12.7 LESSON Describe Triangles FOUS OHERENE RIGOR LESSON AT A GLANE F R Focus: ommon ore State Standards 3.G.A.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. MATHEMATIAL PRATIES (See ematical Practices in GO! in the Planning Guide for full text.) MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP7 Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning. F R oherence: Standards Across the Grades Before Grade 3 After 2.G.A.1 3.G.A.1 4.G.A.2 F R Rigor: Level 1: Understand oncepts...share and Show ( hecked Items) Level 2: Procedural Skills and Fluency...On Your Own, Practice and Homework Level 3: Applications...Think Smarter and Go Deeper Learning Objective Describe and compare triangles based on the number of sides that have equal length and by their angles. Language Objective Students draw and label a four panel cartoon to show how to use sides and angles to help you describe triangles. Materials Board, straws, scissors F R For more about how GO! fosters oherence within the ontent Standards and ematical Progressions for this chapter, see page 695J. About the Professional Development Why Teach This Interactive Student Edition As students describe and compare triangles based on sides and angles, they begin to develop a better understanding of the properties of triangles. Personal Trainer By exploring different triangles students can formulate conjectures about geometric properties and relationships. Students learn to develop and articulate clear mathematical arguments about why geometric relationships are true. For example: You can t make a triangle with two right angles, because if you start with a side across the bottom and make two right angles the sides go straight up and can t meet. These types of mathematical arguments help students verbalize what they have learned and make them better problem solvers. Professional Development Videos 735A hapter 12 on the Spot Video itools: Geometry
Daily Routines ommon ore Problem of the Day 12.7 One box has a mass of 96 grams. Another box has a mass of 115 grams. What is the mass of the boxes together? Vocabulary 211 grams Interactive Student Edition Multimedia Glossary e 12 34 Fluency Builder ommon ore Fluency Standard 3.OA..7 Materials Number Wheels (see eteacher Resources) Multiplication Facts Practice Have students use a number wheel to practice multiplying by 8. Write 8 in the center of the number wheel and the numbers 2 9 in any order in the inner circle. Have students find and record the products in the outer circle. 12 34 Pages 46 47 in Strategies and Practice for Skills and Facts Fluency provide additional fluency support for this lesson. 1 ENGAGE with the Interactive Student Edition Essential Question How can you use sides and angles to help you describe triangles? Making onnections Invite students to tell you what they know about triangles. How many sides does a triangle have? 3 How many angles does a triangle have? 3 Look around the room. Point out any triangles you see. Learning Activity What is the problem the students are trying to solve? onnect the story to the problem. Ask the following questions. How does Lucia want to describe the triangle? using its sides and angles Think of other shapes you've described, like quadrilaterals. How have you described their sides? Possible answers: parallel sides, sides of equal length How have you described angles? Possible answers: more than a right angle, less than a right angle Literacy and ematics View the lesson opener with the students. Then, choose one or more of the following activities to complete after they finish the lesson: Have students use dot paper to draw 3 to 5 different triangles. Have them write about the characteristics of the triangles. Have students draw a triangle on dot paper and then write a description of the triangle. Have students trade their description with a partner. The partner draws the triangle based on the description. Asks students to compare drawings to see if they match. How can you use sides and angles to help you describe triangles? 735B
LESSON 12.7 2 EXPLORE Unlock the Problem MATHEMATIAL PRATIES To introduce the lesson, have students watch the Real World Video, Shapes in Architecture. Activity MP4 Model with mathematics. The purpose of this activity is to help students understand the relationship between the lengths of the sides of a triangle. Students should recognize that if the length of two straws put together is not longer than the third straw, they cannot form a triangle. Make a triangle using 2 straws of equal length. What must be true of the length of the third straw in order to be able to form a triangle? It must be less than the length of the two equal-sized straws put together. Build a triangle using 3 straws of different lengths. Allow students time to share their work. Why could some of you build a triangle with 3 straws of different lengths? Possible answer: I could make a triangle because one of my straws is shorter than the other two straws put together. Use to help students recognize that a triangle can have 3 sides of equal length. MP6 Attend to precision. What types of angles do you see in the triangles that you made? Possible answers: right; greater than a right angle, less than a right angle Hands On 3.G.A.