Picturing numbers grade 6

Picturing Numbers Data Analysis and Probability 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 Boswinkel, N., Niehaus, J., Gravemeijer, K., Wijers, M., Dekker, T., Middleton, J A., Spence, M S., Burrill, G., & Milinkovic, J (2006) Picturing numbers In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in Context Chicago: Encyclopỉdia Britannica, Inc 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 0-03-042402-X 073 09 08 07 06 05 The Mathematics in Context Development Team Development 1991–1997 The initial version of Picturing Numbers was developed by Nina Boswinkel, Jansie Niehaus, and Koeno Gravemeijer It was adapted for use in American schools by James A, Middleton, Mary S Spence, Gail Burrill, and Jasmina Milinkovic 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 Jansie Niehaus Nina Boswinkel Nanda Querelle Frans van Galen Anton Roodhardt Koeno Gravemeijer Leen Streefland Marja van den Heuvel-Panhuizen Jan Auke de Jong Adri Treffers Vincent Jonker Monica Wijers Ronald Keijzer Astrid de Wild Martin Kindt 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 Picturing Numbers was developed by Monica Wijers and Truus Dekker 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, right) © Getty Images; (middle) © Corbis Illustrations v Christine McCabe/© Encyclopỉdia Britannica, Inc.; Holly Cooper-Olds; Christine McCabe/© Encyclopỉdia Britannica, Inc.; 26, 28, 37, 41 Holly Cooper-Olds Photographs © Corbis; © Louis K Meisel Gallery, Inc./Corbis; Sam Dudgeon/HRW; © Bettmann/Corbis; 10 Victoria Smith/HRW; 11 © Corbis; 12 Victoria Smith/HRW; 13 (left to right) © Corbis; Sam Dudgeon/HRW; 16 © Corbis; 17 (left to right) © Comstock, Inc.; Allan Munsie/Alamy; Jenny Thomas Photography/HRW Photo; 19 © Corel; 21 © Brand X Pictures; (left to right) © Corbis; © Joe McDonald/Corbis; 25 Fotosonline/Alamy; 29 © Corbis; 32 © Annie Griffiths Belt/Corbis; 35, 36 © Corbis; 38 Dennis MacDonald/ Alamy; 39 © Corbis; 40 © PhotoDisc/Getty Images Contents Letter to the Student Using Bar Graphs Fliers Marbles How Long? Litter Summary Check Your Work 10 11 12 13 16 16 18 19 17% No Vacation/ Don’t Know r Summer/Winte Resort ple Random sam Visit Family/Friends 30% Stay at Home 21% Travel/ Sightsee 26% of 686 adults North 15% 12% 9% 6% 3% West East A Picture is Worth a Thousand Words Your Profession Cat and Mouse Mean and Mode Throwing Darts Going on Vacation Summary Check Your Work Section D 6% A Piece of the Pie Fuel Gauges How People Spend Their Vacation Data Collection Ways of Traveling to School Television Programs Graphs Tell a Story Summary Check Your Work Section C Spend Their How People Vacation 21 22 25 26 28 30 30 Bars or Lines—Get the Picture? Temperature Three Sports Clubs Combining Graphs Summary Check Your Work 32 38 39 42 42 Additional Practice 44 Answers to Check Your Work 47 South Albatross Island Weather °C 30 mm/m2 500 20 400 10 300 200 –10 100 –20 Rainfall Section B Temperature Section A vi J F M A M J J A S O N D Month Contents v Dear Student, Can you picture numbers? Sometimes a picture of the numbers in a problem makes things easier to understand In this unit, you will discover different ways to picture numbers Pictures can help you tell a story about numbers and c about what they represent such as fliers delivered, fuel in a tank gauges, and TV programs preferred Empty You will look at bar graphs and number line plots You will make a picture called a line graph to show how temperatures change from one time interval to the next Full Number of Babies in Litter You will create charts that look like pies to describe the results of your own surveys Litter A B C You will also begin to describe a distribution of data by some summary numbers—such as the maximum, the minimum, the mean, and the mode You will learn how to think about the word average as the mean of a set of numbers X X X X D E F G H X X X I J K L M X X X X N In the end, you should understand something about how pictures can describe numbers and what types of pictures can help you understand different situations O X X P Q R Sincerely, The Mathematics in Context Development Team vi Picturing Numbers A Using Bar Graphs Fliers The number of whales in the world is decreasing rapidly Peter, Carmenza, and Ann want to something about the situation They earn money by delivering advertising fliers to people’s houses For every flier they deliver, they get five cents, which they donate to the World Wildlife Fund Today, Peter delivered 100 fliers, Carmenza 200, and Ann 50 How much money did the three of them earn? Helen, Amber, John, and Diego also delivered fliers In the table, you can see how many each of them delivered How many fliers were delivered in total? Student Peter Carmenza Ann John Amber Helen Diego Number