MathScape: Seeing and Thinking Mathematically, Course 1, Patterns in Numbers and Shape Student Guide, 2 ed by McGraw-Hill Education


985afe0b6c9e302-261x361.jpeg Author McGraw-Hill Education
Isbn 9780078604669
File size 15.1MB
Year 2005
Pages 381
Language English
File format PDF
Category mathematics


 

MathScape©05 C3 TP 1/15/04 2:20 PM Page 2 mac85 Mac 85:1st shift: 1268_tm:6037a:6037a: Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240 ISBN: 0-07-860468-0 1 2 3 4 5 6 111/058 08 07 06 05 04 iii-1_MS G6_860466-4-TOC 1/8/04 10:43 AM Page iii mac85 Mac 85:1st shift: 1268_tm:6037a:6035a: TABLE OF CONTENTS What Does the Data Say? . . . . . . . . . .2 Graphs and Averages ➤ PHASE ONE Measures of Central Tendency Lesson 1 Class Survey . . . . . . . . . . . . . . . . . . . . . . . . . .6 Lesson 2 Name Exchange . . . . . . . . . . . . . . . . . . . . . . .8 Lesson 3 TV Shows . . . . . . . . . . . . . . . . . . . . . . . . . . .10 ➤ PHASE TWO Representing and Analyzing Data Lesson 4 Animal Comparisons . . . . . . . . . . . . . . . . . . .14 Lesson 5 Double Data . . . . . . . . . . . . . . . . . . . . . . . . .16 Lesson 6 Across the Ages . . . . . . . . . . . . . . . . . . . . . .18 ➤ PHASE THREE Progress over Time Lesson 7 Are You Improving? . . . . . . . . . . . . . . . . . . . .22 Lesson 8 How Close Can You Get? . . . . . . . . . . . . . . .24 Lesson 9 Stories and Graphs . . . . . . . . . . . . . . . . . . . .26 ➤ PHASE FOUR Probability and Sampling Lesson 10 What Are the Chances? . . . . . . . . . . . . . . . .30 Lesson 11 Changing the Chances . . . . . . . . . . . . . . . . .32 Lesson 12 Which Bag is Which? . . . . . . . . . . . . . . . . . .34 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36 ➤ ➤ ➤ TABLE OF CONTENTS • MATHSCAPE iii iii-1_MS G6_860466-4-TOC 1/8/04 10:43 AM Page iv mac85 Mac 85:1st shift: 1268_tm:6037a:6035a: The Language of Numbers . . . . . . . .48 Inventing and Comparing Number Systems ➤ PHASE ONE Mystery Device Lesson 1 Inventing a Mystery Device System . . . . . . . .52 Lesson 2 Comparing Mystery Device Systems . . . . . . .54 Lesson 3 Number Words in Many Languages . . . . . . . .56 Lesson 4 Examining Alisha’s System . . . . . . . . . . . . . . .58 ➤ PHASE TWO Chinese Abacus Lesson 5 Exploring the Chinese Abacus . . . . . . . . . . . .62 Lesson 6 How Close Can You Get? . . . . . . . . . . . . . . .64 Lesson 7 Additive Systems . . . . . . . . . . . . . . . . . . . . . .66 Lesson 8 The MD System . . . . . . . . . . . . . . . . . . . . . .68 ➤ PHASE THREE Number Power Lesson 9 Stacks and Flats . . . . . . . . . . . . . . . . . . . . . . .72 Lesson 10 The Power Up Game . . . . . . . . . . . . . . . . . .74 Lesson 11 Efficient Number Systems . . . . . . . . . . . . . . .76 Lesson 12 A New Number System . . . . . . . . . . . . . . . .78 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80 ➤ iv ➤ ➤ ➤ TABLE OF CONTENTS • MATHSCAPE ➤ iii-1_MS G6_860466-4-TOC 1/8/04 10:43 AM Page v mac85 Mac 85:1st shift: 1268_tm:6037a:6035a: From Wholes to Parts . . . . . . . . . . . . .92 Operating with Factors, Multiples, and Fractions ➤ PHASE ONE The Whole of It Lesson 1 Shapes and Factors . . . . . . . . . . . . . . . . . . . .96 Lesson 2 The Great Factor Hunt . . . . . . . . . . . . . . . . .98 Lesson 3 Multiple Approaches . . . . . . . . . . . . . . . . . .100 Lesson 4 First Things First . . . . . . . . . . . . . . . . . . . . .102 Lesson 5 Putting It All Together . . . . . . . . . . . . . . . . .104 ➤ PHASE TWO Between the Whole Numbers Lesson 6 Designer Fractions . . . . . . . . . . . . . . . . . . .108 Lesson 7 Area Models and Equivalent Fractions . . . .110 Lesson 8 Fraction Lineup . . . . . . . . . . . . . . . . . . . . . .112 Lesson 9 Focus on Denominators . . . . . . . . . . . . . . .114 ➤ PHASE THREE Adding Parts and Taking Them Away Lesson 10 Sums and Differences on the Line . . . . . . . .118 Lesson 11 Numbers Only . . . . . . . . . . . . . . . . . . . . . .120 Lesson 12 Not Proper but Still Okay . . . . . . . . . . . . .122 Lesson 13 Sorting Out Subtraction . . . . . . . . . . . . . . .124 Lesson 14 Calc and the Numbers . . . . . . . . . . . . . . . .126 ➤ PHASE FOUR Fractions in Groups Lesson 15 Picturing Fraction Multiplication . . . . . . . . .130 Lesson 16 Fractions of Fractions . . . . . . . . . . . . . . . . .132 Lesson 17 Estimation and Mixed Numbers . . . . . . . . .134 Lesson 18 Fraction Groups within Fractions . . . . . . . .136 Lesson 19 Understanding Fraction Division . . . . . . . . .138 Lesson 20 Multiplication vs. Division . . . . . . . . . . . . . .140 ➤ ➤ ➤ Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142 ➤ ➤ TABLE OF CONTENTS • MATHSCAPE v iii-1_MS G6_860466-4-TOC 1/8/04 10:43 AM Page vi mac85 Mac 85:1st shift: 1268_tm:6037a:6035a: Designing Spaces . . . . . . . . . . . . . . . . .162 Visualizing, Planning, and Building ➤ PHASE ONE Visualizing and Representing Cube Structures Lesson 1 Planning and Building a Modular House . . .166 Lesson 2 Seeing Around the Corners . . . . . . . . . . . .168 Lesson 3 Seeing All Possibilities . . . . . . . . . . . . . . . . .170 Lesson 4 Picture This . . . . . . . . . . . . . . . . . . . . . . . . .172 ➤ PHASE TWO Functions and Properties of Shapes Lesson 5 String Shapes . . . . . . . . . . . . . . . . . . . . . . .176 Lesson 6 Polygon Paths . . . . . . . . . . . . . . . . . . . . . . .178 Lesson 7 Shaping Up . . . . . . . . . . . . . . . . . . . . . . . . .180 Lesson 8 Assembling the Pieces . . . . . . . . . . . . . . . . .182 ➤ PHASE THREE Visualizing and Representing Polyhedrons Lesson 9 Beyond Boxes . . . . . . . . . . . . . . . . . . . . . . .186 Lesson 10 Drawing Tricks . . . . . . . . . . . . . . . . . . . . . .188 Lesson 11 Mystery Structures . . . . . . . . . . . . . . . . . . .190 Lesson 12 Putting the Pieces Together . . . . . . . . . . . . .192 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .194 ➤ vi ➤ ➤ ➤ TABLE OF CONTENTS • MATHSCAPE ➤ iii-1_MS G6_860466-4-TOC 1/8/04 10:43 AM Page vii mac85 Mac 85:1st shift: 1268_tm:6037a:6035a: Beside the Point . . . . . . . . . . . . . . . . . .206 Operating with Decimals, Percents, and Integers ➤ PHASE ONE Decimals Lesson 1 The Fraction-Decimal Connection . . . . . . .210 Lesson 2 What’s the Point? . . . . . . . . . . . . . . . . . . . .212 Lesson 3 Put Them in Order . . . . . . . . . . . . . . . . . . .214 Lesson 4 Get It Close Enough . . . . . . . . . . . . . . . . . .216 ➤ PHASE TWO Computing with Decimals Lesson 5 Place the Point . . . . . . . . . . . . . . . . . . . . . .220 Lesson 6 More to the Point . . . . . . . . . . . . . . . . . . . .222 Lesson 7 Decimal Pinpoint . . . . . . . . . . . . . . . . . . . .224 Lesson 8 Patterns and Predictions . . . . . . . . . . . . . . .226 Lesson 9 It Keeps Going and Going . . . . . . . . . . . . . .228 ➤ PHASE THREE Percents Lesson 10 Moving to Percents . . . . . . . . . . . . . . . . . . .232 Lesson 11 Working with Common Percents . . . . . . . .234 Lesson 12 Percent Power . . . . . . . . . . . . . . . . . . . . . .236 Lesson 13 Less Common Percents . . . . . . . . . . . . . . .238 Lesson 14 Give It to Me Straight . . . . . . . . . . . . . . . . .240 ➤ PHASE FOUR The Integers Lesson 15 The Other End of the Number Line . . . . . .244 Lesson 16 Moving on the Number Line . . . . . . . . . . . .246 Lesson 17 Taking the Challenge . . . . . . . . . . . . . . . . . .248 