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. Describe Triangles Essential Question How can you use sides and angles to help you describe triangles? Unlock the Problem How can you use straws of different lengths to make triangles? Activity Materials straws scissors Board 1. ompare the lengths of the sides. Describe when you can make a triangle. Possible description: 2. STEP 1 ut straws into different lengths. STEP 2 Find straw pieces that you can put together to make a triangle. Draw your triangle on the Board. STEP 3 Find straw pieces that you cannot put together to make a triangle. when two of the straw pieces put together are longer than the third piece, you can close up the straws to make a triangle. MATHEMATIAL PRATIE 1 Describe when you cannot make a triangle. Possible description: when two of the straw pieces put together are shorter than the third piece, you cannot close up the straws to make a triangle. 3. Explain how you can change the straw pieces in Step 3 to make a triangle. Possible explanation: you can exchange two longer pieces for the two shorter pieces. Geometry 3.G.A.1 MATHEMATIAL PRATIES MP1, MP2, MP6 Hands On MATHEMATIAL PRATIES 2 Reason Abstractly What if you have three straws of equal length? an you make a triangle? Yes; possible explanation: because two equal-length straw pieces put together would be longer than the third equal-length straw piece, you can close them up to make a triangle. hapter 12 735 ELL Strategy: Illustrate Understanding The names of shapes tell students how many sides the shape should have. Restate the meanings of the prefixes in the words: triangle three quadrilateral four pentagon five hexagon six In pairs, have students write the names of the shapes on one side of an index card and write the meaning of the prefix underneath it. Have students draw an example of the shape on the back of the card. 735 hapter 12 Describe Triangles You can describe a triangle by its types of angles. 1 right angle. 1 angle greater than a right angle. 3 angles less than a right angle. Draw a line to match the description of the triangle(s). 1. One angle is a right angle. 2. One angle is greater than a right angle. 3. Three angles are less than a right angle. hapter Resources 2 Reteach 12.7 Enrich 12.7 1 You can describe a triangle by the number of sides of equal length. 0 sides of the same length. 2 sides of the same length. 3 sides of the same length. Reteach 4. No sides are equal in length. 5. Two sides are equal in length. 6. Three sides are equal in length. 12-17 Reteach 3 Differentiated Instruction Sorting Triangles A description of a triangle is on each bucket. Write the letter of the triangle in all buckets that correctly describe it. Each triangle can go in at least two buckets. Some triangles can go in more than two buckets. A D Appears to have 1 right angle A, F 0 sides of equal length E, F B E 1 angle greater than a right angle, E Exactly 2 sides of equal length Exactly 2 angles less than a right angle A,, E, F F 3 sides of equal length A,, D B Sense or Nonsense? Beryl says the shape at the right will go in the first and last buckets because it has 1 right angle and 3 equal sides. Does her statement make sense? Explain. Enrich 3 angles less than a right angle B, D No. Possible explanation: the shape is not a triangle because it is not closed. hapter Resources 12-18 Enrich
Ways to Describe Triangles What are two ways triangles can be described? 736 One Way A A B E Triangles can be described by the number of sides that are of equal length. Draw a line to match the description of the triangle(s). No sides are equal in length. Another Way Triangles can be described by the types of angles they have. Draw a line to match the description of the triangle(s). One angle is a right angle. B A E E Two sides are equal in length. One angle is greater than a right angle. No; possible explanation: if there were two right angles, the shape would have to have more than three sides and would no longer be a triangle. D B D Three sides are equal in length. D Three angles are less than a right angle. MATHEMATIAL PRATIES 2 Use Reasoning an a triangle have two right angles? Ways to Describe Triangles Students will learn how to describe triangles using the lengths of the sides and the types of angles. One Way This way focuses on describing triangles by the lengths of their sides. How can a ruler help you to describe a