Objectives To review types of angles, geometric figures, and the use of the Geometry Template to measure and draw angles. 1 materials Teaching the Lesson Key Activities Students write definitions for acute and obtuse angles, and review other types of angles. They explore the Geometry Template, measure angles using the half-circle and full-circle protractors, and draw angles with the half-circle protractor. Key Concepts and Skills вЂў Define and classify angles according to their measures. [Geometry Goal 1] вЂў Use a full-circle and a half-circle protractor to measure and draw angles. [Measurement and Reference Frames Goal 1] вЂў Explore angle types and relationships. Щ— Math Journal 1, pp. 68 and 69 Щ— Student Reference Book, pp. 162 and 163 Щ— Study Link 3 3 бњ Щ— Teaching Masters (Math Masters, pp. 78 and 79; optional) Щ— Teaching Aid Masters (Math Masters, pp. 414 and 419; optional) Щ— Geometry Template See Advance Preparation [Measurement and Reference Frames Goal 1] Key Vocabulary acute angle вЂў obtuse angle вЂў right angle вЂў straight angle вЂў reflex angle вЂў Geometry Template вЂў arc Ongoing Assessment: Informing Instruction See pages 173 and 174. Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip. [Measurement and Reference Frames Goal 1] 2 materials Ongoing Learning & Practice Students find the landmarks of data represented by a bar graph. Students practice and maintain skills through Math Boxes and Study Link activities. Ongoing Assessment: Informing Instruction See page 175. 3 materials Differentiation Options READINESS Students review the vocabulary of angles and their relationships. ENRICHMENT Students use their knowledge of angle measures to solve a baseball challenge problem. ELL SUPPORT Students define and illustrate angle terms. Additional Information Advance Preparation If the Geometry Templates were not distributed earlier, decide how you will manage them before beginning the lesson. Experienced Everyday Mathematics teachers suggest writing student ID numbers on the templates with a permanent marker before distributing them. 170 Unit 3 Geometry Explorations and the American Tour Щ— Math Journal 1, pp. 70 and 71 Щ— Student Reference Book, p. 121 Щ— Study Link Master (Math Masters, p. 80) Щ— Student Reference Book, p. 141 Щ— Differentiation Handbook Щ— Teaching Masters (Math Masters, pp. 81 and 82) Щ— Geometry Template Technology Assessment Management System Exit Slip See the iTLG. Getting Started Mental Math and Reflexes Math Message Students stand, facing the same direction, and follow directions related to angle measures or fractions of a turn. Direct students to focus on the degree equivalents for quarter (90В°) and half turns (180В°). Use only the information given on journal page 68 to complete Problems 1 and 2. Study Link 3 3 Follow-Up бњ вЂў Rotate 180В° to the right. вЂў Make a half turn to the left. вЂў Make a quarter turn to the right. вЂў Make a 90В° turn to the left. вЂў Turn 360В° to the left. Allow students five minutes to compare their answers. Ask volunteers to describe their strategies for estimating and finding the angle measures. Record strategies on the Class Data Pad for display. 1 Teaching the Lesson б¤ Math Message Follow-Up WHOLE-CLASS DISCUSSION (Math Journal 1, p. 68) Survey the class for their definitions of acute angle, An angle whose measure is greater than 0В° and less than 90 В° and obtuse angle. An angle whose measure is greater than 90В° and less than 180В° For each type of angle, compare the similarities and differences of studentsвЂ™ definitions. Ask students to agree on a common definition for each type of angle and record these definitions on the Class Data Pad. Emphasize the importance of definitions in mathematics. For example, discuss why narrow and wide are not specific enough to be acceptable definitions for acute and obtuse. Ask whether an angle can be both acute and obtuse and why or why not. NOTE Working with mathematical definitions helps students build logical thinking skills that are critical to success in higher mathematics. Definitions are closely related to classification schemes. The types of angles defined in this lesson, for example, make up a classification scheme for all angles. Every angle with a measure greater than 0В° is either an acute