the sum of the angle measures in a polygon and the sum of the measures of the exterior angles of a polygon. Plan a proof and support conjectures with definitions, properties of algebra and equality, and postulates. Write flowchart and paragraph deductive proofs. LONG BEACH UNIFIED SCHOOL DISTRICT 2 Posted 10/16/17

3 Standards for Mathematical Practice SMP 1 SMP 2 SMP 3 SMP 4 SMP 5 SMP 6 SMP 7 SMP 8 Assessed Grade Level Standards Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Standards for Mathematical Content [m] G-CO.B Understand congruence in terms of rigid motions. [Build on rigid motions as a familiar starting point for development of concept of G-CO.6 G-CO.7 G-CO.8 geometric proof.] 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. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. [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. G-CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. [a] G-GMD.B Visualize relationships between two-dimensional and three-dimensional objects. G-GMD.6 Verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer, and the sum of any two side lengths is greater than the remaining side length; apply these relationships to solve real-world and mathematical problems. CA [m] G-SRT.B Prove theorems involving similarity. G-SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Key: [m] = major clusters; [s] = supporting clusters; [a] = additional clusters Indicates a modeling standard linking mathematics to everyday life, work, and decision-making CA Indicates a California-only standard LONG BEACH UNIFIED SCHOOL DISTRICT 3 Posted 10/16/17

4 Evidence of Learning Stage 2 Assessment Evidence 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.B Students will explain how the criteria for the triangle congruence shortcuts follow from the definition of congruence in terms of rigid motion. Students will use the definition of congruence, in terms of rigid motions, to decide if two figures are congruent. G-CO.C Students will use inductive and/or deductive reasoning to make conjectures about the properties of triangles, parallelograms, and polygons. Students will follow informal arguments (proofs) of some of these conjectures. G-GMD.B Students will verify that, in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer. Students will verify the Triangle Inequality Conjecture. G-SRT.B Students will use triangle congruence and similarity criteria to solve problems. Students will use triangle congruence and similarity criteria to complete inductive and/or deductive proofs that confirm relationships in triangles, parallelograms, and polygons. Students will use triangle congruence and similarity criteria to write inductive and/or deductive proofs to confirm relationships in triangles, parallelograms, and polygons. 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 Other Evidence Formative Assessment Opportunities 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.B G-CO.C G-STR.B 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 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 Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website M Mathematics Curriculum Documents LONG BEACH UNIFIED SCHOOL DISTRICT 4 Posted 10/16/17

5 1 day 1 day I will explore writing a wellconstructed and valid argument by participating in the Opening Task. I will verify experimentally properties of triangles by Opening Task- Proofs can be constructed in several styles and from several perspectives. In this opening task, the students are asked to prove whether the given figure is a parallelogram. Allow for student questions and conversation.(smp3) Exploring and making conjectures about the sum of the measures of interior angles and exterior angles of triangles, properties of isosceles triangles, and inequality relationships among the sides and angles of triangles. (SMP3) Using patty paper, compass and straightedge, or algebraic techniques to verify. (SMP5) Using inductive and/or deductive reasoning. (SMP3) Answering questions such as o Is knowing the three angles of a triangle enough to determine the triangle? o What is more convincing: an investigation or a proof? o What is the converse of a converse? o Can you construct a triangle that has two equal sides and three different angle measures? o How would you state the Isosceles Triangle Conjecture to a non-math friend? o Under what conditions can the shorter sides of a triangle actually make a triangle? What would happen if the lengths of the two shorter sides of a triangle added up to exactly the same length of the third side? Added up to less than the third side? Added up to more than the third side? Lesson 4-1 Lesson 4-2 Lesson 4-3 Illustrative Mathematics: Is this a Parallelogram? Illustrative Mathematics: Sum of Angles in a Triangle Task Illuminations: Triangle Classification Task Triangle Inequality Theorem Illuminations: Inequalities in Triangles Activity Triangle Exterior Angle Theorem The Sum of Three Angles Activity Khan Academy: Theorems Concerning Triangle Properties Khan Academy: Working with Triangles Khan Academy: Polygons LONG BEACH UNIFIED SCHOOL DISTRICT 5 Posted 10/16/17

