Number of : 14 1/29/18-2/16/18 Unit Goals Stage 1 Unit Description: Drawing on our knowledge of the definitions of special quadrilaterals and a few of their properties, students will now prove those properties using a variety of proof formats: transformational, analytical, synthetic and paragraph. Students will begin this unit by deriving formulas for the sum of the interior angles in a polygon and the sum of the exterior angles in a polygon. Drawing on our knowledge of triangle congruence from Unit 2, students will now apply that knowledge to proofs involving quadrilaterals. Our family of special quadrilaterals includes parallelograms, rectangles, rhombi, squares, kites, trapezoids and isosceles trapezoids. Materials: Patty paper, rulers, compasses, protractors 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 SMP 5 SMP 6 SMP 7 SMP 8 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 Clusters Addressed [m] G-CO.C Prove geometric theorems [m] G-SRT.B Prove theorems involving similarity. [m] G-MG.A Apply geometric concepts in modeling situations. 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. UNDERSTANDINGS Students will understand that Triangle congruence criteria can be used to prove properties of parallelograms. Simple theorems about coordinates, angles, segments and slope can be used to prove the properties of the subsets of polygons. Coordinates can be used to prove relationships between the lines and angles of quadrilaterals on a coordinate grid. KNOWLEDGE Students will know Properties of a parallelogram and its subsets of quadrilaterals. Properties of trapezoids and kites. Properties of diagonals and opposite angles of quadrilaterals. Making Meaning ESSENTIAL QUESTIONS Students will keep considering How can the properties of a triangle be used to prove the properties of a quadrilateral? What properties of a quadrilateral can you use to identify its shape precisely? How are properties of geometric figures related to their measurable attributes? How does graphing a quadrilateral on a coordinate plane help to identify its shape precisely? Acquisition SKILLS Students will be skilled at and/or be able to Find and use interior and exterior angle measures of polygons. Use properties of parallelograms and special parallelograms to find side lengths and angles. Use the coordinate plane to identify quadrilaterals and solve for their properties. Identify and verify that a quadrilateral is a parallelogram Identify and use properties of trapezoids and kites. 2017-2018 1 Posted 12/20/17
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 [m] G-CO.C G-CO.11 [m] G-SRT.B G-SRT.5 [m] G-MG.A G-MG.1 G-MG.3 Prove geometric theorems. [Focus on validity of underlying reasoning while using variety of ways of writing proofs] 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. Prove theorems involving similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply geometric concepts in modeling situations. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). 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 2017-2018 2 Posted 12/20/17
Evidence of 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: [m] G-CO.C Students will prove theorems about parallelograms, such as: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. [m] G-SRT.B Students will use congruence for triangles to solve problems. Students will use congruence to prove relationships in polynomials. [m] G-MG.A Students will use geometric shapes, measures and properties to solve real-world problems. Students will apply geometric methods to solve design problems. 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 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.C G-SRT.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: G-MG.A 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 Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website M Mathematics Curriculum Documents 2017-2018 3 Posted 12/20/17
OPENING TASK Diagonals to Quadrilaterals This Opening Task provides the students with an opportunity to explore the relationship between a pair of intersecting lines (diagonals) and the resulting quadrilateral. After connecting the endpoints of intersecting segments, the students will be asked to describe/identify the resulting quadrilateral. Ask the students to make generalizations and conjectures. While this activity is best when pairs of students have access to computers/tablets, the teacher can demonstrate the applet for the class or ask students to come up to the front and manipulate the figure. Illuminations: Diagonals to Quadrilaterals 1 day I will identify a quadrilateral based on the given diagonals. The purpose of this task is to encourage student exploration and hypothesizing. This task is an opportunity for the students to begin to exploring and making connections with the properties of quadrilaterals. 2017-2018 4 Posted 12/20/17
3-4 I will investigate the properties and conditions of polygons by Finding and using interior angle measures to identify a Section 7.1 polygon. Section 7.2 Finding and using exterior angle measures to identify a Section 7.3 polygon. Using properties to find side lengths and angles of parallelograms. Identifying and verifying parallelograms. Using a coordinate plane to show that a quadrilateral is a parallelogram. Answering questions such as: o Are the Polynomial Interior and Exterior Angles Theorems valid for both convex and concave polynomials? o Is it possible for a polygon to be equilateral and not equiangular? o What is the measure of the central angle a regular n- gon? o Is it possible for a pentagon to have four right angles? o What is true about the consecutive angles, and sides, of a parallelogram? o How do you prove that a quadrilateral is a parallelogram? o How can you differentiate between parallelograms and other quadrilaterals (rectangles, rhombuses, and/or rectangles)? o How can you construct a parallelogram? o How can you construct a regular hexagon? o How can you use coordinates to write a proof? Illuminations: Perplexing Parallelograms Parallelograms Conjectures and Proofs Is this a Parallelogram? Illuminations: Square Graphs Illuminations: Cutting Corners Geogebra: Polygon Interior Angles Geogebra: Car Drives Through the Exterior Angles of a Polygon Geogebra: Polygons Exterior Angles Is this a Rectangle? Mathematics VisionProject: Under Construction Khan Academy: Properties of Shapes Khan Academy: Classifying Geometric Shapes Khan Academy: Polygons Khan Academy: Angles with Polygons Types Proofs and Angles 2017-2018 5 Posted 12/20/17
3-4 I will investigate the properties of special parallelograms, trapezoids and kites by Using properties of special parallelograms to prove whether Section 7.4 or not the figure is a rhombus, rectangle or square. Section 7.5 Using coordinate geometry to identify special types of STEM Video: parallelograms. Diamonds Using properties of trapezoids to find angle measures and side lengths. Using the Trapezoid Midsegment Theorem to find segment lengths. Using the properties of kites to find angle measures, diagonals and side lengths. Proving properties of kites and trapezoids. Using properties to classify a quadrilateral. Answering questions such as: o How can you differentiate between parallelograms and other quadrilaterals (rectangles, rhombuses, rectangles, kites and/or trapezoids)? o Compare and contrast the diagonal properties of the quadrilaterals. o Always, sometimes or never true: A rectangle is a rhombus; a square is a rectangle; a trapezoid is a parallelogram? o Which quadrilateral could be called a regular quadrilateral? o Is SSASS a valid congruence theorem for kites? o How can you use coordinates to help write a proof? Geogebra: Special Quadrilaterals Geogebra: Properties of Trapezoids Geogebra: Properties of a Kite MathOpenRef: Kite Types Proofs and Angles Khan Academy: Quadrilaterals on the Coordinate Plane Applications: Floor Pattern Task STEM Performance Task: Diamonds 2017-2018 6 Posted 12/20/17
2-3 I will check my knowledge about the properties of quadrilaterals by participating in the FAL. FORMATIVE ASSESSMENT LESSON Guess My Parallelogram FAL: Guess My Parallelogram 1-2 I will prepare for the unit assessment on polygons by... Incorporating the Standards for Mathematical Practice (SMPs) along with the content standards to review the unit. Which One Doesn t Belong: Polygons Sorting Quadrilaterals 1-2 Unit Assessment Unit Assessment Students will take the Synergy Online Unit Assessment. Unit Assessment Resources (Word or PDF) can be used throughout the unit. 2017-2018 7 Posted 12/20/17