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LESSON 10.4 Classify Quadrilaterals FOCUS COHERENCE RIGOR LESSON AT A GLANCE F C R Focus: Common Core State Standards 4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. Mathematical Practices MP2 Reason abstractly and quantitatively. MP4 Model with mathematics. MP6 Attend to precision. F C R Coherence: Standards Across the Grades Before Grade 4 After 3.G.A.1 4.G.A.2 5.G.B.3 5.G.B.4 F C R Rigor: Level 1: Understand Concepts....................Share and Show ( Checked Items) Level 2: Procedural Skills and Fluency.......On Your Own Level 3: Applications..................................Think Smarter and Go Deeper Learning Objective Sort and classify quadrilaterals. Language Objective Students draw quadrilaterals to demonstrate how you can sort and classify quadrilaterals. Materials MathBoard, color pencils F C R For more about how GO Math! fosters Coherence within the Content Standards and Mathematical Progressions for this chapter, see page 547J. About the Math Professional Development Why Teach This Interactive Student Edition In this lesson, the definitions of quadrilaterals are formalized and the relationships among them are emphasized. First, students identify right angles in Professional Development quadrilaterals and recognize that a quadrilateral cannot have exactly three right angles. Next, students examine Professional Development quadrilaterals by the number of pairs of parallel sides and find that some quadrilaterals have 1 pair, some have 2 pairs, while others have none. About the Math About the Math Personal Math Trainer Math on the Spot Animated Math Models Students then classify quadrilaterals in as many ways as possible. For example, a figure can be a quadrilateral, a trapezoid, a parallelogram, and a rectangle. Through this process, students gain a deeper understanding of the properties of quadrilaterals and how they relate to each other. Professional Development Videos 567A Chapter 10 iTools: Geometry HMH Mega Math 1 ENGAGE Daily Routines Common Core Problem of the Day 10.4 Simone makes a quilt square using two triangles as shown. What types of triangles are they? right Common Core Daily Routines with the Interactive Student Edition Essential Question How can you sort and classify quadrilaterals? Making Connections Invite students to tell you what they know about toy kites. What is a toy kite? A kite is a toy that has a light frame with thin material stretched over it, flown in the wind at the end of a long string. How many sides does a basic kite have? 4 sides Learning Activity What is the problem the students are trying to solve? Connect the story to the problem. Vocabulary parallelogram, rectangle, rhombus, square, trapezoid • Interactive Student Edition • Multimedia eGlossary • What problem are you trying to solve? Find the name of the figure given in the problem. • What properties of the given figure do you notice? The figure has four sides and four angles. Literacy and Mathematics Choose one or more of the following activities. Vocabulary Builder Materials magazines, scissors, glue, paper, stapler Identify Quadrilaterals Have students write the name and definition of each type of quadrilateral at the top of separate half-sheets of paper. Then have students look through old magazines and cut out examples of each type of quadrilateral. Students then glue the cutout examples onto the half-sheet of paper for each quadrilateral. When done, have students create a booklet by folding a sheet of paper in half and inserting the half-sheets inside. Staple the pages together. Students can title their booklet “Quadrilaterals” and draw an example of each type on the front cover. • Have students write the problem as a short story. Have them predict what attributes of the figure they could use to determine the name of the figure. • Have students create kites using paper and scissors. Have students compare their kites with each other and note the differences between the kites. How can you sor t and classi fy quadrilaterals? Lesson 10.4 567B LESSON 10.4 2 EXPLORE 4.G.A.2 Classify two-dimensional figures based“File on the presence or absence of parallel or perpendicular lines, or the presence DO NOT EDIT--Changes must be made through info” or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. CorrectionKey=B Lesson 10.4 Name Unlock the Problem Classify Quadrilaterals MP6 Attend to precision. Review naming a quadrilateral by its vertices. Discuss the quadrilaterals shown. Explain that tick marks on a figure indicate sides equal in length. • What are some other possible names for quadrilateral ABCD? Possible answers: Unlock Unlock the the Problem Problem Quadrilateral