Measuring: from Paces to Feet. Used Numbers: Real Data in the Classroom

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AUTHOR Corwin, Rebecca B.; Russell, Susan Jo TITLE Measuring: From Paces to Feet. Used Numbers: Real Data in the Classroom. Grades 3-4. INSTITUTION Lesley Coll., Cambridge, Mass. SPONS AGENCY National Science Foundation, Washington, D.C. REPORT NO ISBN-0-86651-503-8 PUB DATE 90 CONTRACT MDR-8651649 NOTE 151p.; For related documents, see SE 051 929-931. In cooperation with the Technical Education Research Centers (TERC) and the Consortium for Mathematics and Its Applications (COMA?). AVAILABLE FROMAddison-Wesley Publishing Co., The Alternative -Publishing Group, P.O. Box 10888, Palo Alto, CA 94303-0879 ($12.95). PUB TYPE Guides - Classroom Use - Guides (For Teachers) (052)

EDRS PRICE MF01 Plus Postage. PC Not Available from EDRS. DESCRIPTORS Calculators; Computer Assisted Instruction; Cooperative Learning; *Data Analysis; Elementary Education; *Elementary School Mathematics; *Estimation (Mathematics); Graphs; Interdisciplinary Approach; *Learning Activities; Mathematics Education; *Measurement; Pattern Recognition; Problem Solving; Statistics; Student Projects; Teaching Guides; Teaching Methods

ABSTRACT A unit of study that introduces measuring as a way of collecting data is presented. Suitable for students in grades 3 and 4, it provides a foundation for further work in statistics and data analysis. The investigations may extend from one to four class sessions and are grouped into three parts: "Introduction to Measurement"; "Using Standard Measures"; and "A Project in Data Analysis." An overview of the investigation, session activities, dialogue boxes, and teacher notes are included in each investigation. The major goals developed in each part of this guide are:(1) moving through space and counting the movements; (2) comparing units of measure; (3) estimating distances;(4) defining a measurement method; (5) writing directions involving distances;(6) recording and displaying the results of measurement; (7) experiencing a need to standardize;(8) understanding that standard measures were invented to solve real data collection problems;(9) estimating lengths; (10) measuring accurately, using feet and inches; (11) describing the shape of the data; (12) analyzing data through landmarks and features of the data;(13) using standard measures to compare data sets; and (14) experiencing all the phases of a data analysis investigation in which measuring is used to collect data. Seven student sheets are attached. (KR)

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- Used Numbers: Real Data in the Classroom is a project of Technical Education Research Centers (TERC), Lesley College, and the Consortium for Mathematics and Its Applications (COMAP). The Used Numbers materials focus on the processes of data analysiscollecting, organizing, representing, and interpreting data. Through collecting and analyzing real data, students develop and use ideas and tools from key areas of mathematics counting, measuring, sorting and classification, estimation, graphing, computation in context, and statisticsand they are introduced to appropriate uses of computers and calculators for data analysis. The primary goal of the Used Numbers project is to help teachers and students become intelligent and critical users of data: to understand the limitations and the power of using numbers to reveal patterns and trends, to compare and predict, and to make informed decisions about complex issues.

4 Measuring: From Paces to Feet

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A unit of study for grades 3-4 from USED NUMBERS: REAL DATA IN THE CLASSROOM

Developed at Technical Education Research Centers and Lesley College

Rebecca B. Corwin and Susan Jo Russell

DALE SEYMOUR PUBLICATIONS The Used Numbers materials were prepared with the support of National Science Foundation Grant No. MDR-8651649. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily represent the views of the National Science Foundation. These materials shall be subject to a royalty-free, irrevocable, worldwide, nonexclusive license in the United States Government to reproduce, perform, translate, and otherwise use and to authorize others to use such materials for Government purposes. Cover design and illustrations: Rachel Gage Copyright © 1990 by Dale Seymour Publications. All rights reserved. Printed in the United States of America. Published simultaneously in Canada. Limited reproduction permission: The publisher grants permission to individual teachers who have purchased this book to reproduce the blackline masters as needed for use with their own students. Reproduction for an entire school or school district or for commercial use is prohibited. Order number DS01025 ISBN 0-86651-503-8

DAUB SIEVMOUR PUBLICATIONS P.O. BOX 10E3E3E3 PALO ALTO. CA 94303 abcdefghi-MA-9543210

6 USED NUMBERS STAFF

Co-principal investigators Cooperating classrooms for this unit Susan Jo Russell Eric Johnson Technical Education Research Centers (TERC) Boston Public Schools Susan N. Friel Connie Brady Lesley College New York City Public Schools Laurie Friedlander CUITICUILM1development New York City Public Schools Rebecca B. Corwin CTERC and Lesley College) Dolly Davis Tim Barclay (TERC) Clarke County Public Schools, Georgia Antonia Stone (Playing to Win) Jane Holman Research and evaluation Clarke County Public Schools, Georgia Janice R. Mokros CTERC) Alana Parkes (TERC) Debra Gustafson (TERC) Advisory board John Olive (University of Georgia) Joan Akers, California State Department of Education Dehorah Ruff (University of Georgia) Bonnie Brownstein, Institute for Schools of the Future Heide Wiegel (University of Georgia) James Landwehr, AT&T Bell Laboratories Bonnie Brownstein (Institute for Schools of the Future) Steven Leinwand, Connecticut State Department of Ellen Bialo (Institute for Schools of the Future) Education Michele Arsenault (Institute for Schools of the Future) John Olive, University of Georgia Mary Fullmer (University of Chicago) David Pillemer, Wellesley College Andee Rubin, Bolt Beranek and Newman Laboratories Design andpmduction Cindy Stephens, D. C. Heath Elisabeth Roberts (TERC) Marlon Walter, University of Oregon Jessica Goldberg (TERC) Virginia Woo ley, Boston Museum of Science Le Ann Davis (TERC) John Abbe (TERC) Laurie Aragon (COMV) Thanks also to advice and comment from Marilyn Burns, Solomon A. Garfunkel (COMAP), and Bob Willcutt.

s CONTENTS

Preface 1 Teacher Notes Teaching Data Analysis 3 Playing "Giant Steps" 16 Measuring: From Paces to Feet Unit Overview 7 Recording data 17 Line plot: A quick way to show the shape of the data 17 Part 1: introduction to measurement Mathematical discussions: How can we measure the room? 13 Challenging ideas 18 What are data. anyway? Robot paces 25 20 The shape of the data: Clumps, bumps, Paces come in clffrerent sizes 31 and holes 22 Connecting oral and written directions 28 Part 2: Using standard measures Statistical words to introduce as appropriate 33 Are our feet afoot long? 39 Rulers, footsticks, and inchsticks 42 Using a smaller unit 47 Bureau of Standards: Home of the official U.S. measures 43 Sixteen left feet 44 Part 3: A project in data analysis Benchmarks on the body 50 Classroom furniture: Do our chairs _fit us? 61 Sketch graphs: Quick to make, easy to read50 Measuring heights: Using tools How close can you get to a pigeon? 69 56 Phases of data analysis: Learning from the process approach to writing 65 Student Sheets 1-7 73 Goldilocks and the three bears 66 Footstick Pattern 80 Measuring your distance from a pigeon: Inchstick Pattern 81 Some ideas 72 9 PREFACE

In an information-rich society such as area of data analysiscollecting, organiz- Mathematicians and scientists use infor- ours, statistics are an increasingly impor- ing, graphing, and interpreting datapro- mation or data like snapshots to look at, tant aspect of daily life. We are constantly vides a feasible, engaging context in which describe, and better understand the world. bombarded with information about every- elementary grade students can do real They cope with the real-world "messiness" thing around us. This wealth of data can mathematics. Students of all ages are of the data they encounter, which often do become confusing, or it can help us make interested in real data about themselves not lead to a single, clear answer. choices about our actions. and the world around them. Because statistics is an application of real mathematics skills, it provides the oppor- Educators and mathematicians now stress Teaching statistics: Pedagogical issues tunity to model real mathematical behav- the importance of incorporating data anal- iors. As students engage in the study of ysis and statistics into the elementary We introduce students to good literature in statistics, they, like scientists and statisti- mathematics curriculum to prepare stu- their early years. We do not reserve great cians, participate in: dents for living and working in a world literature until they are olderon the con- filled with information based on data. The trary, we encourage them to read it or we y cooperative learning Curriculum and Evaluation Standards for read it to them. Similarly, we can give y theory building School Mathematics, published by the young students experience with real math- National Council of Teachers of ematical processes rather than save the y discussing and defining terms and Mathematics in 1989, highlights statistics good mathematics for later. procedures as one of the key content strands for all grade levels. Through collecting and analyzing real data, y working with messy data students encounter the uncertainty and y dealing with uncertainty Many teachers see the need to support intrigue of real mathematics. Mathemati- students in becoming better problem cians do not sit at desks doing isolated We want elementary school students to solvers in mathematics. However, it is dif- problems. Instead, they discuss, debate, have the opportunity to engage in such real ficult to find problems that give students and arguebuilding theories and collecting mathematical behavior, discussing, the kind of experiences they need, are man- data to support them, working coopera- describing, challenging each other, and ageable in the classroom, and lead to the tively (and sometimes competitively) to building theories about real-world phe- learning of essential mathematics. The refine and develop such theories further. nomena "oased on their work.

Preface 1 1 0 I Data analysis in the mathematics cuniculum Exploring data involves students directly in many aspects of mathematics. Data are collected through counting and measuring; they are sorted and classified; they are represented through graphs, pictures, tables, and charts. In summarizing and comparing data, students calculate, estimate, and choose appropriate units. In the primary grades, work with data is closely tied to the number relationships and measuring processes that students are learning. In the upper elementary grades, students encounter some of the approaches used in statistics for describing data and making inferences. Throughout the data analysis process, students make decisionsabout how to count and measure, what degreeof accuracy is appropriate, and howmuch information is enough; they continually make connections between the numbers and what those numbers represent. Instead of doing mathematics as an isolated set of skills unrelated to theworld of reality, students can understand statistics as the vibrant study of theworld in which they live, where numbers cantell them many different stories about aspects of their own lives.The computation they do is for a purpose, and the analysisthey do helps them to understand how mathematics can function as asignificant tool in describing, comparing, predicting, and making decisions.

2 Preface 1 2 TEACHING DATA ANALYSIS

The nature of data analysis similar. Elementary school students can analysis activities and will be able to anti- both analyze data and use those data to cipate likely questions, confusions, and In data analysis, students use numbers to describe and make decisions about real opportunities. describe, compare, predict, and make deci- situations. sions. When they analyze data, they search for patterns and attempt to understand Because real data are the basis for investi- The importance of discussion what those patterns tell them about the gations in data analysis, there are no prede- In mathematics phenomena the data represent. termined "answers." For example, if your A central activity in data analysis is dia- class collects data on the ages of the stu- logue and discussion. While it is easy for A data analysis investigation generally dents' siblings, the students understand you and your students to become engaged includes recognizable phases: that their Job is more than simply coming and enthusiastic in collecting data and considering the problem up with an answer that you knew all along. making graphs, a significant amount of Not only do you not know the answer in time should also be devoted to reflection collecting and recording data advance, but, without seeing the data, you about the meaning of the data. representing the data may not even know what the most interesting questions are going to bel Since students are not used to talking much describing and interpreting the data during their mathematics work, it is im- While this situation encourages students to portant to support active decisionmaking developing hypotheses and theories do their own mathematical thinking, it can by the students from the very beginning of based on the data also feel risky for you. Many teachers wel- the investigation. Students' participation in framing the initial question, choosing These phases often occur in a cycle: the come a little uncertainty in their mathe- the methods of investigation, and deciding development of a theory based on the data matics classes, when it prods their students often leads to a new question, which may to be more independent thinkers. To on ways to organize their data is essential. begin the data analysis cycle all over again. support you, the authors provide sample Once the data are collected and organized, experiences from teachers who have used the students must grapple with interpreting Elementary students can collect, represent, the activities described here so that you can the results. If you have the outcome of a and interpret real data. Although their be prepared for the kinds of issues that are discussion or the "teaching points" you work differs in many ways from that of likely to arise. You will soon build your want to make too clearly in mind, you may adult statisticians, their processes are very own repertoire of experiences with data guide students' observations too quickly

Teaching Data Analysis 3 13 14 into predetermined channels. When stu- sometimes want their data to be anony- cards (or any paper rectangles): then these dent ideas are ignored, misinterpreted, or mous. Focusing on the patternsand shape can be arranged and rearranged. Unglx rejected, they soon understand that their of the class data, rather than on individual cubes (or other interconnecting cubes) are job is to second-guess the "answer" you had pieces uf data, is particularly helpful, espe- another good material for making repre- in mind. cially for upper elementary students. sentations throughout the grades. We have found that stick-on notes (such as Post-it On the other hand, if students find that notes), with each note representing one anything they say is accepted in the same Small-group work piece of data, are an excellent material for way, if evety contribution is "agood idea" Many of the investigations involve making rough drafts of graphs. They can be and no idea is ever challenged, they can lose students working in teams. At first, keep moved around easily and adhere to tables, motivation to participate. Ask students to small-group sessions short and focused. desks, paper, or the chalkboard. Pencil and reflect on, clarify, and extend their ideas For students not used to working insmall unlined paper should always be available and to listen to and ask questions of each groups, assign specific tasks that encourage for tallies, line plots, and other quick other. Discussions in mathematics should the participation of all the group members. sketch graphs. encourage students to interpret numbers, For example, instead of, "Have a discussion make conjectures, develop theories, con- in your group to decide what you want to Calculators sider opposing views, and support their ask the second graders about their bed- ideas with reasons. times," you might say, "Come up with three Calculators should be available, if possible, possible questions you could ask the second throughout the activities. Their use is graders." specifically suggested in some of the inves- Sensitive issues in data analysis tigations. It is no secret to students that Students of all ages are interested in data calculators are readily available in the about themselves and the issues they care Materials world and that adults use them often. But about. Topics that matter enough to stu- Students need materials to represent their many students do not know how to use a dents to make them compelling topics for data during their investigations. These calculator accurately, do not check their study often have very personal aspects. In- range from Unifix cubes to penciland paper results for reasonableness, and do not vestigations about families, heights, or to computer software. What is most impor- make sensible choices about when to use a students' chores, for example, can all bring tant is that students are able to construct calculator. Only through using calculators up sensitive issues. After trying many multiple views of the data quickly and with appropriate guidance in the context of topics in many classrooms, we have con- easily and that they do not become bogged real problems can they gain these skills. cluded that the potential sensitivity of a down in drawing and coloring elaborate topic is not a reason to avoid it; on the graphs (which are appropriate only at the Computers contrary, these are the very topics that very end of an investigationwhen students most engage student interest. All teachers are ready to "publish" theirfindings). Computers are a key tool in data analysis deal with difficult or sensitive issues in in the world outside of school. Graphing their classroom, and the skills demanded Any material that can be moved easily and software, for example, enables scientists of a teacher in handling issues that arise rearranged quickly offers possibilities for and statisticians to display large sets of during data analysis activities are no looking at data. For example, students data quickly and to construct multiple different. Keep in mind that students may might write or draw their data on index views of the data easily. Some software for

4 Teaching Data Analysis / 5 the elementary grades allows this families. Including parents and other sig- flexibility as well. A finished graph made nificant family members as participants in by the computer may, for some students, be your data analysis investigations can an appropriate illustration for a final stimulate their interest and enthusiasm for report of their findings. But keep in mind the work students are doing in school and, that students also make interesting and at the same time, help students see that the creative graphs by hand that would not be mathematics they do in school is connerted possible with the software available to to their life outside of school. them. Other computer software, including software for sorting and classifying and Interdisciplinary connections data base software, is particularly useful for some data analysis investigations. Many teachers find ways to connect the Where the use of a software tool would data analysis experiences students have in particularly enhance a data analysis mathematics to other areas of the curricu- investigation, recommendations for lum. Data analysis is, after all, a tool for incorporating its use are made in the text investigating phenomena of all kinds. The and noted at the beginning of the session. same approaches that students use in this unit can be called on for an investigation in science or social studies. Making these Home-school connections connections explicit and helping students Many opportunities arise in data analysis transfer what they have learned here to new investigations for communicating with areas wiP'ye them an appreciation of the parents about the work going on in the usefulne. mathematics throughout the classroom and for including them as par- curriculum. ticipants in your data investigations. When you begin this unit, you may want to send a note home to parents explaining that students will be studying data analysis in their mathematics class and that, from time to time, parents can be of assistance in helping students collect data from home. Parents or other family members often provide an available comparison group. Studies of age, family size, height, and so forth can be extended to include parents.If students are studying their own families, they may be interested in collecting comparison data about their parents'

