Expressions and Formulas Algebra 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 Gravemeijer, K.; Roodhardt, A.; Wijers, M.; Kindt, M., Cole, B R.; and Burrill, G (2006) Expressions and formulas In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in Context Chicago: Encyclopædia Britannica 2006 Printed by Holt, Rinehart and Winston Copyright © 2006 Encyclopỉdia 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 0-03-039617-4 073 09 08 07 06 The Mathematics in Context Development Team Development 1991–1997 The initial version of Expressions and Formulas was developed by Koeno Gravemeijer, Anton Roodhardt, and Monica Wijers It was adapted for use in American schools by Beth R Cole and Gail Burrill 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 Expressions and Formulas was developed by Monica Wijers and Martin Kindt It was adapted for use in American schools by Gail Burrill 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 Encyclopædia Britannica, Inc Cover photo credits: (left to right) © PhotoDisc/Getty Images; © Corbis; © Getty Images Illustrations 1, Holly Cooper-Olds; Thomas Spanos/© Encyclopỉdia Britannica, Inc.; Christine McCabe/© Encyclopỉdia Britannica, Inc.; 13 (top) 16 (bottom) Christine McCabe/© Encyclopỉdia Britannica, Inc.; 25 (top right) Thomas Spanos/© Encyclopỉdia Britannica, Inc.; 29 Holly Cooper-Olds; 32, 36, 40, 41 Christine McCabe/© Encyclopỉdia Britannica, Inc Photographs © PhotoDisc/Getty Images; 14 © Corbis; 15 John Foxx/Alamy; 26, 32, 33 © PhotoDisc/Getty Images; 34 SuperStock/Alamy; 43 © PhotoDisc/Getty Images Contents Letter to the Student Section A Arrow Language Bus Riddle Wandering Island Summary Check Your Work Section B 12 14 16 18 22 22 Reverse Operations Distances Going Backwards Beech Trees Summary Check Your Work Section E 10 10 Formulas Supermarket Taxi Fares Stacking Cups Bike Sizes Summary Check Your Work Section D 4 Smart Calculations Making Change Skillful Computations Summary Check Your Work Section C vi 25 28 29 30 30 Order of Operations Home Repairs Arithmetic Trees Flexible Computation Return to the Supermarket What Comes First? Summary Check Your Work 32 34 39 40 42 44 45 Additional Practice 46 Answers to Check Your Work 50 Contents v Dear Student, Welcome to Expressions and Formulas Imagine you are shopping for a new bike How you determine the size frame that fits your body best? Bicycle manufacturers have a formula that uses leg length to find the right size bike for each rider In this unit, you will use this formula as well as many others You will devise your own formulas by studying the data and processes in the story Then you will apply your own formula to solve new problems In this unit, you will also learn new forms of mathematical writing You will use arrow strings, arithmetic trees, and parentheses These new tools will help you interpret problems as well as apply formulas to find problem solutions As you study this unit, look for additional formulas in your daily life outside the mathematics classroom, such as the formula for sales tax or cab rates Formulas are all around us! Sincerely, The Mathematics in Context Development Team vi Expressions and Formulas A Arrow Language Bus Riddle Imagine you are a bus driver Early one morning you start the empty bus and leave the garage to drive your route At the first stop, 10 people get on the bus At the second stop, six more people get on At the third stop, four people get off the bus and seven more get on At the fourth stop, five people get on and two people get off At the fifth stop, four people get off the bus How old is the bus driver? Did you expect the first question to ask about the number of passengers on