6 I will use the Lesson 4-4 definition of Lesson 4-5 congruence to show that two Lesson 4-6 triangles are congruent by 4-6 Using a compass and straightedge to discover that SSS, SAS SSA and ASA are shortcuts for determining congruence of triangles, but SSA and AAA are not. (SMP5) Showing that pairs of angles or pairs of sides are congruent by identifying related triangles and proving them congruent, then applying CPCTC. (SMP3) Using corresponding parts of congruent triangles in paragraph proofs. (SMP3) Using construction skills with a compass and straightedge or patty paper to verify conjectures. (SMP5) Answering questions such as o What is the least amount of information needed to prove that two triangles are congruent? o You know that SSS and SAS are congruence shortcuts. SSA isn t. Why? Are there others? o Is AAA a congruence shortcut? Why or why not? o What is the difference between determining a triangle and being a congruence shortcut? o Why is the word corresponding needed in the statement of the SAA shortcut, but unnecessary in the statement of the ASA shortcut? o When you are looking to prove that two sides or angles are congruent, what are some methods you can use? Illustrative Mathematics: Why does ASA Work? Task Illustrative Mathematics: Why does SSS Work? Task Congruent Triangles Three Sides Equal(SSS) Congruent Triangles Two Sides and Included Angle (SAS) Dynamic Tool Congruent Triangles - Two Angles and Included Side(ASA) Congruent Triangles - Two Angles and an Opposite Side(AAS) Illuminations: Congruence Theorems Illustrative Mathematics: Congruent and Similar Triangles Task LONG BEACH UNIFIED SCHOOL DISTRICT 6 Posted 10/16/17

7 Khan Academy: Transformations and Congruence Khan Academy: Triangle Congruence 2-3 I will check my understanding of congruency proofs by participating in the FAL. FORMATIVE ASSESSMENT LESSON Evaluating Conditions for Congruency (SMP3, SMP5, and SMP7) Evaluating Conditions for Congruency FAL 5-7 I will I will prove theorems about triangles by Using corresponding parts of congruent triangles in flowchart proofs. (SMP3) Knowing that, for isosceles triangles, the angle bisector of the vertex angle and the median and altitude from that angle are the same line. Using properties of isosceles triangles to find the properties of equilateral triangles. Writing supporting statements with definitions, properties of algebra and equality, and postulates. (SMP3) Using deductive reasoning to avoid circular reasoning. Developing a process for planning and writing a proof. (SMP3) Stating conjectures as conditional statements. Using triangle congruence. Answering questions such as o How does the application of logical reasoning facilitate understanding geometric relationships? o Does equilateral imply equiangular for any polygons other than triangles? Lesson 4-7 Lesson 4-8 Lesson 13-1 Lesson 13-2 Lesson 13-3 Illuminations: Joking with Proofs Activity Illuminations: Pieces of Proof Activity Teaching Students First Two-Column Proof Activity Isosceles Triangle Equilateral Triangle Equiangular Triangle LONG BEACH UNIFIED SCHOOL DISTRICT 7 Posted 10/16/17