Teaching Data Analysis 5 MEASURING: FROM PACES TO FEET UNIT OVERVIEW

Measuring: From Paces to Feet is a unit of How to use this unit y Part 2: Using standard measures study that introduces measuring as a way of Are our feet afoot long? collecting data. Suitable for students in Like all the Used Numbers units, Measuring: Using a smaller unit grades 3 and 4, it provides a foundation for From Paces to Feet is organized into in- y Part 3: A project in data analysis further work in statistics and data analysis, vestigations that may extend from one to iNvo options: including the three upper-grade units in the four class sessions. To cover the entire unit Classroom furniture: Do our chairs fit? Used Numbers series. In Measuring: D-om requires approximately 17 class sessions of How close can you get to a pigeon? Paces to Feet, students: about 45 minutes each. Teachers who have used this unit have found that a schedule of The three parts work well as a single five- to experience the iteration of a measure- 2-3 sessions per week works best to main- ment unit, first with their own paces six-week unit. Some teachers have substi- tain continuity while allowing enough time and later with more standardized units tuted this unit for their textbook chapters on for reflection and consolidation between ses- measurement or statistics. Others have used collect real data through measuring, sions. The activities are sequenced so that it late in the year as a way to consolidate using both informal and standard mea- students move gradually from more straight- students' mathematical learning, knowing surement systems forward to more complex investigations. The that it brings together work in measurement, represent measurement data in a variety investigations are grouped into three parts: estimation, computation, graphing, and of ways VPart 1: Introduction to measurement statistics in a problem-solving context. The How can we measure the room? parts can also be spaced over the entire describe landmarks and features of the school year. For example, some teachers use data Robot paces Paces come in dfferent sizes Part 1 in September to start off their work in formulate hypotheses and build theories mathematics. They return to Part 2 in Jan- about the reality represented by the data uary and use Part 3 in May when students

Unit Overview 7 1 8 1 9 have been together for most of the school (2) materials you will need for the investiga- and of collecting and analyzing data, year and are more able to work indepen- tion and any special arrangements you may including ways to graph data and how and dently. Within each part, it is important that need to make, and (3) a list of the important when to introduce special terms. The Teacher 2-3 sessions take place each week so that mathematical ideas you will be emphasizing. Not ls are listed in the contents because the experiences build on each other, allowing Plan to look carefully at this overview a day many are useful as references throughout students gradually to acquire skills and un- or two before launching the investigation. the unit, not just where they first appear. derstanding in data analysis. You might plan to read them all for Session activities. For each session, you background information before starting the In order to understand measurement, stu- will find step-by-step suggestions that out- unit, then review them as needed when they dents need to experience it physically. This line the students' explorations and the come up in particular investigations. unit emphasizes using the body as a teacher's role. Although suggestions for measuring tool; we believe students need to questions and instructions are given, you Goals for students connect the counting of a repeated unit of will of course modify what you say to reflect measure with the repetitive physical your own style and the needs of your The "Important mathematical ideas" listed in movement of stepping and counting. As students. In all cases, the teacher's words the investigation overviews highlight the standard units we use inches and feet, which are intended to be guidelines, notword-for- particular student goals for those sessions. are not only based on human word scripts. Plan to read through this The major goals for Measuring: From Paces to measurements, but are at present in more section before each session to get the general Feet, are as follows: common use in the schools. For teachers flow of the activities in your mind. who want to connect this unit with the Dialogue Boxes. The Dialogue Boxes illus- Part 1: Introduction to measurement metric system, a brief extension into making metric measurements can be added at the trate the special role of discussion in these Moving through space and counting the investigations and convey the natureof typi- end of Part 2. Finding metric benchmarks on movements. Students experience measure- cal student-teacher interactions. Examples the body, taking metric measurements, and ment through their own movements in space; are drawn from the actual experiencesof doing the final projects in metric by actively pacing and counting they learn classes that have used these investigations. measurements should follow naturally when how linear measurements are expressed in They call attention to issues that are likely to numbers. the students realize that any imposed mea- diffi- surement system is a social convention, and arise, typical student confusions and culties, and ways in which you can guide that they can choose one of many systems to Comparing units of measure. As they use and support students in their mathematical use. different units of measure, students learn thinking. Plan to read the relevant Dialogue about the relationship between sizes of mea- Boxes before each session to help preparefor suring units and the results of measuring. Planning the investigations interactions with your students. Students compare different units of measure In this book, you will find four types of in- as they use them, thuslearning their relative formation for each investigation: Teacher Notes. These sections provide im- sizes through use. portant information you will need in Investigation overview. This section in- presenting this unit. Here you will find Estimating distances. Estimation of dis- cludes (1) a summary of the student activity, explanations of key aspects of measuring tance is an important process in 20 Measuring: From Paces to Feet 8 21 Viternalizing a measurement system. A learn that data collected by measuring are Part 3: A project in data analysis measurement system is internalized when not comparable unless the measures are students have stored some mental images or agreed upon and have become a convention. Experiencing all the phases of a data concepts that allow them to estimate fairly analysis investigation in which measuring accurately. Estimating lengths. Students estimate is used to collect data. Parallel to the lengths using feet and inches. They find phases of the writing process, a data Defining a measurement method. Through benchmarks on their bodies that help them analysis investigation includes discussing the methods they will use to mea- make reasonable estimates. "brainstorming* or discussion and definition sure, students learn that defining measure- of data collection methods; rough draft ment procedures is a critical part of data Measuring accurately, using feet and graphs of preliminary results; analyses collection. inches. When students are finding leading to refinement of ideas, and final measurements to answer a question that "publication* through reports of results. The Writing directions involving distances. involves them personally, the level of final project in this unit gives students a Writing and reading symbolic information accuracy they will strive for is greatly chance to experience all phases of this about lengths and distances is an important increased. process. measurement skill. This unit teaches those skills directly through an emphasis on giving Describing the shape of the data. Students and receiving ' ith oral and written direc- learn to use simple statistical terms and con- tions. cepts to describe the distribution of real data collected from their classroom measurement Recording and displaying the results of activities. measurement. Students use a variety of data displays to show the results of their Analyzing data through landmarks and measurement investigations. features of the data. Students begin to analyze data by interpreting the meaning of cer- tain landmarks in the data: Which is the Part 2: Using standard measures middle-sized foot? How can we describe the Experiencing a need to standardize. Stu- typical chair size in this classroom? By dis- dents use standard measurements when cussing and describing their data, students they directly experience the variation that develop beginning skills in data analysis. exists if standards are not developed. Using standard measures to compare data Understanding that standard measures sets. The collection and comparison of two were invented to solve real data collection data s,!tsthe length of student feet vs. the problems. Standardized measures are length of teacher feetgives students experi- needed when measurement information is ence in using, reporting, and recording data transmitted to another person. Students with standard measurement systems.

2 2 Unit Overview 9 Measuring: From Paces to Feet a a OD a NED al al

PART 1 Introduction to measurement

2 4 HOW CAN WE MEASURE THE ROOM?

INVESTIGATION OVERWEW

What happens on which students can write. If you use Oir Save the data gathered in Session 1 for note cards or other pieces of paper, you use in Session 2. See the Teacher Note, What Students participate in activities that link may want to purchase a glue stick with are data anyway? (page 20) for a discussion measurement to physical movement, count- a removable adhesive (such as of what we mean by "data." ing their steps as they measure distances in Dennison's Tack-a-Note), This kind of the classroom. As the students collect data glue lets you fasten pieces of paper and compare their results, they begin to no- temporarily to the chalkboard, Important mathematical ideas tice and analyze the numerical differences converting them to stick-on notes. Keep that are produced when they use units of these materials handy for sessions in Using a nonstandard unit to measure a different sizes. which you are making line plots on the distance. Students use their own giant chalkboard or students are arranging steps, baby steps, and paces as units in The activities take three class sessions of and rearranging their data. measuring the length of the classroom. In about 45 minutes each. making these measurements, they experi- Duplicate Student Sheet 1 (page 73) for ence the iterative nature of measurement, each student (Session 1, homework). counting the number of times an agreed- What to plan ahead of time Duplicate Student Sheet 2 (page 74) for upon unit is used in moving from the begin- y Play the game "Giant Steps" with the each pair of students (Session 3). ning to the end of an object. class. See the Teacher Note. Playing '4Giant Steps" (page 16) for directions. Become familiar with making a line plot. Estimating lnngth. Students estimate See the Teacher Note, Line plot: A quick length using paces. At first they estimate y For Session 1 you will need note cards way to show the shape of the data (page using their own paces or the paces of other or stick-L n notes, such as Post-it notes. 17). members in the class. Students work at

25 How can we measure the room? 13 2G visualizing the unit repeated over a distance SESSION 1 ACTIWTIES You could each measure the length of the and share their methods with the class. room in your own giant steps. Will all your answers be the same? What do you think? Comparing the effects of measurement Preparing for the session: Playing using units of different size. A single dis- "Giant Steps" Allow time for students to talk about whether tance can be described in different ways if we they think there will be any variation in their use different units of measure. Many third Playing the game "Giant Steps" is an impor- results. Third and fourth graders have a wide and fourth graders are confused by the fact tant prerequisite experience for the first ses- range of theories about how and why mea- that for the same distance, they get different sions of this unit (see the Teacher Note, Play- surement results might vary. results from their individual pacing. Re- ing "Giant Steps,"page 16). You may want to peated experiences help them understand do this during class time, or you may prefer Collecting data: Measuring the length that the different results come from using to play the game during a lunch or morning of the classroom somewhat different unitsthat is, not all recess period. "paces" are identical. As they progress Establish student pairs that will pace the thiough these sessions, students begin to length of the classroom in giant steps. While standardize their own units and to under- Considering the problem: Estimating distance in giant steps one partner paces, the other counts. Stu- stand some reasons for using conventional dents then switch roles. Ask students to measurement units. Remember when we played "Giant Steps"? To- record their results on the board. They may day we're going to estimate and then count look like this: distances in the room in giant steps. 13 14 12 13 14 13 Select a student to be the giant. The giant is 11 12 14 12 to stand at the front of the classroom, then take two giant steps and freeze. Ask the Look at the numbers on the board. Nd every- other students to estimate the whole length one get the same results? Is that surprising? of the classroom in those giant steps. Why or why not? Make a y :ture in you,: head and try to imagine how many of [Ricardo's) giant steps it would Recording and organizing the data: take to get to the far wall. Line plots

Write the estimates on the board; then ask Showing your students quick and effective the giant to pace three more giant steps, and ways of recording information gives t.:em im- record any revision of the students' esti- portant tools they will need to investigate mates. Next, as the giant paces the whole and analyze data. As you work together on length of the room, have the students count these investigations, your quick sketching of out loud and record the answer. students' results (using tallies, charts, line 27 14 Measuring: From Paces to Feet 28 plots, and other kinds of informal data dis- everyone felt good about the process. For all talk about this. Why should the littlest steps plays) will help them realize that data can be theories, ask students for their reasons and give us the biggest numbers? Some third and organized and displayed quickly in a variety evidence, and support their thinkirg at this fourth graders are confused by this; they will of ways (see the Teacher Note, Recor stage (see the Teacher Note, Mathematical need to talk about it, try it, and work hard to data, page 17). discussions: Challenging ideas, page 18). understand that the bigger the step, the Some of your students will have an easy time fewer the paces. It's important not to push Because the idea of organizing data is prob- seeing that the, size of a student's step makes the students who don't get it, but to keep ably new to your students, make a line plot t, difference in numerical results; others asking them questions about their ideas. on the board to show the results of their won't. Inverse relationshipsthe smaller the pacing off the room (see the Teacher Note, pace, the bigger the number of pacesare mrSave all the data for the next session; Line plot: A quick way to show the shape of hard to grasp, hold onto, and use. students will be comparing today's results the data, page 17). Draw a number line, label with the results of their next activity. the points to include the smallest and largest Collecting data: Measuring numbers of giant steps, and show students Extensions how to make X's, check marks, or other sym- in "baby steps" bols to record their data along the line. Now you will measure the same distance, but If you have time and your students' inierest you'll use baby steps. Do you think you'll get is high, you can repeat the activity with stu- different results? dent pairs measuring the width of the classroom. Half the class might measure in giant Allow a full discussion of students' predic- steps, the other half in baby steps. tions here, and when you think they're Student Sheet 1, Giant steps and baby steps: Why do you get differe t results? Can you ready, ask someone to demonstrate baby steps. Measuring at home, encourages students to think of some reasons? "measure" using giant steps or baby steps in Imagine, now, how many little steps like that their home environments. Depending on Allow time for students to discuss the varia- it takes to get to the other wall. Try to imagine your homework policy, you may prefer to tion in results (which will surprise some stu- the baby steps in a straight path across the have students conduct a similar activity dents more than others). There is much room. Hov many will it take? at school. room here for theory-building. Many of your students will have theories that do not Record students' pred_ctions (this needn't be match yours, In one room, a student believed done elaborately), then pair up students to that people's steps automatically became pace and count. If they write their results on bimer when they went slowly. Another index cards, they can enter the data quickly believed paces varied because of foot size. onto a line plot when everyone has finished nese children were happy to talk about pacing. their ideas and to debate with their peers. The "odd" theories became the focal point for Baby steps produce much larger numbers hypothesis-testing in the classroom, and than giant steps; your students will want to

How can we measure the room? 15 29 3 0 TEACHER NOTE The repertoire of possible moves includes: anything else your students know (it will vary from one region to another) Playing "Giant Steps" giant steps (the longest steps possible for the individual) Much of the game revolves around the caller imagining how many steps it will take to This game (also called "Mother May 1?") may baby steps (heel to toe steps) reach the finish lineand, of course, playing be one your students know already. It banana twirls (putting hand on head, favorites about who will cross the line first to usually involves 6-12 players. Optimal size is become the next caller. For your purposes, a group of about 10. You may want to divide stepping forward, and turning 360° simultaneously) the point of the game is that students prac- your class and take turns playingor if tice visualizing distances and use a variety of space allows, have two or three groups play- bunny hops (hopping with both feet at steps of different lengths. ing at once. once) Asa variation, you might work specifically on One player is thecaller,who gives directions. lamp post (lying on the ground with feet visualizing measures of distance. Set yourself The caller stands on an imaginary line facing marking current positionplayer up as the caller and have two or three stu- the other players: they stand in a row about reaches as far forward as possible, and dents stand in a row about 6 feet away. As 15 to 20 feet away and move toward the stands at that point) you give directions (i.e., "Take 4 giant steps" caller as directed. The object of the game is y duck steps (squatting, holding onto an- or "Take 3 banana twirls"), ask the rest of the to cross the line on which the caller stands. Ides, walking forward, quacking and class to visualize whether the students will The first person over that line becomes the flapping arms simultaneously) or will not reach you with that move. next caller. The caller gives directions to each player in turn, such as "Take 5 giant steps" or "Take 7 baby steps," typically giving different instructions to each player. After hearing the directions, the player to whom they were directed must ask, "May I?" The caller then gives permission to perform the action with the words, "You may." Gradually, players advance toward the caller. A player who forgets to ask "May I?" must go back to the starting line. In some versions of the game, the player pro- poses a move ("May I take 6 baby stepsn, which the caller can either allow ("Yes, you may") or disallow with a substitution ("No, but you may take six banana twirls").