the bus after the fifth stop? How could you determine the number of passengers on the bus after the fifth stop? Section A: Arrow Language A Arrow Language When four people get off the bus and seven get on, the number of people on the bus changes There are three more people on the bus than there were before the bus stopped Here is a record of people getting on and off the bus at six bus stops Copy the table into your notebook Then complete the table Number of Passengers Getting off the Bus Number of Passengers Getting on the Bus Change more 13 16 16 15 fewer Study the last row in the table What can you say about the number of passengers getting on and off the bus when you know that there are five fewer people on the bus? For the story on page 1, you might have kept track of the number of passengers on the bus by writing: 10 ؉ ؍16 ؉ ؍19 ؉ ؍22 ؊ ؍18 Reflect Do you think that representing the numbers in this format is acceptable mathematically? Why or why not? To avoid using the equal sign to compare amounts that are not equal, you can represent the calculation using an arrow symbol ؉ 16 ⎯⎯→ ؉ 19 ⎯⎯→ ؉ 22 ⎯⎯→ ؊ 18 10 ⎯⎯→ Each change is represented by an arrow This way of writing a string of calculations is called arrow language You can use arrow language to describe any sequence of additions and subtractions, whether it is about passengers, money, or any other quantities that change Why is arrow language a good way to keep track of a changing total? Ms Moss has $1,235 in her bank account She withdraws $357 Two days later, she withdraws $275 from the account Use arrow language to represent the changes in Ms Moss’s account Include the amount of money she has in her account at the end of the story Expressions and Formulas Arrow Language A Kate has $37 She earns $10 delivering newspapers on Monday She spends $2.00 for a cup of frozen yogurt On Tuesday, she visits her grandmother and earns $5.00 washing her car On Wednesday, she earns $5.00 for baby-sitting On Friday, she buys a sandwich for $2.75 and spends $3.00 for a magazine a Use arrow language to show how much money Kate has left b Suppose Kate wants to buy a radio that costs $53 Does she have enough money to buy the radio at any time during the week? If so, which day? Monday It snowed 20.25 inches Tuesday It warmed up, and 18.5 inches of snow melted Wednesday Two inches of snow melted Thursday It snowed 14.5 inches Friday It snowed 11.5 inches in the morning and then stopped Ski Spectacular had 42 inches of snow on the ground on Sunday This table records the weather during the week 10 How deep was the snow on Friday afternoon? Explain your answer Wandering Island Wandering Island constantly changes shape On one side of the island, the sand washes away On the other side, sand washes onto shore The islanders wonder whether their island is actually getting larger or smaller In 1998, the area of the island was 210 square kilometers (km2) Since then, the islanders have recorded the area that washes away and the area that is added to the island Year Area Washed Away (in km2) Area Added (in km2) 1999 5.5 6.0 2000 6.0 3.5 2001 4.0 5.0 2002 6.5 7.5 2003 7.0 6.0 11 What was the area of the island at the end of 2001? 