ABCD is a possible name for the figure shown at the right. Quadrilateral ACBD is not a possible name, since points A and C are not endpoints of the same side. • What can you tell about the sides of a rhombus? Explain. All four sides have one tick A D The tick marks on the line segments show that they have the same length. Sides AD and BC have the same length. Sides AB and CD have the same length. Assume that line segments that appear to be parallel are parallel. mark each, so all four sides are equal in length. C B A quadrilateral is a polygon with four sides and four angles. You can name a quadrilateral by the vertices of its angles. BCDA, CDAB, DABC, ADCB, DCBA, CBAD, BADC Geometry— 4.G.A.2 MATHEMATICAL PRACTICES MP1, MP2, MP6 Essential Question How can you sort and classify quadrilaterals? MATHEMATICAL PRACTICES Common Quadrilaterals ELL Strategy: Illustrate Understanding Trapezoid • at least 1 pair of parallel sides • On separate index cards, have students draw and label a quadrilateral, a trapezoid, a parallelogram, a rhombus, a rectangle, and a square. Rectangle • 2 pairs of parallel sides • 2 pairs of sides of equal length • 4 right angles Square • 2 pairs of parallel sides • 4 sides of equal length • 4 right angles Materials ■ color pencils Use the Quadrilateral Color Guide to color the quadrilaterals. Quadrilateral Color Guide RED: R © Houghton Mifflin Harcourt Publishing Company Remind students to look for the number of right angles in each quadrilateral. • How do you know which angles are right angles? Right angles are marked with square corners. • How can you describe the two sides of a quadrilateral that form a right angle? The Rhombus • 2 pairs of parallel sides • 4 sides of equal length Activity 1 Identify right angles in quadrilaterals. • On the back of the card, have students draw a line down the center and write the book’s definition on the left and their own description on the right. Activity 1 Parallelogram • 2 pairs of parallel sides • 2 pairs of sides of equal length exactly 4 right angles BLUE: exactly 2 right angles ORANGE: exactly 1 right angle O B No; possible explanation: if a quadrilateral has 3 right angles, the fourth angle must also be a right angle. Math Talk MATHEMATICAL PRACTICES 6 Can a quadrilateral have exactly 3 right angles? Explain. Chapter 10 567 sides are perpendicular. • Do all parallelograms have right angles? Explain. No; possible explanation: squares and rectangles are parallelograms that have right angles, but other parallelograms do not have right angles. Math Talk Use Math Talk to help students understand that a quadrilateral cannot have exactly three right angles. • Look at the rectangle in Activity 1. Could you have redrawn it to show a quadrilateral with only 3 right angles? Explain. Possible answer: no; 4_MNLESE342286_C10L04.indd 567 Reteach 10.4 the acute angle to a right angle, I would have had to change the obtuse angle to a right angle to still have 4 sides. 567 Chapter 10 2 DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B 1 Lesson 10.4 Reteach Name Differentiated Instruction Enrich 10.4 3 DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B Lesson 10.4 Enrich Name Classify Quadrilaterals Quad Logic Read each statement carefully. Write true or false. A quadrilateral is a polygon with 4 sides and 4 angles. Some quadrilaterals have special names: Quadrilateral 4 sides 4 angles 2. Alltrapezoidsareparallelograms. 3. Allsquaresarerectangles. 4. Somequadrilateralsaretrapezoids. Parallelogram 2 pairs of parallel sides Rectangle 4 right angles 2 pairs of parallel sides true false true true true false true false 1. Someparallelogramsarerectangles. Trapezoid At least 1 pair of parallel sides if I changed one right angle, I would have had to change another right angle to still have 4 sides. • Look at the trapezoid with 2 right angles in Activity 1. Could you have redrawn it to show a quadrilateral with 3 right angles? Explain. Possible answer; no; if I would have changed 1/8/15 12:31 PM 5. Somerectanglesarerhombuses. 6. Allrhombusesaresquares. 7. Someparallelogramsaretrapezoids. Rhombus 4 sides of equal length 8. Allrectanglesaresquares. Make each statement true. Write All, No, or Some. Square 4 right angles 4 sides of equal length All All All Some 9. 10. 11. Classify each figure as many ways as possible. Write quadrilateral, trapezoid, parallelogram, rhombus, rectangle, or square. 12. 1. 2. quadrilateral, trapezoid quadrilateral, quadrilateral, trapezoid, trapezoid, parallelogram, parallelogram rectangle, rhombus, square Chapter Resources © Houghton Mifflin Harcourt Publishing Company 4_MNLEAN343153_C10R04.indd 11 3. 