16 Measuring: From Paces to Feet Comparing their results by analyzing simple %1TEACHER NOTE 'ITEACHER NOTE line plots or tables of data becomes an im- Recording data portant analytic tool for these students. Line plot: A quick way to show the shape of the data Graphs are traditionally the focus for Unit graphs are most often used in the primary grades. These representations use one instruction in statistics in the elementary A line plot is a quick way to organize numeri- grades, even though the process of doing item to represent each piece of data. Because of this, each student can keep track of his or cal data. It clearly shows the range of the statistics involves much more than simply data and how the data are distributed over making and reading graphs. her own piece of data as it is classified, moved, re-sorted, and discussed. Line plots that range. Line plots work especially well for Most of us think of graphs as the endpoint of done "on the fly" to organize data while they numerical data with a small range. the data analysis process. Statisticians, how- are being discussed and analyzed do not need titles, labels, or vertical axes. These This representation is often used as a work- ever, use pictures and graphs during the ing graph during data analysis. It is an initial process of analysis as tools for understand- graphs are truly rough drafts and are quickly put on the board as data are collected. Such organizing tool for beginning work with a ing the data. Representing data in a picture, data set, not a careful, formal picture used to table, or graph is an attempt to discover fea- quickie graphs provide the focus for beginning data analysis (see the Teacher Note, The present the data to someone else. Therefore, tures of the data, to array the data in such a it need not include a title, labels, or a vertical shape of the data: Clumps, bumps, and holes, way that the shape of the data and the rela- axis. A line plot is simply a sketch showing page 22). tionships among the data can be seen. the values of the data along a horizontal aids Our students must gain facility with creating When you ask your students to talk about and X's to mark the frequency of those representations that provide a first quick the shape of the data, the most important values in the data set. For example, if 15 look at the data. Encourage students to cre- idea is that they should pay attention to the students have just collected data on the ate quick pictures, diagrams. tables, and overall shape of the data. Where are data number of paces it takes to walls the length graphs in order to get a feel for the data. clumped? Is there more than one clump? Are of the classroom, a line plot showing these Younger students use more concrete and pic- there tall, tall peaks? Are there holes? Is it data might look like this: torial representations for this workinter- an even distribution? Questions like these X will help the students talk about what they locking cubes, real objects, their own X see and learn to describe the distributions of pictures. In the upper grades, more work will X X be done with numerical representations. data that they gather by measuring. Grad- ually the traditional data analysis landmarks X X The numbers, or data, collected by measur- can be introduced as students' experiences X X X ing and counting are the "stufP of this mea- prepare them for such numerical conven- 12 13 14 15 16 17 18 surement unit. Those numbers must repre- tions as the mode, the median, the range, sent real events and concrete experiences for and the mean. II From this display, we can quickly see that students. For this reason. the early part of two-thirds of the students took either 18 or the unit stresses the physical act of pacing. 19 paces. Although the range is from 11 to

How can we measure the room? 17 33 34 19, the interval in which most data falls is Yes but that might give us a problem since we from 16 to 19. The outlier, at 11, appears to 'ITEACHER NOTE don't know for sure that those really are in the be an unusual value, separated by a Mathematical discussions: middle. Could we do it two different ways and considerable gap from the rest of the data. see if the results are the same? (For further information, see the Teacher Challenging ideas KYLE: What if we do it two ways and get dif- Note,Stattstical words to introduce as ferent middle people? appropriate, page 33.) One of the most important ideas in mathematics is that one's assertions should be What do you think will happen? One advantage of a line plot is that we can subject to scrutiny and challenge. The his- ELENA: An argument! record each piece of data directly as we col- tory of mathematics is the history of debate lect it. To set up a line plot, start with an ini- and discussion, yet we do not see much dis- Indeed, such a situation will likely produce a tial guess from students about what the cussion in most mathematics classes. conflict with the two methods yielding two range of the data is likely to be: What do you sets of results. But this kind of argument lies think the lowest number should be? How In fact, many students are convinced that at the heart of mathematical discussion. En- high should we go? Leave some room on there is always one right answer and one couraging students to make assertions, to each end of the line plot so that you can "best" procedure in mathematics class. This base their arguments on data, to state their lengthen the line later if the range includes idea often leads them to be nervous if their reasons, and to ask others clarifying ques- lower or higher values than you expected. answers seem to conflict. That feeling is re- tions is a vital aspect of teaching mathemat- flected in the following discussion in a third ics. By quickly sketching data in line plots on the grade classroom, which occurred when the chalkboard, you provide a model of how such students were working on a later investiga- Challenging students' ideas is a delicate mat- plots can provide a quick, clear picture of the tion in this unit, Paces come in different ter, yet it can be a very effective way of shape of the data. sizes. probing to find out what your students are thinking and to help them clarify and extend Students have measured their paces and their own ideas. Many teachers find that the recorded their results. Now they are trying to best way of examining students' ideas is to decide how to find their middle-sized pace, ask questions that invite them to explain and the teacher is helping them decide on a their reasons. method. Say more about that. We still don't know what the middle-sized pace is. What do you suggest we do? Can you give me an example? JENNIFER: Put them in order from largest to How do the data tell you that? smallest. MICHAEL: Compare them with the ones we Another technique is ask students to re- decided were all in the middle. late their ideas to other students' ideas.

30 18 Measuring: From Paces to Feet 3; Is this like Su-Mei's idea? Let's record our results in a line plot. We need SESSION 2 ACTIVITIES to decide on the beginning and end of the Are your reasons the same as Jeremy's? number line we'll use. Any ideas? Considering the problem: Pacing A third way teachers sometimes probe stu- What can we see from this line plot? What can the classroom dents' ideas is to give counterexamples. you say about the data? Some other third [fourth] graders say that it's Recently we measured the length of our class- Help students express their ideas as they de- not the one that's chosen the most, but the room in giant steps and baby steps. The scribe the distribution of the data (see the stretch of taking giant steps and the finicki- one in the middle. What would you tell them? Teacher Note.The shape of the data: Clumps, nese of taking baby steps can be uncomfort- bumps, and holes, page 22). Do you think that would work if we measured able, though. A more widely used measurement something else? is the pace, which is closer to a regular step. What do you see in the graph? Do you see any How can we measure someone's pace? Let's clusters of data? Did most of you get the same You may have other techniques that work for try with a volunteer. number of paces when you measured the you. Once ideas are flowing, you may find length of the room? Is the range of data very that the students themselves make many Ask one student to take 3 or 4 regular wide? Are there any unusual values? suggestions, ask each other probing ques- walking steps, then freeze. Ask the class how tions, and help to formulate ideas. Until long they think the room is in paces of that For an example of such a discussion, see the then, however, it is important to keep dis- size. After they estimate the distance, have Dialogue Box, Describing the shape of the cussion alive. Researchers have found that the same student pace off the length of the data (page 21). simply waiting 3 seconds after asking a room while the class counts to keep track. question gives students time to organize In another line plot, put the results of the their thinking and to develop some of their previous session on the board and ask stu- Collecting and describing data: ideas before answering. Three seconds seems dents to compare those data with their new like a very long time when you're used to Comparing different steps results. much faster answers, but students need to Working in the same pairs as in the previous What can we say about our paces, our giant feel that they have time to think and time to session, the students estimate and count the make suggestions. Hunying them to an steps, and our baby steps? How do they com- number of paces it will take them to measure pare? answer defeats the whole purpose of the length of the classroom. Record their mathematical discussions. data on the chalkboard. One technique that This is likely to produce a long discussion of works well is simply to record the data in a the relationship between step size and num- list, and then model a quick, effective way to ber of steps. Students will argue with each organize it. A line plot is a good representa- other about this relationship and about rea- tion for these data (see the Teacher Note, sons for the difference in results. They may Line plot: A quick way to show the shape of have trouble articulating their ideas, and the data. page 17). some will not be able to put their ideas into

37 How can we measure the room? 19 38 words. Students may assert that it takes TEACHER NOTE Should "brown" be a single category, or more steps for bigger people; others will should it be divided into several shades? assert that it's feweragain, the inverse re- What are data, anyway? lationship (the bigger the person the fewer Once data are collected, recorded, counted, and analyzed, we can use them as the basis the steps) may confuse some of your stu- I'm 48 inches tall. dents. Some teachers who have worked with for making decisions. In one school, for example, a study of a,;cidents on a particular this unit believe that the fact that smaller This bottle holds 2 liters of soda pop. numbers "win" here is unusual and impor- piece of playground equipment showed that most of the accidents involved children who tant, so that students do not just look at She is wearing sneakers. large numbers without evaluating what they were in first, second, or third grade. Those represent. (See the Dialogue Box, Discovering These shoes weigh 150 grams. who studied the data realized that the children's beliefs about numbers, page 23, for younger students' hands were too small to another example of a mathematical side trip Each of these statements contains descrip- grasp the bars firmly enough. A decision to away from data analysis into more general tive information or data about some person keep primary grade children off this equipment was made because the data, or the beliefs.) or thing. Data is a plural noun; one datum is a single fact. Data are the facts, or the facts, led to that conclusion. Questions like the following can produce information, that differentiate and describe When students collect data, they are collect- fruitful discussion: people, objects, ot other entities (e.g., countries). Data may be expressed as num- ing facts. When they interpret these data, Which are the biggest steps? Did you take bers (e.g., he is 48 inches tall; she has 4 they are developing theories or generaliza- more baby steps or more giant steps? Why? people in her family) or attributes (her fa- tions. The data provide the basis for their If I take 6 giant steps to walk along here, will I vorite flavor is chocolate; his hair is curly). theories. take about 30, or about 10, or more like 3 baby steps to cover the same distance? Data are collected through surveys, observa- tion, measurement, counting, and experi- Make sure that students give reasons for ments. If we study rainfall, we might collect their statements. Just as in the previous rain and measure the amount that falls each session, some students will not grasp all of day. If we study hair color, we might collect the connections between size of step and and record the color of each person's hair. number of steps taken. At this point, be sat- isfied that you are providing experiences that Collecting data involves detailed judgments will lay the foundation for understanding. about how to count, measure, or describe. Should we record rainfall data for each day or for each rainstorm? Should we roundoff to the nearest inch? Should we count just one color for a person's hair?Should we record "yellow" or "dirty blond" or "gold"?

39 20 Measuring: From Paces to Feet 4 0 MICHAEL: Well, there's nothing at 12 or 13 description of how many paces long our room "MIALOGUEBOX either. is? Describing the shape Yes, 14 is the lowest count and there's nothing CHRIS: I think so, because most of us took below it. But this situation, that Susan noticed 16 paces. of the data up here, is a little different. What can you say about that? Any other ways to say this? Or any different The students have just counted the number ideas? of paces it takes each of them to walk the SU-MEI: Mostly, the paces go from 14 to 18, length of the classroom. They have recorded but sometimes you get something higher. PEGGY: Well, I wouldn't say just 16. their data on a line plot and are beginning to Can anyone add to that? Why not? talk about what they can see. SEAN: You must have really small paces if it PEGGY: Well, there's really not that much takes you 23. difference between 16, 17, and 18. They're X X all really close together. I'd say 14 to 18, X X X In fact, mathematicians have a name for a cause the 20 and 23 aren't what you'd X X X X X piece of data that is far away from all the rest. usually get. 16 17 18 19 20 21 2223 24 They call it an outlier. An outlier is an unusual piece of datasometimes it might actually be So Peggy is saying she'd use an interval to de- So what can you say about these pace data? an error, but sometimes it's just an unusual scribe the pace-length of our room, from 14 to Let's hear a few of your ideas. piece of data. It's usually interesting to try to 18, and Chris said he'd say the length was SEAN: Well, there are a lot at 16. find out more about an outlier. Who had the about 16. What do other peoi le think about outlier in this case? that? SHANTA: There was only one at 20 and one at 23. BOBBY: I did. And I counted twice, and Michael checked me, too, so I know it was Or In this discussion, the class has moved VALERIE: There are two each at 14 and 15. 23 paces. gradually from describing individual features What else do you notice? of the data to looking at the shape of the JENNIFER: Maybe he's got smaller feet. data as a whole. The teacher introduced the ARACELI: Fourteen is the lowest. Any other theories about Bobby's pace? ideas of interval, range, and outlier because So no one took fewer than 14 paces? they came up in the discussion and were (Later)... appropriate in describing these data (see the BENJAMIN: Yeah. And 23 was the highest. So if someone asked you, "What's the typical Teacher Note: Statistical words to introduce So the range was from 14 to 23. What else? number of paces to cross our classroom'?" as appropriate, page 33). Throughout the what would you say? conversation, the teacher tries to have stu- SUSAN: There's nothing at 19, 21, or 22. dents give reasons for their ideas and pushes TAMARA: Well, I'd say 16. Susan's noticing that there are a lot of holes in them to think further by asking for additions this part of the data. Can anyone say any more [Addressing the class as a whole, not Just the stu- or alternatives to ideas students have about that? dent who answered) Would 16 be a reasonable raised. I

How can we measure the room? 21 41 42 Where are the clumps or clusters, the gaps, In the first stage of discussion, students ob- %TEACHER NOTE the outliers? Are the data spread out, or are served the following special features: lots of the data clustered around a few val- The shape of the data: Clumps, VThere is a clump of lions between 400 ues? Second, we decide how we can interpret and 475 pounds (about a third of the bumps, and holes the shape of these data: Do we have theories lions). or experience that might account for how the Describing and interpreting data is a skill data are distributed? y There is another cluster centering that must be acquired. Tao often, students around 300 pounds (another third). simply read numbers or other information As an example, consider the following sketch from a graph or table without any interpre- graph of the weights (in pounds) of 23 lions y There are two pairs of much lighter tation or understanding. It is easy for stu- in U.S. zoos. lions, separated by a gap from the rest dents to notice only isolated bits of informa- of the data. tion (e.g., "Vanilla got the most votes," "Five 25-49 In the second stage of discussion, students people were 50 inches tall") without develop- 50-74 considered what might account for the shape ing any overall sense of what the graph 75-99 of these data. They immediately theorized shows. Looking at individual numbers in a 100-124 data set without looking for patterns and that the four lightest lions must be cubs. 125-149 They were, in fact, one litter of 4-month-old trends is something like decoding the indi- 150-174 vidual words in a sentence without compre- cubs in the Miami Zoo. The other two clus- 175-199 ters turned out to reflect the difference be- hending the meaning of the sentence. 200-224 If tween the weights of adult male and female To help students pay attention to the shape 225-249 If lions. of the datathe patterns and special fea- 250-274 Throughout this unit, we strive to steer stu- tures of the datawe have found it useful to 275-299 use such words as clumps, clusters,bumps, dents away from merely reading or calculat- together, 300-324 1 I ing numbers drawn from their data (e.g., the gaps, holes, spread out, bunched range was 23 to 48, the median was 90, the and so forth. Encourage students to use this 325-349 I If biggest height was 52 inches). These num- casual language about shape to describe 350-374 I bers are useful only when they are seen in where most of the data are, where there are 375-399 I the context of the overall shape and patterns no data, and where there areisolated pieces 400-424 I of the data set and when they lead to ques- of data. 425-Aitd I tioning and theory-building. By focusing in- A discussion of the shape of the data often ,450-474 stead on the broader picturethe shape of (Source: Zoos in Atlanta. Cleveland, Little Rock, Memphis. the datawe discover what those data have breaks down into two stages. First, we decide Miami. thc Bronx. Philadelphia. Rochester, San Antonio. and what are the special features of the sbape: Washington. DC. Data collected in 1987.) to tell us about the world.

22 Measuring: From Paces to Feet 4-1 those paces. As the pacer walks to the target, both students count to keep track; they record the results on Student Sheet 2.