12 a Was the island larger or smaller at the end of 2003 than it was in 1998? b Explain or show how you got your answer Section A: Arrow Language A Arrow Language Arrow language can be helpful to represent calculations Each calculation can be described with an arrow starting number action ⎯ ⎯→ resulting number A series of calculations can be described by an arrow string ؉ 16 ⎯⎯→ ؉ 19 ⎯⎯→ ؉ 22 ⎯⎯→ ؊ 18 10 ⎯⎯→ Airline Reservations There are 375 seats on a flight to Atlanta, Georgia, that departs on March 16 By March 11, 233 of the seats were reserved The airline continues to take reservations and cancellations until the plane departs If the number of reserved seats is higher than the number of actual seats on the plane, the airline places the passenger names on a waiting list The table shows the changes over the five days before the flight Date Seats Requested Seats Cancellations 3/11 233 3/12 47 3/13 51 3/14 53 3/15 12 3/16 16 Expressions and Formulas Total Seats Reserved E Order of Operations What Comes First? Arithmetic trees are useful because they resolve any question about the order of the calculation The problem is that they require a lot of room on your paper Copy the first tree below 15 ؉ ؊ Since the ؉ is simplified first, circle it on your copy 15 ؉ ؊ The tree can then be simplified 15 10 ؊ Instead of the second arithmetic tree, you could write: 15 ؊ ؉ 27 What does the circle represent? The whole circle is not necessary People often write 15 ؊ (6 ؉ 4) This does not require as much space, but the parentheses show how the numbers are grouped together 42 Expressions and Formulas Order of Operations E 28 a Rewrite the tree using parentheses to indicate which numbers are associated 12 ؉ ، b Make a tree for ؋ (6 ؉ 4) c What is the value of ؋ (84 ؊ 79)? d Rewrite the tree using parentheses 16 40 ؉ 32 ؉ ، 29 Use parentheses to find the correct total bill for Karlene’s problem in the Home Repair section ؋ 37 ؉ 25 ؋ Section E: Order of Operations 43 E Order of Operations The beginning of this unit introduced arrow language to represent formulas There are several ways to write formulas You can express formulas with words cost ؍tomatoes ؋ $1.50 ؉ grapes ؋ $1.70 ؉ green beans ؋ $0.90 (in lb) (in lb) (in lb) You can express formulas with arithmetic trees tomatoes 1.50 grapes ؋ 1.70 green beans ؋ 0.90 ؋ ؉ ؉ Arithmetic trees show the order of calculation If a problem is not in an arithmetic tree and does not have parentheses, there is a rule for the order of operations: Complete multiplication and division before addition and subtraction ؋ ؉ ، ؊ is represented in this tree ؋ ، ؉ ؊ You can use parentheses to convert an arithmetic tree to an expression that shows which operations to first (5 ؋ 4) ؉ (3 ، 2) ؊ ؍20.5 44 Expressions and Formulas a Use the mathematicians’ rule to simplify this expression 24 ، ؉ ؋ ؊ 10 You may use an arithmetic tree if you wish b Write 24 ، ؉ ؋ ؊ 10 ؍using parentheses so that the expression reflects the mathematicians’ rule for order of operations Design an arithmetic tree that makes each of the following problems easier to solve using mental calculation a 17 ؉ ؉ ؉ ؉ 1 ᎑᎑ ؉ 2᎑ ؉ 10 ᎑᎑ ؉ 4᎑ ؉ 10 ᎑᎑ c 10 b 4.5 ؉ 8.9 ؉ 5.5 ؉ 1.1 You can estimate your ideal weight using the following rule: weight (in kilograms) ؍ height (in cm) ؊ 100 ؉ (4 ؋ circumference of wrist in cm) This general rule applies for adult men For women, the rule is slightly different: Replace 100 with 110 a Write an arithmetic tree to represent the general rule for women Matthew is 175 cm tall The circumference of his wrist is 17 cm b Use the rule to estimate Matthew’s ideal weight Andrew is 162 cm tall The circumference of his wrist is 16 cm His weight is 54 kilograms (kg) c Does Andrew weigh too much or too little, according to the general rule? How might you modify the mathematician’s rule for the order of operations to calculate (3 ؋ ؊ ؉ 7) ، 4? Section E: Order of Operations 45 Additional Practice Section A Arrow Language Here is a record for Mr Kamarov’s bank account Date Deposit Withdrawal Withdrawal 10/15 $210.24 10/22 $523.65 $140.00 10/29 $75.00 $40.00 a Find the totals for October 22 and October 29 b Write arrow