triangle based on its sides? I can use a ruler to measure the sides to see if any of them are the same length. Another Way This way focuses on describing triangles by the types of angles. Remind students of the types of angles. How can you use the corner of a sheet of paper to help you describe an angle in a triangle? I can use the square corner to find out if any of the angles are right angles. Draw a triangle like this one on the board. Have students describe the triangles they see. Possible answers: one triangle has three angles that are less than a right angle, and two triangles have one right angle. Use to explain that a shape with two right angles and three sides cannot be a closed shape. Do you think a triangle could have two angles that are greater than a right angle? Explain. No, if there were two angles greater than a right angle the shape would have to have more than three sides and would no longer be a triangle. Advanced Learners Materials ruler, Dot Paper (see eteacher Resources) Draw the following polygons on the board and have students draw them on square dot paper. Verbal / Linguistic Individual / Partner Then have students draw line segments from one vertex of each polygon to all other vertices. Have students describe and compare the triangles formed. Then challenge students to find the relationship between the number of triangles a polygon can be divided into and the number of sides the polygon has. The number of triangles is equal to the number of sides minus 2. OMMON ERRORS Error Students may not be able to identify sides of equal length or the types of angles. Example Students may not be able to identify sides of equal length in Exercise 1 or the types of angles in Exercises 2-4 on page 737. Springboard to Learning Remind students that they can always use a ruler to check the lengths of sides and they can always use a square corner to check the types of angles. 736
3 If Then EXPLAIN Share and Show MATH BOARD Quick heck MATH Use to check students understanding of how to describe triangles. an you explain another way to describe triangle G? Possible answers: triangle G has 2 sides of equal length. Use the checked exercises for Quick heck. a student misses the checked exercises 3 2 1 RtI Differentiate Instruction with Reteach 12.7 Personal Trainer 3.G.A.1 RtI Tier 1 Activity (online) On Your Own If students complete the checked exercises correctly, they may continue with the On Your Own section. For Exercises 5 7, have students first analyze the angles in each triangle then focus on how the three side lengths compare. MP1 Make sense of problems and persevere in solving them. Have students look at the triangles labeled L and M used in Exercises 5 7. Identify the largest angle and the longest side in each of the triangles. What do you notice about the relationship between the largest angle and the longest side? The longest side is opposite the largest angle. Share and Show F G H On Your Own K L MATH BOARD 1. Write the number of sides of equal length the triangle appears to have. 2 sides Use the triangles for 2 4. Write F, G, or H. 2. Triangle _ F has 1 right angle. 3. Triangle _ H has 1 angle greater than a right angle. 4. Triangle _ G has 3 angles less than a right angle. Use the triangles for 5 7. Write K, L, or M. Then complete the sentences. 5. Triangle _ M has 1 right angle and appears to have _ 2 sides of equal length. 6. Triangle _ K has 3 angles less than a right angle and appears to have _ 3 sides of equal length. 7. Triangle _ L has 1 angle greater than a right angle and appears to have _ 0 sides of equal length. M MATHEMATIAL PRATIES 8 Generalize Explain the ways you can describe a triangle. Possible explanation: triangles can be described by their sides or by their angles. hapter 12 Lesson 7 737 737 hapter 12
MATHEMATIAL PRATIES OMMUNIA E ONSTRUT ARGUMENTS MATHEMATIAL 8. PRATIE 1 Make Sense of Problems Martin said a triangle can have two sides that are parallel. Does his statement make sense? Explain. No; possible explanation: a triangle has only 3 sides. If 2 sides were parallel, the shape would be an open shape. 4 ELABORATE Problem Solving Applications MATHEMATIAL PRATIES 9. DEEPER ompare Triangles R and S. How are they alike? How are they different? Possible explanation: alike both have 3 sides and 3 angles. Three angles are less than a right angle. Different Triangle R appears to have 3 sides of equal length, and R S MP1 Make sense of problems and persevere in solving them. For Exercise 8 remind students that parallel lines do not intersect. Triangle S appears to have 2 sides of equal length. SMARTER 10. SMARTER Use a ruler to draw a straight line from one corner of this rectangle to the opposite corner. What shapes did you make? What do you notice about the shapes? 