10-11 rectanglesareparallelograms. squaresaretrapezoids. parallelogramsarequadrilaterals. quadrilateralsareparallelograms. 13. Stretch Your Thinking Writethreeofyourown Reteach quad-logicstatements.Thenexchangethemwithaclassmate andcompleteeachother’sstatements. Check students’ work. Chapter Resources © Houghton Mifflin Harcourt Publishing Company 1/8/15 5:24 PM 4_MNLEAN343153_C10E04.indd 12 10-12 Enrich 1/8/15 4:24 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B Activity 2 Direct students to look for pairs of parallel sides. • How can you determine which sides are parallel? Possible answer: I extended the sides, they Activity 2 Use a Venn diagram to sort quadrilaterals. Write the names of the quadrilaterals in the Venn diagram. F G E H B C A D R would never cross. Q • What do you notice about the side lengths of quadrilaterals with two pairs of parallel sides? Possible answer: the opposite sides are of P S U K J equal length. N L V W M Z Q T • If a quadrilateral has exactly one pair of parallel sides, do the other sides have to be of equal length? Explain. No, not necessarily; R for example, quadrilateral JKLM has exactly one pair of parallel sides, but the other sides are not of equal length. Quadrilaterals MP8 Look for and express regularity in repeated reasoning. • What names can you use to classify all of the quadrilaterals that you identified as having exactly 1 pair of parallel sides? 2 Pairs of Parallel Sides At least 1 Pair of Parallel Sides No Parallel Sides quadrilateral PQRS quadrilateral JKLM quadrilateral ABCD quadrilateral QRNZ quadrilateral TUVW quadrilateral EFGH quadrilateral and trapezoid • What names can you use to classify all of the quadrilaterals that you identified as having 2 pairs of parallel sides? quadrilateral, trapezoid, and parallelogram b. c. quadrilateral quadrilateral quadrilateral, trapezoid, trapezoid trapezoid parallelogram, rhombus, parallelogram rectangle, square 568 4_MNLESE342286_C10L04.indd 568 1/10/15 1:18 PM Visual Advanced Learners Advanced Advanced Learners Learners Partners Materials Venn Diagram (see eTeacher Resources) •Distribute a Venn diagram to each pair of students. •Have partners use attributes of quadrilaterals to show the relationships of the following quadrilaterals in a Venn diagram: quadrilateral, trapezoid, parallelogram, rhombus, rectangle, and square. •Have students explain their Venn diagrams. Quadrilaterals Trapezoids Parallelograms Rectangles Rhombuses Squares © Houghton Mifflin Harcourt Publishing Company a. © Houghton Mifflin Harcourt Publishing Company Try This! Classify each figure as many ways as possible. Write quadrilateral, trapezoid, parallelogram, rhombus, rectangle, or square. Try This! • Is a trapezoid always a quadrilateral? Explain. Yes; possible explanation: a trapezoid is a polygon with four sides and four angles, so it is always a quadrilateral. • How are a square and a rectangle alike, and how are they different? Possible answer: both are quadrilaterals with two pairs of parallel sides and four right angles. A square always has four sides of equal length, but a rectangle does not. A rectangle always has two pairs of sides of equal length. Teacher Note: Some mathematicians define a trapezoid as a quadrilateral with exactly one pair of parallel sides. This book does not use this definition. COMMON ERRORS COMMON ERRORS Error Students may not identify all possible ways to classify a figure. Example quadrilateral, parallelogram, square, rhombus Springboard to Learning Have students systematically go through the properties for each type of quadrilateral and determine if the properties are true for the figure. Lesson 10.4 568 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B Name 3 EXPLAIN Share Share and and Show Show MATH BOARD 1. Tell whether the quadrilateral is also a trapezoid, parallelogram, Share and Show rhombus, rectangle, or square. MATH BOARD B Think: C A 3 2 31 2 1 Rt I Rt I If Then 2. 3. quadrilateral quadrilateral quadrilateral, trapezoid trapezoid, parallelogram parallelogram rectangle rhombus • How many parallel sides does a trapezoid have? at least 1 pair • How many parallel sides do a parallelogram, a rhombus, a rectangle, and a square have? • What kind of 4-sided figure has no parallel sides? a quadrilateral; a quadrilateral can have 1 pair of Classify each figure as many ways as possible. Write quadrilateral, trapezoid, parallelogram, rhombus, rectangle, or square. 5. 