Figuring All the Angles Geometry and Measurement Mathematics in Context is a comprehensive curriculum for the middle grades It was developed in 1991 through 1997 in collaboration with the Wisconsin Center for Education Research, School of Education, University of Wisconsin-Madison and the Freudenthal Institute at the University of Utrecht, The Netherlands, with the support of the National Science Foundation Grant No 9054928 The revision of the curriculum was carried out in 2003 through 2005, with the support of the National Science Foundation Grant No ESI 0137414 National Science Foundation Opinions expressed are those of the authors and not necessarily those of the Foundation deLange, J.,van Reeuwijk, M., Feijs, E., Middleton, J A., and Pligge, M A (2006) Figuring all the angles In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in Context Chicago: Encyclopỉdia Britannica, Inc Copyright © 2006 Encyclopædia Britannica, Inc All rights reserved Printed in the United States of America This work is protected under current U.S copyright laws, and the performance, display, and other applicable uses of it are governed by those laws Any uses not in conformity with the U.S copyright statute are prohibited without our express written permission, including but not limited to duplication, adaptation, and transmission by television or other devices or processes For more information regarding a license, write Encyclopædia Britannica, Inc., 331 North LaSalle Street, Chicago, Illinois 60610 ISBN 0-03-039622-0 073 09 08 07 06 The Mathematics in Context Development Team Development 1991–1997 The initial version of Figuring All the Angles was developed by Jan deLange, Martin van Reeuwijk, and Els Feijs It was adapted for use in American schools by James A Middleton, and Margaret A Pligge Wisconsin Center for Education Freudenthal Institute Staff Research Staff Thomas A Romberg Joan Daniels Pedro Jan de Lange Director Assistant to the Director Director Gail Burrill Margaret R Meyer Els Feijs Martin van Reeuwijk Coordinator Coordinator Coordinator Coordinator Sherian Foster James A, Middleton Jasmina Milinkovic Margaret A Pligge Mary C Shafer Julia A Shew Aaron N Simon Marvin Smith Stephanie Z Smith Mary S Spence Mieke Abels Nina Boswinkel Frans van Galen Koeno Gravemeijer Marja van den Heuvel-Panhuizen Jan Auke de Jong Vincent Jonker Ronald Keijzer Martin Kindt Jansie Niehaus Nanda Querelle Anton Roodhardt Leen Streefland Adri Treffers Monica Wijers Astrid de Wild Project Staff Jonathan Brendefur Laura Brinker James Browne Jack Burrill Rose Byrd Peter Christiansen Barbara Clarke Doug Clarke Beth R Cole Fae Dremock Mary Ann Fix Revision 2003–2005 The revised version of Figuring All The Angles was developed by Els Feijs and Jan de Lange It was adapted for use in American Schools by Margaret A Pligge Wisconsin Center for Education Freudenthal Institute Staff Research Staff Thomas A Romberg David C Webb Jan de Lange Truus Dekker Director Coordinator Director Coordinator Gail Burrill Margaret A Pligge Mieke Abels Monica Wijers Editorial Coordinator Editorial Coordinator Content Coordinator Content Coordinator Margaret R Meyer Anne Park Bryna Rappaport Kathleen A Steele Ana C Stephens Candace Ulmer Jill Vettrus Arthur Bakker Peter Boon Els Feijs Dédé de Haan Martin Kindt Nathalie Kuijpers Huub Nilwik Sonia Palha Nanda Querelle Martin van Reeuwijk Project Staff Sarah Ailts Beth R Cole Erin Hazlett Teri Hedges Karen Hoiberg Carrie Johnson Jean Krusi Elaine McGrath (c) 2006 Encyclopædia Britannica, Inc Mathematics in Context and the Mathematics in Context Logo are registered trademarks of Encyclopỉdia Britannica, Inc Cover photo credits: (left to right) © Comstock Images; © Corbis; © Getty Images Illustrations James Alexander; 8, Holly Cooper-Olds; 26 Jerry Kraus/© Encyclopỉdia Britannica, Inc.; 34 © Encyclopỉdia Britannica, Inc Photographs © Corbis; © Charles E Rotkin/Corbis; 14 © Corbis; 17 © Roger Ressmeyer/Corbis; 21 (top) © Tim Boyle/Newsmaker/Getty Images; (bottom) © PhotoLink/PhotoDisc/ Getty Images; 23 ImageGap/Alamy; 26 © Corbis; 28 © PhotoDisc/Getty Images; 30 © Corbis; 42 (top) Historic Urban Plans, Inc.; (bottom) Design Pics; 43 ©PhotoDisc/ Getty Images Contents A Sense of Direction 20 340 31 30 290 70 80 270 280 60 260 100 110 12 0 24 250 S 15 160 170 180 190 23 13 SW 90 10 11 12 12 0 200 22 21 14 18 20 20 21 24 24 26 31 32 32 From Turns to Angles Sled Tracks Regular Polygons Summary Check Your Work 33 35 36 36 N Angles and Their Measures 40 14 30 52 15 Answers to Check