angle, right angle, obtuse angle, straight angle, or reflex angle. Adjusting the Activity Consider assigning groups to combine their individual definitions into a group proposal for the class definition. A U D I T O R Y бњ K I N E S T H E T I C бњ T A C T I L E бњ V I S U A L Draw a right, a straight, and a reflex angle on the Class Data Pad. Ask students to name these angles. A 90В° angle is called a right angle; a 180В° angle is called a straight angle; an angle whose measure is greater than 180В° and less than 360В° is called a reflex angle. Draw a small square in the corner of the right angle and explain that the symbol represents the square corner formed by perpendicular lines. The symbol indicates that the measure of the angle is 90В°. Lesson 3 4 бњ 171 Ask: Why do you think the 180В° angle is called a straight angle? Sample answer: Because the two sides of the angle form a straight line Assign volunteers to label the angles on the Class Data Pad. Ask partners to demonstrate angle types (right, straight, acute, and obtuse) using their arms as physical models. Sample answers: think of the elbow as the vertex of the angle and bend one arm to form a 90В° angle, an angle smaller than 90В°, an angle between 90В° and 180В°, and a 180В° angle. Students should name each modeled angle in turn. (See margin.) Ask partners to illustrate each of the following terms using their arms: parallel, perpendicular, and intersecting. Assign volunteers to add these examples to the Class Data Pad. To summarize, review the concepts and objects from this lesson listed on the Class Data Pad with a focus on the vocabulary. Demonstrating a right angle б¤ Introducing the Geometry WHOLE-CLASS ACTIVITY Template (Student Reference Book, pp. 162 and 163; Math Masters, p. 419) Transition to this activity by displaying the Geometry Template, either by placing the template on the overhead or by using a transparency of Math Masters, page 419. Ask students to turn to pages 162 and 163 of the Student Reference Book. Consider assigning groups one each of the paragraph topics on page 162. Groups should read their assigned paragraph(s), identify the important information, and present it to the class. As groups present their information, make sure that the following features of the Geometry Template are noted: бџ There is an inch ruler along the left edge and a centimeter ruler along the right edge. бџ As part of the geometric figures, there are six pattern-block shapesвЂ”equilateral triangle, square, trapezoid, two rhombuses, and regular hexagonвЂ”labeled PB. The regular pentagon and octagon are not pattern-block shapes. бџ Some of the other shapes have sides that are the same length as the pattern-block shapes. бџ The squares and circles of various sizes can be used to draw figures quickly. The Geometry Template appears on both Math Masters, page 419 and Student Reference Book, page 163. 172 Unit 3 Geometry Explorations and the American Tour Ask students to find the relationship between the template shapes and actual pattern blocks. The sides of the template shapes are half as long as the edges of the pattern blocks. Links to the Future The triangles on the Geometry Template will be discussed in Lesson 3-6. The Percent Circle is used to read and make circle (pie) graphs and will be discussed in Lessons 5-10 and 5-11. Ask students to use their templates to make tracings that identify different figures on the Geometry Template that have sides of the same length. Have volunteers record this list of figures on the Class Data Pad. The sides of the pentagon, octagon, five pattern-block shapes, and three sides of the pattern-block trapezoid are the same length; the fourth side of the trapezoid is twice as long. Comparing each figure with the octagon shows that all regular polygons on the Geometry Template have sides of the same length. Other shapes can be compared in the same way. Adjusting the Activity Remind students that any two figures can be compared by tracing them so they are adjacent; that is, they share a common side. (See margin.) A U D I T O R Y бњ K I N E S T H E T I C бњ T A C T I L E бњ NOTE The key to using the half-circle protractor is knowing which scale to read. The key to using the full-circle protractor is knowing that the scale runs clockwise. The 0В° mark