of Fliers Delivered 100 200 50 50 100 250 250 Section A: Using Bar Graphs A Using Bar Graphs Peter showed the information about the number of fliers in the graph below He called his display a one-bar graph Helen Diego Carmenza Peter Amber Ann John a Name one advantage of Peter’s one-bar graph compared to the table b Name one disadvantage of Peter’s one-bar graph compared to the table Number of Fliers Delivered Number of Fliers Delivered 300 You can also present the data from the table in a bar graph, which uses a bar for each category rather than one bar as Peter did In a bar graph, all bars have the same width 250 200 150 100 50 er Pet a enz rm Ca n An n Joh ber Am len He igo De Students a Which way of presenting the number of fliers delivered you prefer, the table or the bar graph? Why? b Reflect Think of another way to arrange the bars in the graph What is one advantage of your new graph over the one above? A bar graph has two axes, a vertical axis and a horizontal axis In this graph, the vertical axis shows you how many fliers were delivered The horizontal axis shows you who delivered them A graph needs labels and a title in order to tell its story You could also make a bar graph showing how much money each student earned a What would this graph look like? b How would it compare to the graph showing the number of fliers delivered? Picturing Numbers Using Bar Graphs A Marbles Bar graphs can be used to compare sizes or amounts of different things On this graph, the horizontal axis identifies the things that are being compared (students) The vertical axis tells how many (marbles) Students’ Marbles 40 30 20 Monica Helen Jane Alice Peter Mark Rosalie 10 Steve Number of Marbles 50 Students Here you can see that Steve has five marbles Suppose you also know that: • • • • • • • Rosalie has twice as many marbles as Steve Mark has four times as many marbles as Rosalie Peter has four marbles more than Steve Alice has as many marbles as Steve Jane has five fewer marbles than Rosalie Helen lost all of the marbles she had Monica has as many marbles as Rosalie Section A: Using Bar Graphs A Using Bar Graphs Using the information on page 3, complete the bar graph on Student Activity Sheet a Who has the most marbles? How many is that? b Who has the fewest? How many is that? c Explain how you got your answers Jose found another way to show how many marbles each person has Look at his drawing called a picture graph Steve Rosalie Make a graph to show how many marbles each student has Make your graph different from José’s graph and the one on Student Activity Sheet Be sure to label everything on your graph How Long? Elena is sure she can stand on one foot a really long time She challenges Tomas, and he agrees to see who can stand longer But he says they have to stand on one foot with their eyes shut! Reflect Do you think it makes a difference whether their eyes are open or closed? Why or why not? Picturing Numbers D Bars or Lines—Get the Picture? This table shows the typical average temperatures for each month in 15 vacation spots Typical Average Temperatures (°F) Vacation Spot J F M A M J J A S O N D Acapulco 88 88 88 88 89 90 91 91 90 90 90 89 Antigua Aruba 80 83 80 84 80 84 82 86 90 88 90 88 90 88 90 91 89 91 89 90 89 89 83 86 Cancún 84 85 88 91 94 92 92 91 90 88 86 82 Cozumel 84 85 88 91 94 92 92 91 90 88 86 82 Grand Cayman 88 87 86 88 88 89 90 91 91 89 88 88 Ixtapa 89 90 92 93 89 88 89 90 91 91 90 89 Jamaica 86 87 87 88 90 90 90 90 89 89 89 87 Los Cabos 73 74 79 83 88 93 95 93 92 89 82 74 Manzanillo 77 78 82 86 83 88 93 95 93 92 89 74 Mazatlán 73 74 79 83 84 92 94 92 92 90 85 71 Nassau 76 76 78 80 84 88 89 90 88 84 81 79 Puerto Vallarta 76 77 81 85 83 88 93 95 93 92 89 75 St Martin/St Kitts 80 81 82 83 86 86 86 87 87 86 85 84 U.S Virgin Islands 80 81 82 83 88 88 90 90 88 87 86 86 13 Based on the temperatures in the table above, where would you prefer to go on a vacation in January? Why? 14 a Use the table to make separate number-line plots for the temperatures of Los Cabos and Ixtapa b Compare the two plots Explain the differences in the temperatures at these two resorts 15 Was it important to use the same scale to create the number-line plots in problem 14a? Why or why not? 16 a In the title of the table, you see the words typical average temperatures How you think the average temperature for a month is calculated? b How the mean yearly temperatures in Los Cabos and Ixtapa compare? 36 Picturing Numbers Bars or Lines—Get the Picture? D You could plot the typical mean temperatures for each month in a graph Since separate dots could be confusing if more than one resort is presented in the graph, the dots are connected by straight lines and different kinds of dots are used The graph shown here is a line graph of the temperatures in three of the resort areas from the table on page 36 Typical Average Temperatures December November October September July June May August = ? April = ? 