Your Work 20 46 10 Additional Practice 80 90 100 10 70 80 100 12 70 60 110 20 60 13 50 50 38 40 42 44 45 170 180 160 0 15 20 10 30 14 Angles Measures of Angles Palmanova Summary Check Your Work 180 170 160 Section G E W Changing Directions: Turns Flight Instructions Plane Landing Activity Summary Check Your Work Section F E N Navigation and Orientation Traffic Control Summary Check Your Work Section E 40 Directions San Francisco Bay Area Sun Island Summary Check Your Work Section D 30 SE Section C 20 50 Finding Your Way Sunray (1850) Sunray (1900) Sunray Today Downtown Provo, Utah Summary Check Your Work 10 W N Section B 6 N Getting Your Sense of Direction Summary Check Your Work 350 33 Section A vi 14 Letter to the Student Q Contents v Dear Student, Welcome to the unit Figuring All the Angles In this unit, you will learn how planes and boats navigate their way around the globe You will build upon the cardinal directions of north, east, south, and west to give a better indication of where you want to go Along the way, you will pay close attention to the turns made along a route Turns relate to angles, and you will use these to solve some geometry problems involving shapes Sincerely, The Mathematics in Context Development Team N W E S airport bridge 5 10 miles 10 Map from the Road Atlas © 1994 by Rand McNally vi Figuring All The Angles 16 km A A Sense of Direction Getting Your Sense of Direction Point toward north Is everyone in your class pointing in the same direction? a Sketch a top view of your classroom Include the desktops in your sketch Draw an arrow pointing north on each of the desks b Do all the arrows point in the same direction? c Will lines from the arrows ever meet? In what direction is south? The position of the sun in the sky is related to the direction south Record the sun’s position in the sky before noon, at noon, and after noon In which direction from the classroom is your school’s playground? Name a town about 50 miles away and point in the direction of that town Describe this direction If you traveled north from your school, which towns would you pass through? Section A: A Sense of Direction A A Sense of Direction Use a compass to answer problems and N W a Sketch the room where you sleep Be sure to include windows in your sketch b Designate where north is in the sketch E Compare the sketches of all the students in your class Count how many of the windows in the sketches face south S Here is a partial map of the United States to answer the next questions N W E S Mt Rushmore Monument Valley West Virginia 400 miles 400 kilometers Figuring All The Angles A Sense of Direction A Monument Valley, in Arizona, has many spires and mesas that did not erode as fast as the land around them You may have seen the picture above in Western movies Western movies often describe how the West was settled 10 Why is the West called the West? On one side of Mount Rushmore, the heads of four United States presidents have been carved out of the mountain Mount Rushmore is located in South Dakota 11 Is South Dakota in the South? Why is it called South Dakota? You have probably read about the North Pole 12 Explain why the word North in North Pole has a different meaning from West in West Virginia (West Virginia is labeled on the partial map of the United States on page 2.) Section A: A Sense of Direction A A Sense of Direction The state capitol of Wisconsin is located in Madison The capitol building is unusual because of the four identical wings radiating from the huge central dome The four wings of the building point in the four compass directions: north, south, east, and west S O U T H The south wing is indicated in the drawing below 13 Label the directions of the remaining three wings on Student Activity Sheet Figuring All The Angles G Angles Angles are formed by two sides and a point where the sides meet: the vertex e Sid Vertex Side If you give the vertex a name, for example P, you call the angle “angle ∠P” and write it as ∠P You can classify angles according to their size: Acute angles are less than 90° Right angles are 90° Obtuse angles are between 90° and 180° A straight angle is 180° Sometimes you can find the size of an angle by reasoning, sometimes by estimating, and sometimes by measuring You can measure an angle by using a compass card or a protractor 44 Figuring All The Angles Measure ∠A, ∠B, and ∠C A B C a Estimate the size of ∠P, ∠Q, and ∠R Q P R b Which angles, if any, are obtuse? Draw an angle of 40° and one of 110° Label the vertices A pilot needs to draw a heading of 330° on her map She only has a protractor Explain how she can make this drawing Describe differences between the compass card and the protractor and the ways you can use them Section G: Angles and Their Measures 45 Additional Practice Section A A Sense of Direction N W E S © 1996, Encyclopædia Britannica, Inc Use the world map above to answer the following questions Write the number that corresponds to each of the following places: Middle East South America South Africa North Korea East China Sea How you think South Africa got its name? In what direction people from South America have to travel to get to North Korea? 