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CHAPTER 7 4 STUDENT BOOK PAGES 198–200 Classifying Triangles by Side Lengths Guided Activity Goal Investigate side lengths of triangles. Prerequisite Skills/Concepts Expectations • Use a protractor to measure the angles of shapes. • Know that triangles have three angles. 5m76 classify two-dimensional shapes according to angle and side properties 5m83 recognize and explain the occurrence and application of geometric properties and principles in the everyday world 5m85 discuss ideas, make conjectures, and articulate hypotheses about geometric properties and relationships Assessment for Feedback What You Will See Students Doing… Students will When Students Understand If Students Misunderstand • classify triangles according to their side lengths as isosceles, scalene, or equilateral triangles • Students will measure the sides of a triangle and classify it according to their measurement results. • Students may not be able to classify triangles. Provide students with the following guidelines: Measure the sides of the triangle and look at the side lengths. If all the sides are equal, it is an equilateral triangle. Identify the part of the word that means equal. If two sides are equal, it is an isosceles triangle. If all sides are unequal, it is a scalene triangle. Scalene means uneven. • classify triangles according to their side lengths and angle measure • Students will relate their knowledge of classifying according to side lengths to the classification of triangles according to angle measures. • Students may be confused when classifying triangles according to both side lengths and angle measures. Prepare and post a chart showing how to classify triangles. Encourage students to consult it regularly. The chart could have the following headings: Triangle; Side length measurements; Type (equilateral, isosceles, or scalene); Angle measurement(s); Type (right-angled, obtuse-angled, or acute-angled). As an example, the first row would read: ABC; 3 cm, 4 cm, 5 cm; scalene; 90 degrees; right-angled. Preparation and Planning Pacing 5–10 min Introduction 20–30 min Teaching and Learning 15–20 min Consolidation Materials •ruler (1/student) •protractor (1/student) •set of coloured pipe cleaners or straws: red (5 cm), yellow (7 cm), and blue (9 cm) (10 of each colour/small group) •set of pencil crayons (1/small group) • Optional: Cuisenaire rods • Optional: transparent mirror Masters • Optional: Chapter 7 Mental Math p. 62 Workbook p. 65 Vocabulary/ Symbols equilateral, isosceles, scalene Key Assessment of Learning Question Question 5, Understanding of Concepts Copyright © 2005 by Thomson Nelson Meeting Individual Needs Extra Challenge • Students can classify triangles jointly according to their side lengths and angle measures. They can list all the combinations in the following categories: Set A is obtuse-angled, acute-angled, and right-angled triangles, and set B is isosceles, scalene, and equilateral triangles. Encourage students to make all the combinations by choosing one from each set. Ask students to explain why it may not be possible to construct a triangle for every pair. Extra Support • Have students play a similar game to that described in Lesson 3. This time students visually match the triangles to the names of the triangles. Students are dealt name cards: equilateral, isosceles, and scalene. Then they take turns picking one card from a deck of triangle cards. The triangles on these cards are drawn with the same colour and proportional length as the pipe cleaners used in the lesson. • Assign a triangle, such as an isosceles triangle, of the day. Ask students to look for isosceles triangles in their environment. Lesson 4: Classifying Triangles by Side Lengths 25 1. Introduction (Whole Class) ➧ 5–10 min Draw an isosceles triangle on the board and ask students if they can find the line of symmetry. Have one student draw the line of symmetry using a different colour. Point to the hypotenuse of both congruent halves and ask students what they know about the lengths of the sides. Students should understand that the sides are equal because the triangles are congruent. Draw examples of an equilateral triangle and a scalene triangle. Ask students how they are the same and how they are different. Encourage them to talk about the side lengths. Tell students that they will be classifying triangles according to their side lengths. 2. Teaching and Learning (Whole Class/Small Groups) ➧ 20–30 min Ask students to turn to Student Book page 198. As a class, read the goal, the central question, and Camille’s Pipe Cleaner Models. Have students work in small groups to complete prompts A to F. It is very important that the students be given enough time to explore and construct different triangles with the pipe cleaners (or straws or Cuisenaire rods). For prompt A, make sure students understand that their task is to construct as many different triangles as they can using only one pipe cleaner for each side. They will be able make ten different triangles altogether. For prompt B, they need to record their findings on chart paper using the same colour pencils as the pipe cleaners are (you may want to substitute orange for yellow because yellow is hard to see). For prompt F, students will look at their findings and classify the triangles. Guide their thinking into looking at the similarities between the sides. For example, equilateral triangles are triangles made of the same colour pipe cleaners; isosceles triangles are made of two different colour pipe cleaners; and scalene triangles are made of different colours. Since each colour of pipe cleaner has a specific length, it’s easy for the students to transfer their thinking from classifying by colours to classifying by lengths. 