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Suggested new non-essential art: page 11 page 12 will replace with new cover file 2nd Pass (B file) Expressions and Formulas Algebra FPO 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., Boon, P.; Kindt, M.; Cole, B. R.; and Burrill, G. (2006). Expressions and formulas using algebra arrows. In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in context. Chicago: Encyclopædia Britannica. 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. check for ISBN { ISBN 0-03-000000-0 1 2 3 4 5 6 073 09 08 07 06 05 The Mathematics in Context Development Team Development 1991–1997 The initial version of Expressions and Formulas Using Algebra Arrows 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 Formula Using Algebra Arrows was developed by Monica Wijers, Peter Boon, and Martin Kindt. It was adapted for use in American schools by Gail Burrill and Teri Hedges. 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, 6 Holly Cooper-Olds; 7 Thomas Spanos/© Encyclopædia Britannica, Inc.; 8 Christine McCabe/© Encyclopædia Britannica, Inc.; 13 (top) 16 (bottom) Christine McCabe/© Encyclopædia Britannica, Inc.; 25 (top right) Thomas Spanos/© Encyclopædia Britannica, Inc.; 29 Holly Cooper-Olds; 32, 36, 40, 41 Christine McCabe/© Encyclopædia Britannica, Inc. Photographs 3 © PhotoDisc/Getty Images; 14 © Corbis; 15 John Foxx/Alamy; 26, 32, 33 © PhotoDisc/Getty Images; 34 SuperStock/Alamy; 43 © PhotoDisc/Getty Images These are the credits from EF. They will need to be checked for this unit. Contents Letter to the Student Section A Arrow Language Bus Riddle Wandering Island Summary Check Your Work Section B 8 14 14 The TOC folios will be corrected on the next pass if page breaks are OK’d. Art for TOC will also be picked up then. 16 18 20 22 28 28 Reverse Operations Distances Going Backwards Beech Trees Summary Check Your Work Section E 6 Formulas Supermarket Taxi Fares Stacking Cups Bike Sizes Summary Check Your Work Section D 1 3 4 4 Smart Calculations Making Change Skillful Computations with Algebra Arrows Summary Check Your Work Section C vi 32 35 36 38 38 Order of Operations Home Repairs Arithmetic Trees Home Repairs Again Flexible Computation (Without Computer) Return to the Supermarket What Comes First? Summary Check Your Work 40 41 46 48 50 52 54 55 Additional Practice 57 Answers to Check Your Work 62 Contents v Dear Student, Welcome to Expressions and Formulas Using Algebra Arrows. Imagine you are shopping for a new bike. How do 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 Request from Michael to add something about the applets to the letter. 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 sixth stop, four people get off the bus. 1. How old is the bus driver? 2. Did you expect the first question to ask about the number of passengers on the bus after the sixth stop? 3. How could you determine the number of passengers on the bus after the sixth stop? Section A: Arrow Language 1 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. 4. 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 5 8 3 more 9 13 16 16 15 8 9 3 5 fewer 5. 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 6 16 3 19 3 22 4 18 6. 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. 6 16 ⎯⎯→ 3 19 ⎯⎯→ 3 22 ⎯⎯→ 4 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. 7. 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. 8. 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. 2 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. 9. 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 2000 2001 2002 2003 5.5 6.0 4.0 6.5 7.0 6.0 3.5 5.0 7.5 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 3 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. 6 16 ⎯⎯→ 3 19 ⎯⎯→ 3 22 ⎯⎯→ 4 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 0 3/13 51 1 3/14 53 0 3/15 5 12 3/16 16 2 4 Expressions and Formulas Total Seats Reserved 1. Copy and complete the table. 2. Write an arrow string to represent the calculations you made to complete the table. 3. On which date does the airline need to form a waiting list? 4. To find the total number of reserved seats, Toni, a reservations agent, suggests adding all of the new reservations and then subtracting all of the cancellations at one time instead of using arrow strings. What are the advantages and disadvantages of her suggestion? 5. a. Find the result of this arrow string. 1.40 12.30 ⎯⎯⎯→ ⎯⎯ 0.62 ⎯⎯⎯→ ⎯⎯ 5.83 ⎯⎯⎯→ ⎯⎯ 1.40 ⎯⎯⎯→ ⎯⎯ b. Write a story that could be represented by the arrow string. 6. Write a problem that you can solve using arrow language. Then solve the problem. 7. Why is arrow language useful? Juan says that it is easier to write 15 3 18 6 12 2 14 than to make an arrow string. Tell what is wrong with the string that Juan wrote and show the arrow string he has tried to represent. Section A: Arrow Language 5 B Smart Calculations Making Change At most stores today, making change is easy. The clerk just enters the amount of the purchase and the amount received. Then the computerized register shows the amount of change due. Before computerized registers, however, making change was not quite so simple. People invented strategies for making change using mental calculation. These strategies are still useful to make sure you get the right change. 1. a. When you make a purchase, how do you know if you are given the correct change? b. Reflect How might you make change without using a calculator or computerized register? A customer’s purchase is $3.70. The customer gives the clerk a $20 bill. 2. Explain how to calculate the correct change without using a pencil and paper or a calculator. It is useful to have strategies that work in any situation, with or without a calculator. Rachel suggests that estimating is a good way to begin. “In the example,” she explains, “it is easy to tell that the change will be more than $15.” She says that the first step is to give the customer $15. Rachel explains that once the $15 is given as change to the customer, you can work as though the customer has paid only the remaining $5. “Now the difference between $5.00 and $3.70 must be found. The difference is $1.30—or one dollar, one quarter, and one nickel.” 3. Do you think that Rachel has proposed a good strategy? Why or why not? 6 Expressions and Formulas Smart Calculations B Many people use a different strategy that gives small coins and bills first. Remember, the total cost is $3.70, and the customer pays with a $20 bill. The clerk first gives the customer a nickel and says, “$3.75.” Next, the clerk gives the customer a quarter and says, “That’s $4.” Then the clerk gives the customer a dollar and says, “That’s $5.” The clerk then gives the customer a $5 bill and says, “That makes $10.” Finally, the clerk gives the customer a $10 bill and says, “That makes $20.” This method could be called “making change to twenty dollars.” This method could also be called “the smallcoins-and-bills-first method.” 4. a. Does this method give the fewest possible coins and bills in change? Explain. b. Why do you think it might also be called the small-coins-andbills-first method? These methods illustrate strategies to make change without a computer or calculator. They are strategies for mental calculations, and they can be illustrated with arrow strings. Another customer’s bill totals $7.17, and the customer pays with a $10 bill. 5. a. Describe how you would make change using the small-coinsand-bills-first method. b. Does your solution give the customer the fewest coins and bills possible? Section B: Smart Calculations 7 B Smart Calculations Arrow language can be used to illustrate the small-coins-and-bills-first method. This arrow string shows the change for the $3.70 purchase. $0.05 $0.25 $1.00 $5.00 $10.00 $3.70 ⎯ ⎯⎯→ $3.75 ⎯ ⎯⎯→ $4.00 ⎯ ⎯⎯→ $5.00 ⎯ ⎯⎯→ $10.00 ⎯ ⎯⎯→ $20.00 6. a. What might the clerk say to the customer when giving the customer the amount in the third arrow? b. What is the total amount of change? c. Write a new arrow string with the same beginning and end but with only one arrow. Explain your reasoning. Now try some shopping problems. For each problem, write an arrow string using the small-coins-and-bills-first method. Then write another arrow string with only one arrow to show the total change. 