Extension: Estimating other distances The student teams may develop some good strategies for estimating distances in paces. To practice these strategies, follow up with a whole-class activity. Select various points in the classroom and ask everyone to estimate the number of a specific student's paces between the points. Write down all estimates, then check by pacing to see how close the class estimates are. Visual estimation is an important component of measurement, and practice does make a diffe-ericel

4 7

24 Measuring: From Paces to Feet don't need to count the other halves because SESSION 3 ACTIVITIES "MIALOGUEBOX nobody has those numbers. VALERIE: I get it. I think we should count Discovering children's beliefs Collecting and recording data: them too, but only if they have people. about numbers Estimates and actual distances

Children have many ideas about numbers Ask one student to stand in a fairly open that are not always apparent in the class- The discussion clearly had two threadsthe part of the classroom. Select a target that is room. One of the benefits of mathematical "reality" of the fractional numbers, and the a moderate distance away in a straight line. discussions is that those ideas can surface search for the middle-sized foot. The stu- Have the student take three or four paces to and be talked about. dents resolved it by granting reality to the help the others visualize the length of a pace. halves when they represented real values Here, for instance, is a discussion that (i.e., if they "had people"). You may want to dramatize the visualization surfaced in a third grade classroom while process by "thinking through" your own way students were analyzing a data display The teacher did not take an active role in this of making an estimate: discussion, preferring that the students talk produced in the investigation Using a smaller Let's seeI can see how long Shanta's pace is, unit. The display showed the lengths of their among themselves about whether the numbers were real or not. She was surprised by so I'll try to imagine: 2 ... 3 .. 4 ... 5 ... 6. feet in inches. They were looking for the About 6 paces to the desk, I think, Let's try it! middle-sized foot. their idea that halves weren't numbers, and decided to follow up in a different context at .. Now, how many pacesis it from Shanta to a later date. the globe? X X X Gaps between children's intuitive, informal Students estimate, the pacer paces the dis- X X X X X understanding of numbers and their school tance, and everyone counts. Repeat this two 718 819 91 10101 11 2 2 2 2 knowledge do not always arise unless there or three times, selecting different objects. You may want to use more tiaan one student VALERIE: I think 8 is in the middle because is some reason to talk in a fairly open-ended 9-1/2 and 10-1/ 2 aren't numbers, theyre way about numbers used in a real context. as a pacer. One of the important benefits of mathemati- halves. Then if you match 6 and 10, and 7 Next break the class into two-person teams cal discussions is the chance to hear and 9, 8 is in the middle. that will work in different parts of the room. students expressing their intuitive beliefs Give each team a copy of Student Sheet 2, TAMARA: I think you should count 9-1/2 and about numbers. Informal diagnosis of chil- Pacing in pairs. On cad team, one student is 10-1/2 too, because they're numbers. dren's understanding is often based on such the caller and the other is the pacer. Each RICARDO: I think so too. conversations. team marks a starting place, and the caller VALERIE: I still think that 1/2 numbers selects an object or location in the classroom aren't regular numbers. that can be reached on a straight line. The pacer then demonstrates a "typical" pace. ELENA: But we have to count those numbers and both students estimate the distance in because those numbers have people. We

How can we measure the room? 23 4 4 6 ROBOT PACES

fM INVESTIGATION OVERVIEW

What happens What to plan ahead of time dents to practice their estimation skills at home and on the playground as well. Students give directions to move from one yProvide paper plates as targets, one for point to another, taking turns being the each pair of students. Write a number Comparing lengths of routes. Students direction-giver or the robot who follows direc- or letter on each plate to differentiate compare the lengths of routes and learn that tions. At first these directions involve stating them. the segments of a distance, when added, yield the whole distance traveled. a number of paces to move forward. Gradu- y Duplicate Student Sheet 3 (page 75), ally the directions become more complex and Student Sheet 4 (page 76), and Student Giving directions to another person. First include right and left turns. Partners com- Sheet 5 (page 77) for each pair of stu- orally, and then in writing, students give di- pare the number of paces they take from one dents (Sessions 2 and 3). rections that involve nonstandard measure- spot to another, learning to estimate the y Have calculators available (Session 3). ment of distance. The progression from oral number of their partner's paces it takes to go to written directions is important to from A to B. Differences in pace size and the students' understanding of written effects of those differences on numerical re- Important mathematical ideas information about measurement. Help them sults are experienced and discussed. Learn- to read each other's directions by making ing to count off units (in this case, paces) Estimating distance to a point in space. Students will be giving directions to each "mental maps" of the routes being along an unmarked distance provides stu- described. dents the opportunity to measure with famil- other that involve careful estimation of dis- iar units. tance, given a particular unit. Some will need help in finding ways to make effective esti- The activities take three class sessions of mates: they may need more experience than about 45 minutes each. these sessions alone provide. Encourage stu-

Robot paces 25 4 8 SESSION 1 ACTIVITIES this twice, then return to their seats. dents to give just one direction at a time. Was this difficult? Were your estimates of dis- Considering the problem: Preparing Considering the problem: How can we tance fairly accurate? written directions give good directions? Encourage students to talk about their ex- After students have had the chance to both If someone asks you for directions to another periences, calling attention to the issue of estimating distance in paces. give and respond to oral directions, call them place, it's important to gilt e them good direc- back into the whole group for some discus- tions, Today you will pretend to be robots and sion before starting the next phase of this practice giving directions by telling each other Considering the problem: Making turns activity. how far to move. It's usually harder to give directions in real Let's hear from the direction-givers first, and Demonstrate the task in front of the class. life, because you can't always go in a straight then the robots. What was hard about this Ask for a volunteer robot, who must do ex- line to get somewhere. The person who's task? Whet was easy? actly what you say. Put a paper plate some asking for directions will probably have to turn distance away from the robot; this is your at least once to reach the goal. We'll practice The nezt time we give directions to robots, we target. Give directions that will move the making turns and measuring distances that will still be doing pacing and turning, but robot to the target paper plate. Use only aren't in a straight line. First, I need another you'll be writing down your directions first. so "forward" or "backward" commands at this volunteer robot. you can keep a record of them. time; you will be making turns a little later. 'Say, for example: Give directions to the robot that involve Demonstrate this with a volunteer robot. making turns. Write directions on the board that will get Robot, go 5 paces forward. the robot to some unnamed target. Use just Robot, turn left. ...Robot, go 4 paces forward. two forward commands and one turn; for Assess the effect of this movement with the ... Robot, turnright. instance: Forward 9 paces. Turn right. class. Forward 3 paces. Help students establish a working definition Does the robot need to be given new direc- of a turn. Most classes decide that a turn is a Ask students to imagine the robot starting tions? Did my estimate as the direction-giver 900 turn, or a square corner. Point out a tar- and pacing, and to tell where they think the work for this particular robot? get in the classroom and give directions to robot will end up. Encourage them to the robot as a demonstration. The robot is to visualize a robot's pace and to place that Students work in pairs, taking turns being move as directed after each command. pace down over and over. This is very hard the robot. Give each team a paper plate mental work and requires much concentra- identified by a number or letter. The direc- Students again pair up, one being the dinc- tion. When most of the students have made tion-giver positions the plate, tells the robot tioli-giver and the other the robot. As before, their estimates of the finishing point, ask the how many paces to take toward the target, they use the paper plates to mark target robot to follow the directions. and assesses the success of his or her direc- points, but this time the directions will re- tions. Then students swap roles. They do quire the robot to make turns. Remind stu- Did the robot end up where you expected?

26 Measuring: From Paces to Feet 5 1 Were you close? How did you make your esti- kitchen is forward 3 paces, turn right. forward SESSION 2 ACTIVITIES mate? How far did the robot travel in all? How 18 paces, turn left; then how would you tell many paces did the robot take altogether? someone how to get from the kitchen to your bedroom? Preparing for the next session: Writing One important aspect of measuring is the directions with turns ability to combine many small You and the students might create a work- measurements into a total. Adding the sheet for recording the distances they have Remember when I wrote directions on the smaller.distances to find the full length of decided to pace off at home. board, telling a robot how to move? Today the robot's trip will help to lay the foundation you'll be doing that in your teams. When you for understanding relationships between write, you will be writing down the same thing small distances and longer ones. you would say to the robot. Choose a volunteer robot again. Place a Extension: Using the computer and paper plate on the floor fairly near you, the Logo language making sure that your demonstration robot can get to it with only one turn. Ask the If you and your students have access to whole class to visualize how the robot could Logo, using the Logo language on the com- get to the target. Write the directions on the puter is a logical extension and reinforce- board again: Fbrward 3 paces. Turn left. ment of these activities. Giving and respond- Forward 2 paces. ing to directions about turns and distances is similar to moving the Logo turtle around Ask students if they think these directions the screen. You can organize the task so that will get the robot to the paper plate. Then one student points to a place on the screen have the robot follow the directions to find and the other moves or programs the turtle out whether they work. Repeat the process to get to that spot. You can save the pictures with the whole class until everyone is clear or the programs and have students share about the task (see the Teacher Note: Con- and compare their solutions. necting oral and written directions, page 28). lir Some third and fourth graders, as you Extension: Practicing at home well know, aren't sure about which direction is left and which is right. Some may need a Encourage students to pace off distances at good deal of experience making turns before home to practice their estimation and mea- they can function smoothly on this task. surement skills. Have students pair up. Give each pair a copy What's the length of your bed in paces? What's of Student Sheet 3, Pacing and turning, on your path to the kitchen sink from your bed? which students will write robot directions for Suppose the path from your bedroom to the each other. Each team works cooperatively to

5 2 Robot paces 27 5 "" position the paper plate targets, and to write them dictate one direction at a time, write it the directions for the robot. The robot then on the board, and ask the robot to follow just %1TEACHER NOTE paces the distance as directed. At this time, that part of the directions. Connecting oral and the directions should :nvolve no more than one turn. When you think the class has had Give each pair of students a copy of Student written directions enough practice, call them back together for Sheet 4, Making many turns. a discussion. When students give oral directions and Working in pairs, write a set of directions to commands to a "robot," they can see the ef- Was it hard to write down the directions? I get your robot from the same place our robot fect of their words immediately. Creating a suspect for some of you it was harder than started to the same target where our robot series of written directions, however, involves saying them. What are some good strategies ended up. Try to find some different routes to taking on another person's orientation in for deciding what to write down? get from here to there. The only rule is that space. Telling the robot whether to turn right you have to make the robot turn more than or left when the robot is facing in a different It's important that students talk about their twice. direction is very difficult. strategies. Let them share their methods with their classmates. Students are to write directions that involve Sometimes students need time to realize that more than one turn. Some may have trouble the directions they write down are just the doing this; it is a complicated task. Further preparation: Making many turns same as those they would have said out loud. It may help them to think through the Now we're going to make up some more com- effects of one move at a time, jotting down plicated directions. Let's pick a starting point the directions as they think them. in the classroom and a target. Michael, be our Sometimes students may need to write out robot and start from here (the corner of my and then check their directions. desk]. We're going to write directions to get the robot to (the flsh tank], and we're going to Orientation in space is a complicated issue make him turn were than twice along the way. for students of this age. Some are better able OK? Araceli, write ow directions on the board than others to visualize the movement of as we decide what they should be. What should someone else in space. Many students will we tell the robot first? Forward how many need more experience before they can visual- paces? And turn which way? And now? ize turning left and right from another's point of view. These variations in ability are ty0cal Lead the students to dictate directions that for third and fourth graders. Working care- they think will get the robot to the target. If fully through this investigation and listening they seem ready for it, have them dictate all to other students describe their visualization the directions first, and then ask the robot to methods will help students who are more follow those directio Is and see if they lead to hesitant. the target. If, on the other hand, your students seem to need more experience, have

28 Measuring: From Paces to Fee SESSION 3 ACTIWTIES TEAM ROUTE TOTAL PACES

Collecting data: Planning and Elena and Kelly Forward 2Right Forward 7 Left Forward 9Right Forward 5 23 comparing robot routes Barbara and Kim Forward 4Right Forward 2 LeftForward 4Right Forward 10 20 Today we're going to write directions for lots of robot routes. When we're finished we'll com- Shanta and Jeremy Forward 6Right Forward 3 LeftForward 6RightForward 10 25 pare those routes.

Give a copy of Student Sheet 5, Mapping and or on the overhead projector so that the Students' strategies for getting the longest measuring, to each pair of students. They whole class can see it, as shown above. distance 'nvolve overshooting the target and will work together to generate three routes Record only the shortest route for each pair. turning a lot, and will produce agoodlaugh from designated starting points to designated for the whole class. This part of the session targets. Before they begin, help them decide, Whose route is longest? Whose is the shortest? as a class, on the three starting points and should result in some shared enjoyment, but Is there something special about the shortest more important, it will give the students a three ending points. Write these on the board route of all? as they are confirmed, and be sure each pair sense that they know how to give directions, with turns, using measurements to describe records the starting and ending points on the Follow the same procedure to find the student sheet before devising their routes. distance. This activity is a good place to use longest route any pair has developed. calculators: have one or two students pace a Compare and discuss these routes. Students then add up the total number of route while the others add up the paces they've used. paces in each route. Let them use calculators We've spent time finding the very shortest to check and compare the distances of their route and the very longest route you wrote routes. down. Can you find an even longer one? Yes, Ray, come up and show us. Carl, you come too. Describing the data: ... Can anyone think of an even longer one? Finding the shortest route Talk with your partner and together write down the very longest route from [the desk] to We've found a lot of routes! Let's look first at [the globe] that you can think of. routes between [the office] and [the library]. Choose your team's shortest route, and then You'll find that the students can generate we'll compare to see whose is the shortest of even longer routes once they start to have all. One of you read your route aloud. fun with the task. Just as you can always find a larger number by adding one more, As one student reads the pair's shortest you can always find a longer route by adding route, record it in a large grid on the board some turns or some backward steps.

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What happens tape comes in thin rolls of different pace, and students need to agree on a single widths, available in stationery stores, method. In this investigation, students spend Students pace routes to distant landmarks office supply stores, and many drug a good deal of time arguing about and defin- in the school and compare their results. stores and five and dimes. A roll ing their preferred methods of measurement. Differences in results lead to an attempt at typically costs less than a dollar, and Support them by asking for demonstrations standardization of measure by finding the three will probably be enough for all the of their proposed methods, by recording on middle-sized pace of the students in the activities in this unit. A width of about the board their final decisions about a classroom, and using that as a standard one inch works well. method of measuring, and by calling their pace. attention to questions that may arise during the data collection because of the method The activities take two class sessions of Important mathematical ideas they have chosen. about 45 minutes each. Finding a standard of measure. The Displaying data and finding the middle. "middle-sized pace" is proposed to students After students compare their paces by direct What to plan ahead of time as a standard that will reduce the variation measure, they need to find ways of in their measurement of distances. The focus displaying and organizing their data so they VDuplicate Student Sheet 6 (page 78) for each pair of students (Session 1). of these sessions is on finding the advan- can determine the middle-sized pace. In this tages of standardizing their paces a ld investigation, students consider methods for Have calculators available (Session 1). achieving more uniform results. finding the middle of a data set. Help the V students talk about these methods and y Provide adding machine tape and Defining and refining measurement meth- refine their ideas about what is the 'imiddle" scissors (Session 2). Adding machine ods. There are different ways to measure a of the data.