46 Figuring All The Angles Finding Your Way N NW 94 NE W E SW SE S Women’s Museum Train Station Goldstein Park Water Park ??? 1st Ave East Madison Ave 1: 2: 3: 4: 3rd Ave West Section B Scale 0.5 mi Use the map above to answer the following questions In what direction does Interstate 94 run? There are question marks by a street name that is missing Use the given street names to assign an appropriate name for this street Explain your reasoning A tourist has arrived at the train station and plans to visit the Women’s Museum and Goldstein Park Give him directions Use the scale line to find the distance from Women’s Museum to Train Station, as the crow flies Additional Practice 47 Additional Practice Section C Directions M T H A N W scale E S km Using your compass card and the scale line, write directions to get from point M to point A, to point T, and, finally, to point H (Directions in degrees, distances in kilometers.) Section D Navigation and Orientation N o Use the grid to the left to help you answer the following questions o 330 30 o 300 One plane is at 30°/20 miles Another plane is at 210°/25 miles What is the distance between the planes? o 60 o o 90 270 10 mi mi o 240 o 120 20 mi mi o 30 mi mi 210 o 180 48 Figuring All The Angles o 150 One plane is at 150°/30 miles Another plane is at 330°/35 miles What is the distance between the planes? Additional Practice Section E Changing Directions: Turns Suppose you are traveling at a heading of 126° NW N W NE E SW SE S a If you turn 98° to the left, what is the new heading? b If you then turn 35° to the right, what is the new heading? Write instructions, using headings or turns, to direct the airplane so that it can drop a parachutist in the target area Avoid the areas marked with an X; these are dangerous mountains You not need to give distances Additional Practice 49 Additional Practice Section F From Turns to Angles Here is a top-view drawing of some sled tracks, showing two angles N NW NE W 143° E SW SE S 122° The angles that are known are indicated in the picture Which turns did the sled driver make (to left or right)? In the star shown below, the heading of leg (1) is 20° The star is a regular figure What is the measure of ∠A? N NW (1) NE W E SW SE S A 50 Figuring All The Angles Additional Practice Section G Angles and Their Measures a How many different right angles are in the picture below? b How many different obtuse angles are in the picture below? N NW NE W E SW SE S Use a protractor to draw an angle of 35° Properly mark the vertex Nathalie says, "When it is the same time, the number of degrees between the hands of the clock tower is larger than the number of degrees between the hand on my wristwatch!" Is Nathalie right? Explain your answer Here are the mosaic tiles you worked with in Section G on page 40 45˚ 60˚ 45˚ 90˚ 60˚ 60˚ Create three new shapes consisting of these two mosaic tiles You may use more than one of each shape to create their new shape Label the size of each angle in your shapes Additional Practice 51 Section A A Sense of Direction The North Pole is unique because of its fixed position It is north of every location Another such unique place is the South Pole Hawaii is located southwest of the mainland Alaska is located northwest of the contiguous United States South America is located southwest of Europe, or Europe is located northeast of South America South America is located west of Australia, or Australia is located east of South America Section B Finding Your Way a Here is one possible drawing Your drawing might be different b Here is one possible response: In Sunray, all of the blocks are square because there are no streets in the directions NW, NE, SW, SE, and it is easy to find a place if you know its address In the town I drew, the streets are named after my pets and my friends’ pets 52 Figuring All The Angles Answers to Check Your Work Here