70 March = ? 80 February Key 90 January Typical Average Temperatures (°F) 100 Month 17 The key on this graph is incomplete Figure out which resort areas these lines represent Write them in your notebook 18 Which of these three resort areas has the best climate? Why you think so? Describe the temperature changes over one year for that area 19 Look in an almanac or check with your local weather bureau to find the typical temperatures for your city Describe the temperature changes over one year for your city Section D: Bars or Lines—Get the Picture? 37 D Bars or Lines—Get the Picture? Three Sports Clubs Bar graphs and pictographs can give a quick overview of the data, but they are not very helpful if you want to answer questions such as “Which grew the most over the years?” Newville has a number of sports clubs The table shows the numbers of members in three sports clubs for different years Number of Club Members Per Year Sports Club Year Softball Soccer Swimming 1970 45 130 60 1980 100 200 90 1990 135 240 120 2000 150 250 130 On Student Activity Sheet 9, you see the beginning of a line graph 20 a What you think the three dots on the graph represent? b Plot the other points for the club represented here Connect the dots with line segments 21 Plot the data for the other clubs on Student Activity Sheet and draw the connecting line segments You may want to use different colors for each graph or use different types of dots 22 Does the graph help you decide which club grew the fastest? 23 If you look at how many new members joined a club in this 30-year period, which club grew the most? 24 If you look at how many times larger the total number of members in 2000 is than the total in 1970, which club grew the most? 25 a Based on your answers to problems 22, 23, and 24, which club you think grew the most? b Reflect Why was a plot-over-time chosen to represent the information? 38 Picturing Numbers Bars or Lines—Get the Picture? D Combining Graphs Sometimes different types of graphs are combined to represent information This may also help you to see a relationship between different data sets This graph shows the mean temperature and amount of rainfall for each month of the year on Murre Island, situated in the Pacific Northwest The curve represents temperature in degrees Celsius The bars indicate rainfall in millimeters per square meter (mm/m2) mm/m2 500 20 400 10 300 200 –10 100 –20 Rainfall Temperature Murre Island Weather ؇C 30 J F M A M J J A S O N D Month Section D: Bars or Lines—Get the Picture? 39 D Bars or Lines—Get the Picture? 26 a Do you see a relationship between the temperature and rainfall on Murre Island? Explain mm/m2 500 20 400 10 300 200 –10 100 –20 Rainfall Temperature Murre Island Weather ؇C 30 b Do graphs made with double vertical axes have the potential to be confusing? Explain your thinking J F M A M J J A S O N D Month 27 Write a short description of the weather on Murre Island Give reasons to support your description 28 What time of the year would you advise tourists to visit Murre Island if you knew they wanted to hike around the island? Another island, Albatross Island, is rocky and wind-swept It is covered in tussock grass that can grow up to nine feet high The island is home to a large breeding colony of Black-browed albatross, as well as a number of Rockhopper penguins 40 Picturing Numbers Bars or Lines—Get the Picture? D The graph shows the mean temperature and amount of rainfall for each month of the year on Albatross Island The curve represents temperature in degrees Celsius The bars indicate rainfall in millimeters per square meter mm/m2 500 20 400 10 300 200 –10 100 –20 Rainfall Temperature Albatross Island Weather °C 30 J F M A M J J A S O N D Month 29 Compare the weather on Albatross Island with the weather on Murre Island 30 Based on weather conditions, which island you think would attract more tourists as a nature refuge, Murre Island or Albatross Island? Explain Section D: Bars or Lines—Get the Picture? 41 D Bars or Lines—Get the Picture? In this section, you learned how to make a line graph to show change over time A line graph or plot-over-time graph shows how numbers change in different circumstances When a line in a graph goes down as the time advances, something else (like the temperature or the number of people) is getting smaller; when a line in a graph goes up as the time advances, something else is getting larger Graphs can help you see patterns and trends and are useful in making decisions The smallest number in a set of data is called the minimum The largest number is called the maximum Sometimes different types of graphs are combined, which may help you see a relationship between different data about the same situation What are some differences between a line graph or plot-over-time and a bar graph? 