must be aligned with one side of the angle. Students measure from 0В° in the clockwise direction to find the answers. One way to reinforce this habit is for students to estimate whether the angle is more or less than 90В° before measuring it. V I S U A L Ask students to identify the half-circle protractor and the full-circle protractor on the Geometry Template. Then ask a volunteer how they would measure one of the angles from Problem 1 on journal page 68. Ask another volunteer to use the full-circle protractor to measure one of the angles in Problem 2. Student Page Ongoing Assessment: Informing Instruction Date Watch for students who find it difficult to align the protractor when the Geometry Template spans across two pages. Have students try turning the page and/or aligning a different leg of the angle with the 0В° mark on the protractor. Time LESSON 3 4 бњ Acute and Obtuse Angles Math Message 1. Acute Angles NOT Acute Angles 91В° 90В° 28В° 65В° б¤ Measuring Angles 120В° 88В° PARTNER ACTIVITY (Math Journal 1, p. 68) Sample answer: An acute angle is an angle whose measure is less than 90В°. Write a definition for acute angle. 2. Obtuse Angles NOT Obtuse Angles 91В° 120В° Partners work on Problem 3 at the bottom of the journal page. Remind students to estimate the angle measures before they use a protractor. (See margin note.) Explain that estimating first will help them confirm whether theyвЂ™re reading the correct scale or moving in the correct direction on the protractor. Allow five to ten minutes, then discuss studentsвЂ™ answers. TheresaвЂ™s and DevonвЂ™s answers are incorrect. Devon interpreted the angle to be a reflex angle (the angle larger than 180В°). Theresa used the wrong scale on her half-circle protractor. 170В° 55В° 89В° Sample answer: An obtuse angle is an angle whose measure is greater than 90В°. Write a definition for obtuse angle. Measuring and Drawing Angles with a Protractor Sarah used her half-circle protractor to measure the angle at the right. She said it measures about 35РЉ. Theresa measured it with her half-circle protractor. Theresa said it measures about 145РЉ. Devon measured it with his full-circle protractor. Devon said it measures about 325РЉ. E 3. a. Use both your template protractors to measure the angle. Do you agree with Sarah, Theresa, or Devon? Sample answer: I agree with Sarah. Sample answer: The measure of the angle with the arc is about 35В°. b. Why? 68 Math Journal 1, p. 68 Lesson 3 4 бњ 173 Student Page Date Time LESSON Measuring and Drawing Angles with a Protractor 34 бњ Use your half-circle protractor. Measure each angle as accurately as you can. E Explain that in Everyday Mathematics when students measure an angle, they will normally measure the smaller angle, not the reflex angle. When they are supposed to measure the reflex angle, it is indicated with an arc. D A S 4. a. mР„A is about 56В° . T b. mР„EDS is about 115В° . c. mР„T is about 88В° . Use your full-circle protractor to measure each angle. 330В° angle U C E G 5. a. mР„G is about 35В° . b. mР„LEC is about 121В° . c. mР„U is about 78В° . б¤ Practicing Measuring and Draw and label the following angles. Use your half-circle protractor. 6. a. Р„CAT: 62В° b. Р„DOG: 135В° C A PARTNER ACTIVITY Drawing Angles D O T 30В° angle Students will practice measuring angles throughout the year in Math Boxes problems and in Ongoing Learning & Practice activities. L G (Math Journal 1, p. 69; Math Masters, pp. 419, 78, and 79) 69 Math Journal 1, p. 69 NOTE Students used the full-circle and the half-circle protractors in Fourth Grade Everyday Mathematics. With half-circle protractors expect students to be able to measure to within about 2В° of the measured approximation. With full-circle protractors, expect that students might be less precise, measuring to within about 5В° of the approximation. Remind students that all measurements are approximations. They should always measure carefully but recognize that a measure obtained with a measuring tool does not give an exact size for the object being measured. The goal is to be as precise as possible, but the degree of precision is dependent on the accuracy of the measuring tool and how effectively it is used. Assign journal page 69. Remind students to use the half-circle protractor to complete