Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Multilateration wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Perceived visual angle wikipedia , lookup
History of trigonometry wikipedia , lookup
Rational trigonometry wikipedia , lookup
Euler angles wikipedia , lookup
Trigonometric functions wikipedia , lookup
Pythagorean theorem wikipedia , lookup
CHAPTER 7 3 Classifying Triangles by Angles STUDENT BOOK PAGES 194–196 Guided Activity Goal Investigate angle measures in triangles. Prerequisite Skills/Concepts Expectations • Understand the definition of a polygon. • Use a protractor to measure the angles within shapes. 5m76 classify two-dimensional shapes according to angle and side properties 5m78 measure and construct angles using a protractor 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 • estimate and measure different types of angles: obtuse, acute, and right • Students will be able to look at angles and estimate obtuse or acute angles. They may need to measure right angles because they are exact. • Students may not be able to estimate the different angles. Provide students with opportunities to measure to get an “angle sense.” Students can use the square corner (90°) from a piece of cardboard as a referent. For example, if the angle they are measuring is wider that the square corner, it is an obtuse angle. If the angle is smaller than the square corner, it is an acute angle. Students can then measure and confirm their estimate. If students have difficulty using a protractor, review the activities from Lesson 2. • classify triangles according to their angles as obtuse-angled and right-angled • Students will be able to classify obtuse-angled triangles or right-angled triangles once they measure one angle in a triangle. They will first identify the angle to be measured through estimation. • Students might think that they need to measure all the angles in a triangle to classify it. Guide them to look for either a right angle (square corner) or an obtuse angle (wider than a square corner), and then measure that angle. • classify an acute-angled triangle • Students will understand that all angles have to be acute for a triangle to be classified as acuteangled. They may also realize that if the largest angle is acute, then the triangle is acute-angled because the other angles will be smaller. • Students might think that they need to measure all angles in a triangle to classify it. Guide them to look for the largest angle in the triangle. If the largest angle is narrower than a right angle (square corner), then the triangle is an acute-angled triangle. 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) Masters • Optional: Chapter 7 Mental Math p. 61 Workbook p. 64 Vocabulary/ Symbols acute, obtuse, right, acute-angled triangle, obtuse-angled triangle, right-angled triangle Key Assessment of Learning Question Question 5, Understanding of Concepts Copyright © 2005 by Thomson Nelson Meeting Individual Needs Extra Challenge • Encourage students to find as many applications and uses for different types of angles and triangles in the environment as they can. • Ask students to explain why there can only be one right angle in a rightangled triangle or one obtuse angle in an obtuse-angled triangle. Extra Support • Have students play a game to visually match the angles to the names of the angles. Students are dealt name cards of angles: acute, obtuse, or right. Then they take turns picking one card from a deck of angle cards. If they correctly identify the angle, they keep the card. The player with the most angle cards at the end of the game wins. • Post a triangle of the day on the board or wall, out of reach of the students so that they will not be able to measure the angles using a protractor. Label the vertices A, B, and C. Ask students to classify the triangle by estimating the angle or angles. Lesson 3: Classifying Triangles by Angles 21 1. Introduction (Whole Class) ➧ 5–10 min Ask students to look around the classroom and identify some triangular shapes, such as a coat hangar or a triangle shape in a geometry set. Find at least one example each of an acute-angled triangle, right-angled triangle, and obtuseangled triangle, and sketch these triangles on the board. Point out that some angles in the room have square corners (right), some angles are wider than a square corner, and some angles are narrower than a square corner. Sample Discourse “How are these triangles the same and how are they different? • They have three sides, three vertices, and three angles. But the sides are different lengths. • The angles are different sizes. Tell students that they will be classifying triangles by their angle measures. 2. Teaching and Learning (Pairs/Whole Class) ➧ 20–30 min Ask students to turn to Student Book page 194. As a class, read about the roof angles and the central question. Discuss Teresa’s Solution. Draw the roof triangles on an overhead or on the board, and demonstrate how to measure and classify the angles. Explain that the classification of the angles enables us to classify triangles. Talk about how we can estimate obtuse angles and acute angles because of their sizes, but that we need to measure right angles. Have students complete prompts A to E. To complete prompt B, guide students to draw a right angle for the peak of the roof for one of the triangles. Talk about how the peak measure relates to steepness. Reflecting Students will reflect on their experiences with classifying triangles and draw conclusions about how many angles measures are needed. 22 Chapter 7: 2-D Geometry Sample Discourse 1. a) • I would measure angle A because it is the biggest. If the angle is a right angle, then I know the triangle is a right-angled triangle. • I would measure angle A because it looks like a right angle. b) • I would measure angle D because it is the biggest and it looks like an obtuse angle. If it is, then the triangle is an obtuse-angled triangle. • I would measure angle D because it looks like an obtuse angle. c) • I would measure angle I because it looks close to 90°. If it is less than 90°, then I know that the other angles are acute and it is an acute-angled triangle. 