Lesson 18 The Meaning of the Sign . . . . . . . . . . . . . . .250 Lesson 19 The Cube Model . . . . . . . . . . . . . . . . . . . .252 Lesson 20 Write It Another Way . . . . . . . . . . . . . . . . .254 ➤ ➤ ➤ Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .256 ➤ ➤ TABLE OF CONTENTS • MATHSCAPE vii iii-1_MS G6_860466-4-TOC 1/8/04 10:43 AM Page viii mac85 Mac 85:1st shift: 1268_tm:6037a:6035a: Small Town 2:3 (.67:1) Quarterville 1:4 (.25:1) Micropolus 3:8 (.375:1) Gulliver’s Worlds . . . . . . . . . . . . . . . . .276 Measuring and Scaling ➤ PHASE ONE Brobdingnag Lesson 1 The Sizes of Things in Brobdingnag . . . . . . .280 Lesson 2 A Life-Size Object in Brobdingnag . . . . . . . .282 Lesson 3 How Big is “Little” Glumdalclitch? . . . . . . . .284 Lesson 4 Telling Tales in Brobdingnag . . . . . . . . . . . . .286 ➤ PHASE TWO Lilliput Lesson 5 Sizing Up the Lilliputians . . . . . . . . . . . . . . .290 Lesson 6 Glum-gluffs and Mum-gluffs . . . . . . . . . . . . .292 Lesson 7 Housing and Feeding Gulliver . . . . . . . . . . .294 Lesson 8 Seeing Through Lilliputian Eyes . . . . . . . . . .296 ➤ PHASE THREE Lands of the Large and Lands of the Little Lesson 9 Lands of the Large . . . . . . . . . . . . . . . . . . .300 Lesson 10 Lands of the Little . . . . . . . . . . . . . . . . . . . .302 Lesson 11 Gulliver’s Worlds Cubed . . . . . . . . . . . . . . .304 Lesson 12 Stepping into Gulliver’s Worlds . . . . . . . . . .306 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .308 ➤ viii ➤ ➤ ➤ TABLE OF CONTENTS • MATHSCAPE ➤ iii-1_MS G6_860466-4-TOC 1/8/04 10:43 AM Page 1 mac85 Mac 85:1st shift: 1268_tm:6037a:6035a: Patterns in Numbers and Shapes . .320 Using Algebraic Thinking ➤ PHASE ONE Describing Patterns Using Tables Lesson 1 Calendar Tricks . . . . . . . . . . . . . . . . . . . . . .324 Lesson 2 Painting Faces . . . . . . . . . . . . . . . . . . . . . . .326 Lesson 3 Crossing the River . . . . . . . . . . . . . . . . . . .328 ➤ PHASE TWO Describing Patterns Using Variables and Expressions Lesson 4 Letter Perfect . . . . . . . . . . . . . . . . . . . . . . .332 Lesson 5 Tiling Garden Beds . . . . . . . . . . . . . . . . . . .334 Lesson 6 Chocolates by the Box . . . . . . . . . . . . . . . .336 ➤ PHASE THREE Describing Patterns Using Graphs Lesson 7 Gridpoint Pictures . . . . . . . . . . . . . . . . . . .340 Lesson 8 Points, Plots, and Patterns . . . . . . . . . . . . . .342 Lesson 9 Payday at Planet Adventure . . . . . . . . . . . . .344 ➤ PHASE FOUR Finding and Extending Patterns Lesson 10 Sneaking Up the Line . . . . . . . . . . . . . . . . .348 Lesson 11 Something Fishy . . . . . . . . . . . . . . . . . . . . .350 Lesson 12 The Will . . . . . . . . . . . . . . . . . . . . . . . . . . .352 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .354 ➤ ➤ ➤ ➤ ➤ TABLE OF CONTENTS • MATHSCAPE 1 001-003_MS G6 U1_866792-5 12/19/03 9:30 AM Page 2 mac85 Mac 85:Desktop Folder: PHASEONE Measures of Central Tendency Super Data Company collects data, shows data on graphs, and analyzes data. As an apprentice statistician, you will conduct a survey, create a graph of the data you collect, and analyze it. These are important skills for statisticians to know. How can we use data to answer questions about the world around us? WHAT DOES THE DATA SAY? 001-003_MS G6 U1_866792-5 12/19/03 9:31 AM Page 3 mac85 Mac 85:Desktop Folder: PHASETWO PHASETHREE PHASEFOUR Representing and Analyzing Data You will explore bar graphs. You will create single bar graphs and identify errors in several bar graphs. You will also investigate how the scale on a bar graph can affect how the data is interpreted. At the end of the phase, you will conduct a