Problems 4 and 6, and the full-circle protractor to complete Problem 5. Ask them to estimate each angle before measuring. It is important that students practice with both types of protractors. You might want to display or use overhead transparencies of the Geometry Template and journal pages (Math Masters, pp. 419, 78, and 79). Circulate and assist. Student Page Date Time LESSON Watching Television 3 4 бњ Ongoing Assessment: Informing Instruction Adeline surveyed the students in her class to find out how much television they watch in a week. She made the following graph of the data. Watch for students who have difficulty measuring Angle T because the figure is so small. Suggest that students extend the angleвЂ™s sides with a straightedge before measuring the angle with a protractor. Demonstrate this approach and illustrate the pitfalls of extending the sides without using a straightedge. Hours of Television Watched per Week Number of Students 8 7 6 5 4 3 2 1 0 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Hours per Week Find each data landmark. 1. a. minimum: d. median: 13 22 29 20.96 b. maximum: c. range: e. mean: f. mode: 16 23 Sample answer: I listed all of the values in order from smallest to largest, and then found the middle value. 2. Explain how you found the median. 3. a. Which data landmark best represents the number of hours a typical student watches televisionвЂ”the mean, median, or mode? b. Why? Answers vary. Answers vary. 70 Math Journal 1, p. 70 174 Unit 3 Geometry Explorations and the American Tour Incorrect Correct Extending the sides of an angle Student Page Ongoing Assessment: Recognizing Student Achievement Exit Slip а¬™ Date Time LESSON Math Boxes 34 бњ 1. Estimate and solve. 2. Find the landmarks for this set of numbers: 99, 87, 85, 32, 57, 82, 85, 99, 85, 65, 78, 87, 85, 57, 85, 99 463 П© 2,078 2,500 2,541 Estimate: Ask students to respond to the following question on an Exit Slip (Math Masters, page 414) or half-sheet of paper. Which is easier to useвЂ”the full-circle protractor or the half-circle protractor? Why? Students are making adequate progress if their answers demonstrate an understanding of how to use both protractor types. [Measurement and Reference Frames Goal 1] Solution: Minimum: Range: 5,046 ПЄ 2,491 Median: 2,500 2,555 Estimate: Solution: 13вЂ“17 119 3. Solve. 2 Ongoing Learning & Practice 4. Estimate and solve. 23 П© x П 60 xП 36 П p вЂ« ШЎвЂ¬4 pП 200 П 50 вЂ« ШЎвЂ¬m mП 55 П© t П 70 tП 28 ПЄ b П 13 bП 37 9 4 15 15 473.894 П© 59.235 a. 530 533.129 Estimate: Solution: b. 78.896 ПЄ 29.321 50 49.575 Estimate: Solution: 219 5. Write the name of an object in the room б¤ Interpreting a Bar Graph 99 32 67 85 Maximum: 6. Solve. that is about 10 inches long. INDEPENDENT ACTIVITY 5.8 вЂ« ШЎвЂ¬76 П Write the name of an object in the room that is about 10 centimeters long. 159 вЂ« ШЎвЂ¬7 П 0.4 вЂ« ШЎвЂ¬231 П 185 Students find landmarks for data represented by the bar graph on journal page 70. They describe how to find the median, and tell whether the mean, median, or mode best represents the data for a typical student. 2,108 440.8 1,113 92.4 634.12 34 вЂ« ШЎвЂ¬62 П Answers vary. Answers vary. (Math Journal 1, p. 70; Student Reference Book, p. 121) 34вЂ“36 76.4 вЂ« ШЎвЂ¬8.3 П 19 20 38вЂ“40 71 Math Journal 1, p. 71 Ongoing Assessment: Informing Instruction Watch for students who have difficulty finding the mean. It might be helpful to review the steps on Student Reference Book, page 121. Ask students to summarize the process for finding the mean in their own words and have them list the steps on an index card. For example: 1. Count the numbers in the data set; 2. Add the numbers in the data set; and 3. Divide the sum by the count. Students should check their steps by using them to solve the Check Your Understanding problem at the bottom of the Student Reference Book page. Study Link Master Date STUDY LINK 34 бњ measure of Р„CAT П 2. mР„BAR П 3. 4. 5. 6. 70РЉ 50РЉ mР„RAT П 110РЉ mР„CAB П 130РЉ mР„BAT П 60РЉ mР„CAR П 180РЉ T B 80 90 100 70 100 90 80 110 1 70 20 60 0 110 60 1 2 50 0 1 50 30 13 C 0 10 180 170 1620 3 01 0 50 4 14 0 0 1. 180 0 170 0 0 16 10 15 20 0 30 14 0 4 Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 3-2. The skill in Problem 6 previews Unit 4 content. A R Find the approximate measure of each angle at the right. mР„MEN П 8. mР„DEN П 9. 