is one possible response: Taxicab distances are usually longer than as-the-crow-flies distances The distances could be the same if you have to go in one direction and a road goes in that direction a The distance from A to E “as the crow flies” is about 35 miles b The “taxicab” distance from A to E is also about 35 miles since you can get there in a straight line, without having to make turns c The distance from A to D “as the crow flies” is about 55 miles d The “taxicab” distance from A to D is about 75 miles since you cannot get there in a straight line but will have to make turns Directions The heading should be about 70° a 33° b 110° ad in g 21 5° N he Section C Drawing a dotted line for north may help you make an accurate heading Don’t forget to use your compass card Answers to Check Your Work 53 Answers to Check Your Work Section D Navigation and Orientation The distance between the planes is 20 miles N 20 miles 15 miles The distance between the planes is 30 miles 90؇ 35 miles N 270؇ 90؇ 15 miles 15 miles 30 miles The best way to report a fire would be (c); it uses both a heading as well as a distance There is only one possible location for the fire Using a compass direction or a heading, (b) is better than (a) because you have at least the exact direction Both (a) and (b) not give enough information because you are not given the distance in that direction You could be wandering in the specified direction for a long time until you found the fire Section E Changing Directions: Turns N The new heading is 180° Reasoning: left 40°, then right 20° means left 20° Looking at your compass card, and starting from a heading of 200°, turning left 20°, results in a heading of 180° • Reasoning: 200° minus 40° results in a heading of 160°; then 160° plus 20° results in a heading of 180° 40° 54 Figuring All The Angles Reasoning by making the drawing shown on the right in d hea heading 180˚ • 0° g 16 • hea din g 200 ° Here are three solution strategies: 20° Answers to Check Your Work The second turn is a turn of 5° to the right Here are two solution strategies: • Reasoning: The first turn results in a new heading of 55° In order to get a heading of 60°, you have to make a right turn of 5° • In a drawing: N turn of 5° to the right headin g 100° g din 55 ° g din a e h 60° a he 45° Make sure you start your drawing with a line directing north Use your compass card Note that the length of the lines in your drawing may differ from those in the sample drawing The directions, however, should be the same as in the sample drawing N present heading 180° 160 ° 180° Answers to Check Your Work 55 Answers to Check Your Work Section F From Turns to Angles a The new heading is 140° Here is one solution strategy: By looking at the compass card, I found that for a right turn I need to add 90° to 50°, which is 140° b Here is the drawing for problem 1a N 90° w ne g in 0° ad he 14 l 0° tia g i in din a he c The angle between the two legs of the track is 90° The initial heading was 310° Here is one solution strategy: 31 0° final heading 0° If ∠Q ؍130°, then the turn to make a 0° heading has to be 50° to the right I used the compass card to see that this would make an initial heading of 310° 50° 130° Q in iti al he ad in g 56 Figuring All The Angles Answers to Check Your Work a The final heading is 240° (180° ؉ ؋ 15°) This drawing helped me to reason about the four 15° right turns 0° angle of the track initial heading 180° N 15° 15° 15° g 0° 24 adin e h al fin ° 15 b The angles formed by the sled are 165° (180° ؊ 15°) Since the turns are equal, the other angles of the track are also 165° Answers to Check Your Work 57 Answers to Check Your Work Section G Angles and Their Measures ∠A is 69° ∠B is 100° ∠C is 90° a ∠P is less than 90°; it is about 80° ∠Q is between 40° and 90°, so about 60° ∠R is between 90° and 180°, so about 120° b ∠R is an obtuse angle ∠S ؍40°, ∠T ؍110° S T 58 Figuring All The Angles ... in the corner: The corner of any angle is called the vertex The two partial lines or line segments that form the angle are called the sides of the angle 38 Figuring All The Angles Angles and Their... back in the original direction The result of all the turning is 360 ° (12 ؋ 30° ؍ 360 °) Explain this reasoning 34 Figuring All The Angles From Turns to Angles F Regular Polygons 150° The drawing... 150° START 12 There are also 12 angles on the inside, each of them measuring 150° 11 Angles inside the polygon are called interior angles 10 What is the sum of all the 12 interior angles of a 12-gon?