7. a. You give $10.00 for a $5.85 purchase of some cat food. b. You give $20.00 for a $7.89 purchase of a desk fan. c. You give $10.00 for a $6.86 purchase of a bottle of car polish. d. You give $5.00 for a $1.76 purchase of pencils. A customer gives a clerk $2.00 for a $1.85 purchase. The clerk is about to give the customer change, but she realizes she does not have a nickel. So the clerk asks the customer for a dime. 8. Reflect What does the clerk give as change? Explain your strategy. Skillful Computations with Algebra Arrows Icon for tech questions. They haven't all been placed. Will do upon approval. With the applet Algebra Arrows, you can make arrow strings and let the computer do the computations. This is similar to the logo used on the MiC Web site. 8 Expressions and Formulas Smart Calculations B Arrow strings made in Algebra Arrows look like the following: In/Output Operations 3 3 6 10 3 16 /3 1 /… 3 18 冑苳苳苳 … 7 9 23 …2 Back Forth 5 15 Table Graph 18 18 18 Clear expression value 9. Use Algebra Arrows to make the following arrow strings and write down the result. On page XX, there is an explanation of the applet. 83 a. 278 ⎯⎯→ b. 17 34 ⎯⎯→ 15 ? ⎯⎯→ ? ⎯⎯→ ⎯⎯ ⎯⎯ 69 ? ⎯⎯ ? ⎯⎯ Use the applet to make the following calculations: 10. a. 12.75 23.32 b. 498 29 67 Section B: Smart Calculations 9 B Smart Calculations How To Use Algebra Arrows 1 Drag the boxes to the workspace. In/Output Operations 3 3 3 3 /3 1/ In/Output 2 Click on the input box and fill in the starting number. Operations 3 10 3 3 3 /3 1/ 3 Click on the operation box and fill in the number for the operation. In/Output Operations 3 10 3 3 /3 1/ In/Output 4 Connect the boxes with arrows by dragging them to the next box. Operations 3 3 3 /3 1/ 10 Expressions and Formulas 10 6 Smart Calculations B In problem 7, you wrote two arrow strings for the same problem. One had many arrows, and the other had only one arrow. 11. Shorten the following arrow strings so that each string has only one arrow. Use the applet to check your answer. ? a. 375 50 50 ? ? 375 158 1 is the same as ? b. ? 100 is the same as ? 158 ? c. 1000 1274 ? 2 is the same as ? 1274 Suggested nonessential art here to help fill page 13. When you use the applet, it does not matter if the computations in an arrow string are difficult or not. The computer does the hard work for you! When you have to do it without the computer, it can make a difference. Some arrow strings, such as the one in problem 11a, are easier to calculate when they have fewer steps. Others, such as the one in problem 11b, are easier to calculate when they are longer. You can make some arrow strings easier to use by making them shorter or longer. 12. For each arrow string below, make a longer string that is easier. Then use the new arrow string to find the result. You can check this result with the applet by making both strings. 99 a. 527 ⎯⎯→ 98 b. 274 ⎯⎯→ ? ⎯⎯ ? ⎯⎯ 13. Change each calculation into an arrow string with one arrow. Then make a longer arrow string that is simpler to use to do the mental calculations. Use the longer arrow strings to calculate the answers. a. 1,003 999 b. 423 104 c. 1,793 1,010 Section B: Smart Calculations 11 B Smart Calculations 14. a. Make the arrow string below with the applet, and write down the result. ? 273 100 99 b. If the first number, 273, in part a is changed to 500, what is the new result? c. What if 273 is changed to 1,453? d. What if 273 is changed to 76? e. What if 273 is changed to 112? f. Use a short string with a one-arrow calculation to show the result for any first number. Monica: Add non-essential art of arrow applet on a computer display. If done, show an updated LCD computer monitor. 15. Make the arrow string below in the applet, and find out which numbers you could use in the operation boxes. Make two versions: one that is difficult to calculate using mental computation and one that is easy to calculate without the use of a computer or calculator. 36 … … … … 177 Numbers can be written using different combinations of sums and differences. Some of the ways make it easier to perform mental calculations. To calculate 129 521, you can write 521 as 500 21 and use an arrow string. 21 500 129 ⎯⎯→ 150 ⎯⎯→ 650 ⎯⎯ Sarah computed 129 521 as follows: 500 129 ⎯⎯→ 12 Expressions and Formulas ⎯⎯ 20 ⎯⎯→ ⎯⎯ 1 ⎯⎯→ ⎯⎯ Smart Calculations B 16. a. Is Sarah’s method correct? b. What other method using an arrow string could Sarah have used to compute 129 521 mentally? 17. How could you rewrite 267 – 28 to make it easier to calculate using mental computation? Section B: Smart Calculations 13 B Smart Calculations The small-coins-and-bills-first method is an easy way to make change. The applet Algebra Arrows, which is a small computer program, can be used to make arrow strings. The applet does the computations for you. If a computer is not available, sometimes an arrow string can be replaced by a shorter string that is easier to calculate mentally. ⎯⎯ 64 ⎯⎯→ ⎯⎯ 36 ⎯⎯→ ⎯⎯ becomes ⎯⎯ 100 ⎯⎯→ ⎯⎯ Sometimes an arrow string can be replaced by a longer string that makes the calculation easier to compute mentally without changing the result. ⎯⎯ 99 ⎯⎯→ ⎯⎯ becomes ⎯⎯ 1 ⎯⎯→ ⎯⎯ 100 ⎯⎯→ or ⎯⎯ 99 ⎯⎯→ ⎯⎯ becomes 100 1 ⎯⎯→ ⎯⎯→ ⎯⎯ ⎯⎯ 1 a. Complete each arrow string. 1 i. 15 20 ⎯⎯→ ⎯⎯ 0.03 ii. 6.77 ⎯⎯⎯→ iii. This was 12.20 in the ms ? ⎯⎯→ iii. 12.28 ⎯⎯ 8 ⎯⎯→ ⎯⎯ ⎯⎯ – 2 ⎯⎯ ? ⎯⎯ ⎯⎯→ ? ⎯⎯ ⎯⎯→ ⎯ ⎯→ ⎯⎯ ⎯⎯ ⎯⎯ ? ⎯⎯ ⎯⎯→ 20 ? ⎯⎯ ⎯⎯→ 20 b. Use the applet Algebra Arrows to check your answers for part a and to make two more arrow strings, one different arrow string for the calculation as in string ii and one different arrow string for the same calculation as in string iii. 14 Expressions and Formulas 2. For each arrow string, either write a new string that will make the computation easier to calculate mentally, and explain why it is easier, or explain why the string is already as easy to calculate as possible. 237 a. 423 ⎯⎯⎯→ 24 b. 544 ⎯⎯⎯→ 54 c. 29 ⎯⎯⎯→ 34 ⎯⎯ ⎯⎯ ⎯⎯ d. 998 ⎯⎯⎯→ 25 ⎯⎯⎯→ ⎯⎯ ⎯⎯ 3. Give two examples in which a shorter string is easier to calculate mentally. Include both the short and long strings for each example. 4. Give two examples in which a longer string would be easier to calculate mentally. Show both the short and long strings for each example. 5. Explain why knowing how to shorten an arrow string can be useful in making change. You can use the applet Algebra Arrows to solve this problem. Write an arrow string that shows how to make change for a $4.15 purchase if you handed the cashier a $20.00 bill. Show how you would change this string if the cashier had no quarters or dimes to use in making change. Explain why making an arrow string shorter or longer does not make much sense when using the applet. Section B: Smart Calculations 15 C Formulas Supermarket Tomatoes cost $1.50 a pound. Carl buys 2 pounds (lb) of tomatoes. 1. a. Find the total price of Carl’s tomatoes. b. Use the applet Algebra Arrows to make an arrow string that can compute the price of the tomatoes. At Veggies-R-Us, you can weigh fruits and vegetables yourself and find out how much your purchase costs. You select the button on the scale that corresponds to the fruit or vegetable you are weighing. APPLES BANANAS CABBAGE CARROTS CELERY CORN CUCUMBERS GRAPES GREEN BEANS LETTUCE LEMONS ONIONS ORANGES PEAS PEPPERS POTATOES TOMATOES PRICE & STICKER The scale’s built-in calculator computes the purchase price and prints out a small price sticker. The price sticker lists the fruit or vegetable, the price per pound, the weight, and the total price. weight ⎯⎯→ 16 Expressions and Formulas ⎯⎯→ price The scale, like an arrow string, takes the weight as an input and gives the price as an output. Formulas C Once you have built an arrow string with the applet Algebra Arrows, you can use it as a “machine” that gives you the price when you fill in the weight. You do not have to make the computations yourself. price weight 1.5 How to Make a Label 1 Click on the right mouse button (or CRTL + click) to open the pop-up menu and choose Show label. You can make labels for the input and output boxes, to make the meaning of the numbers visible and show what the “machine”can do. 2. Find the price for each of the following weights of tomatoes using the arrow string in the applet. a. 4 lb b. 0.5 lb 36 c. 2.5 lb Show label Hide label Show table Hide table Show Arrow string Hide Arrow string The prices of other fruits and vegetables are calculated in the same way. Green beans cost $0.90 per pound. 