survey of two different age groups. Then you will make a double bar graph to represent the data and compare the opinions of the two groups. Progress over Time Probability and Sampling Start off with a Memory Game in which you find out if your memory improves over time and with practice. You will create a broken-line graph to show your progress. Then you will analyze the graph to see if you improved, got worse, or stayed the same. The phase ends with a project in which you measure your progress at a skill of your choice. You will investigate the chances of choosing a green cube out of a bag of green and yellow cubes. Then you will explore how changing the number of cubes in the bag can change the probability of picking a green cube. At the end of the phase, you will apply what you’ve learned so far by helping one of Super Data Company’s clients solve the Jelly Bean Bag Mix-Up. 12/19/03 PHASE ONE 004-005_MS G6 U1_866792-5 6:51 AM Page 4 mac85 Mac 85:1st shift: 1268_tm:6037a: ny job ans Compa i c i t new s a i r t t u a a yo rD and e St ntic t, Supe y! In onduct e n r a p en mp Ap ll c can To: : Presid a Co you wi ers. t a you that D , m m t r n o o a t e r a h F cus st Sup ion stici e to ce stati r many question format r first m o Welc pprenti s for ou nds of lot of in . In you t data ey na t ki oa eys d collec as a ze surv ifferen e is als of surv d y an r ly ana ere are ey. The results a surve Th a surv m the duct in fro will con t k e s g a you can es. you nment, assmat l g assi t your c abou 004-005_MS G6 U1_866792-5 12/19/03 6:51 AM Page 5 mac85 Mac 85:1st shift: 1268_tm:6037a: A statistician collects and organizes data. Surveys, questionnaires, polls, and graphs are tools that statisticians use to gather and analyze the information. In Phase One, you will begin your new job as an apprentice statistician by collecting data about your class. You will learn ways to organize the data you collect. Then you will analyze the data and present your findings to the class. Measures of Central Tendency WHAT’S THE MATH? Investigations in this section focus on: COLLECTING DATA ■ Conducting surveys to collect data ■ Collecting numerical data GRAPHING ■ Making and interpreting frequency graphs ANALYZING DATA ■ Finding the mean, median, mode, and range of a data set ■ Using mean, median, mode, and range to analyze data mathscape1.com/self_check_quiz WHAT DOES THE DATA SAY? 5 006-011_MS G6 U1_866792-5 12/19/03 1 ANALYZING DATA TO FIND MODES AND RANGES 7:00 AM Page 6 mac85 Mac 85:1st shift: 1268_tm:6037a: Class Survey How well do you know your class? Taking a survey is one way to get information about a group of people. You and your classmates will answer some survey questions. Then you will graph the class data and analyze it. You may be surprised by what you find out about your class. Find the Mode and Range How can you find the mode and range for a set of data? The data your class tallied from the Class Survey Questions is a list of numbers. The number that shows up most often in a set of data like this is called the mode. The range is the difference between the greatest number and the least number in a set of data. For the data in the class frequency graph you see here, the mode is 10. The range is 9. Look at the frequency graph your class created for Question A. Find the mode and the range for your class data. How Many Glasses of Soda We Drink 6 X X X X X X X X X 1 2 X X X X 3 X X X X X X 4 5 6 7 8 Number of glasses WHAT DOES THE DATA SAY? • LESSON 1 X X X X X X X X X X 9 10 006-011_MS G6 U1_866792-5 12/19/03 7:00 AM Page 7 mac85 Mac 85:1st shift: 1268_tm:6037a: Analyze the Class Data Your teacher will give your group the class’s responses for one of the survey questions you answered at the beginning of the lesson. Follow these steps to find out everything you can about your class. 