26 Chapter 7: 2-D Geometry Reflecting Here students reflect upon how to classify triangles according to their side lengths. Sample Discourse 1. • My strategy for deciding whether a triangle is equilateral, isosceles, or scalene is to measure the side lengths. If there are different side lengths, then it is a scalene triangle. If two sides are the same length, then it is an isosceles triangle. If all the sides are the same length, then it is an equilateral triangle. 2. a) • I noticed that in equilateral triangles, all the angles were equal. b) • I noticed that in isosceles triangles, the angles next to the two equal sides were also equal. The angle that is between the two equal sides was different from the others. c) • I noticed that in scalene triangles, none of the angles were the same. Copyright © 2005 by Thomson Nelson 3. Consolidation ➧ 15–20 min Checking (Whole Class) For intervention strategies, refer to Meeting Individual Needs or the Assessment for Feedback chart. Practising (Individual) 6. Students might need review in looking for lines of symmetry with a transparent mirror. Related Questions to Ask Ask Possible Response About Question 5: • Can a right-angled triangle also be an equilateral triangle? • Can an obtuse-angled triangle also be an equilateral triangle? • What angles must an equilateral triangle have? • No, a right-angled triangle cannot also be an equilateral triangle. It has to have at least two unequal sides. • No, an obtuse-angled triangle cannot also be an equilateral triangle. It has to have at least two unequal sides. • An equilateral triangle can only have acute angles so it is also an acute-angled triangle. Key Assessment of Learning Question (See chart on next page.) Closing (Whole Class) Answers A. & B. For example, C. For example, the red and yellow triangles are similar to the blue triangle because all the sides are the same length. Also, the triangles are the same shape, but different in size. D. For example, a triangle that has one red side and two blue sides, and a triangle with two blue sides and one yellow side are like the triangle with two yellow sides and one blue side. The triangles are alike because two of the sides are the same colour, which means that they are the same length. E. For example, the three-coloured triangle is the only triangle that is made of different-coloured pipe cleaners, which means that the lengths are different on all sides. Copyright © 2005 by Thomson Nelson Have students summarize their learning by explaining how they classify triangles according to the lengths of sides. As an extension, ask, “How is classifying triangles according to the lengths of their sides related to classifying by lines of symmetry?” • An equilateral triangle has three lines of symmetry, an isosceles triangle has one line of symmetry, and a scalene triangle has no line of symmetry. So we can classify the triangles by the number of lines of symmetry and get the same results as when we classify by side lengths. F. For example, the equilateral triangles are the ones with all sides the same colour and length, the isosceles triangles are the ones with two sides the same colour and length, and the scalene triangle is the one that has all different side colours and lengths. 1. For example, my strategy for deciding whether a triangle is equilateral, isosceles, or scalene is to measure the side lengths. If the sides are different lengths, then it is a scalene triangle. If two sides are the same length and one is different, then it is an isosceles triangle. If all the sides are the same length, then it is an equilateral triangle. (Lesson 4 Answers continued on p. 80) Lesson 4: Classifying Triangles by Side Lengths 27 Assessment of Learning—What to Look for in Student Work… Assessment Strategy: short answer Understanding of Concepts Key Assessment Question 5 Classify the triangles by the lengths of their sides. (Score correct responses out of 6.) Extra Practice and Extension At Home • You might assign any of the questions related to this lesson, which are cross-referenced in the chart below. • Students might enjoy looking for specific triangles in their home environment. Ask students to make a list of the equilateral, isosceles, and scalene triangles that they find in their home and indicate where they are found. Mid-Chapter Review Student Book p. 201, Question 4 Mental Imagery Student Book p. 203 Skills Bank Student Book p. 211, Question 5 Problem Bank Student Book p. 212, Questions 2 & 3 Chapter Review Student Book p. 214, Question 4 Workbook p. 65, all questions Nelson Web Site Visit www.mathK8.nelson.com and follow the links to Nelson Mathematics 5, Chapter 7. Optional: Chapter 7 Mental Math p. 62 Math Background Both scalene and isosceles triangles can contain all three types of angles: acute, obtuse, and right. It’s entirely possible to have acute-angled, obtuse-angled, and right-angled scalene and isosceles triangles. However, the equilateral triangle has all three sides are equal, which means that all three angles are equal. Since the angle sum of triangles is equal to 180°, it follows that each angle in an equilateral triangle has to be 60°. Therefore, an equilateral triangle can only be an acute-angled triangle. 28 Chapter 7: 2-D Geometry Copyright © 2005 by Thomson Nelson