3. a. Make a “machine” for computing the price of green beans. 2 Click on the label. weight ? price 1.5 3 Fill in the label. 4 Press ENTER. 36 price b. Use it to calculate the price for the following weights of beans: 3.5 lb, 4.5 lb, and 5.5 lb. Section C: Formulas 17 C Formulas The Corner Store does not have a calculating scale. The price of tomatoes at The Corner Store is $1.20 per pound. Siu bought some tomatoes, and her bill was $6. 4. What was the weight of Siu’s tomatoes? How did you find your answer? Check your answer with the applet. Taxi Fares In some taxis, the fare for the ride is shown on a meter. At the Rainbow Cab Company, the fare increases during the ride depending on the distance traveled. You pay a base amount no matter how far you go, as well as a price for each mile you ride. The Rainbow Cab Company charges these rates. The base price is $2.00. The price per mile is $1.50. 5. What is the fare for each ride? a. From the stadium to the railroad station: 4 miles b. From the suburbs to the downtown area: 7 miles c. From the convention center to the airport: 20 miles The meter has a built-in calculator to find the fare. The meter calculation can be described by an arrow string. 6. Which of these “machines” will give the correct fare? Explain your answer. FPO 18 Expressions and Formulas Formulas C The Rainbow Cab Company changed its rates. The new ones are shown: The base price is $3.00. The price per mile is $1.30. 7. Is a cab ride now more or less expensive than it was before? To compare the new price and the old price, you can build the machine below: FPO 8. a. Build this machine with the applet and complete it. b. Use it to compare the prices before and after the rate change by filling in several distances. c. Did you answer problem 7 correctly? If not, try it again. Siluh is a taxi driver. Now that the new prices are in effect, he wants to post a rate chart in his taxi. The rate chart will show the customers the new fares for typical distances. 9. Use the applet to make such a rate chart for Siluh. After the company changed its rates, George slept through his alarm and had to take a taxi to work. He was surprised when it cost $18.60! 10. a. Use the machine that you made in problem 8 to calculate the distance from George’s home to work. b. It’s also possible to find the answer to part a by making a calculation instead of using the machine. Write an arrow string for this calculation. Section C: Formulas 19 C Formulas Stacking Cups Materials: Each group will need a centimeter ruler and at least four cups of the same size. Plastic cups from sporting events or fast-food restaurants work well. rim hold base Measure and record the following: • the total height of a cup • the height of the rim • the height of the hold (The hold is the distance from the bottom of the cup to the bottom of the rim.) • Stack two cups. Measure the height of the stack. • Without measuring, guess the height of a stack of four cups. • Write down how you made your guess. Tell a classmate your guess and how you arrived at it. • Make a stack of four cups and measure it. Was your guess correct? 11. Calculate the height of a stack of 17 cups. Describe your calculation with an arrow string. The cups will be stored in a space under a counter. The space is 50 centimeters high. 12. a. How many cups can be stacked to fit under the counter? Show your work. b. Use arrow language to explain how you found your answer to part a. 20 Expressions and Formulas Formulas C Sometimes a formula can help you solve a problem. You can write a formula to find the height of a stack of cups if you know the number of cups. You can also write an arrow string for a formula. 13. Complete the following arrow string for a formula using the number of cups as the input and the height of the stack as the output. ? ? ? ⎯⎯→ height of stack number of cups ⎯⎯→ ⎯⎯ Suppose another class has cups of a different size. The students use this formula for finding the height of a stack of their cups: 1 number of cups ⎯⎯→ ⎯⎯ 3 ⎯⎯→ ⎯⎯ 15 ⎯⎯→ height of stack You can use the applet to make this arrow string. 14. a. How tall is a stack of 10 of these cups? b. How tall is a stack of five of these cups? c. Sketch one of the cups. Label your drawing with the correct height. d. Explain what each number in the formula represents. Now consider the following arrow string. 3 12 number of cups ⎯⎯→ ⎯⎯ ⎯⎯→ height 15. Could this arrow string be used for the same cup from problem 14? Show your work. FPO To compare both arrow strings, you can also build this machine and try several numbers. 16. Reflect Compare using this machine for solving problem 15 to the way you solved it. What is the same? What is different? Section C: Formulas 21 C Formulas 3 We can write the arrow string number of cups ⎯⎯→ ⎯⎯ 12 ⎯⎯→ height as a formula, like this: number of cups 3 12 height 17. Will the formula number of cups 12 3 height also work for these cups? Explain your reasoning. These cups will be stored in a space 50 cm high. 18. How many of these cups can be put in a stack? Explain how you found your answer. Bike Sizes You have discovered some formulas written as arrow strings. On the next pages, you will use formulas that other people have developed. saddle height frame height inseam height Bike shops use formulas to find the best saddle and frame heights for each customer. One number used in these formulas is the cyclist’s inseam. This is the length of the cyclist’s leg, measured in centimeters along the inside seam of the pants. The saddle height is calculated with this formula. inseam (in cm) 1.08 saddle height (in cm) 19. a. Do you think you can use just any numbers at all for inseam length? Why or why not? b. Write an arrow string for this formula, and with Algebra Arrows, make a machine that can calculate the saddle height for you. FPO Does this go with b or c? 22 Expressions and Formulas Formulas C c. Use the arrow machine to complete the table: Inseam (in cm) 50 60 70 80 Saddle Height (in cm) ..... 64.8 ..... ..... d. How much does the saddle height change for every 10-cm change in the inseam? How much for every 1-cm change? 140 Saddle Height (in cm) 120 100 80 A 60 40 20 0 20 40 60 80 100 120 140 Inseam (in cm) To get a quick overview of the relationship between inseam length and saddle height, you can make a graph of the data in the table. In this graph, the point labeled A shows an inseam length of 60 cm with the corresponding saddle height of 64.8 cm. For plotting this point, 64.8 is rounded to 65. 20. a. Go to Student Activity Sheet 1. Label the point for the inseam of 80 cm with a B. What is the corresponding saddle height in whole centimeters? b. Choose three more lengths for the inseam. Calculate the saddle heights with your machine, round to whole centimeters, and plot the points in the graph on Student Activity Sheet 1. c. Why is it reasonable to round the values for saddle height to whole centimeters before you plot the points? Section C: Formulas 23 C Formulas If you plotted the points accurately, the points in the graph can be connected by a straight line. 21. a. Go to Student Activity Sheet 1. Connect all points in the graph with a line. b. If you extend your line, would it intersect the point (0, 0) in the bottom left corner? Why or why not? c. A line goes through an infinite number of points. Does every point you can locate on the line you drew provide a reasonable solution to the bike height problem? Explain your reasoning. 22. Write a question you can solve using this graph. Record the answer to your own question. Exchange questions with a classmate. Then answer the question, and discuss the questions and the answers with your classmate. Look at the formula for the frame height of a bicycle. inseam (in cm) 0.66 2 frame height (in cm) 23. a. Write an arrow string for this formula, and make a machine with the applet. b. Use the machine to complete the table (round the frame height to a whole number), and draw the graph for this formula on Student Activity Sheet 2. Since the graph that was shown here is a blank graph, can it be eliminated to make the section fit? (Graph is placed on the following page for reference.) Inseam (in cm) 50 60 70 80 Frame Height Calculated (in cm) ..... ..... ..... ..... ..... ..... Frame Height Rounded (in cm) c. If you connect all points in this graph and extend your line, does the line you drew intersect the origin (0, 0)? Why or why not? d. Use the graph to find the frame height for Ben, whose inseam length is 75 cm. 24 Expressions and Formulas 140 Frame Height (in cm) 120 100 80 60 40 20 0 20 40 60 80 Inseam (in cm) 100 120 140 Formulas C Margit did not use the applet to do problem 23, she used the formula instead. She just used it to find the first two frame heights in the table. She did not round the heights. Then she used the first two values to calculate the third value. 