Pacescome In different sizes 31 3s 59 SESSION 1 ACTIWTIES other pairs share those difficulties? cations, they will have many theories about why their results are different. Some of them Ask students questions that encourage them may want to make a standard pace and use Considering the problem: Pacing to reflect on their experiences. In some cases, that; others won't be bothered by the varia- to distant targets they will have found that their numbers of tion in results. paces did not match over long distances. We've been measuring distances in our class- Students use a variety of methods to solve room by pacing them. Today you'll be pacing that problem, including writing down each Describing the data: Comparing routes person's paces and trying to figure out why much longer distances within the school build- Your students may be interested in compar- differences occurred. ing. We'll all agree on three targets to pt ce to. ing their actual routes. The routes them- but they can be anywhere in the school. How selves can be recorded (as in the previous will this be different from pacing inside the In order to compare results within the whole sessions) and compared on a large grid. One classroom? class, ask students to count their total paces to each of the locations. Be sure that third grade teacher used the following format Have students pair up; give each pair a copy calculators are readily available. in her classroom. Her students had used the of Student Sheet 6, Pacing to distant targets. Logo language in their computer classes, and As the class chooses three targets in the What is the closest target location? The far- they suggested using the Logo abbreviations thest? The middle location? Did the number of school, write their choices on the board. for Forward, Right, and Left: paces surprise you? Students then write each target in a box on Rayi Kim FD 19 RT FD 52 RTFD 19 RT FD 28 Student Sheet 6. One of the important aspects of statistics is Susan/ChrisFD 24 RT FD 63 RTFD 15 RT FD 30 the study of variation, or the differences In each pair, one of you will be the pacer and among related data. For one of the target lo- Bobby/SeanFD 29 RT FD 80 RTFD 46 RT FD 45 one the recorder. As you make your way to the cations, make a line plot ind record the target, you will record both paces and turns. How are your routes all the same? How are number of paces that each pair of students they different? Do you have some ideas about Once you get to the target, return to the class- took to get to that location. room with your Jobs swappedthe one who why the results might be different? Suppose paced will read the directions to the pacer, and What do you notice? How can you describe we wanted to give directions to a school visi- the one who wrote will follow those directions these data? Are the distances pretty much the tor, explaining how to get from the front door and d the pacing. Your right And left turns same? Or are they spread out? Are there of the school to our classroom. Ray and Kim will be different, but see whether your paces clumps? Outliers? What might explain the might tell our visitor to take 125 paces, while work. things we notice about the data? Susan and Chris might say to take 138 paces. How could we make these direztions more Help students talk about the data and the similar, so that our visitor would be sure to Recording and describing data: variation (see the Teacher Note,Statistical find our classroom? How many paces did we take? words to introduce as appropriate, page 33). Chances are your students will suggest some What happened as you paced these longer dis- As students look at the variation in the total rules for pacing; if they don't think of using a tances? Did you run into any problems? Did number of paces they took to get to these lo- standard pace, you can suggest it. If

32 Measuring: From Paces to Feet 60 6 .1 students propose using standard measures students must judge whether there are such as inches or feet, you may want to TEACHER NOTE values that really don't seem to fit with the respond, "yesinches or feet would be a rest of the data. For example, in the pace good way. But let's focus on paces for today, Statistical words to introduce data, the measure of 11 paces for crossing and look at other measurements a little as appropriate the room seems to be an outlier. later.* Students may have better ideas than we do, though, so be sure not to shut off Range and outlier are two statistical ideas Both range and outliers are ideas that will discussion. that come up naturllly while discussing data come up naturally as students work with with students. different data sets in this unit. They can be introduced as soon as they arise in the stu- The range of the data is simply the interval dents' descriptions of their data. Students from the lowest value to the highest value in easily learn the correct terms for these ideas the data set. The range of the data in the line and are particularly interested in outliers. plot below, showing how many paces it took Outliers should be examined closely. Some- each of 15 students to walk the length of the times they turn out to be mistakessomeone classroom, is from 11 to 19: counted, measured, or recorded incorrectly but other times they are simply unusual val- X ues. Students are usually very interested in

X building theories about these odd values:

X X What might account for them? X X Because of the nature of the data students X X X collect in this unit on measuring, you are not 12 13 14 15 16 17 18 very likely to find significant outliers in most of the data sets. You will, however, need to An outlier is an individual piece of data that be alert to the possibility of outliers in data has an unusual value, much lower or much collected over greater distances, since small higher than most of the data. That is, it "lies differences in measurement units become outside" the overall shape and pattern of the large differences when the distance itself is data. There is no one definition of how far greater. Outliers may also appear in data away from the rest of the data a value must collected at home. Extremely tall or ex- be to rate mention as an outlier. Although tr-nely young brothers and sisters can skew statisticians have rules of thumb for finding the foot-size data, and students will enjoy outliers, these are always subject to judg- talking about why someone's foot size might ment about a particular data set. As you be so small or so very large. Third and fourth view the shape of the data, you and your graders enjoy contemplating extremes.

Paces come in different sizes 33 6 3 SESSION 2 ACTIVITIES Collecting data: Measuring paces Organizing and describing data: Looking at everyone's pace How could we find the length of everyone's Considering the problem: pace? Let's start out by finding how long my As the groups finish, tape their pace-tapes Finding a standard pace pace is. How could we use this paper tape? randomly to the chalkboard. Ask students how they propose to find the middle-sized When we talked about giving directions to help Solicit ideas for measuring your pace di- pace. Among their suggestions, they may in- a visitor find our classroom, a number of you rectly. Take two or three paces and freeze clude ordering the tapes by size (smallest to wanted to find one pace for everyone to use Have two student volunteers mark your pace largest, or vice versa). Try not to accept that when they walked, so that our directions on the tape and use scissors to cut it to size, suggestion too quickly. Wait to hear many would always come out the same. Today we following directions from the class. There is ideas, and ask students to demonstrate or will look at everyone's pace and try to find lots of opportuinity for discussion here, and explain so that others understand their the middle-sized pace to use as a standard. students need to agree on a single method methods. When the discussion has run its That way you'll all be able to give similar before the volunteers cut the tape. Will they course, enlist volunteer help in ordering the directions. How could we go aboutfinding the measure toe to toe or heel to heel? (A sur- pace-tapes by size. middle-sized pace? prising number want to measure toe to heel!) Be sure that they give reasons for choosing a Analyzing the data: Finding Listen to all their ideas about solving this particular method. You may need to drama- problem (see the Dialogue Box, Ftndtng the tize the differences between suggested meth- the middle-sized pace ods by having the volunteers cut many dif- middle-sized pace, page 35). List ideas on the Now, how cou1.1 we find themiddle-sized pace? board as they come up so that students can ferent tapes for your pace, follow4,..g the dif- develop and modify each others' plans. This ferent methods, and then compare them. Third and fourth graders have some good should be considered an introductory dis- ideas about finding the middle of a set of Once the class has settled on a method, cussion to get them thinking about the data. Probe their ideas and question them so establish working groups of two or three and problem: they need not settle on a plan at that their reasoning becomes clear. For an give each group a pair of scissors, a pencil, this time. example of the discussion in one classroom, and a length of adding machine tape. Helping one another, students in a group cut see the Dialogue Box, Ftnding the middle- Nir Some of your students will suggest that sized pace (page 35). you standardize these measurements by us- the adding machine tape to the length of ing inches or feet. They will "get it"! Others, each individual's pace. Writing names on the Elena says to look for the middle. But how though, won't even think about standard pace-tapes will help later when students could we do that? Which one is in the middle? units of measurement until later in this unit. compare sizes. How can you tell? Can you be sure? What is For those students, it's important to hold off the middle-sized pace? on the standard units of measure for a while. Elena If someone suggests inches or feet, you might r\Taierie After your class has agreed on a method, respond, 'That's a good idea, Jeremy, but mir identify the middle-sized pace. Allow some let's wait. We'll use them later." time to talk about any surprises. Is the

34 Measuring: From Paces to Feet

4 65 middle-sized pace very different from the SAMIR: Yeah, that sounds neat! others? Are the pace-lengths clumped? "MIALOGUEBOX What, Sarnir? Finding the middle-sized pace Using the data: A standard pace SAMIR: Likemine, Elena's and Valerie's. This discussion varies from classroom to Mine is smaller than Elena's and Valerie's is Now that we have agreed on the middle-sized classroom, and teachers make decisions bigger than Elena's, we can put them in a pace, let's make copies of it with adding ma- spontaneously about what ideas to support. row like that. chine tape so that all of you can use it to mea- Because classes are so different, there is no From smallest to largest? sure distances in the classroom. If we measure "best" way to proceed: we include here ex- the length of the room with the middle-sized amples from two classrooms to illustrate dif- SAMIR: Yeah. pace, will our results vary? Let's try it. ferent approaches, both workable. How would that help us find the middle size? Students may want to mark the middle-sized The first conversation happened after stu- TAMARA: Because the smallest one, you can pace on the floor so they can practice stan- dents had paced to predetermined targets in start from there and go until you get to the dardizing their paces. Keeping copies of it the school. They found that while the direc- one in the middle, and that could be the will allow students to settle debates by refer- tions they wrote down worked for the one middle size. ring to the tape itself. In this way the middle- who originally paced the distance, they did Any other ide&.o help us find the middle- sized pace will truly function as a standard not always work for the other partner. The measure for your class. teacher asked the students about that expe- sized pace? Set some other measuring tasks that involve rience. BARBARA: I was going to measure them and then try to find the middle. pacing distances with the new standard. How many of you had the problem that your partner's paces weren't the same size as yours, So actually write down how long the longest Let's see how far it is to the school office in so the number didn't come out right? What do one was and how long the shortest one was standard paces, and how far to the bathrooms. you think we could do about that? and all the ones in between? So it's kind of the At lunchtime, see how many standard paces it same thing, isn't it? takes to get to the cafeteria, TAMARA: You can measure our paces, measure all of them, and find the biggest and In this classroom the teacher expects stu- the smallest and then you get the middle dents to give reasons for their ideas and to Extension: Measuring objects one and try and make everybody take that listen to each other so that they can relate with the pace-tape size of a pace. their ideas to those of classmates. She does Students can also use the standard pace- BENJAMIN: The ones with the middle-sized not force a method onto the class, but solic- tape as a ruler to measure objects in the pace come to the front of the room and walk. its methodology from them, repeating and classroom. They might measure things such Then we can see who has the middle. sometimes asking questions to help them clarify and expand on their own ideas. as the teacher's desk, the chalk tray. the CHRIS: Rearrange it, the tall ones on one end, bulletin board, the windowseven the and it goes down to the other ones that way. (Dialogue Box continued) heights of their classmates.

66 Pacescome in different sizes 35 67 In a second classroom, the students did not And how would we say that? want to select one pace as the middle one, SARAH: In between here and here. (Points to but instead kept selecting a range of values 14 and 18.1 as their middle. Because this was so important to them, the teacher did not push for Can we be more specific? the selection of one middle-sized pace. JONATHAN: Say it's 17-1/2 inches, because it is. During this discussion, the students were looking at a line plot that showed how many How do you know that? paces each of them took to cross the room. JoNAMAN: It's mine and I measured it to be sure. X

X X How do you know yours is the middle? X X X X X X JONAITIAN: It's in the middle group. X X X X X X X X X SARAH: But mine is too and it's shorter than 12 13 14 15 16 17 18 19 20 yours. JANE: These paces are the m4ririle ones. (Marks under 14, 15, and 16 on the chalk- lir Most of the students really did not want board.) to select one and only one pace for their Is there one pace that's in the middle? There middle-sized pace, and the teacher was are 18 paces in all. Can we use that to find the willing to let them define their own terms. Like the teacher in the first classroom, she is middle? supportive and challenging, asking students SARAH: This is the middle one. (She points to to give their reasons and explain their state- 16.) ments; although she makes suggestions, she lets the students set the terms of the task. KEVIN: Here. 1, 2, 3. 4, 5, 6 ...these are the big paces (indicates 12 to 14) and 1, 2, 3, 4, The student who had measured his own pace 5, 6...these are the small ones (indicates was unable to see that his pace might not be 18 to 20) and 1, 2, 3, 4. 5. 6...these are the only middle value in the distribution. the middle ones (indicates 15 to 171, all six of The teacher let this go for the time being, them here. knowing that other methods would probably What if we wanted to tell someone how long appeal to him more in the long run. our middle-sized pace was? MAR1A: Just tell them in between our small- size ones and the big-size ones.

36 Measuring: From Paces to Feet 68 Measuring: From Paces to Feet

PART 2 Using standard measures

69 ARE OUR FEET A FOOTLONG?

INVESTIGATION OVERVIEW

surement data at home and display it at What happens y Have index cards or stick-on notes available for every student. school. The combined data give them a great Students look for foot-long feet at home, deal to work with, and they will need to spend time thinking about what it shows and comparing household members' feet to a Important mathematical ideas foot-long measure. these data are displayed talking about it in class. Encourage them to and analyzed to see how people's feet com- Using a standard measure. Students use an use sketch graphs to organizetheir data. pare to a "measuring foot." Inthese sessions uncalibrated foot-ruler (a "footstick") to Theory-building. Third and fourth grade of other peo- they learn an imaginazy story of the devel- compare their feet and the feet students often generate theories about the opment of the foot as a measure and specu- ple in their households. The emphasis in this data they collect. Sometimes these are data- late aboat reasons for its relatively large size. investigation is on comparison with a stan- based, sometimes not. It is important that dard (12-inch) foot. In the data collection ac- they learn to ask questions about data. What The activities take two class sessions of tivity, students internalize the size of the do the data tell us? What do they not tell us? about 45 minutes each. foot-ruler by using it frequently and by com- of While encouraging students to build paring objects with it. Rather than think theories, help them keep their theories the foot as a length that is broken up into What to plan ahead of time grounded in the data they collect and smaller pieces, we concentrate here on the analyze. III VObtain or prepare unmarked foot foot as a unit. Support student.s who decide measures for each student,following the to measure other objects in addition to hu- guidelines in the Teacher Note, Rulers, man feet. footsticks, and inchsticks (page 42). y Duplicate Student Sheet 7(page 79) for Collecting, displaying, and analyzing each student (Session 1). compaTative data. Students collect mea-

Are our feet a foot long? 39 70 7 SESSION 1 ACTIVITIES The King's Foot The king knew that he would have to tell the royal carpenter how large to make the stall. ONCE UPON A TIME there was a king who So, using heel-to-toe baby steps, the king Introducing the problem: The story kept ponies. His daughter, the princess, had carefully walked around the mare. "...5, 6, of the king's foot a little pony of her own that she dearly 7, 8, 9 feet long,* he murmured, "and 3. 4. 5 loved. As the princess grew older she grew Earlier we measured distanceslike the feet wide." The king jotted down the bigger, but the pony did not. The day came numbers: 9 feet long and 5 feet wide. The length of our classroom, and how far it is from when she climbed on her pony and her feet our room to the front door of the schoolby message was sent to the carpenter. and she dragged on the ground. That was the day the set to work at once. pacing them off. We also tried using giant king decided that he would surprise his steps and baby steps. Here's a story about a daughter with a beautiful new full-size Soon the stall was ready and the king sent king who measured the same way. Listen to horse. for the mare. He thought he would have a find out how well it worked for him. little fun with the princess, so he had the The king went himself to the best stable in royal groom hide the mare behind the barn. Read the story in the box (at right) as a the kingdom and chose a sleek Arabian discussion-starter. Then he said to the princess. "Come with me mare. "Because it's a surprise," the king and see if you can guess your surprise." said, "I want to leave the mare here at your stable until I can get a new stall built in the Together they walked into the royal barn, royal barns to fit such a grand, large horse." past all the stalls of little ponies, and stopped in front of the empty new stall. But no sooner had the princess inspected the new stall than she burst into tears. "I truly hoped that my surprise would be a horse. because I have outgrown my little pony. But now that I see the size of the stall. I know that you are just giving me another little pony, no larger than the first." The king was puzzled. He saw that indeed. the new stall was much too small for a full- size horse. The groom quickly brought the new Arabian mare out of hiding, and as soon as the princess laid eyes on her, she forgot her tears. Only the king did not forget. He called angrily for the royal carpenter to account for her terrible mistake.