30 E 12 0 24 11. 0 90 0 0 21 Home Connection Students practice measuring angles with half-circle and full-circle protractors. 0 33 60 10. M T 30 0 (Math Masters, p. 80) 120РЉ 90РЉ mР„MET П 50РЉ mР„MED П 150РЉ mР„TEN П 170РЉ 7. 270 INDEPENDENT ACTIVITY 180 бњ 204вЂ“205 Find the approximate measure of each angle at the right. (Math Journal 1, p. 71) б¤ Study Link 3 4 Time Angle Measures 0 бњ INDEPENDENT ACTIVITY 15 б¤ Math Boxes 3 4 Name N D Practice 12. 5,844 П© 2,399 13. 8,243 15. 60 П¬ 5 П 238 ПЄ 129 109 12 16. 50 П¬ 6 в†’ 14. 234 вЂ« ШЎвЂ¬22 5,148 8 R2 Math Masters, p. 80 Lesson 3 4 бњ 175 Teaching Master Name Date LESSON Time Points, Lines, and Angles 34 бњ 3 Differentiation Options 141 Identify the terms and objects in the riddles below. Use the words and phrases from the Word Bank to complete the table. Word Bank point line segment ray line angle parallel lines parallel line segments intersecting lines vertices perpendicular lines perpendicular line segments READINESS б¤ Identifying Points, Lines, vertex Clues What Am I? 1 I am a location in space. It takes only one letter to name me. 2 My length cannot be measured, but I am named by two of my points. (Student Reference Book, p. 141; Math Masters, p. 81) ray angle 3 I do not curve. I have only one end point. I am measured in degrees. I have a vertex. My sides are two rays. 5 Perpendicular line segments There are always at least two of us. We have endpoints. Parallel line We always stay the same distance apart. segments I am the point where two rays meet to form an angle. vertex Two of us meet. Intersecting lines, perpendicular lines, or perpendicular line segments Our lengths cannot be measured. When two of us meet, Perpendicular we form right angles. lines I am the endpoint where two sides of a polygon meet. vertex My length can be measured. I have two endpoints. Line segment Our lengths cannot be measured. There are always at Parallel lines To review vocabulary and concepts related to angles, have students solve riddles about points, lines, line segments, and angles. We have endpoints. When two of us meet, we form one or more right angles. 7 8 9 10 11 12 5вЂ“15 Min and Angles point Line or ray 4 6 SMALL-GROUP ACTIVITY ENRICHMENT б¤ Measuring Baseball Angles least two of us. We always stay the same distance apart. INDEPENDENT ACTIVITY 15вЂ“30 Min (Math Masters, p. 82) Math Masters, p. 81 To apply studentsвЂ™ understanding of angle properties and angle measurements, have students solve a baseball problem that involves addition and subtraction of angle measures. As students complete the assignment, discuss answers and strategies. The field has a 90В° angle within which a batted ball is put in play. Each of the four infielders covers 13В°, for a total of 4 вЂ« ШЎвЂ¬13В°, or 52В° and the pitcher covers 6В°. That leaves 90В° ПЄ 52В° ПЄ 6В°, or 32В°, uncovered, which suggests that on average a little more than one-third of hard-hit ground balls should get past the infield. Ask the baseball players and fans in the class whether that conclusion is consistent with their experiences. ELL SUPPORT Teaching Master Name Date LESSON б¤ Building a Math Word Bank Time Baseball Angles бњ p rtsto sho over c can lin ul e lin fo ul fo 3r d ca bas n e co ma ve n r 1s t ca base nc m ov an er can cover pitcher 13В° 6В° To provide language support for angles, have students use the Word Bank Template found in the Differentiation Handbook. Ask students to write the terms acute angle, right angle, obtuse angle, straight angle, and reflex angle, draw pictures relating to each term, and write other related words. See the Differentiation Handbook for more information. 13В° 13В° 13В° batter The playing field for baseball lies between the foul lines, which form a 90РЉ angle. Suppose that each of the four infielders can cover an angle of about 13РЉ on a hard-hit ground ball, and that the pitcher can cover about 6РЉ. (See the diagram above.) Source: Applying Arithmetic, Usiskin, Z. and Bell, M. В© 1983 University of Chicago 1. How many degrees are left for the batter to hit through? 32РЉ Math Masters, p. 82 176 30+ Min (Differentiation Handbook) 2n db can asem cov an er e 34 SMALL-GROUP ACTIVITY Unit 3 Geometry Explorations and the American Tour