24. a. Explain how Margit might have used the first two values to find the third value. b. Check to see if her method also works to find the next frame heights. c. How could Margit find the frame height for an inseam of 65 cm? Formulas are often written with the result first, for example: saddle height (in cm) inseam (in cm) 1.08 frame height (in cm) inseam (in cm) 0.66 2 81 cm 54 cm 25. Study this bike. a. What is the frame height? b. What is the saddle height? c. Do both of these numbers correspond to the same inseam length? How did you find your answer? FPO Section C: Formulas 25 C Formulas A formula shows a procedure that can be applied over and over again for different numbers in the same situation. With the applet Algebra Arrows, you can make a machine for a formula. This kind of machine can carry out the procedure for different numbers you use for input. Bike shops use formulas to fit bicycles to their riders. inseam (in cm) 0.66 2 frame height (in cm) The formula can also be written with the result first. frame height (in cm) inseam (in cm) 0.66 2 Many formulas can be described with arrow strings; for example: 0.66 2 inseam ⎯⎯→ ⎯⎯ ⎯⎯→ frame height An arrow string made in the applet Algebra Arrows is a machine that will do the calculations. FPO You can use the machine to complete a table for the formula. From the data in the table, you can draw a graph. Some problems are easier to solve with a graph, some are easier to solve using the formula some with an arrow string or a machine made with Algebra Arrows, and some are easier to solve using a table. You have studied many strategies to solve problems. 26 Expressions and Formulas 1. a. Write the following formula about taxi fares as an arrow string: total price = number of miles $1.40 $1.90 b. Why is it useful to write a formula as an arrow string? c. Use the applet Algebra Arrows to make a list of prices for several distances. The manager at The Corner Store wants customers to be able to estimate the total cost of their purchases. She posts a table with prices next to a regular scale. 2. a. Help the manager by copying and completing this table. Weight Tomatoes $1.20/lb Green Beans $0.80/lb Grapes $1.90/lb 0.5 lb 1.0 lb 2.0 lb 3.0 lb b. The manager would like to have one machine to calculate the exact prices of tomatoes, green beans, and grapes for all weights. Build such a machine in the applet Algebra Arrows. FPO Section C: Formulas 27 C Formulas This picture shows a stack of chairs. Notice that the height of one chair is 80 centimeters, and a stack of two chairs is 87 centimeters high. 87 cm 80 cm Damian suggests that the following arrow string can be used to find the height of a stack of these chairs: 1 7 80 number of chairs ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ height 3. a. Explain the meaning of each number in the arrow string. b. Alba thinks a different arrow string could also solve the problem. ___ ___ ⎯⎯→ height number of chairs ⎯⎯→ ⎯⎯ What numbers should Alba use in her arrow string? Explain your answer. c. Use the applet Algebra Arrows to see if your string for part b is the same as the one Damian suggested. Show how you did this. 28 Expressions and Formulas This graph represents the height of stacks of chairs. The number of chairs in the stack is on the horizontal axis, and the height of the stack is on the vertical axis. Height (in cm) 250 200 A 150 100 50 0 5 10 15 20 25 30 Number of Chairs 4. a. What does the point labeled A represent? b. Does each point on the line that is drawn have a meaning? Explain your reasoning. c. Explain why the graph will not intersect the point (0, 0). d. Use the graph to determine the number of chairs that can be put in a stack that will fit in a storage space that is 116 cm high. e. Check your answer for d using an arrow string or a machine in the applet Algebra Arrows. State your preferences for using a graph or an arrow string to display the saddle height for a bicycle. Explain why you think this is the better way to describe the data. Section C: Formulas 29 D Reverse Operations Distances Marty is going to visit Europe. He wants to prepare himself to use the different currencies and units of measure. He knows distances in Europe are expressed only in kilometers and never in miles. He looks on the Internet for a way to convert between miles and kilometers. The computer uses an estimate for the relationship between miles and kilometers. 1. Reflect Think of a problem that Marty might do using the converter while he is traveling in Europe. Enter miles or kilometers and click the “Calculate” button. Miles: Kilometers: Conversion 1 1.609 _____ Marty wants to convert miles to kilometers to get a better understanding of distances in kilometers. He decides to build his own mile–kilometer converter with the applet Algebra Arrows by making a machine. FPO 2. a. Write down an arrow string that Marty can use to convert miles to kilometers, and build the machine with the applet. b. Marty lives 30 miles from his office. About how many kilometers is that? c. Marty’s parents live about 200 miles away. About how many kilometers is that? d. About how many kilometers is a distance of 10,000 miles? 30 Expressions and Formulas Reverse Operations D 3. a. Copy this table, and complete it using the machine you have built in the applet. Miles 10 20 30 40 50 60 70 80 90 100 Kilometers b. Use the table to estimate how many kilometers there are in 35 miles. Explain your reasoning. In Europe, Marty travels 70 km from the airport to his hotel. 4. a. Use the table to estimate the distance in miles. Show your work. b. It is not easy to get a precise answer to part a using the table. Check your answer with the machine. If it’s not very accurate, try to improve it. While traveling in Europe, Marty wants to estimate the distances in miles. A mile–kilometer converter on the Internet can be used for this by entering the kilometers instead of the miles. In the machine you made with the applet, you can only enter miles. 5. Try to make a machine in the applet Algebra Arrows that can be used for calculating miles by entering kilometers, and check to see if it works correctly. Write down how you made and checked your machine. FPO Marty uses the fact that 80 kilometers is about 50 miles and thinks of the following rule to estimate distances in miles: 80 number of kilometers ⎯ ⎯→ ⎯⎯ 50 ⎯ ⎯⎯→ number of miles 6. Will this rule give reasonable estimates? Explain. Section D: Reverse Operations 31 D Reverse Operations Marty thinks the rule is a little difficult to use with large numbers. He wonders if he can change the numbers and use this rule instead. 8 number of kilometers ⎯⎯→ ⎯⎯ 5 ⎯⎯→ number of miles 7. Will this new rule result in reasonable estimates? Explain. 8. a. Combine the machines for problems 5, 6, and 7, and check to see if they all give the same answers. Write down your conclusions, and explain why the answers are the same or different. FPO b. Can you make a fourth machine that gives the same answers? Write down the rule for this machine. Marty uses the machine to convert the distances he will travel from kilometers to miles. 9. Copy the table with Marty’s travel plans. Convert the distances into miles, and record them in your table. Amsterdam–Paris 514 kilometers .... miles Paris–Barcelona 839 kilometers .... miles Barcelona–Rome 879 kilometers .... miles 1185 kilometers .... miles 576 kilometers .... miles Rome–Berlin Berlin–Amsterdam 32 Expressions and Formulas Reverse Operations D Marty’s friend Pascale from Paris will be visiting Marty in the United States next summer. She wants to be able to convert miles into kilometers when she is traveling in the United States. 