42 Picturing Numbers North 15% 12% 9% In this section, you have seen different types of graphs about the weather Here is a new one 6% 3% West East Wind Speed (in m/s) >11.06 8.49 – 11.06 5.40 – 8.49 3.34 – 5.40 1.80 – 3.34 0.51 – 1.80 Source: http://www.wcc.nrcs.usda.gov South The wind rose shows how often the wind comes from a certain direction The longer a bar, the more often the wind comes from the corresponding direction a During the 31-day period shown in the graph, what percent of the time did the wind come from the north? The color of the bars gives an indication of the wind speed Wind speed is expressed in the legend in meters per second (m/s) b What direction did the strongest winds come from during this period—North (N), East (E), South (S), or West (W)? What was the wind speed of the strongest winds? c Find some other information based on the graph Think of an advantage of using a line graph instead of another representation Give an example Section D: Bars or Lines—Get the Picture? 43 Additional Practice Section A Using Bar Graphs Students at Roosevelt Middle School are raising money by selling raffle tickets for $5 each Students Number of Raffle Tickets Sold Juan 12 Suzanne 27 Briana 18 David 35 a Describe what the table shows b Make a bar graph showing how much money each student raised Make sure that your bar graph is clearly labeled Use the following information to create a bar graph • • • • • Section B Daren has started a coin collection He has two foreign coins Hector has six coins more than Daren Elory has half as many coins as Hector Gina has three fewer coins than Elory Loritz has as many coins as Elory and Gina put together A Piece of the Pie Favorite Amusement Park Rides Tilt-a-Whirl 26% Tea Cup 18% Flying Swings 10% Roller Coaster 36% Octopus 10% The graph represents the results of a survey of 150 students a What does this graph tell you? b How many students prefer to ride on the Octopus? c About what fraction of the students prefer the Tilt-a-Whirl? Create a stacked bar graph that tells the same story as the pie chart 44 Picturing Numbers Section C A Picture is Worth a Thousand Words Tia James Gerado Phil Carol Key ‫ ؍‬5 basketball cards Write an appropriate title for the pictograph a What is the largest number of basketball cards any student has? b How many more cards does Gerado have than Tia? a If Jody is 23, Mary is 31, and LaShanda is 27, what is their mean age? b If Jody is 2, Mary is 31, and LaShanda is 57, why is it not a good idea to compute the mean age? Additional Practice 45 Additional Practice Section D Bars or Lines—Get the Picture? The table shows the typical average temperatures each day for one week in six cities City S M T W T F S 101 96 101 102 101 103 101 Springfield 86 80 90 89 91 93 92 Orlando 99 88 95 96 99 97 98 Chicago 100 99 102 103 102 104 103 Aspen 80 80 79 82 83 81 82 Boston 94 91 92 94 95 92 93 Dallas a Which city had the highest average temperature on Thursday? The lowest? b What is the difference between the maximum average temperature and the minimum average temperature on Thursday? a What was the mean temperature in Dallas for the week? b Which city had the highest average temperature for the week? a Create a graph that shows two line graphs: one for the average daily temperatures of Boston, the other for the average daily temperatures of Springfield b Compare the two graphs Explain the differences in the temperatures 46 Picturing Numbers Using Bar Graphs a Have a classmate look at your bar graph and the description to see if they make sense b The data you found can probably be presented in a table as well Whether or not the data can be presented in a different type of graph depends on the example you found You might want to discuss your answers with a classmate a You can tell that there are 19 pieces of data, but without labels on the axes and without a title on the graph, it is almost impossible to tell something about the data Neville and Sonia gathered b Your answer will be different from the ones shown here Sample responses: The number line graph represents the number of baseball hats owned by each student in the sixth grade Baseball Hats Owned by Sixth Graders Number of Students 70 x x x x x x x x x 60 50 x x x x 40 30 20 10 x x x x x 10 x 11 12 13 Number of Baseball Hats The number line plot represents the shoes sizes of 19 students Number of Students Section A x x x x x x x x x 5.5 6 x x x x 4 4.5 x x x x x 6.5 7.5 8.5 x 9.5 10 Shoe Sizes Answers to Check Your Work 47 Answers to Check Your Work Favorite Sports Among 6th