6. quadrilateral trapezoid parallelogram In Exercises 5–7, students classify quadrilaterals in as many ways as possible. DEEPER MP7 Look for and make use of structure. • Is it possible to draw a rhombus that is also a rectangle? Explain. Yes; possible explanation: 7. quadrilateral, trapezoid, parallelogram, rhombus, quadrilateral trapezoid rectangle, square Chapter 10 • Lesson 4 parallel sides, 2 pairs of parallel sides, or no parallel sides On Your Own MATHEMATICAL PRACTICES 2 Use Reasoning How would you classify a figure with 4 sides, none of which are parallel? Explain. On On Your Your Own Own © Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company Use Math Talk to focus on students’ understanding that a quadrilateral without any parallel sides can only be named a quadrilateral. 569 Chapter 10 4. Math Talk Math Talk draw a square. A square is a rhombus and a rectangle. 0 right angles _ Classify each figure as many ways as possible. Write quadrilateral, trapezoid, parallelogram, rhombus, rectangle, or square. Differentiate Instruction with • Reteach 10.4 • Personal Math Trainer 4.G.A.2 • RtI Tier 1 Activity (online) 2 pairs D parallelogram, rhombus, and Quadrilateral ABCD is also a _____ a trapezoid ____ . a student misses the checked exercises If Math Talk: quadrilateral; possible explanation: because a parallelogram, a square, a rectangle, and a rhombus all have 2 pairs of parallel sides, and a trapezoid has at least one pair of parallel sides 4 sides of equal length _ Use the checked exercises for Quick Check. Quick Check Quick Check 2 pairs of parallel sides _ 4_MNLESE342286_C10L04.indd 569 569 1/8/15 12:44 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B MATHEMATICAL PRACTICES MODEL • REASON • MAKE SENSE Problem Problem Solving Solving •• Applications Applications 8. SMARTER Explain how a rhombus and square are alike, and how they are different. 4 ELABORATE 9. SMARTER Classify the figure. Select all that apply. Problem Solving • Applications Possible explanation: a rhombus and square are both quadrilaterals, trapezoids, and parallelograms. Both have 4 sides of equal length and 2 pairs quadrilateral rectangle trapezoid rhombus parallelogram square MATHEMATICAL PRACTICES SMARTER Exercise 8 requires a student to understand that a rhombus may or may not have 4 right angles. When it does, it is also a square. However, a rhombus and a square both always have 4 sides of equal length. of parallel sides. A rhombus does not always The Louvre Museum is located in Paris, France. Architect I. M. Pei designed the glass and metal structure at the main entrance of the museum. This structure is called the Louvre Pyramid. WRITE Below is a diagram of part of the entrance to the Louvre Pyramid. 10. MATHEMATICAL PRACTICE 1 Math • Show Your Work see in the diagram. DEEPER How many triangles do you see in the diagram? Explain. Possible answer: there are trapezoids 11 triangles; 4 small triangles, 3 triangles with 1 pair of parallel sides. There are made of 2 triangles and 1 rhombus, also trapezoids with 2 pairs of parallel sides that also look like rhombuses. If you combine the smaller shapes in different ways, you can also see other trapezoids, rhombuses, and parallelograms. 2 triangles made of 3 triangles and 3 rhombuses, 1 triangle made of 4 triangles and 6 rhombuses, and 1 large outer triangle. total triangles 5 4 1 3 1 2 1 1 1 1 = 11 Describe the quadrilaterals you 11. © Houghton Mifflin Harcourt Publishing Company • Image Credits: (cr) ©Robert Harding Picture Library Ltd/Alamy Images have 4 right angles, but a square does. 4_MNLESE342286_C10L04.indd 570 3/9/15 3:30 PM DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIES Differentiated Centers Kit Students complete purple Activity Card 13 by drawing and classifying polygons. Use this video to help students model and solve this type of Think Smarter problem. Math on the Spot videos are in the Interactive Student Edition and at www.thinkcentral.com. SMARTER Exercise 9 assesses students’ ability to classify quadrilaterals and identify subgroups based on their properties. Students who have difficulty explaining the relationship of a square and rectangle may not know their definitions. More likely, however, is that those students do not understand the heirarchy in the classification of quadrilaterals. Connect to Art 570 Activities Connecting Vertices Math on the Spot Video Tutor Literature A New Angle on Trains and Train Stations Students read about lines, angles, and plane figures that can be found in train stations. MP1 Make sense of problems and persevere in solving them. • What type of quadrilateral is formed by a rhombus and a small triangle in the diagram? a trapezoid 5 EVALUATE Formative Assessment Essential Question Using the Language Objective Reflect Have students draw quadrilaterals and demonstrate to answer the Essential Question. How can you sort and classify quadrilaterals? Possible answer: by the number of pairs of parallel sides, number of equal length sides, and number of right angles Math Journal WRITE Math Draw and label an example of each type of quadrilateral: trapezoid, parallelogram, rhombus, rectangle, and square. Lesson 10.4 570 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B Practice and Homework Name Lesson 10.4 Classify Quadrilaterals COMMON CORE STANDARD—4.G.A.2 Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Classify each figure as many ways as possible. Write quadrilateral, trapezoid, parallelogram, rhombus, rectangle, or square. 