10. a. How can Marty’s last rule be changed to convert miles into kilometers? Write the new rule as an arrow string, and make it into a machine using the applet Algebra Arrows. b. Use the applet to check whether the machine you built in part a gives the same results as the machine you built in problem 2a. Write down how you checked this, and give your conclusions. Pascale wants to tour the United States. She wants to visit some interesting places, such as national parks, theme parks, and major cities. 11. Reflect Make a list of five interesting places Pascale might like to visit. Find the distances in miles between them (use the Internet or an atlas), and convert them to kilometers using one of the machines you built in the applet Algebra Arrows. Going Backwards Pat and Kris are playing a game. One player writes down an arrow string and the output (answer) but not the input (starting number). The other player has to determine the input. Here are Pat’s arrow string and output. +4 x 10 –2 ÷2 _?_ ____> __ ____> __ ____> __ ____> 29 12. a. What should Kris give as the input? Explain how you found this number. b. One student found an answer for Kris by using a reverse arrow string. What number should go above each one of the reversed arrows? ? ⎯⎯ ←⎯⎯ ⎯⎯ ←⎯⎯ ⎯⎯ ←⎯⎯ ←⎯⎯ 29 ⎯⎯ Section D: Reverse Operations 33 D Reverse Operations When it was her turn, Kris wrote this. ? ⎯⎯ 3 ⎯⎯→ ⎯⎯ 6 ⎯⎯→ ⎯⎯ 5 ⎯⎯→ ⎯⎯ 2 ⎯⎯→ 6 13. a. What will Pat give as the input? Explain how you found this number. b. Write the reverse arrow string that can be used to find the input. To make a reverse arrow string with Algebra Arrows, it is possible to choose the opposite direction for the arrows. FPO 14. a. Use the applet to make the arrow string in the picture above, and complete and write down the reverse arrow string. b. Use the reverse arrow string to find the unknown input. Use the first arrow string to check it. Beech Trees The park near Jessica’s house is full of beech trees. Some botanists have observed that a beech tree grows pretty evenly when it is between 20 and 80 years old. They have developed two formulas that describe the growth of beech trees if the age is known. They are written as arrow strings. 0.4 age ⎯⎯→ ⎯⎯ 2.5 ⎯⎯→ thickness 0.4 age ⎯⎯→ ⎯⎯ 1 ⎯⎯→ height In these arrow strings, age is in years, and height is in meters; thickness (diameter) is in centimeters and is measured at 1 meter (m) above the ground. To answer the next questions, you can use the applet Algebra Arrows to make machines fitting the arrow strings and if necessary, the reverse arrow strings. 34 Expressions and Formulas Reverse Operations D 15. Find the heights and thicknesses of trees that are 20, 30, and 40 years old. 16. Jessica wants to know the age of a tree. How can she find it? Jessica estimates the height of a beech tree as about 20 meters. 17. Use this estimate of the height to find the age of the tree. Jessica uses some straight sticks to help her measure the thickness of another tree. She finds that that the tree is 25.5 centimeters thick. 18. About how old is the tree? Show your work. Jessica realizes that she can make a new formula. Her new formula gives the height of a tree if the thickness is known. 19. Write Jessica’s formula as an arrow string. Make the machine with Algebra Arrows, and then use it to fill in the table below. FPO Thickness (in cm) Height (in cm) 30 34 45 52 20. a. Find a (beech) tree in your area and measure its thickness. b. Use the formula, or the machine, to estimate its height. c. How well do you think the formula predicted the height? Give reasons for your thinking. Section D: Reverse Operations 35 D Reverse Operations Every arrow has a reverse arrow. A reverse arrow represents the opposite operation. 4 4 For example, the reverse of ⎯⎯→ is ←⎯⎯. Reverse arrows can be used to make reverse arrow strings. For example, ⎯⎯ ⎯⎯ 4 ←⎯⎯ 3 ⎯⎯ ⎯⎯ ⎯⎯→ 4 ⎯⎯→ 3 ←⎯⎯ ⎯⎯ 4 ⎯⎯ ⎯⎯ ⎯⎯→ 3 ⎯⎯→ ⎯⎯ reverses to , which is the same as . ⎯⎯ In the applet Algebra Arrows, you can also reverse the arrows. From Amsterdam in the Netherlands to Chicago, Illinois, the distance is 4,090 miles. Marty wants to convert this distance to kilometers. The Internet converter, shown in the beginning of this section, uses 1.609 as the number of kilometers per mile. Marty wonders if he could use the rounded number 1.6 too. 1. a. Use the applet Algebra Arrows to find out how many kilometers Marty will find for the distance if he uses the Internet conversion number. b. How many kilometers will Marty find if he applies the rounded number 1.6? c. What is the difference between the answers for parts a and b? Why is the difference not really important? 36 Expressions and Formulas Carmen and Andy are at the store buying ham and cheese for sandwiches. Carmen sees some Swiss cheese that costs $4.40 per pound. She decides to buy 0.75 lb, but she wants to calculate the cost before she orders it. 2. Write an arrow string to show the cost. Carmen wrote this arrow string. 4 3 $4.40 ⎯⎯→ $1.10 ⎯⎯→ $3.30 3. Is Carmen’s arrow string correct? Why or why not? 4. Write the reverse arrow string for each of these strings. 2 a. input ⎯⎯→ 2 b. input ⎯⎯→ ⎯⎯ ⎯⎯ 3 ⎯⎯→ 5 ⎯⎯→ ⎯⎯ ⎯⎯ 4 ⎯⎯→ output 7 ⎯⎯→ output 5. Try sample numbers to test your reverse arrow strings. You might want to use the applet Algebra Arrows. Show your work. Explain when it may be important to have exact calculations and when a reasonable estimate is acceptable. You can use examples in your explanation. Section D: Reverse Operations 37 E Order of Operations Home Repairs Jim is a contractor specializing in small household repairs that require less than a day to complete. For most jobs, he uses a team of three people. For each one of the three people, Jim charges the customer $25 in travel expenses and $37 per hour. Jim usually uses a calculator to calculate the bills. He uses a standard form for each bill. 1. Use the forms on Student Activity Sheet 3 to show the charge for each plumbing repair job. a. Replacing pipes for Mr. Ashton: 3 hours b. Cleaning out the pipes at Rodriguez and Partners: 212– hours c. Replacing faucets at the Vander house: 34– hour 38 Expressions and Formulas Order of Operations E People often call Jim to ask for a price estimate for a particular job. Because Jim is experienced, he can estimate how long a job will take. He then uses the table to estimate the cost of the job. Labor Cost per Worker Travel Cost per Worker Cost per Worker Total for Three Workers (in dollars) (in dollars) (in dollars) (in dollars) 1 37 25 62 186 2 74 25 99 297 3 111 25 136 408 4 148 25 173 519 Hours 5 6 7 2. a. What do the entries in the first row of the table represent? b. Copy the table, and add the next row for five hours to the table. 3. a. Reflect Explain the regularity in the column for the labor cost per worker. b. Study the table. Make a list of all of the regularities you can find. Explain the regularities. 4. a. Draw an arrow string that Jim could use to make more rows for the table. b. Use your arrow string to make two more rows (for 6 and 7 hours) on the table. Section E: Order of Operations 39 E Order of Operations Arithmetic Trees While working on the home repair cost problems, Enrique writes this arrow string to find the cost of having three workers for two hours of repairs. 37 25 3 2 ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ Karlene is working with Enrique, and she writes this expression. 2 37 25 3 Karlene finds an answer of 149. Enrique is very surprised. 5. a. How does Karlene find 149 as her answer? b. Why is Enrique surprised? Karlene and Enrique decide that the number sentence 2 37 25 3 is not necessarily the same as the arrow string: 37 2 ⎯⎯→ ⎯⎯ 25 ⎯⎯→ ⎯⎯ 3 ⎯⎯→ ⎯⎯ There is more than one way to interpret the number sentence. The calculations can be completed in different orders. 6. Solve the problems below. Compare your answers with your classmates’ answers. a. 1 11 11 b. 10 10 1 10 c. 10 ⎯⎯→ ⎯⎯ 2 ⎯⎯→ ⎯⎯ d. How can you be sure that everyone will get the same answer? Sometimes the context of a problem helps you understand how to calculate it. For example, in the home repair problem, Karlene and Enrique know that the 3 represents the number of workers. So it makes sense to first calculate the subtotal of 2 37 25 and then multiply the result by 3. Sometimes people are not careful how they write the calculations for a problem. 2 37 74 25 99 3 297 7. Why is this a not a good way to write the calculations? 