strings to show how you found the totals c When does Mr Kamarov first have a minimum of $600 in his account? Find the results for these arrow strings ؊3 a 15 ⎯⎯→ ؉ 1.9 b 3.7 ⎯⎯→ ⎯⎯ ⎯⎯ ؊ 1,520 c 3,000 ⎯ ⎯⎯→ Section B ؉ 11 ⎯⎯→ ؉ 8.8 ⎯⎯→ ⎯⎯ ⎯⎯ ⎯⎯ ؊ 600 ؊ 1.6 ⎯⎯→ ⎯⎯⎯→ ⎯⎯ ⎯⎯ ؉ 5,200 ⎯ ⎯⎯→ ⎯⎯ Smart Calculations For each shopping problem, write an arrow string to show the change the clerk should give the customer Be sure to use the small-coins-and-bills-first method Then write another arrow string that has only one arrow to show the total change a A customer gives $20.00 for a $9.59 purchase b A customer gives $5.00 for a $2.26 purchase c A customer gives $16.00 for a $15.64 purchase 46 Expressions and Formulas Rewrite these arrow strings so that each has only one arrow ؉ 35 a 750 ⎯⎯→ ؊3 b 63 ⎯⎯→ ؉1 c 439 ⎯⎯→ ⎯⎯ ⎯⎯ ⎯⎯ ؉ 40 ⎯⎯→ ؉ 50 ⎯⎯→ ؊ 20 ⎯⎯→ ⎯⎯ ⎯⎯ ⎯⎯ Rewrite each arrow string with a new string that will make the computation easier to calculate Explain why your new string makes the computation easier, or why it is not possible to simplify the string ؉ 66 a 74 ⎯⎯→ ⎯⎯ Section C ؊ 58 b 231 ⎯⎯→ ⎯⎯ ؉ 27 c 459 ⎯⎯→ ⎯⎯ Formulas If Clarinda is connected to the Internet for a total of three hours one month, she pays $15 plus three times $2, or $21, for the month Which string shows the cost for Clarinda’s Internet service through Tech Net? Explain your answer ؉ $2 ؋ number of hours → total cost a $15 ⎯⎯→ ⎯⎯ ⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯ ؉ $15 ؋ $2 b number of hours ⎯⎯→ ⎯⎯ ⎯⎯→ total cost ؉ $15 ؋ $2 c number of hours ⎯⎯→ ⎯⎯ ⎯⎯→ total cost How much does it cost Clarinda for these monthly usage amounts? a hours b 20 hours c 2᎑ hours Another Internet access company, Online Time, charges only $10 per month, but $3 per hour Write an arrow string that finds the cost of Internet access through Online Time If Clarinda uses the Internet approximately 10 hours a month, which company should she use—Tech Net or Online Time? Additional Practice 47 Additional Practice Carlos works at a plant nursery that sells flower pots One type of flower pot has a rim height of cm and a hold height of 16 cm cm 16 cm How tall is a stack of two pots? Three pots? Write a formula using arrow language that can be used to find the height of any stack if you know the number of pots Carlos has to stack these pots on a shelf that is 45 cm high How many pots can he place in a stack this high? Explain your answer Compare the following pairs of arrow strings and determine whether they provide the same results ؋8 ،2 ،2 ؋8 ؉5 ؋3 ؋3 ؉5 ،2 ؉1 a input ⎯⎯→ ⎯⎯ ⎯⎯→ output input ⎯⎯→ ⎯⎯ ⎯⎯→ output b input ⎯⎯→ ⎯⎯ ⎯⎯→ output input ⎯⎯→ ⎯⎯ ⎯⎯→ output ؉6 c input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ output ،2 ؉6 ؉1 input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ output Section D Reverse Operations Ravi lives in Bellingham, Washington He travels to Vancouver, Canada, frequently When Ravi was in Canada, he used this rule to estimate prices in U.S dollars ،4 ؋3 number of Canadian dollars ⎯⎯→ ⎯⎯ ⎯⎯→ number of U.S dollars Using Ravi’s formula, estimate U.S prices for these Canadian prices a a hamburger for $2 Canadian b a T-shirt for $18 Canadian c a movie for $8 Canadian d a pair of shoes for $45 Canadian 48 Expressions and Formulas Additional Practice Write a formula that Ravi can use to convert U.S dollars to Canadian dollars Write the reverse string for each of these strings ؊1 ؋ 2.5 ؉4 a input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ output ؉6 ؊2 ،5 b input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ output Find the input for each string ؉ 10 ،2 ؊3 a input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ ؋4 ؊5 ،3 ؉1 b