1. Think: 2 pairs of parallel sides 4 sides of equal length 0 right angles Practice and Homework Use the Practice and Homework pages to provide students with more practice of the concepts and skills presented in this lesson. Students master their understanding as they complete practice items and then challenge their critical thinking skills with Problem Solving. Use the Write Math section to determine student’s understanding of content for this lesson. Encourage students to use their Math Journals to record their answers. quadrilateral, trapezoid, parallelogram, rhombus 2. 3. quadrilateral, trapezoid, 4. quadrilateral, trapezoid quadrilateral parallelogram, rectangle Problem Problem Solving Solving 5. Alan drew a polygon with four sides and © Houghton Mifflin Harcourt Publishing Company four angles. All four sides are equal. None of the angles are right angles. What figure did Alan draw? PROFESSIONAL DEVELOPMENT PROFESSIONAL PROFESSIONAL DEVELOPMENT DEVELOPMENT 7. 6. Teresa drew a quadrilateral with 2 pairs of parallel sides and 4 right angles. What quadrilateral could she have drawn? Possible answers: a quadrilateral, Possible answers: a square, rhombus, trapezoid rectangle, trapezoid Math Draw and label an example of WRITE each type of quadrilateral: trapezoid, parallelogram, rhombus, rectangle, and square. Check students’ drawings. Mathematical Practices in Your Classroom Chapter 10 571 4_MNLESE342286_C10P04.indd 571 Mathematical Mathematical Practices Practices in in Your Your Classroom Classroom CCSS.Math.Practice.MP7 Look for and make use of structure. Ask questions such as the following to help students focus on the structure of geometric figures in order to classify them: In order to classify quadrilaterals, students must understand the structure of geometric figures. How many parallel lines does it have? Does it have right angles? How many sides of equal length does it have? By analyzing the structure of the figure, students can describe and classify it. This analysis deepens students’ understanding of properties of two-dimensional figures and how they are related. • How can identifying pairs of parallel lines help you classify quadrilaterals? Possible answer: I can check if the figure has 1/8/15 1:26 PM 2 pairs, exactly 1 pair, or no pair of parallel lines. • How can identifying the number of right angles help you classify quadrilaterals? Answers will vary. Possible answer: if the quadrilateral has 4 right angles, it is a rectangle, a square, a parallelogram, and a trapezoid and sometimes it is a rhombus. • How can identifying the number of sides of equal length help you classify quadrilaterals? Answers will vary. Possible answer: if the quadrilateral has 4 sides of equal length, it is a rhombus, a square, a rectangle, a parallelogram, and a trapezoid. But not all rectangles, parallelograms, and trapezoids are rhombuses or squares. 571 Chapter 10 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B TEST PREP Lesson Check (4.G.A.2) 1. Joey is asked to name a quadrilateral that is also a rhombus and has 2 pairs of parallel sides. What should be his answer? 2. What quadrilateral has at least one pair of parallel sides, but cannot be called a parallelogram? Continue concepts and skills practice with Lesson Check. Use Spiral Review to engage students in previously taught concepts and to promote content retention. Common Core standards are correlated to each section. trapezoid square Spiral Review (4.OA.B.4, 4.OA.C.5, 4.NF.B.3d, 4.G.A.1) equal groups. What are all the possible numbers of eggs that Terence could put in each group? 4. In a line of students, Jenna is number 8. The teacher says that a rule for a number pattern is add 4. The first student in line says the first term, 7. What number should Jenna say? 1, 2, 3, 4, 6, 8, 12, 24 6 of a pizza. What fraction of the 5. Lou eats __ 8 pizza , in simplest form, is left over? 1 __ 4 35 6. Name a capital letter that appears to have parallel lines. Possible answer: N © Houghton Mifflin Harcourt Publishing Company 3. Terrence has 24 eggs to divide into FOR MORE PRACTICE GO TO THE 572 4_MNLESE342286_C10P04.indd 572 Personal Math Trainer 1/8/15 1:28 PM Lesson 10.4 572