40 Expressions and Formulas Order of Operations E So that everyone gets the same answer to a string of calculations with different operations, mathematicians have agreed that multiplication and division should be completed before addition and subtraction in an expression. 8. Use the mathematicians’ rule to find the value for each expression: a. 32 5 20 b. 18 3 2 5 c. 47 11 6 8 Calculators and computers nearly always follow the mathematicians’ rule. Some old or very simple calculators, however, do not use the rule. 9. a. Use the mathematicians’ rule to find 5 5 6 6 and 6 6 5 5. b. Does your calculator use the mathematicians’ rule? How did you decide? c. Reflect Why do you think calculators have built-in rules? To make sure that everyone agrees on the value of an expression, it is important to have a way to write expressions so that it is clear which calculation to do first, which next, and so on. 6 4 A B C D This is a very simple map, not drawn to scale. Suppose the distance from A to D is 15 miles. In the drawing, you can see that the distance from A to B is 6 miles, and the distance from B to C is 4 miles. 10. What is the distance from C to D? Write down your calculations. Section E: Order of Operations 41 E Order of Operations Telly found the distance from C to D by adding 6 and 4. Then she subtracted the result from 15. She could have used an arithmetic tree to record this calculation. • To make an arithmetic tree, begin by writing down all of the numbers. In/Output 15 6 4 Operations …… …… …… …/… …2 冑苳苳苳 … … • Then pick two numbers. Telly picked 6 and 4. (Sometimes there is no choice in picking the numbers, and sometimes there is. This depends on the problem and the calculations to be performed.) In/Output 15 6 4 Operations …… …… …… …/… …2 冑苳苳苳 … … … • Telly added the numbers and found the sum of 10. In/Output 15 6 4 Operations …… …… …… …/… …… 10 … 2 … … 42 Expressions and Formulas 冑苳苳苳 … Order of Operations E • Telly selected the 15 and the new 10. 15 In/Output 6 4 Operations …… …… …… …/… …… 10 … 冑苳苳苳 … 2 … … Graph Clear • She subtracted to find a difference of 5. 15 In/Output 6 4 Operations …… …… …… …/… …… 10 … 冑苳苳苳 … 2 … …… … 5 Graph Clear 11. Complete Student Activity Sheet 4. 0.8 10 0.6 16 1.2 4 5 7 3 0.6 In/Output …… Operations …… …… …… …/… …2 冑苳苳苳 … …… …… …… …… …… …… … Section E: Order of Operations 43 E Order of Operations You can make arithmetic trees using the applet Algebra Trees. This applet works in almost the same way as the applet Algebra Arrows, which you used before. When you build a tree for a calculation and fill in the input boxes, the computer does the computation for you. In what picture? Do you mean the image below? In the picture, you see a screenshot of the applet Algebra Trees with the tree for the calculation of problem 10. FPO 12. a. Start the applet Algebra Trees, and build the tree shown in the screenshot. b. In the bottom-left corner, you can choose expression or value. Try these and describe what they do. c. Change the tree in such a way that it becomes the tree for the calculation: 15 6 4. By activating expression, you can check to see if the tree is correct. In/Output 25 2 37 3 Operations …… …… …… …/… …2 冑苳苳苳 … …… … … …… Graph Clear Should this be highlighted to point it out? expression value 44 Expressions and Formulas …… …… Order of Operations E 13. a. Make a tree in the applet Algebra Trees that shows the proper calculation of 1 2 3 4. Remember you should multiply first, and then add the products. FPO Until this art is redrawn, ignore the arrow string beneath the tree. This is how the manuscript appears. FYI Because this arrow string only appears beneath the other piece of art, I can’t really read the numbers. This may not be correct. b. Make the tree that shows the same calculation as this arrow string. 2 1 ⎯⎯→ ⎯⎯ 3 ⎯⎯→ ⎯⎯ 4 ⎯⎯→ ⎯⎯ c. With the same four input numbers and the same three operations, you can make other trees. Make at least two other trees with different results from the trees you made in parts a and b. Write down the corresponding expressions and their results. Home Repairs Again Alex feels that the calculation for home repair bills that Jim the contractor makes (see page xx) should begin with the travel costs. Travel costs are part of the base rate; the customer always has to pay them. Alex wants to write the calculation as an arrow string. Flo thinks that it is impossible to begin an arrow string for the home repair bills with the travel costs. 14. Is it impossible, as Flo thinks, to begin an arrow string for the home repair bills with the travel costs? Explain your reasoning. Section E: Order of Operations 45 E Order of Operations Alex decides to make an arithmetic tree with the applet Algebra Trees for calculating the bill for a two-hour home repair job expressing the travel costs first. 15. Does this arithmetic tree give the costs (as shown in the table for problem 2 on page 41)? Explain why or why not. FPO It is possible to build an arithmetic tree using words instead of numbers. The picture shows how this can be done on paper. hours wage per hour labor costs In the applet Algebra Trees, you can do this by using labels (as you did in the applet Algebra Arrows). In the table, you can see how such a tree is built step by step. FPO 46 Expressions and Formulas Order of Operations E By making some of the “branches” longer, the tree looks like the one from problem 15, with all of the input boxes on the top. FPO Once you have built this tree, you have created a machine that can do the computations for all kinds of bills. You only have to change the input numbers. 16. a. Make this tree for the home repair bills in the applet Algebra Trees. b. Use the tree to investigate what would happen to the total costs of a job when the travel costs double. Would the total costs double too, or does something else happen? Explain how you found out. c. Think of something else you can investigate using the tree. Investigate it, and write a few lines about your investigation and the results. Section E: Order of Operations 47 E Order of Operations Flexible Computation (Without Computer) Arithmetic trees are another strategy to make it easier to mentally calculate some addition and subtraction problems. 18 23 7 18 ____ 7 23 ____ ____ ____ 17. a. Compare the two trees. b. Design a tree that makes adding the three numbers easier to do. Addition problems with more numbers have many possible arithmetic trees. Here are two trees for 12 14 43 32 . 18. a. Copy the trees, and find the sum. 1 2 1 4 3 4 3 2 ____ ____ 1 2 1 4 3 4 3 2 ____ ____ ____ ____ b. Design two other arithmetic trees for the same problem, and find the answers. c. Which arithmetic tree makes 12 14 43 32 easiest to calculate? Why? 19. Design an arithmetic tree that makes each problem here easy to calculate. a. 17 3 22 8 b. 4.5 8.9 5.5 1.1 4 1 1 3 2 10 4 c. 10 20. How are different arithmetic trees for the same problem the same? How could they differ? 48 Expressions and Formulas Order of Operations E You may have noticed that if a problem has only addition, the answer is the same no matter how you draw the arithmetic tree. You might wonder if this is true for subtraction. 21. Do the following trees give the same result? What can you conclude about different arithmetic trees for subtraction? 18 7 4 18 7 ____ 4 ____ ____ ____ Return to the Supermarket The automatic calculating scale at Veggies-R-Us is out of order. Ms. Prince buys 0.5 lb of grapes and 2 lb of tomatoes. 22. a. What is the total cost for the grapes and tomatoes? b. Can you write an arrow string to show how to calculate Ms. Prince’s bill? Why or why not? c. Can Ms. Prince’s bill be calculated with an arithmetic tree? If so, make the tree. If not, explain why. Section E: Order of Operations 49 E Order of Operations Dr. Keppler buys 2 lb of tomatoes, 0.5 lb of grapes, and 12 lb of green beans. 23. Make an arithmetic tree for the total bill for the tomatoes, grapes, and green beans. The store manager provides calculators for the cashiers. The calculators use the rule that multiplication is calculated before addition. Then the store manager wrote these directions. amount of tomatoes x 1.50 + amount of grapes x 1.70 + amount of green beans x 0.90 = (in pounds) (in pounds) (in pounds) FPO 24. a. If the cashiers punch in a calculation using these directions, will they find the correct total for the bill? b. Make a tree with the applet Algebra Trees that calculates the total if you enter the weights. 