and 7th Graders You might have different graphs to show the comparison One example is below Share your answer with a classmate Number of Students 6th Graders 7th Graders Football Basketball Section B Tennis Swimming None of These A Piece of the Pie Yes You might say that each bar in the bar graph matches the part with the same label in the pie graph For example, 20 dogs is one-fourth of the total of 80 animals, and in the pie chart, the dogs’ part is one-fourth of the pie Sample questions: a How many more people chose dogs than fish as their favorite pet? b Do more than half the people surveyed prefer cats? a Here are two sample graphs that tell the story of the data Your graph may be different Favorite Computer Game Favorite Computer Game 200 19% 180 20% Hit the Safe Tic Tac Go 5% 11% Tilings Island Falling Problems 15% 30% Number of Students Book of Magic 160 140 120 100 80 60 40 20 Book of Magic Hit the Safe Tic Tac Go Tilings Computer Game 48 Picturing Numbers Island Falling Problems Answers to Check Your Work You might say you chose the pie chart because you wanted to show how the parts were related to the whole thing You might say you chose the bar graph because you wanted to show the numbers as close as possible Or you could say: b The pie chart shows the that Tic Tac Go is the most popular because it is the largest slice of the pie The bar graph shows that Tic Tac Go is the most popular because it is the tallest bar Section C A Picture is Worth a Thousand Words a A number-line plot shows exactly how many pushpins were in each box and how the numbers varied from box to box b 100.4 is the mean You can calculate the mean by adding up the number of pushpins in each box and dividing the total by 10 104 ؉ 100 ؉ 98 ؉ 99 ؉ 103 ؉ 97 ؉ 100 ؉ 102 ؉ 98 ؉ 103 ‫ ؍‬1,004; 1004 ، 10 ‫ ؍‬100.4 pins per box You can also use the compensation method: 97 ؉ 103 ➞ 100 ؉ 100 98 ؉ 102 ➞ 100 ؉ 100 98 ؉ 99 ؉ 103 ➞ 100 ؉ 100 ؉ 100, which makes a sum of 700 pins 100 ؉ 100 ؉ 700 ‫ ؍‬900; 900 ؉ 104 ‫ ؍‬1,004; since 1,004 is pins away from an even 1,000 pins (which divided by 10 boxes is 100 pins per box), that leaves pins to be distributed among the 10 boxes c Even if numbers of pins are whole numbers, the mean can be a decimal number Sample explanations: A mean of 100.4 means that on average, ten boxes have 100 pins each, and four of them have one pin extra A mean of 100.4 means that there are 1004 pins divided over 10 boxes Answers to Check Your Work 49 Answers to Check Your Work Since the mean or average number of pushpins for the boxes Kay looked at is a little more than 100, but not 101 (100.4), she could have said that there were about 100 pushpins per box a You can figure out different ways to divide up the money For example, you might start by having everyone with $0.75 (the smallest amount to start) and then have them pool the rest to make $2.55 + $0.65 + $0.90 + $1.40, or $5.50, which would give everyone another $1 and another $0.10, so each would have $1.00 + $0.10 + $0.75, or $1.85 b The amount they would each spend if they spent the same amount would be the mean, $1.85 Section D Bars or Lines—Get the Picture? You may have noted that a bar graph is used to present categories of data—like the number of brochures each student has delivered— in a visual way A line graph gives a picture of how something like temperature changes over time Even though there is no data for the intervals between the times the temperature was measured, the change is continuous Therefore, the temperature in between measurements lies on the connecting line A bar graph describes data for separate categories, and it does not make sense to connect them because there is no data in between the bars representing people and how many brochures they delivered a A little less than 4% of the time b From the four directions mentioned, the strongest winds came from the West The speed was greater than 11.06 m/s However, the strongest winds of all came from a direction West by Southwest c Sample information you may have found from the graph: Winds more often come from West to East and less often from North or South The most common speeds were from 3.34 m/s to 8.49 m/s 50 Picturing Numbers ... 11 12 13 16 16 18 19 17% No Vacation/ Don’t Know r Summer/Winte Resort ple Random sam Visit Family/Friends 30% Stay at Home 21% Travel/ Sightsee 26% of 68 6 adults North 15% 12% 9% 6% 3% West... Street, Chicago, Illinois 60 610 ISBN 0-03-042402-X 073 09 08 07 06 05 The Mathematics in Context Development Team Development 1991–1997 The initial version of Picturing Numbers was developed by... Milinkovic, J (20 06) Picturing numbers In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in Context Chicago: Encyclopỉdia Britannica, Inc Copyright © 20 06 Encyclopædia