40 Measuring: From Paces to Feet 72 Collecting data at home The carpenter was shocked. She knew she After your reading, allow time for discussion. was good at her trade; her work alwaysdrew Some students may connect the story with Do you think anyone's foot is really one foot high praise. And she had made the stall just their own recent measurement experiences. long? Whose might be? as the king had said-9 feet longand 5 feet wide. What could have happened? Tell me again why the new stall was too small Third and fourth graders have feet that vary at first. How did the carpenter solve the in length. Most will be less than a foot long [Pause and ask the students what could have problem? Why did she call the sticks she made in fact, most adults' feet are less than a foot happened; then continue.] "rulers"? Does this remind you of anything long. Provide time for students to speculate that's happened in the classroom? Have we about whose feet might be much longer than The carpenter stared sadly at her work. She had some experiences like the carpenter's? theirs. Be sure to allow time for them to give paced thoughtfully around the little stall, their reasons and to talk about their ideas. carefully counting her foot-lengths. Then she Collecting and analyzing data: sat down beside the king to thins. Tonight you'll be looking for a foot-long foot at Measuring our feet home. To check foot length, you will compare Sitting there, staring at her feet, that was Pass out the uncalibrated foot-long rulers everyone's foot to the foot-ruler. I'm going to when the carpenter noticed something give you a form for recording what you Ilnd. when she saw the king's foot next to hers. and explain that these "footsticks" are one official measuring foot long. After their recent "That's it!" she cried. "Your foot is much Give out copies of Student Sheet 7, Data longer than mine! I made the stall 9 feet experiences in measuring, students should have a solid idea about the advantages of a from home: Foot size. Explain how to use the long, but I used 9 of my feet instead of 9 columns to record the name and age of each standard unit of measure. king's feet." person whose foot they measure. They can Ask students to compare their own feet with start by entering the data they found when Then the carpenter had a truly remarkable they compared their own feet with the foot idea. She took a flat stick of wood, and she these rulers and find out whether their feet are shorter than, the same length as, or ruler. That is, they should write their own cut it just exactly the same length as the name and age in the appropriate column. king's foot. "This way," she told the king, 1 longer than the ruler. Display their results can always know exactly howbig you want on the chalsboard in a format like this: things made." Shorter than Equal to the Longer than Extension the foot ruler foot ruler the foot ruler Now the carpenter made a stall for the new Benjamin Susan The history of measurement is fascinating to horse that was 9 king's feet long and 5 Shanta some students, and they may want to find king's feet wide. This time the stall fit Elena out more about how our measurements have perfectly. So the king was happy, and the Chris been developed and are enforced. Two princess was happy, and the carpenter was What do you see from these data? What can we Teacher Notes, Bureau of Standards (page happiest of all. She started a factory and say about your feet? Does that meanthat 43) and Sixteen left feet (page 44) provide made lots of sticks just as long as the king's everyone's feet are the same? What does this background information: you may want to foot, which she called rulers. Selling these display tell us? What doesn't it tell us? share some of it with your students. sticks, she became rich and famous.

Are our feet a foot long? 41 7 4 ad\ or shorter than a foot. Indeed, this footstick fourths of an inch sometimes add useful TEACHER NOTE makes simple comparison much easier. information. When your students measure and compare Rulers, footsticks, and the lengths of human feet, both in class and We encourage you to keep the measurement inchsticks at home, they can make a simple "shorter/ tools simple at the start so that they engage longer/equal to" comparison without the all the students in your class. The National A typical school ruler can be visually confus- distraction of the inch-marks on a typical Council of Teachers of Mathematics' Curricu- ing to many students because of the profu- ruler. You can help your students internalize lum and Evaluation Standards (Reston, VA: sion of lines. When a skill is being developed the length of a foot by using the footsticks to 1989) suggests that teachers observe or re-taught, it is important not to introduce measure a variety of objects in the classroom students' work in mathematics in order to confusing extra information. Although the after finishing this Used Numbers unit. learn what they know and how they think usual ruler is a good tool when students un- about mathematics. The measurement activ- derstand the concept of measurement and Similarly, when inchsticks were used in one ities in this unit allow you to engage your see a need for fractional markings, the sim- third grade class, the students found they students in conversations that will help you pler the better at the beginning. could easily tell the lengths of objects by to assess their measurement skills in a real counting inch-squares on the inchsticks. context. As they work, talking with small For that reason, you will need for this unit They were relieved not to have the "extra" groups about the task and their approaches unmarked foot-long measures (footsticks) lines found on the school ruler, and the al- to it will give you a great deal of valuable ob- and foot measures marked in whole inches ternating colors of the inch squares made servational information. Keep an eye out for only (inchsticks). The footsticks are used in counting even easier. Fractional parts of an the students who can easily handle the more the investigation, Are our feet a foot long?, inch were introduced gradually, as needed, sophisticated tools as well as those who are and the inchsticks in the investigation, Using when the teacher felt that students were struggling. You will find that the knowledge a smaller unit. ready. Working with whole inches before in- you gain from these observations will help troducing fractions gave students a firm base you to plan follow-up measurement activi- Appropriate rulers are commercially available for their understanding of measurement. ties. from Dale Seymour Publications. Alterna- aradually they came to see that halves and tively, use the patterns on pages 80-81 to make your own. A 9-by-12-inch piece of tagboard can be cut into nine footsticks. Inch-square graph paper is useful for making inchsticks. When laminated, these rulers can be used over and over. The footsticks were named by a fcurth grader who enjoyed the fact that they were exactly one foot long, and that he could tell immediately whether something was longer 1- COSCK

42 Measuring: From Paces to Feet 77 and Technology, Office of Weights and Now, however, all linear measurements TEACHER NOTE based on the yard have been calibrated to Measures, Room A-617, Gaithersburg, MD 20899; phone (301) 975-2000. Bureau of Standards: Home of the meter. Even though our country has not entirely converted to the metric system, the the official U.S. measures yard is now measured in metric terms. And exHIrr Room although there is a physical model of a meter Bureau-of17`anciaredS Irb This background information is provided bar (the Official Meterl at the Bureau of for your interest; share it with your class as Standards, the meter is actually defined in you deem appropriate. terms of the speed of light! That is, a meter is the distance light travels in a vacuum in The Bureau of Standards in Washington, 1/300 millionth of a second. Clearly we've DC, sets the standards that regulate mea- moved a long way from the original, time- surements. Within the bureau, the Division honored measurements based on the body. of Weights and Measures regulates all the measurements used in tradepounds, me- Weights are still kept at the Bureau as ters, kilograms, yards, gallons, and so forth. physical models, so it is possible to see the Official Kilogram. Just as the yard is now If there is a disagreement about the amounts measured in terms of the meter, the pound is being weighed or measured, the people who calibrated to the kilogram. It is ironic that we disagree can seek help from the Division of claim not to be converting to the metric Weights and Measures in their own state. system when our English measures are now The federal office coordinates the states and determined metrically. enforces interstate regulations. The goal is to make sure that items sold (ranging from fuel The history of weights and measures oil to oatmeal) meet the packaging standards includes many interesting tidbits. For set by the Division. example, Napoleon mandated the metric system because it was more systematic and In order to enforce the law, there has to be a scientific than the body-based systems used place to settle disagreements about what a in France. Its abstraction and rationality fit pound really weighs. At one time, physical perfectly with the growth of rational science models of the official yard, foot, and pound at that time in history. were all kept at the Bureau of Standards, so { FIRST OFFICIAL YARD I that various measuring devices could be English measures, too, have an interesting *set* against them. These measures were history. Your students ;nay want to write for housed in special vaults at the Bureau. information about the history of weights and Copies of them were found in the Museum of measures in the United States. Address such the National Institute of Standards. requests to National Institute of Standards

43 78 Are our feet a foot long? 79 TEACHER NOTE: their left feet one behind the other, and the SESSION 2 ACTIVITIES length thus obtained shall be a right and law- Sixteen left feet ful rood to measure and survey the land with, and the 16th part of it shall be a right and Recording the homework data: Although people have been measuring in one lawfulfoot." * Foot sizes way or another for many thousands of years, the use of standard measurement is rela- Your students may want to explore the Ask how many students found a foot-long tively recent when all of human history is length of their own rood and compare theirs foot at home. Count the number and ask taken into account. That is, measures have with another class's rood. How much whose feet those were. been standardized only in the last two thou- variation in result is theie if they choose six- sand years. Measurements were originally teen students by drawing names from hats? We're going to be putting all our data together based in the human body and became Fourth graders particularly might want to so we can compare evezyone's results at once. standardized when people needed a clearer play with some of these possibilities. You need to make a card for each person way to communicate their results (often, whose feet you measured. If you have students who are particularly because they were written down and seat to Students write the name and age of each different places). For example, directions for interested in history, the history of mea- suremen .s opens a real door to the past. person whose foot they measured on a cooking once relied on "handfuls" and stick-on note or an index card. If some "pinches," but these have evolved into "cups" They may want to look up other early defini- tions of measures. How did the "foot" as we students have measured a lot of people, and "teaspoons"measures whose volume is those who measured fewer can help them specified and standardized. know it evolve, really? How was an acre defined? Many encyclopedias offer information record all the names and ages. While Measures of length have likewise evolved on the evolution of measurements. students are doing this, replicate the standards that are not subject to human columns of Student Sheet 7 on the board. variability. In this excerpt from an old ency- 'From Compton's Pictured Encyclopedia, 1929 edition. Chicago: Shorter than Equal to the Longerthan F.E. Compton & Company. We are grateful to Margaret W. the foot ruler loot ruler the foot ruler clopedia, you can get a flavor of the problems Maxfield of Kansas State University, whose article "Sixteen Left Feet" in The Best of Teaching Statistics, r ter Holmes, ed. Shawn4 Tara 17 Daddy 34 of living in a non-standardized world (and (Sheffield, England: Teaching Statistics Trust. 1986). brought Janie 12 Rolf 24 some of its joys!): this to our attention. Students then stick each name card in the A good idea of the inconvenience caused by correct column on the board, showing the lack of absolute and invariable standards whether the feet they measured are shorter is furnished by a German treatise on survey- than, the same length as, or longer than a ing of the 16th century, which instructs the foot ruler. surveyor to establish the length of a rood thus: "Stand at the door of a church on a Describing the data: Looking at Sunday and bid 16 men to stop, tall ones and :he display small ones, as they happen to pass out when the service is finished; then make them put Count the number in each category. Ask

44 Measuring: From Paces to Feet 81 student pairs to discuss the chart together size. Is there another group whose feet they and to make a statement about what this might measure? They may be interested in chart shows them and what it doesn't show. some tall people's foot sizes. Ask the pairs to share their statements with the class. Some students may have powerful Robert Pershing Wadlow, the world's tallest beliefs about what they can see from these man at 8 feet 11.1 inches, wore size 37AA data. (See the Dialogue Box, Measuring feet: shoes. His feet were 18-1/2 inches long. Are older feet bigger feet?, page 46.) Zeng Jinlian, the world's tallest woman, had The next step may be to tally the results. If 14-inch-long feet. you think it would be helpful to your students, rwlace each card with a tally mark in These facts come from the Guinness Book of the appropriate column. You could use this World Records, edited by Alan Russell and opportunity to demonstrate the convention Norris McWhirter (New York: Bantam Books, 1988). for tallying by fives (144.n.

Analyzing the data: What do we know about foot size? What can we tell from these data? Are there any surprises? Are there any patterns? 'pa you see anything you consider unusual? Encourage students to talk about what they see in this collection of data. Some have commented on the large number of people in the "smaller than a foot" section. Others have generated theories about foot growth. This discussion can go in many different directions. Your students theories and ideas may surprise you!

Extensions Some students may want to conduct more Sneakers research and collect more data about foot

8 Arnr feet a foot long? 45 Me. too. them. Her final question helps students "M1ALOGUEBOX focus their ideas and allows them to look at a SHANTA: But the ones that are older have new question in this context. The misconcep- bigger feet. Measuring feet: Are older feet tions about growth and foot size are tabled, bigger feet? BENJAMIN: But the oldest person has the as the teacher decides to deal with them in a smallest foot. later science lesson. This third grade class has brought in a great SU-MEI: When you are in the middle of your deal of data from home. They compared age you have the biggest foot. people's f.tet with the footsticks and have recorded each person's foot as longer than, RAY: Whenever you get older and the body shorter than, or equal to one foot. They have stops growing, the foot stops growing. spent time putting their data together. Here BARBARA:When you're young, your foot is is a partial table of their results. (Note that short; when you're in the middle, your foot is they actually looked at a huge display of all longer; and when you're older, your foot is the names and ages of the people they mea- shorter. sured, and then counted to find the totals.) RICARDO: The body never stops growing. It Foot length Total Name and age makes new cells all the time, and that shorter than a foot 74 Jamie 3, Peggy 10, Bobby 9, makes it grow. Making cells makes It grow. Sean 9, Mom 27, ... So what do you think happens to your foot equal to a foot 21 Benjamin 11, Joseph 18, size? Serena 37,... RICARDO: It'sstill growing but it just doesn't longer than a foot 33 Dad 34, Grampy 67, Bobby's show that it's growing. brother 19,... OK, so haw do you feel about thia zategory? The teacher wants to see what the students Where would you fit when you wtr2 real old? find in these data. KIM: In "longer than a foot." Do you have comments about these results? Let me ask you all a question. Could we have real old people in every category? MICHAEL: Wemeasured 128 people in all. SEVERALSTUDENTS: Yeah, you could. How do you get that number? MICHAEL: I added 74 and 21 and 33. lir In this discussion, students are expressing their ideas about relationships TAMARA: I'm surprised that there are more between age and foot size. Even though some people in the "longer than a foot" group than of the ideas sound very strange to adults, the in the "same as a foot" group. teacher allows the students to talk about

46 Measuring: From Paces to Feet 84 85 USING A SMALLER UNIT

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INVESTIGATION OVERVIEW

What happons Avoid using rulers with half- and y Provide material for making presentation quarter-inch marks, since those are graphs, including graph paper, con- Students establish a method for measuring confusing to many students at this age struction paper, stick-on dots, markers, their feet in inches, then analyze the data to level. scissors (Sessions 2 and 3). decide on a typical foot length for their class. They compare their class data with data on y The day before Session 1, remind stu- y Have available adding machine tape, the foot lengths of teachers in their school dents to wear clean socks the next day scissors, yardsticks, or measuring tapes anci describe the differences in these because they will be measuring their (Session 4). distributions. In order to practice using feet with shoes off. inches to measure objects longer than one y Provide blank paper and pencils to use Important mathematical ideas foot, they measure their heights in inches for sketch graphs (Session 1). and find the height of the middle- st:ed third Measuring with inches as a standard. Stu- [fourth] grader. y Before Session 2, make arrangements dents use calibrated rulers for more accurate with the other teachers tn rur school measures of foot length as they practice us- The activities take four class sessions of so that your students can to them ing inches as a measurement unit. about 45 minutes each. to measure their feet. You qit make a schedule for your students' visits and Measuring objects shorter and longer than double-check it with the faculty. the foot-ruler, and specifying lengths in What to plan aheau of time small unita. Objects smaller than a ruler yPrepare inchsticks (see page 81) or y After you have a line plot of the data can be measured mor- easily than those that provide rulers calibrated in whole inches students collect in Session 1, make are larger; with a small object, students can for every student (Sessions 1 and 2). copies for small-group work (Session 2). simply line up Me end of the ruler with the

Using a smaller unit 47 86 87 end of the object and read off the number on SESSION 1 ACTIWTIES about halfway, you may call it 3-1/2. It's all the ruler at the other end of the object. They right to approximate. can do this mechanically, without an understanding of what these numbers mean. Being Exploring a new tool: The inchstick Collecting data: Deciding how specific about a length that is longer than a to measure our feet foot involves re-placing the ruler, adding to- You've compared your feet with a foot-ruler tals, and understanding the relationship of and most of you found that your foot is not a Now we're going to use inches to measure our inches to feet. Repeated repositioning of the foot long. So we know that your foot is shorter feet. We'll write down everyone's results to see ruler echoes the iterative movements of the than a footbut just how long is it? How could what we can learn about foot size in our class. pacing students did to measure distances in we find out? The rulers you'll be using today earlier investigations. are marked off in inches, which have been Scientists collect data from measuring things, used for hundreds of years to measure things. and sometimes they hare to redo a whole Collecting and analyzing data to find An inch is about the same size as the first experiment because different people measured what's typical of a data set. Students joint on your thumb. things in different ways. Let's make some define their methods, collect data, and decisions about how we'll measure. display them, looking for ways to describe Pass out the inchsticks or rulers. Encourage the typical foot lengths in their class. As they students to find their own personal bench- Should we measure with our shoes on or off? look for ways to describe these data, they marks corresponding to one foot or one inch Would it make any difference? Where should develop a more extensive vocabulary for data (see the Teacher Note, Benchmarks on the we put the ruler? Should we stand up or sit description, including ways to define what's body, page 50). They can record this infor- down while we're measuring? Would that make typical of those data. mation as a group by making two posters: a difference? What if the length is a little more Benchmarks: One foot long and Benchmarks: than an inch-measure? Should we round down, Developing methods of displaying and One inch long. Another way to record the round up, or use fractional parts of an inch? comparing data. Students first display one information is to give students index cards set of datatheir foot lengthsand find the for writing down their personal benchmarks. Encourage students to try a variety of meth- typical foot length (for their class) in inches. Students might illustrate their personal ods to see how their results for the same foot Next they display and compare these results measurements to keep for future reference. differ according to the method used. Give with teachers' foot lengths. They draw from them a chance to talk about what they think previous experiences to develop techniques Ask students to try using their benchmarks gives a reasonable measure of foot length. for comparing sets of data. to measure some small things: a pencil, a Help them decide how to make the measure- paper clip, an index card, a pocket. They can ment standard for everyone. Write their then use inchsticks to check. If your stu- agreed-on methods on the board for dents get answers that are between two inch- reference. marks, you may want to discuss what to do. Ask students to measure their feet again, If it's between 3 inches and 4 inches, you following the established method, and write might pick the number it's closest to. If it's their results on a stick-on note or index card.