50 Expressions and Formulas Order of Operations E What Comes First? Arithmetic trees are useful because they resolve any question about the order of the calculation. The problem is that they take up a lot of room on your paper. 15 6 4 ____ ____ 25. a. Copy the first tree. b. Since the 6 4 is simplified first, circle it on your copy. 15 6 4 ____ ____ The tree can then be simplified. 15 10 ____ Instead of the second arithmetic tree, you could write: 15 6 4 c. 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. Section E: Order of Operations 51 E Order of Operations If you make a tree with the applet Algebra Trees, you can see the expression with parentheses when you click on expression in the bottom-left corner. FPO 26. a. Rewrite the tree as an expression using parentheses to indicate which numbers are grouped together. b. Check your answer with the applet. 52 Expressions and Formulas Order of Operations E c. Use the applet to make a tree for 6 4 2. d. Use the applet to make a tree for 30 5 (84 79). e. Choose a set of numbers as input in the same tree that you made in part d that will give you a negative answer. Now choose a set of numbers that will give you an answer between 0 and 1. 27. Use parentheses in the expression 2 37 25 3 to find the correct total for Karlene’s problem in the Home Repair section. Section E: Order of Operations 53 E Order of Operations The beginning of this unit introduced arrow language and the applet Algebra Arrows to represent formulas. This section introduced arithmetic trees and the applet Algebra Trees to represent formulas that cannot be made using arrow language. There are several ways to write these types of 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, either on paper or in the applet Algebra Trees. If you make a formula in the applet, you can easily investigate what happens if you change the numbers. The applet does the calculations for you. 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. 5 4 3 2 1 is represented in this tree. 5 4 3 ____ 2 ____ ____ ____ 54 Expressions and Formulas 1 Check for first instance of these terms. You can use parentheses to convert an arithmetic tree into an expression that shows which operations to do first. (5 4) (3 2) 1 20.5 If you make this tree in the applet Algebra Trees and click on expression, the expression with parentheses is shown in the output box. 1. Design an arithmetic tree on paper that makes each problem easier to solve using mental calculation. a. 17 6 3 7 4 3 1 1 12 10 43 10 c. 10 b. 4.5 8.9 5.5 1.1 2. a. Use the mathematicians’ rule to simplify this expression. 24 3 5 8 10 You can use an arithmetic tree and the applet Algebra Trees if you wish. b. Write 24 3 5 8 10 ____ using parentheses to show in which order the operations should be performed. 3. a. Use the applet Algebra Trees to make a tree that can calculate the height of a stack of cups. FPO b. Use this tree or machine to calculate the height of a stack of 16 cups with a rim of 3.5 cm and a hold of 9.5 cm. Section E: Order of Operations 55 E Order of Operations Adult men can use the following rule to estimate their ideal weight. Can "kilograms" be abbreviated? weight (in kilograms) height (in cm) 100 (4 circumference of wrist in cm) 2 For women, the rule is slightly different: change 100 to 110. 4. a. Use the applet Algebra Trees to make an arithmetic tree to represent the general rule for men. b. Matthew is 175 cm tall. The circumference of his wrist is 17 cm. Use the tree to estimate Matthews ideal weight. c. Andrew is 162 cm tall. The circumference of his wrist is 16 cm. His weight is 54 kilograms (kg). Does Andrew weigh too much or too little, according to the general rule? Show your work. d. Investigate using your tree to find how much the difference in “ideal” weight is if the circumference of the wrist changes by 1 cm. Some formulas can be written using an arrow string, and some cannot. What is the difference between these two types of formulas? Find some examples of each type to illustrate this difference. 56 Expressions and Formulas Additional Practice Section A Arrow Language 1. Here is a record for Mr. Kamarov’s bank account. Date Deposit Withdrawal Total 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? 2. 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 1. For each shopping problem, write an arrow string to show the change the cashier owes to 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. Additional Practice 57 Additional Practice 2. Rewrite these arrow strings so that each one has only one arrow: 35 a. 750 ⎯⎯→ 3 b. 63 ⎯⎯→ 1 c. 439 ⎯⎯→ ⎯⎯ ⎯⎯ ⎯⎯ 40 ⎯⎯→ 50 ⎯⎯→ 20 ⎯⎯→ ⎯⎯ ⎯⎯ ⎯⎯ 3. Use the applet Algebra Arrows to make two machines for each arrow string: one machine that has the one arrow that is shown and another machine that will make the computation easier to calculate mentally. 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 Clarinda has a personal computer at home, and she subscribes to Tech Net for Internet access. Tech Net charges $15 per month for access plus $2 per hour of usage. So if Clarinda is connected to the Internet for a total of 3 hours one month, for example, she pays $15 plus 3 times $2, or $21, for the month. 1. Which string shows the cost for 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 2. In the applet Algebra Arrows, make a machine for Clarinda’s Tech Net costs, and use it to calculate the costs for these monthly usage amounts. a. 5 hours b. 20 hours c. 6 12 hours 58 Expressions and Formulas Additional Practice Another Internet access company, Online Time, charges only $10 per month, but $3 per hour. 3. a. In the applet Algebra Arrows, make another machine for the cost of Internet access through Online Time. b. Use the machines to find out for what numbers of hours of Internet use per month Clarinda should use which company—Tech Net or Online Time—if she wants to pay the least in monthly charges? Show your work. Carlos works at a plant nursery that sells flower pots. One type of flower pot has a rim height of 4 cm and a hold height of 16 cm. 4 cm 16 cm 4. a. How tall is a stack of two pots? Three pots? b. Write a formula using arrow language that can be used to find the height of any stack if you know the number of pots. c. Carlos has to stack these pots on a shelf that is 45 cm high. How many pots can he place in a stack that high? Explain your answer. 5. Compare the following pairs of arrow strings, and determine whether they provide the same results. You may use the applet Algebra Arrows if you wish. 8 2 2 8 5 3 2 1 a. input ⎯⎯→ ⎯⎯ ⎯⎯→ output input ⎯⎯→ ⎯⎯ ⎯⎯→ output b. input ⎯⎯→ ⎯⎯ ⎯⎯→ output 3 5 input ⎯⎯→ ⎯⎯ ⎯⎯→ output 6 c. input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ output 2 6 1 input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ output Additional Practice 59 Additional Practice Section D Reverse Operations Ravi lives in Bellingham, Washington. He travels to Vancouver, Canada, frequently. When Ravi is in Canada, he uses this rule to estimate prices in U.S. dollars. 4 3 number of Canadian dollars ⎯⎯→ ⎯⎯ ⎯⎯→ number of U.S. dollars 1. With the applet Algebra Arrows, make a machine for Ravi’s formula, and use it to 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 2. a. Write a formula and make the machine for it that Ravi can use to convert U.S. dollars to Canadian dollars. b. Using the applet, how can you check to see if the machine you made in part a is correct? 3. Write the reverse string for each one of these strings. You may use the applet to check your strings. 1 2.5 4 a. input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ output 6 2 5 b. input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ output 4. Find the input for each string, and show your work. 10 2 3 a. input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ 9 4 5 3 1 b. input ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ ⎯⎯ ⎯⎯→ 10 60 Expressions and Formulas Additional Practice Section E Order of Operations 1. In your notebook, copy and complete the arithmetic trees. a. 12 b. 3 2 ____ ____ 24 4 c. 1.5 3.5 3 7 ____ ____ ____ ____ 8 2 ____ ____ 2. In the applet Algebra Trees, make trees to find the answers to the following: a. 10 1.5 6 b. (10 1.5) 6 c. 15 (2 2 1) 3. a. 