8 o Using the Vertex Angle Bisector Conjecture, what all can you prove? o Why is it important that conjectures become theorems? o Why is it important to prove theorems within a deductive system? o What makes a good proof and are they difficult to write? o Is it enough to only reason inductively? o What information do you need to write a proof successfully? o What is a planning/thought process that you can use to help you write a proof successfully? Is there a process that is common in all proof writing? o How is writing a proof different from solving an equation? o Why would you choose to use one proof format over another: two-column, flowchart, or paragraph? Khan Academy: Theorems Concerning Triangle Properties Khan Academy: Triangle Congruence 6-8 I will verify experimentally properties of polygons by Finding the sum of the angle measures in a polygon and the sum of the measures of a set of exterior angles of a polygon. Writing and testing formulas for the measure of the interior angle of an equiangular polygon. (SMP8) Using patty paper and/or compass and straightedge, discover properties of kites and trapezoids. (SMP5) Defining properties of midsegments in triangles and trapezoids. Finding edge length and angle measures of parallelograms. Generalizing from the properties of parallelograms to those of rectangles, rhombuses, and squares. (SMP7) Using paragraph and flowchart proofs to support conjectures with deductive reasoning. (SMP3) Answering questions such as o When you collect data, what can cause answers to differ slightly? o Can you derive a formula for the measure of the angle of an n-gon? Can you use patty paper to help you with this? Lesson 5-1 Lesson 5-2 Lesson 5-3 Lesson 5-4 Lesson 5-5 Lesson 5-6 Finding the Measure of an Interior Angle of a Regular Polygon Task Interior Angles of a Parallelogram Dynamic Tool Diagonals of a Parallelogram Midsegment of a Triangle Patty Paper Kite Investigation LONG BEACH UNIFIED SCHOOL DISTRICT 8 Posted 10/16/17

9 o How can the Exterior Angle Sum Conjecture be derived algebraically from the Polygon Sum Conjecture? o Which regular polygons (from three sides to twenty) can be constructed using a compass and straightedge? o What are the differences between the measures of exterior angles in a concave polygon and the measures of exterior angles in a convex polygon? o How can you find the number of sides of an equiangular polygon by measuring one of its interior angles? By measuring one of its exterior angles? o Can the properties of a kite be seen from its lines of symmetry? o Can we use symmetry to see the properties of other polygons? o What strategies are you using to write your proofs? o What do trapezoids and triangles have in common? Can you make a triangle form a trapezoid? o The median of a trapezoid or the midsegment of a trapezoid, which is the better name for that segment? o Using coordinate geometry, how would you confirm that the midsegment of a triangle is parallel to the third side? o How can you use the Triangle Midsegment Conjecture to find a distance between two points that you can t measure directly? o Is the statement In a rectangle, adjacent sides are congruent true or false? Why? o Does a parallelogram have lines of symmetry? How do these show the properties of that parallelogram? o Are there any false conjectures that a person might make about parallelograms? What would lead a person to make these conjectures? o Can a kite be a parallelogram? o Do trapezoids have the Consecutive Angle Property? Any other properties of parallelograms? Khan Academy: Polygons Khan Academy: Theorems Concerning Quadrilateral Properties LONG BEACH UNIFIED SCHOOL DISTRICT 9 Posted 10/16/17

10 o o What can you say about a quadrilateral if it has perpendicular diagonals? Exactly one line of symmetry? Perpendicular diagonals and exactly one line of symmetry? Exactly two lines of symmetry? Perpendicular diagonals and two lines of symmetry? Perpendicular diagonals and four lines of symmetry? Which of the properties for rhombuses, rectangles and squares hold for all parallelograms? What special features of these figures allow for this? 2-3 I will check my understanding of how to apply angle theorems by participating in the FAL. FORMATIVE ASSESSMENT LESSON Applying Angle Theorems (SMP3 and SMP7) Applying Angle Theorems 2-3 I will prove theorems about quadrilaterals by Writing flowchart and paragraph proofs. (SMP3) Using triangle theorems to prove quadrilateral theorems. (SMP3) Answering questions such as o Sometimes working backward can help with writing a proof. What are other times when starting at the end has helped you complete a task? o What premises in the deductive system so far allow you to prove that a quadrilateral is a parallelogram? o How can you prove that the opposite sides of a parallelogram are parallel? o If you have a difficult time with a proof, what are some things you can do to get started? o What are the differences between postulates, definitions, lemmas, and theorems? Lesson 5-7 Lesson 13-4 Khan Academy: Theorems Concerning Quadrilateral Properties LONG BEACH UNIFIED SCHOOL DISTRICT 10 Posted 10/16/17

11 I will prepare for Incorporating the Standards for Mathematical Practice the unit (SMPs) along with the content standards to review the unit. assessment on the triangle and polygon 2 properties by day WODB Triangle and Polygon Properties Prove It! Two-Column Proofs Dynamic Practice Tool 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 11 Posted 10/16/17