input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ 10 Section E Order of Operations In your notebook, copy and complete the arithmetic trees a 12 b ؋ ؊ 24 c 1.5 3.5 ، ؉ ؋ ؋ ؊ ؋ Make or design an arithmetic tree and find the answer a 10 ؉ 1.5 ؋ b (10 ؉ 1.5) ؋ c 15 ، (2 ؋ ؉ 1) Suzanne took her cat to the veterinarian for dental surgery (Her cat had never brushed his teeth!) Before the surgery, the veterinarian gave Suzanne an estimate for the cost: $55 for anesthesia, $30 total for teeth cleaning, $18 per tooth pulled, $75 per hour of surgery, and the cost of medicine Draw an arithmetic tree to represent the total cost of Suzanne’s bill from the veterinarian Use words in your arithmetic tree when necessary Additional Practice 49 Section A Arrow Language Date Seats Requested Seats Cancellations Total Seats Reserved 3/11 233 3/12 47 280 3/13 51 330 3/14 53 383 3/15 12 376 3/16 16 390 Arrow strings will vary Sample response: ؉ 47 ؊0 ؉ 51 ؊1 ؉ 53 ؊0 ؉5 ؊12 ؉ 16 ؊2 3/12 233 ⎯ ⎯⎯→ 280 ⎯ ⎯⎯→ 280 3/13 280 ⎯ ⎯⎯→ 331 ⎯ ⎯⎯→ 330 3/14 330 ⎯ ⎯⎯→ 383 ⎯ ⎯⎯→ 383 3/15 383 ⎯ ⎯⎯→ 388 ⎯ ⎯⎯→ 376 3/16 376 ⎯⎯⎯→ 392 ⎯ ⎯⎯→ 390 The airline needs to begin a waiting list on March 14 Answers will vary Sample response: One advantage is that it quickly tells you how many people are booked for the flight on the 16th One disadvantage is that you not know on what day the waiting list was started ؉ 1.40 ؊ 0.62 ؉ 5.83 ؊ 1.40 a 12.30 ⎯⎯⎯→ 13.70 ⎯ ⎯⎯→ 13.08 ⎯ ⎯⎯→ 18.91 ⎯⎯⎯→ 17.51 b Discuss your answer with a classmate Sample response: Vic had $12.30 in his pocket His mom gave him $1.40 for bus fare On the way to the bus stop, he bought a pen for $0.62 Then he sold his lunch to Joy for $5.83 He paid the bus driver $1.40 How much did Vic have left? 50 Expressions and Formulas Answers to Check Your Work Discuss your answer with a classmate Sample response: Fourteen people got on the empty bus at the first stop At the second stop, two got off and eight got on How many were still on the bus? [20 people, or 21 people if you count the driver] ؊2 ؉8 14 ⎯⎯→ 12 ⎯ ⎯→ 20 Sample response: Arrow language shows all the steps in order so that you can find answers that are in the middle of a series of calculations Section B Smart Calculations ؉ 15 – ؉2 ؊8 a 20 ⎯⎯⎯→ 35 ⎯ ⎯→ 27 ⎯ ⎯→ 27.5 ؉ 0.03 ؉ 0.20 ؉ 13 b 6.77 ⎯⎯ ⎯⎯→ 6.80 ⎯⎯ ⎯⎯→ ⎯⎯ ⎯→ 20 ؉ 0.20 ؉ 0.70 ؉7 c 12.10 ⎯⎯ ⎯⎯→ 12.30 ⎯⎯ ⎯⎯→ 13 ⎯⎯→ 20 a Sample response: ؉7 ؉ 30 ؉ 200 423 ⎯⎯→ 430 ⎯⎯⎯→ 460 ⎯⎯⎯→ 660 This string is easier because you can add the numbers in the ones place, then add the numbers in the tens place, and finally add the numbers in the hundreds place b Sample response: This string is already easy because you can easily subtract 24 from 44 to get 20, so the answer is 520 You can this mentally in one step, so the arrow string is already the easiest one c Sample response: ؊ 25 ؉ 54 29 ⎯ ⎯→ ⎯⎯⎯→ 58 This string is easier because when you subtract 25 first, it leaves an easy number to work with Answers to Check Your Work 51 Answers to Check Your Work d Sample response: ؉2 ؉ 32 998 ⎯⎯→ 1,000 ⎯⎯⎯→ 1,032 This string is easier because when you add to 998, you get lots of zeros that are easy to work with It’s easy to add numbers to 1,000 Check your answer with a classmate Sample response: (long) ؉ 31 ؉ 19 232 ⎯⎯⎯→ 263 ⎯⎯⎯→ 282 ؉ 50 (short) 232 ⎯⎯⎯→ 282 Short strings are easier when the total of the numbers over the arrows is a multiple of 10 or a number between and 10 Check your answer with a classmate Sample response: ؉ 98 (short) 232 ⎯⎯⎯→ 330 (long) ؉ 100 ؊2 232 ⎯ ⎯⎯→ 332 ⎯⎯⎯→ 330 Longer strings are easier when the total of the numbers over the arrows is not a multiple of 10 or a number between and 10 Sample explanation: The