1 How can you use mode and range to analyze data? Create a frequency graph of the data. a. Include everyone’s answer on your graph. b. Don’t forget to label the graph and give it a title. 2 Analyze the data from your graph. a. Find the mode. b. Find the range. Write About the Class Data Write a summary that clearly states what you learned about your class from the data. Be sure to include answers to the following questions: ■ What does the data tell you about the class? Make a list of statements about the data. For example, “Only one student in the class has 7 pets.” ■ What information did you find out about the class from the mode and range? hot words mode range frequency graph W Homework page 36 WHAT DOES THE DATA SAY? • LESSON 1 7 006-011_MS G6 U1_866792-5 12/19/03 2 EXPLORING MEANS AND MEDIANS 7:00 AM Page 8 mac85 Mac 85:1st shift: 1268_tm:6037a: Name Exchange One of the questions often asked about a set of data is, “What is typical?” You have learned to find the mode of a list of numbers. Two other measures of what’s typical are the mean and the median. Here you will use mean, median, and mode to analyze data on the names in your class. Find the Mean How can you find the mean length of a name in the class? One way to find the mean length of a set of first names is to do the Name Exchange. Follow these steps to find the mean, or average, length of the first names of members in your group. 1 Write each letter of your first name on a different sheet of paper. 2 Members of your group should exchange just enough letters so that either: a. each member has the same number of letters, or b. some group members have just one letter more than other members. You may find that some members of the group do not need to change letters at all. 3 Record the mean, or average, length of the first names in your group. How does the mean for your group compare to the mean for the class? 8 WHAT DOES THE DATA SAY? • LESSON 2 006-011_MS G6 U1_866792-5 12/19/03 7:00 AM Page 9 mac85 Mac 85:1st shift: 1268_tm:6037a: Find the Median When the numbers in a set of data are arranged in order from least to greatest, the number in the middle is the median. If there is an even number of numbers in a set of data, the median is the mean of the two middle numbers. Use the frequency graph your class made for the lengths of names to answer these questions. ■ What is the median length of a first name for your class? ■ What does the median tell you? How can you find the median for a data set? How does the mean compare to the median for the class? Write About the Class Data You have learned about the mean, median, mode, and range. Think about what you have learned to answer the following questions about your class: ■ What do each of the measures (mean, median, mode, and range) tell you about the lengths of the first names in the class? ■ Which of the measures (mean, median, or mode) do you think gives the best sense of what is typical for the class? Why? ■ What are some situations where it would be helpful to know the mean, median, mode, or range for the class? Ms. Bryan’s Class 1 2 3 4 5 6 7 8 9 Number of letters 10 11 12 hot words mean median W Homework page 37 WHAT DOES THE DATA SAY? • LESSON 2 9 006-011_MS G6 U1_866792-5 12/19/03 3 INVESTIGATING RATINGS AND DISTRIBUTIONS 7:00 AM Page 10 mac85 Mac 85:1st shift: 1268_tm:6037a: TV Shows Rating scales are often used to find out about people’s opinions. After your class rates some television shows, you will look at some data on how another group of students