2 triangles; Possible answer: the triangles are the same size and same shape; each triangle has 1 right angle. Possible drawing. Exercise 10 requires students to decompose a polygon. on the Spot Video Tutor Use this video to help students model and solve this type of Think Smarter problem. 11. SMARTER Write the name of each triangle where it belongs in the table. Some triangles might belong in both parts of the table. Some triangles might not belong in either part. 738 Has 1 Right Angle L N P R Has at Least 2 Sides of Equal Length J O P Q R P J K M R N O Q L on the Spot videos are in the Interactive Student Edition and at www.thinkcentral.com. SMARTER This item assesses a student s ability to classify triangles by their angles and sides. Students should examine each triangle individually and determine whether it belongs in either or both parts of the table. Students first should look for a right angle, and then look to see whether it has two or more equal sides. Some students might place a triangle in only one part of the table when it belongs in both parts if they do not clearly understand the directions of the problem. DIFFERENTIATED INSTRUTION INDEPENDENT ATIVITIES 5 EVALUATE Formative Assessment Activities lassification Act Students complete orange Activity ard 18 by classifying two-dimensional shapes based on their attributes. Differentiated enters Kit Activities Figure It Out Students complete blue Activity ard 18 by identifying two-dimensional shapes by their attributes. Essential Question Using the Language Objective Reflect Have students draw and label a four panel cartoon to answer the Essential Question. How can you use sides and angles to help you describe triangles? I can measure the sides of triangles to find out which sides, if any, have equal lengths. I can also find out if there is 1 right angle, 3 angles less than a right angle, or 1 angle greater than a right angle to describe a triangle. Journal WRITE Draw a triangle that has two sides of equal length and one right angle. 738
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 section to determine student s understanding of content for this lesson. Encourage students to use their Journals to record their answers. Describe Triangles Use the triangles for 1 3. Write A, B, or. Then complete the sentences. A B 1. Triangle _ B has 3 angles less than a right angle and appears to have _ 3 sides of equal length. 2. Triangle _ has 1 right angle and appears to have _ 0 sides of equal length. 3. Triangle _ A has 1 angle greater than a right angle and appears to have _ 2 sides of equal length. Practice and Homework OMMON ORE STANDARD 3.G.A.1 Reason with shapes and their attributes. Problem Solving 4. Matthew drew the back of his tent. How many sides appear to be of equal length? 5. Sierra made the triangular picture frame shown. How many angles are greater than a right angle? 2 sides 6. WRITE Draw a triangle that has two sides of equal length and one right angle. heck students work. 0 angles hapter 12 739 OMMON ON ORE PROFESSIONAL DEVELOPMENT in Action The class is discussing ways to describe triangles. Amad: Lena: How can you describe a triangle? You can describe it by the number of sides of equal length. Very good. Besides using sides, is there another way to describe a triangle? By its angles. A triangle can have 1 right angle, 3 angles that are less than a right angle, or 1 angle that is greater than a right angle. Lena: Jonah: Lena, you just said that a triangle may have a right angle. Would a triangle ever have two right angles? Well, it might. No, wait. That wouldn t work. Two right angles would be like part of a square with a side missing. Good thinking. Is there any kind of angle that a triangle could have two of? It could have 2 angles less than a right angle. You re right! 739 hapter 12
Lesson heck (3.G.A.1) 1. How many angles less than a right angle does this triangle have? 3 angles 2. How many sides of equal length does this triangle appear to have? 2 sides ontinue concepts and skills practice with Lesson heck. Use Spiral Review to engage students in previously taught concepts and to promote content retention. ommon ore standards are correlated to each section. Spiral Review (3.NF.A.1, 3.MD.D.8, 3.G.A.1) 3. A quadrilateral has 4 right angles, 2 pairs of sides of equal length, and 2 pairs of opposite sides that are parallel. The pairs of opposite sides are not the same length. What quadrilateral could it be? 4. Mason drew a quadrilateral with only one pair of opposite sides that are parallel. What quadrilateral did Mason draw? Possible answers: rectangle or trapezoid trapezoid 5. Draw a rectangle that has an area of 8 square units and a perimeter of 12 units. What are the side lengths of the rectangle? 740 heck students drawings; 2 units and 4 units 6. What fraction of the square is shaded? 3_ 8 FOR MORE PRATIE GO TO THE Personal Trainer 740