shortened arrow string shows the total amount of change Section C Formulas ؋ 1.40 ؉ 1.90 a number of miles ⎯⎯⎯⎯→ ⎯⎯⎯⎯→ total price b Using an arrow string makes calculations easier 52 Expressions and Formulas Answers to Check Your Work Weight Tomatoes $1.20/lb Green Beans $0.80/lb Grapes $1.90/lb 0.5 lb $0.60 $0.40 $0.95 1.0 lb $1.20 $0.80 $1.90 2.0 lb $2.40 $1.60 $3.80 3.0 lb $3.60 $2.40 $5.70 a The ؊ means that one chair is subtracted from the total number of chairs in the stack; ؋ means that for each chair that is added, the height of the stack will grow cm The ؉ 80 represents the height of the first chair in the stack b Alba should use ؋ and ؉ 73 above the arrows She wrote ؋ because each chair adds cm to the height of the stack Next, ؉ 73 is added for the height of the first chair minus the cm that was already added in the first step a The point on the graph labeled A represents a stack of 15 chairs with a total height of about 175 cm b Not every point on the graph has a meaning For example, you cannot add “half a chair,” or the total height of the stack cannot be 100 cm c The arrow strings can be used to find the height of one or more of these chairs Zero chairs makes no sense Also, if is used for the number of chairs in Damian’s arrow string, the result is: ؊1 ؋7 ؉ 80 ⎯⎯→ ؊1 ⎯⎯→ ؊7 ⎯⎯⎯→ 73 cm 0⎯ d About six chairs ؋7 ؉ 73 e ⎯ ⎯⎯→ 35 ⎯⎯⎯→ 108 cm A stack of five chairs requires 108 cm of space ؋7 ؉ 73 6⎯ ⎯⎯→ 42 ⎯⎯⎯→ 115 cm So six chairs will fit ؋7 ؉ 73 7⎯ ⎯⎯→ 49 ⎯⎯⎯→ 122 cm Seven will not Answers to Check Your Work 53 Answers to Check Your Work Section D Reverse Operations a 4,090 ؋ 1.609 ഠ 6,581 km Your answer should not contain decimals because the original measurement is rounded to the nearest mile b 4,090 ؋ 1.6 ഠ 6,544 km c 6,581 ؊ 6,544 ഠ 37 Discuss your answer with a classmate Sample responses: • You not know how 4,090 miles was measured As an airplane flies? Making computations using a model of the earth? Using and converting sea miles? They will all result in different outcomes • 37 km compared to 6,500 (or 6,581 or 6,544) is less than 1% That is not a very big difference • If you are traveling that far, 37 km is not a very big difference ؋ 0.75 $4.40 ⎯⎯⎯⎯→ $3.30 3 Carmen’s string is correct because 0.75 is the same as Ϫ By dividing by 4, she found one-fourth of the price Next she multiplied by three, which results in three-fourths of the price ؉4 ،3 ؊2 ؉5 ؊2 a output ⎯⎯→ ⎯ ⎯⎯→ ⎯ ⎯→ input (divided by) ؊7 b output ⎯⎯→ ⎯⎯⎯→ ⎯⎯→ input Check your answers with a classmate You should have used several numbers to check if your arrow string works Section E Order of Operations a 38 b (24 ، 3) ؉ (5 ؋ 8) ؊ 10 _ ؍ 54 Expressions and Formulas Answers to Check Your Work Sample answers: a 17 ؉ 20 ؉ 10 ؉ 30 ؉ 37 b 4.5 5.5 8.9 ؉ 10 1.1 ؉ 10 ؉ 20 c 10 10 10 ؉ 10 ؉ 10 ؉ ؉ 13 Answers to Check Your Work 55 Answers to Check Your Work a height 110 ؊ wrist circumference (cm) ؋ ؉ ، b Matthew should weigh 71.5 (or 72) kg c Andrew’s ideal weight is 63 kg according to the general rule So he does not weigh enough 56 Expressions and Formulas ... Chicago, Illinois 60610 ISBN 0-03-0396 17- 4 073 09 08 07 06 The Mathematics in Context Development Team Development 1991–19 97 The initial version of Expressions and Formulas was developed by Koeno Gravemeijer,... strings ؉ 15 a 20 ⎯⎯→ ⎯⎯ ؉ 0.03 b 6 .77 ⎯⎯⎯→ ؉ 0.20 ⎯⎯→ ⎯⎯ c 12.10 ⎯⎯⎯→ 10 Expressions and Formulas ؊8 ؉– ⎯⎯ ؉ 0.20 ⎯⎯⎯→ ⎯⎯ ⎯ ⎯→ ؉ 0 .70 ⎯⎯ ⎯⎯⎯→ ⎯⎯ ⎯⎯ ؉ 13 ⎯ ⎯⎯→ 20 ? ?7 ⎯⎯→ 20 For each of these arrow... so each has only one arrow ؉ 50 a 375 ⎯⎯→ ؊1 b 158 ⎯⎯→ ⎯⎯→ ? ⎯⎯→ ⎯⎯ ⎯⎯ ؊ 1,000 c 1, 274 ⎯⎯ ⎯ ⎯⎯→ Expressions and Formulas ؉ 50 ? ؉ 100 ? ⎯⎯ is the same as 375 ⎯⎯→ ? is the same as 158 ⎯⎯→ ⎯⎯ ⎯⎯