2. • No, I don’t think you need to measure all three angles. If you measure one angle and it is 90°, then you can classify the triangle as a right-angled triangle. If you can see that all the angles are less than 90°, then you can say it is an acute-angled triangle. If the largest angle is more than 90°, then you can classify the triangle as an obtuseangled triangle. 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. 4. Encourage students to estimate the size of the angles and to measure if necessary. Practising (Individual) 5. & 6. Encourage students to construct and classify triangles according to the angles. This will allow students to familiarize themselves with the definitions and to gain experience in estimating angles as acute or obtuse. Related Question to Ask Ask Possible Response About Question 5: • How does the length of the name on the name tag affect whether the name tag is an obtuse-angled, right-angled, or acute-angled triangle? Key Assessment of Learning Question (See chart on next page.) Answers A. Triangle A is a right-angled triangle, triangle B is an obtuse-angled triangle, and triangle C is an acuteangled triangle. B.–C. For example, 45° 1 70° 65° 45° 90° 2 45° 115° 32.5° 3 32.5° D. For example, triangle 1 is an acute-angled triangle, triangle 2 is a right-angled triangle, and triangle 3 is an obtuse-angled triangle. E. For example, as the roofs get steeper, the measures of the peak angles decrease (the angles get smaller). 1. a) For example, I would measure angle A because it looks like a right angle. If it is a right angle, then I know the triangle is a right-angled triangle. b) For example, I would measure angle D because it is the greatest angle and it looks like an obtuse angle. If it is, then the triangle is an obtuse-angled triangle. Copyright © 2005 by Thomson Nelson • I think that the longer the name, the more likely the triangle will be an obtuse-angled triangle because the peak angle would have to be bigger than 90° so that the triangle is flattened out to fit the name. For a short name, the peak angle will be smaller than 90° so that all angles would be acute, and the name tag would be an acute-angled triangle. Closing (Whole Class) Have students summarize their learning by thinking about how they can classify triangles according to their angle sizes. Ask, “When do you need to measure angles in a triangle to classify it according to angle size?” • You need to measure angles only if you are not sure what type of angle it is. • You need to measure angles only if you are not sure if the angle is 90º. If it is 90º, then the triangle is a right-angled triangle. • You do not need to measure angles if it is obvious that one of the angles is obtuse because then the triangle is an obtuseangled triangle. • You do not need to measure angles if it is obvious that all three angles are acute because then the triangle is an acuteangled triangle. c) For example, I would measure angle I because it is the largest angle and it looks close to 90°. If it is less than 90°, then I know that the other angles are acute and it is an acute-angled triangle. (Lesson 3 Answers continued on p. 80) Lesson 3: Classifying Triangles by Angles 23 Assessment of Learning—What to Look for in Student Work… Assessment Strategy: short answer Understanding of Concepts Key Assessment Question 5 Kumiko’s class used triangle name tags for open house. Classify the triangles. (Score correct responses out of 4.) 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 and classifying angles and triangles in their homes. Students could list all the obtuse, right, and acute angles that they find, and then see if they can find them in triangles as well. Curious Math Student Book p. 197 Mid-Chapter Review Student Book p. 201, Question 3 Mental Imagery Student Book p. 203 Skills Bank Student Book p. 210, Question 4 Problem Bank Student Book p. 212, Question 1 Chapter Review Student Book p. 214, Question 3 Workbook p. 64, all questions Nelson Web Site Visit www.mathK8.nelson.com and follow the links to Nelson Mathematics 5, Chapter 7. Math Background In an obtuse-angled or right-angled triangle, there needs to be one obtuse angle or right angle, respectively. However, there cannot be more than one of these defining angles since the angle sum of any triangle is 180º. It follows that acute angles are the most common angles found in triangles. Not only are they found in acute-angled triangles in which all three angles are acute, but they are also found in rightangled triangles and obtuse-angled triangles. STUDENT BOOK PAGE 197 Curious Math: Diagonal Angles Using Curious Math Materials: protractor (1/student), ruler (1/student) When diagonals cross in rectangles, they bisect each other. The angles formed by the diagonals at the point of intersection are all equal in a square. In a rectangle, the diagonals form angles that are close in measurement if the rectangle approximates the dimensions of a square. There are two pairs of angles of the same size at the intersection of two diagonals in a rectangle. The less the rectangle looks like a square, the greater the difference in size for the angle pairs in the intersection. Answers to Curious Math 1. Angle measures Square Rectangle A 90° B 90° E 115° F 65° C 90° D 90° G 115° H 65° 2. a) For example, I notice that all of the angles where the diagonals of the square cross are the same. b) For example, I notice that where the diagonals cross in the rectangle, the pairs of angles that are opposite are the same. 3. For example, where the diagonals cross, the opposite angles are equal. 4. For example, for the square, the distances from the vertices to the point where the diagonals cross are all the same. The same is true for the rectangle. 24 Chapter 7: 2-D Geometry Copyright © 2005 by Thomson Nelson