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. Use the applet Algebra Trees to make an arithmetic tree to represent the total cost of Suzanne’s bill from the veterinarian. Use labels in your arithmetic tree. b. Use your tree to make a list of the costs Suzanne has to pay for different possible treatments. c. Write an expression using parentheses to calculate the costs for pulling 3 teeth and for a total time in surgery of half an hour. The cost of the medicine is $13. Additional Practice 61 Section A Arrow Language 1. Date Seats Requested Cancellations Total Seats Reserved 3/11 233 3/12 47 0 280 3/13 51 1 330 3/14 53 0 383 3/15 5 12 376 3/16 16 2 390 2. 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 3. The airline needs to begin a waiting list on March 14. 4. 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 do not know on what day the waiting list was started. 1.40 0.62 5.83 1.40 5. 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? 62 Expressions and Formulas Answers to Check Your Work 6. 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 7. 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 1. a. 15 8 1 – 2 i. 20 ⎯⎯⎯→ 35 ⎯ ⎯→ 27 ⎯ ⎯→ 27.5 ii. Your arrow string may be different from the one shown here. Check to see if you get $20 as the output; also see part b. 0.03 0.20 13 6.77 ⎯⎯ ⎯⎯→ 6.80 ⎯⎯ ⎯⎯→ 7 ⎯⎯ ⎯→ 20 iii. Your arrow string may be different from the one shown here. Check to see if you get $20 as the output; also see part b. 0.20 0.70 7 12.10 ⎯⎯ ⎯⎯→ 12.30 ⎯⎯ ⎯⎯→ 13 ⎯⎯→ 20 b. To check if your strings are correct, you can make them into “machines” in the applet Algebra Arrows. They will look like this: FPO Note that some numbers look different; for example, 0.20 has been changed to 0.2, and no dollar signs are used. Answers to Check Your Work 63 Answers to Check Your Work You can make a number of different strings for ii and iii. Here you see an example for each. FPO 2. 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. c. Sample response: 25 54 29 ⎯ ⎯→ 4 ⎯⎯⎯→ 58 This string is easier because when you subtract 25 first, it leaves an easy number to work with. d. Sample response: 2 32 998 ⎯⎯→ 1,000 ⎯⎯⎯→ 1,032 This string is easier because when you add 2 to 998, you get an easy number with a lot of zeros that are easy to work with. It’s easy to add numbers to 1,000. 3. Check your answer with a classmate. Sample answer: (long) 31 19 232 ⎯⎯⎯→ 263 ⎯⎯⎯→ 282 50 (short) 232 ⎯⎯⎯→ 282 Shorter strings are easier when the total of the numbers above the arrows is a multiple of 10 or a number between 1 and 10. 64 Expressions and Formulas Answers to Check Your Work 4. Check your answer with a classmate. Sample answer: 98 (short) 232 ⎯⎯⎯→ 330 (long) 100 2 232 ⎯ ⎯⎯→ 332 ⎯⎯⎯→ 330 Longer strings are easier when the number above the one arrow is not a multiple of 10 or a number between 1 and 10. 5. Sample explanation: The shortened arrow string shows the total amount of change. Section C Formulas 1.40 1.90 1. a. number of miles ⎯⎯⎯⎯→ ______ ⎯⎯⎯⎯→ total price If you use the applet Algebra Arrows, your arrow string (or machine) might look like this. FPO b. Using an arrow string makes calculations easier. c. Your lists may be different. Compare your list with a classmate. You can make the list by filling in a different number of miles each time. In the applet, instead of filling in a number of miles, if you click on table (in the menu on the left), you will see lists (or a table) with miles and prices. FPO Answers to Check Your Work 65 Answers to Check Your Work 2. a. 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 b. FPO 3. a. The – 1 means that one chair is subtracted from the total number of chairs in the stack; the 7 means that for every chair that is added, the height of the stack will grow 7 cm. The 80 represents the height of the first chair in the stack. b. Alba should use 7 and 73 above the arrows. She wrote 7 because every chair adds 7 cm to the height of the stack. Next, 73 is added for the height of the first chair minus the 7 cm that was already added in the first step. c. To check if the arrow strings are the same, you can make this machine in the applet algebra arrows. If you check for some different numbers of chairs, both strings should give the same answers. (You need to check several numbers; one is not enough.) FPO 66 Expressions and Formulas Answers to Check Your Work Note that in the arrow strings in this machine, the intermediate solutions are not shown. You can leave out or add extra input/ output boxes as you wish. You can also use the table option in the menu on the left to check several numbers at once. FPO 4. 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 line has a meaning. For example, you cannot add half a chair, and the total height of the stack cannot be 100 cm. c. The graph will not intersect (0, 0) because a stack of zero chairs make no sense. Also if 0 is used for the number of chairs in Damian’s arrow string, the result is 73 cm, which is impossible of course. So the line should start at one chair. d. About six chairs e. Checking with the arrow string in the applet will show the heights when entering the number of chairs (see the illustration for problem 3c.): for five chairs, the required space is 108 cm, and they will fit; for six, it is 115 cm, and these will probably fit too; and for seven, it is 122 cm, but these will clearly not fit. Answers to Check Your Work 67 Answers to Check Your Work Section D Reverse Operations 1. a. FPO Your answer should not contain decimals because the original measurement is rounded to the nearest mile, so your answer has to be rounded to 6,581 km. b. FPO c. 6581 6544 37 km You may have given several reasons why this difference is not important. Discuss your answer with a classmate. Sample answers: • You do not know how 4,090 miles was measured. Was it as an airplane flies? Making computations using a model of the earth ? Using and converting sea miles? They will all result in different outcomes, and so 4,090 will most likely be a rounded number itself. • 37 km compared to 6,500 (or 6,581 or 6,544) is less than 1%. That is not a important difference. • If you are traveling that far, 37 km will not make as important difference. 0.75 2. $4.40 ⎯⎯⎯⎯→ $3.30 3 3. Carmen’s string is correct because 0.75 is the same as 4. By dividing by 4, she found one-fourth of the price. Next she multiplied by three, which gave her three-fourths of the price. 4 3 2 7 5 2 ⎯⎯→ ______ ⎯ ⎯→ input 4. a. output ⎯⎯→ ______ ⎯ b. output ⎯⎯→ ______ ⎯⎯⎯→ ______ ⎯⎯→ input 68 Expressions and Formulas Answers to Check Your Work 5. You can check your arrow strings in several ways using the applet Algebra Arrows. Two different ways are shown here. Check your answers with a classmate. You should have used several numbers to see if your arrow string works. Note that the intermediate results are not shown in these strings. FPO FPO Section E Order of Operations 1. a. 17 3 6 20 4 7 10 30 37 b. 4.5 5.5 8.9 10 1.1 10 20 Answers to Check Your Work 69 Answers to Check Your Work c. 3 10 1 10 1 10 1 2 3 4 4 10 5 10 1 13 4 2. a. You can simplify this expression to 8 40 10 by calculating the division and multiplication first. The result will then be 48 10 38. Using the applet Algebra Trees, you may have made this tree. FPO 70 Expressions and Formulas Answers to Check Your Work b. (24 3) (5 8) 10 _____ You can find this expression using the applet by clicking on expression in the menu on the left once you have made the tree. In the output box, the expression will be presented this way. FPO 3. a. Your tree may look like this. FPO b. The height of this stack is 65.5 cm. FPO Answers to Check Your Work 71 Answers to Check Your Work 4. a. FPO b. If you enter the data for Matthew, you find out he should weigh 71.5 kg. c. If you enter 192 cm for Andrew’s height and 16 cm for his wrist circumference, his ideal weight will come out as 63 kg. According to the rule, he does not weigh enough. d. You can choose one weight and fill in different numbers for wrist circumference, with a difference of 1 cm between them, to see what happens. If you do this with a height 180 cm and wrist circumferences of 14, 15, 16, 17 cm, this is the result: Wrist Circumference (cm) Ideal Weight (kg) 14 15 16 17 68 70 72 74 So if the wrist circumference is 1 cm larger/smaller, the ideal weight is 2 kg more/less. 72 Expressions and Formulas

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