9 48 Measuring: From Paces to Feet Displaying and describing the data: finding patterns in the data: What's Although no answer is "wrong," some an- What do we see in the data? typical of our feet? swers are more typical of this distribution than are others. Record their results on the board as a What's typical of your feet? How can we say simple, unordered list: what we see? Are there clumps or clusters of When the students have reached some con- data? Are there any outliers? What else can clusions about typical foot length in the 7 inches you tell me about the lengths of your feet? class, write these down and keep them with 9 inches the data for use in the next session. Remem- Discussing what they see in their sketch ber to duplicate the data for small-group 8 inches graphs may provide the opportunity to review work, and to speak with your colleagues to 9i. inches statistical terms. Refer to the Teacher Note, find good times to measure their feet. Statistical words to introduce as appropriate 8 inches (page 33). Extensions We're trying to find what the typical foot Looking at our data, what would you say is the If time allows and you want to continue the length is in our classbut looking at all thew.: length of a typical third [fourth] grader's foot? activitc, ask students what shoe size they numbers, it's hard to tell what would be typi- wear (those who don't know may be able to cal. If we wanted to organize these numbers This discussion requires the teacher to be find it inside their shoes). Compare the dis- better, how could we do it? flexible, yet it is important that students' tribution of foot length with the distribution misconceptions be addressed. This is a deli- of shoe size. Are they identical? What might Working in pairs or small groups, students cate balance. As students talk about what explain some differences? make sketch graphs to organize the data, they think is typical, you will hear many showing the distribvtion of foot lengths in different theories and many reasons. Some You might also ask students whether they the classroom. These need not be painstak- students will choose the mode (the most can make a rule about the relationship being, but can be quite sketchy. Think of them frequent). Some will look for a range of tween foot length and shoe size. This pro- as first draft graphs (for details, see the values. Some will find a middle value (the videsgoodpractice in finding and describing Teacher Note, Sketch graphs: Easy to make. median). For typical problems that arise, see number patterns. quick to read, page 50). the two Dialogue Boxes, Describing data.: Getting caught in the range (page 51) and When they have finished making their sketch Discussing invented methods for finding graphs, ask them to think about what the typical values (page 54). graphs show, and write one sentence about their sketch graphs on the same sheet of pa- It is important to help students sort out per. When every pair is done, call them to- methods and answers that seem sensible gether for a discussion. and for which they have strong reasons.

Using a smaller unit 49 9 1 "How full?" and the second responded, while TEACHER NOTE looking at her thumb. "About two inches." %TEACHER NOTE Benchmarks on the body Once benchmarks on the body are estab- Sketch graphs: Quick to make, lished, it's important to use these measure- easy to read Carpenters used to make marks on their ments as much as possible every day in workbenches to indicate lengths that they school. Teachers have found ways to weave Graphing is often taught as an art of presen- often usedsuch as 2 inches, or 4 inches, or measurement into lining up for recess tation, as the endpoint of the data analysis 1 foot. These "benchmarks" speeded up the ("Everyone whose little finger is less than process, as the means for communicating measuring process, since the carpenter did 3 inches can line up now"), selecting what has been found. Certainly, a pictorial not then need to find a measuring tape or students for special jobs ("Anyone with a representation is an effective way to present ruler every time. In a similar way, handspan of more than 7 inches can help data to an audience at the end of an investi- "benchmarks" on the body are useful be- here"), or doing tasks at the end of the school gation. But graphs, tables, diagrams, and cause we often need to estimate measure- day ("Find an object that's between 5 and 6 charts are also data analysis tools. A user of ments when we don't have a ruler handy. inches long and put it away"). If you have statistics employs pictures and graphs fre- your students first estimate with their quently during the process of analysis as a You can help your students find such benchmarks and later use a ruler to check means of better understanding the data. benchmarks; for instance, an inch is about their estimates, you will be helping them the size of the first joint on the thumb. Ask internalize a system of measurement. That Many working graphs need never be shown students to check this out with their skill will help them tremendously, both in to anyone else or posted on the wall. Stu- inchsticks and see if it's true far them. They mathematics classes and in their daily dents can make and use them just to help can use the footsticks to find a place on the lives. uncover the story of the data. We call such body that is 1 foot long. One possibility is the representations used during the process of forearm. Or, they may find somethinglike a data analysis "sketch graphs" or "rough draft handspanthat measures 6 inches; two of graphs." these would equal 1 foot. 6v g7vvvvv Teachers who have done thishad students find benchmarks on their bodies and then qvvvv use those benchmarks to estimate other MC.71::1 lengthshave found that it helps their stu- 10 ty dents internalize the measures. They are able to estimate measurements with We want students to become comfortable reference to their own bodies and no longer with a variety of such working graphs. feel the need to rely on a ruler all the time. Sketch graphs should be easy to make and One third grade teacher overhead his easy to read; they should not challenge stu- students giving each other directions to pour dents' patience or fine motor skills. Unlike water into a glass. The first student asked, graphs for presentation, sketch graphs do J3 50 Measuring: From Paces to Feet not require neatness, careful measurement SAMIR: There are seven of those (marks or scaling, use of clear titles or labels, or 6690DIALOGUEBOX above the 9) and seven of the numbers on decorative work. the number line, three on this side and three Describing data: Getting on that side, so 9 is right in the middle. Sketch graphs: caught in the range VALERIE: I was thinking that 6 was in the can be made rapidly middle. These students have just finished measuring reveal aspects of the shape of the data the length of their feet in inches. Their col- Why do you think that? are clear, but not necessarily neat lection of data looks like this: VALERIE: Because if you count from 1 to 10, 9 10 7 9 9 8 9 8 8 9 6 is in the middle. don't require labels or titles (as long as 9 9 9 6 1010i students are clear about what they are Does it matter to you that there are more looking at) To organize the data, they decide to make a marks above the 9 than above the 6? don't require time-consuming attention line plot. VALERIE: No. I'm just looking at the numbers to color or design and it doesn't matter if there are more marks above one number than above the Encourage students to invent different forms other. until they discover some that work well in PEGGY: How can it be 6? 'Cuz 6 is way down organizing their data. Sketch graphs might at theleastpart, it's the last one, and the be made with pencil and paper, with Uniflx line goes to 10-1/2, sohowcan 6 be in the or other connecting rlubes, or with stick-on middle? So 6 is thelowestwe have in the notes. Cubes and stick-on notes offer flexibil- 6 7 8 9 10 10- 2 2 class, not the middle. ity because they can easily be rearranged. Looking at the shape of the data, they ob- I'm curious about some of your ways to find serve that it has a range from 6 to 10-1/2 the middle-sized foot. Let's try something to x inches. that most of the data are around 8 or help me understand. Would it change things if X Xx 9 inches (a clump), and that almost half the we showed the data this way?

X X X XX I 1 I feet are 9 inches long. At this point they X begin to focus on finding the typical foot, X 7 q /0 which they decide is the middle-sized foot. X

/o, X How can we find the middle-sized foot from us- One standard form of representation that is X X ing this line plot? particularly useful for a first look at the X X X datathe line plot--is suggested for use SAMIR: It's 9, because most of th c. people X X X XX X X 3 5 6 7 8 9 9 1 10 10111 throughout this unit. have 9. 4 2 2 Why do you say 9 is the middle size? (Dialogue Box continued)

9 ding a smaller unit 51 Does this change your ideas about how big the of numerals along the bottom and to focus SESSION 2 AND 3 ACTIWTIES middle-sixed foot is? instead on the overall distribution of data displayed above those numbers. To help CHRIS: Yes. It's 8 now because that's the them, this teacher shifted gears, and the Posing the question: How much longer middle number. Three matches with 11 and class ended up putting their foot lengths in 4 with 10-1/2,and 5 with 10, and 6 with are teachers' feet than ours? 9-1/2, and 7 with 9, and that leaves 8 in order from shortest to longest foot, finding the middle foot in the list, and calling that Remind students about the conclusions they the middle. the middle-sized foot. reached in their last session about the typi- ELENA: No, I don't agree. We don't have cal foot length of members of this class. people at 3 and 4 and 5 and 11. And there's This teacher was aware that students had lots of people at 9. So I think it's really more ideas to be shared, but she was not afraid to Today we'll be comparing your feet with like 9. push the students beyond what they initially another group of feet to see how the lengths believed. This combination of sensitivity and are similar and different. You said that the Well, we seem to have a lot of controversy. I willingness to challenge often seems to pro- way you'd describe the typical foot length in think it's important for you to think about duce the most thought-provoking class dis- our class was [9 inches). Do you think the whether we're looking for a middle number or cussions. teachers' feet in this school will be much a middle-sizedfoot. Which one were we looking longer or shorter than yours? What would you for? predict? Do you have an estimate of the length RAY: The middle-sized foot. of a middle-sized teacher-foot? Then it's important for us to think about how Collect and record students' estimates of to do that. What did we do when we found the typical teacher foot length. Ask for some of middle-sized pace? their reasons. MICHAEL: We put them in order and we found the ones in the middle. Collecting data: Measuring Can we do something like that here? teachers' feet What data do we need to collect to answer our wirThis teacher is trying to lead students question? How can we collect those data to away from looking only at the number line find out? and the range of numbers on it. Many in the class don't understand how to find the Generate ideas from the class fairly quickly. middle from looking at the graph (it may be They will need to specify a method before that this idea develops fairly slowly over collecting data, and they may want to do as time). The teacher realizes that they are they did when they measured their own foot- caught in the range, the numbers along the lengths. bottom of the graph. It is difficult for students to ignore the compelling, simple list Share your data collection schedule so that

52 Measuring: From Paces to Feet 9 6 97 students know when to measure teachers' or bar graphs. This is not easy work. Making Ask groups to share their conclusions and feet, Select and determine teams to measure decisions about how to uisplay and how to their data displays informally. Help them see and collect data, As students return, help compare two data sets is very complex. Allow where their ideas are similar and where they them to consolidate their results, a good deal of time for students to talk about are different. It may help to ask questions their ideas, to share their data displays, and designed to probe their understanding to draw some conclusiots in their small Organizing and displaying data: How groups (see the Teacher Note, Phases of data What U we added a class of kindergarteners? long is a typical teacher's foot? analysis, page 65). How would you show their feet too? What kind of results would you expect? You may want to talk about ways of finding As you circulate among the groups, ask stu- out what's typical. See the Dialogue Box, Dis- dents questions about their methods and Extensions cussing invented methods for finding typical their reasons. Bring their focus back to the values (page 54) for some ideas. question of comparison when you think it is Your students may be interested in making a Put all the teachers' foot length data up on needed. Help them relate their displays of giant chart displaying the foot size in one the board. Ask students to organize it to look data to their conclusions. You may need to class at each grade level in the school, or in for patterns. Cot spare methods and then ask spend time asking groups to tell you what comparing high school students with students to describe the typical teacher's foot their displays show, because some lose track themselves. Some may want to talk with length. Think about clumps and clusters of of the whole picture as they begin to display shoe store personnel about the distribution data, and whether there are some clear the details. of foot size in the adult population. patterns.

Analyzing the data: Comparing teachers' feet and students' feet How can we compare these two sets of data? rs-L, We have the data on teachers' feet now, and you have the data on your feet, Can you think of ways of comparing these data? Your task in f?, small groups is to find some ways of comparing these data sets and answering the question, "How much longer are teachers' feet than ours?" rs-L, Students can invent some wonderful com- 1 I I I I I I I I 1 1 parative techniques! Some will want to com- 6 61/2 7 7S 10 10/2. / pare mediansothers will compare line plots

Using a smaller unit 53 98 SU-ME1: Yeah, the clump seems like it's BENJAMIN: Yeah, but maybe more people "99DIALOGUEBOX crowded around 10-1/2. really have 12. We didn't measure all the teachers in the school. Discussing invented methods Yes, that's an interesting reason. I can see your reasons for both of these methods. Does So you're asking a new question: If you mea- for typical anyone have a good argument for choosing one sured more teachers' feet, would more of them over the other for the most typical value for come out around 12 inches? the teachers' feet? BENJAMIN: Yeah. X BOBBY: Well, even though 9 inches has the Ben noticed that four teachers had 12-inch X most, there are still only five teachers with X X feet; that's an important part of the shape of 9, but there are 12 teachers bunched these data. But Chris and Michael are arguing 10 11 111 12 88992 2 around 10, 10-1/2. and 11. that with these data, a typical value wouldn't So, looking at these data, how can we find the So you'd choose the biggest clump of data? be that high. typical length of a teacher's foot? [Later in the same discussion] ... JEREMY: But can we try measuring Mr. Crane and Mr. Diaz? They're bot.h really tall. CHRIS: We think you should pick a numbei' BARBARA: Our group chose 12. that comes up the most, so we got 9 because more teachers had 9-inch feet than What were your reason's? (IIIrb This kind of discussion can be a difficult any other number. BARBARk. A lot of teachers had 12, and also one for the teacher because while there is certainly not a single right answer, students it's the highest numbc.r. What does everyone else think about that may come up with unreasonable approaches. method? You chose the highest number? Encourage students' invented methods; VALERIE: We did the same thing. There are a BARBARA: Yes, because a bunch of teachers many of their ideas will lead to understand- lot of 95, so that seemed like what was had it. ing the idea of choosing a reasonable typical typical. value and will prepare them for later work What do some of you who picked numbers with standard measures of center, such as Did anybody mei" a different choice? around 9 or 10 inches think about that? median and mode. However, students should JENNIFER: We came out with 10-1/2. JENNWER: I don't think you should pick the not get the message that any method is as good as any other; expect students to reflect So your choice is a little higher than what highest, because that's not like what's typi- on whether or not their results are reason- Chris's group picked. Why do you think that's cal. Most teachers don't have feet that big. able. Juxtaposing one student method with reasonable? CHRIS: Yeah, like with our paces, the typical another, as the teacher does in this discus- SU-MEI: I don't know. It just seemed like that pace wasn't the shortest or the longest, but sion, is often a good way to help students would be it. somewhere in the middle. think about the reasonableness of their own a middle value is more typical. But I'm interested in your reason for 10-1/2. So you think methods. What do you think about thatargument,Ben? (Pause. Sttl: no response from the student.)I see a big clump ofdata between 10 and 11 inches. 101 54 Measuring: From Paces to Feet 00 SESSION 4 ACTIVITIES y Stand against the wall and have a friend Reporting on the data: Writing hold a pencil where the top of your head conclusions comes; then measure up the wall. After a discussion of their findings, have stu- Considering the problem: Finding the y Lie down and have a friend mark where middle-sized thircl [fourth] grader dents work in pairs to write the story of these your feet and head are; then measure data. Some may want to draw their class- the distance between those two marks. We've been measuring foot length in inches. mates; others may want to write a pen-pal Could we use inches to measure our height as Many confusions about standard measures letter describing their heights. Some well? How many inches tau do you think you will surface when students are measuring students may want to include foot data as are? Could we figure out how many inches tall distances longer than the ruler being used. well. Ask them to make a display of the data a typical third [fourth] grader is? How could we The Dialogue Box, I'm 48 inches and 1 cen- on their paper, and to write some sentences do that? timeter tall! (page 57), offers some ideas for that describe their conclusions and handling typical misunderstandings. observations. Think of the graph as the Allow time for a discussion of this idea. picture that prompts the story, and ask They'll be measuring in incheshow can students to tell as much as they can about Displaying data: What can we say the overall shape of the data. they do that? Can they connect this task about our heights? with any other measurement problems they When these stories are finished, give stu- have worked on? Will they use the same Draw a line plot on the chalkboard to orga- dents a chance to share them, perhaps procedures? nize and display the data. As you set it up, through oral reports, or a tulletin board ask students to talk you through the pro- display, or a class booklet of the collected Collecting data: Measuring cess: they should establish the range, and stories. tell you what values to write below the num- our own heights ber line. Have them put their data on the line plot themselves. Let's figure out what procedures we could use to mes.aure how tall you are. As they put their data on the board, encourage students to make predictions. What will Students may find that measuring height is the shortest height be? The tallest? The more difficult than they had anticipated (see middle-sized height? the Teacher Note. Measuring heights: Using tools, page 56). Some methods that students Now, let's talk about the heights of the third have used successfully are as follows: (fourth) graders in this room. What can you see from our display? Do you see any clusters? y Stand with your back against the chalk- What's the middle of the data? board and have a friend make a mark where the top of your head is, then mea- What can you say about your heights from this sure up the wall. display? What doesn't this display tell you?