rated other television shows. Then you will apply everything you have learned so far to conduct and analyze a survey of your own. Analyze Mystery Graphs What information can you get by analyzing the distribution of data in a graph? The graphs below show how some middle school students rated four TV shows. Use the information in the graphs to answer the following questions: ■ Overall, how do students feel about each show? ■ Do the students agree on their feelings about each show? Explain your answer. ■ Which TV shows that you watch might give the same results if your classmates rated the shows? Mystery Graphs TV Show A X X 1 Terrible 10 X X X X 2 X X X X X X X X X X 3 4 Okay TV Show B X X 5 Great X X X X X X X 1 Terrible X X X X 2 3 4 Okay TV Show C X X X X X X X 5 Great WHAT DOES THE DATA SAY? • LESSON 3 X 1 Terrible 2 X X X X X X X X X 3 4 Okay TV Show D X X X X X X X X 5 Great X X X 1 Terrible X X X X 2 X X X X X X X X 3 4 Okay X X X 5 Great 006-011_MS G6 U1_866792-5 12/19/03 7:00 AM Page 11 mac85 Mac 85:1st shift: 1268_tm:6037a: Collect and Analyze Data, Part 1 Now it’s your turn! You will apply what you have learned about collecting, representing, and analyzing data to find out about a topic of your choice. Follow these steps. 1 2 3 How can you conduct and analyze a survey? Make a data collection plan Choose a topic On what topic would you like to collect data? Choose a population to survey Whom do you want to survey? For example, do you want to ask 6th graders or 1st graders? How will you find at least 10 people from your population to survey? Write survey questions Write four different survey questions that can be answered with numbers. At least one of the questions should use a rating scale. Make sure that the questions are easy to understand. Identify an audience Who might be interested in the information you will collect? Why might they be interested? Collect and represent data Collect data Collect data for just one of the survey questions. Ask at least 10 people from the population you chose. Record your data. Graph data Create an accurate frequency graph of your data. Analyze the data Write a report that answers these questions: What are the mean, median, mode, and range? How would you describe the distribution, or shape, of the data? What did you find out? Make a list of statements that are clearly supported by the data. hot words frequency graph distribution W Homework page 38 WHAT DOES THE DATA SAY? • LESSON 3 11 12/19/03 P H A S E TWO 012-013_MS G6 U1_866792-5 7:04 AM Page 12 mac85 Mac 85:1st shift: 1268_tm:6037a: ny Data ans Compa i r c e i t p Su ata tatis with ce S Super D i k t r n it o , re xhib ur w e App esident o : y w o T ne for : Pr ying on a t easy s enjo g From e n i i r t k a wor ll make ting fac you s i e i p any tw to eres I ho y! omp exhibi are int ul way n C a will p a Com per Dat oo. The d comp a usef nt, you Su al z h is t an nme e loc find ou ar grap t assig . h t b ex als to for ors mals. A your n o anim t i s i i n v zo t an ta. I out abou sent da phs ab e a repr bar gr e mak

Author McGraw-Hill Education Isbn 9780078604669 File size 15.1MB Year 2005 Pages 381 Language English File format PDF Category Mathematics Book Description: FacebookTwitterGoogle+TumblrDiggMySpaceShare NSF-funded program helps you meet your state standards in an engaging and student friendly format.     Download (15.1MB) Rich and Engaging Mathematical Tasks: Grades 5-9 California Algebra 1: Concepts, Skills, and Problem Solving California Algebra 2: Concepts, Skills, and Problem Solving Elementary Number Theory Category Theory, 2 edition Load more posts

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