102 Using a smaller unit 55 103 :1.,: . rei4.,1:.,11,4; . '171

TEACHER NOTE All these skills depend to some degree on prior measurement experience. Measuring heights: Using tools For many teachers, it helps to think of stu- Measuring heights seems simple enough, but dents' initial measurements as first approxi- for elementary students it can pose a real mations to their answers, rather than final challenge. Even though students can do results. A vital part of their learning is the measurement worksheets and manipulate opportunity to discuss the reasonableness of measurement data on paper, they may not their measurements, to measure several have had much experience using rules and times, and to correct their measuring mis- other measurement tools. Students who have takes. When students feel the results matter, done woodworking, who have built things at they become much more precise. home, who have played with and built models (including dollhouses) will be the most A teacher's role in this process is delicate. expert at this activity. They have some physi- Your students will need to discuss their cal experience to draw fromthey are famil- methods (Should they use the yardstick? the iar with tools and imow how to use them, foot rulei? Should they stand against the and they may have internalized the sizes of wall?) as well as talking about their results. the measurement units. Asking questions that focus on whether their results make sense (Is Jeremy about the Conducting measurement activities in small same height as Elena? Is Su-Mei that much groups allows you to take the observer's role, taller than Susan?) will help students think to see what understandings and misconcep- about what their results mean. tions your students have. Listening to their comments and questions as you circulate Measurement work provides a natural place will give you some real insight into the me- to use the calculator. If some students mea- chanical and conceptual problems that measure in inches and others in feet and inches, surement often presents to them. you have a good opportunity to work with conversion methods. Although conversion Some predictable problems arise when stu- seems straightforward to adults, it is hard dents use measurement tools. The need to work for elementary students. Students will line up the ruler at zero is not always obvi- have a variety of methods and will want to ous. Additionally, students may start from spend some time trying them out. Be sure the wrong end when they pick up and move that they estimate first, so that they can a ruler. They may combine units, using ooth check the calculator results with common metric and English systems. They may not sense. This step will further reinforce the role notice that their "yardstick" is a meterstick. of estimation in the measurement process. 1 0 5 56 Measuring: From Paces to Feet 104 This dialogue occurred in a fifth grade class ROBERTO: Yeah, we got confused about the 6690DIALOGUEBOX doing another Used Numbers unit, Statistics: two sides of the yardstick. I'm 48 inches and The shape of the data. However, it highlights Anyone else come up with a problem they had issues that are likely to occur when third to solve when they were measuring? 1 centimeter tall! and fourth graders try to measure their heights. JANE: Yeah, we did. We meas. red me and During this investigation, you will probably we came out to 57 inches, and I know I'm discover that many students have difficulty What were some of the problems you noticed not 57 inches. with linear measurement. Teachers com- in trying to get accurate measurements of monly find that some of their upper elemen- your height? How did you know? tary students: sARAH: Well, in our group we got 50 inches JANE: Because my morn keeps a chart at for Mary, and then we got 125 inches for home of all the kids and we mark it on our combine metric and English systems birthdays, and I know I'm shorter than that. ("I'm 48 inches and 1 centimeter tall.") Karen, and we knew that didn't make any sense because Karen and Mary are about On my birthday I wi.s 51. do not distinguish between metric and the same. KIM: And also Jane's one of the shortest in English units ("I'm 125 inches tall.") KAREN: And, anyway, no one could be 125 our class, and 57 wouldn't be near to the y do not know how to combine two parts inches! shortest height in the class. of the measurement ('Tm 6 inches taller So you realised that 57 inches wasn't reason- than one yardstickhow do I do that?") sUE: Yeah, that would be past the ceiling. (Laughter.) able? Did you figure out what happened? measure from the wrong end of the ruler JANE: Yeah. When we measured the first or yardstick I'm not sure it would be past the ceiling, but you're rightit doesn't sound like a fifth part and then we moved the ruler up to Don't be alarmed! Such difficulties are com- grader? So you paid attention to whether your measure the rest, we flipped it around and mon among students who have had little measurements seemed reasonable. Before you we had the ruler going in the wrong direc- experience with measurement in a real tell us what was wrong, can anyone else guess tion. context. Use this investigation as an how they got 125 inches for someone's Oh, you mean you measured from the end of opportunity for students to gain f:xperience height? the yardstick that says 35? and to help each other be accurate. Having DAVE: I know, because we did the same students discuss and check accuracy as they KIM: Yeah, we got to 36 and then we swung thing. Because on one side is centimeters it around like this, so we were measuring work in small groups is often more effective and on one side is inches, and we were than teaching a lesson on linear measure- from 36 up, and we marked the second part using the centimeter side Lt first. Then we at 21, so we had 36 plus 21 inches and we ment. Have students share the difficulties realized it couldn't be over 100 inches for they encounter; support students who use got 57. But we were going the wrong way. somebody's height. their own experience and knowledge to check the reasonableness of their measurements. Is Dave right? Was that the problem?

Using a smaller unit 57 106 107 Measuring: From Paces to Feet a Wilb gip ND 1

PART 3 A project in data analysis A note on the final projects

Two final projects are suggested here. You The second investigation uses data collected may do both, but most teachers have se- outside of the classroom. Students try to get lected the one that best fits their situation measurements of the "comfort distance" for and the interests of their students. pigeons, or some other ubiquitous neighbor- hood animal. They work in teams, getting as The first project focuses on classroom furni- close to a pigeon as they can before it flies ture, using measurement to analyze whether off, then measuking the distance and the classroom furniture fits the students and recording it. The data are recorded and making recommendations about the optimal analyzed, and students look for patterns to distribution of chairs and desks for the class. see whether they can make assertions about If your students are interested in measure- pigeons' comfort distance. This has been ment as a tool to analyze their own sizes appealing to students who are particularly (which will have emerged in the foot-size and interested in animal behavior, and it sparks height investigations), this investigation will many students' interest in interdisciplinary be very appealing. science work.

60 Measuring: From Paces to Feet 09 CLASSROOM FURNITURE: DO OUR CHAIRS FIT US? (

INVESTIGATION OVERWEW

What happens bears, page 66). If you prefer to use your Important mathematical ideas own version, be aware that for this in- In this investigation students explore vestigation, it is important to stress size whether their chairs fit their bodies. After rather than hardness or softness for de- Defining a question in data collection. hearing the story of Goldilocks and the Three termining chair comfort. Defining and discussing the problem is an Bears, they determine how they might mea- important first step in data collection and sure themselves and their chairs: Which VHave on hand chairs of three sizes: one data analysis. Students engage in lively dis- body dimensions affect the fit? They estab- very small, one very large, and one "just cussions trying to decide what data they lish what methods of measurement to use: right" for your students (Session 1). need to collect and how to make sense of then they measure themselves and the these data. chairs in the classroom. Finally, they write a y Have adding machine tape, scissors, letter to the principal about their findings. pencils, and yardsticks available Experiencing the phases of data collection (Sessions 1 and 2). and analysis. In doing this project, students The activities take three class sessions of experience all the phases of data analysis, as about 45 minutes each. y Duplicate a class list for each small group to use for recording their data described in the Teacher Note, Phases of (Sessions 1 and 2). data analysis: Learning from the process ap- What to plan ahead of time proach to writing (page 65). Working with a y Provide materials for presentation real problem, students' analytical skills be- VPrepare to read "Goldilocks and the graphs and written work, including come sharpened as they investigate the im- Three Bears" (Session 1). We provide an writing paper, pencils, graph paper of plications of their data collection. Do their adapted version that stresses size as a various sizes, markers, stick-on dots, chairs fit them? Can they reshuffle them to criterion of how well a chair fits (see the scissors, and construction paper make them fit? Do they need some different Teacher Note, Goldilocks and the three (Session 3). chairs?

110 Classroom furniture: Do our chairs fit us? 61 .1 I Building theories based on data. Using the SESSION ACTIVITIES While most third or fourth graders know the data they collect, students can develop some story of Goldilocks, it is important te reread aloud a version that raises the issuf: of fit" theories about human measurements and Children's school desks and chairs are often human growth. They may decide that their sizes of things vary, and some things fit us ill-fitting. Use of furniture tends to evolve while others don't. When you finish reading, chairs do not fit because they have all grown over time in a classroom, and children accept too much or too little; they may find that allow some time for general response to the what's there without much concern over its story. Then ask students to focus on there are no overall patterns to their chair fit. Often, more attention is given to the sizes; they may decide that chairs should be Goldilocks' feelings of discomfort in the chair appearance of the chair and desk that a that was too big. Did she seem comfortable? assigned to classrooms based on middle-of- student is to use for the year. Adults' office the-year measurements. It is impossible to furniture is taken more seriously. Fitting Set three chairs before the classpreferably predict their theories in advance; a teacher's chairs and desks to people has become an one that's much too small, one that's about role is to support and probe the students' important field of work. Designing chairs, right, and one that's much too big. Ask a ideas. desks, and other parts of a working very small student to sit in the biggest one. Making presentation graphs and reporting environment is called ergonomics. Does it fit? How can people tell by looking? on data analysis activities. This is a chance What matches? What doesn't? for your students to review what they have Children can be educated to improving the fit of their working furniture. Measuring them- Have a large student sit in the smallest done in the investigation and write about selves and the available chairs is a good chair. You might continue to dramatize the their process and their findings. The oppor- matter of "fit" in a chair until you think your tunity to write and to illustrate that writing starting'ace. In this investigation students ;hat dimensions should be taken students have some ideas about what is an exciting one for many students. determi into account. Do their chairs fit? How can dimensions might make a difference. they tell? How could they analyze their body measurements and make some recommen- Defining the problem: How big are we? dations about chair size? During this project, How big are our chairs? students will begin to develop their skills at measurement in a problem-solving context In the project we're going to do, you will be that is immediately relevant to them. working in small teams. You'll have the chance to conduct a careful study of your chairs to see whether they are the right size. You'll take Establishing a context for the problem: careful measurements to see whether the Goldilocks tries the bears' chairs chairs are a good fit. You'll need to talk first about exactly which measurements you'll Today we're going to start work on a real-life need, and how the measuring should be done. measurement problem. First I want to read you a familiar story to get us startedthinking Divide the class into small groups of three or about the problem. four that will work as teams for the duration

62 Measuring: From Paces to Feet 113 1 1Z of the project. Deciding what data to collect both pieces of data in the first session and What would the principal need to know about in order to evaluate chair sizes will need then graph or display them in another what you've been doing? How could you de- some serious discussion, either as a whole session. In either case, they will need adding scribe it? What would a lette r to the principal class or in the small groups that report their machine tape, yardsticks, or tape measures sound like? decisions back to the whole group. Most stu- and access to graphing materials. Distribute dents have to work hard to verbalize what copies of a class list for students to use to The production of a final report and accom- they can see. This work is certainly worth the record their data. panying graphs will take the last session of time spent; as they talk, they see how to de- this data analysis project. The important fine the task with more and more precision. point is that the students draw some con- For a typical discussion, see the Dialogue Displaying and analyzing data: clusions from their work and make some Box, Building a structure for thinking: Does What sizes do we have? recommendations based on the data they this chatrfit? (page 64). Measuring methods have collected. are also an issue: should they measure the Working in their small groups to prepare rough draft graphs, students will be looking front or the back legs of the chairs? The out- Extensions side or the inside of the human legs? Does it into two questions: What size chairs do we matter? What's the most important dimen- have available? What size legs do we have? Groups should make a rough draft graph for Some students may want to pursue further sion: width of the seat? height of the back? the topic of analyzing the design and the size height of the scat? each kind of data and write one sentence describing the data they have recorded. of school furniture. They may want to study another classroom's furniture and see Once students have defined how they will whether those chairs fit the intended users. measure the size of the chairs, they will have When everyone has finished this preliminary Or they might study the desk heights in their to decide how to define and measure the analysis, call the class together to look at the class and see whether they need or want to crucial dimension of the human leg that de- ways the information was graphed and to make adjustments. termines how a chair fits. hear each other's comments. Help students focus on the conclusions they are reaching. Many third and fourth graders are interested Do they feel the chair situation is adequate? in designing environments. To tap this inter- Collecting data: Measuring students Do they want to make some recommenda- and chairs est, suggest that they design a school desk tions to the school administrators? and chair (or a working environment) that Students need to collect two pieces of they would like to use. This could involve numerical data: the sizes of the chairs and Making presentation graphs: measuring their appropriate dimensions and the critical dimension of their fellow Do our chairs fit? including the measurements in the designs. students. Decide how they should proceed. Or, it might involve a more functional analy- You may want the small groups to collect After the class has talked about its conclu- sis of their needs and their preferences. As a and display the information about the chair sions and made some recommendations, the cooperative project, some students might sizes in one session and the information students rejoin their small groups to make survey the class to learn their preferences. about their leg dimensions in another; final presentation graphs to accompany a then draw up a report for the designers to alternatively, you may want them to collect written report of their work. use.

I 14 Classroom furrtiule Co our chairs fit us? 63 114 1 1 5 too, but he doesn't fit well ill that chair. BENJAMIN: When it's uncomfortable. 6699DIALOGUEBOX (Ricardo sits in Maria's chair. He's too big. OK. When it's uncomfortable. Building a structure for Evenjbody giggles.) SUSAN: When your feet are right on the floor. thinking: Does this chair fit? JENNIFER: He doesn't fit. My feet are right on the floor, but I'm not com- Why, but why? fortable. When ideas are complicated and students BOBBY: He's bigger. TAMARA'. The same size. The same size as seem reluctant or unable to grasp them you. quickly, teachers have to take an active role But what does that mean, he's bigger? in posing problems, clarifying ideas, and The same size as what? What should be the KIM: Because his legs and his knees are helping students figure strategies. It is a very same size as what? familiar kind of teaching, one that Jerome sticking up. Bruner calls "scaffolding" children's ideas. If I come sit in this chair .. Or In this conversation the teacher pushes the students to clarify, extend, and defend Building a structure to support the KYLE: It's just right. children's thinking does feei like building a their statements. While he does have an scaffold. How do you know it's just right? outcome in mindidentifying what parts of MICHAEL: Cause you're big. the chair and body could be compared and Finding the "right" human dimension to measured in order to determine fithe match with the height of a chair is a difficult But where am I big? You mean my belly? repeatedly reflects students' own words and problem. Students have to isolate one vari- ILaughter)... statements back to them and challenges able from a welter of information. In a situa- [Later. Now Maria and Ricardo are both sit- them to take their conjectures further: But tion like this, a teacher has to decide how ting in chairs that are too big for therm and what does that mean, he's bigger? How do much scaffolding to provide. In the following the teacher is in a chair that is too small.) you know it's just right? What should bethe dialogue, the teacher is working hard to help same size as what? students isolate the information they need to CHRIS: Maria's feet aren't touching the floor, know without providing it for them. He is and Ricardo's feet aren't, and yours are. Even when he gets a saUsfactory description of what "just right" means ("When your dramatic, energetic, and very active in this So, if her feet are off the floor, does it mean process. knees are not sticking up"), he does not use the chair is too big or too small? that to simplify the complexity of the issue. We've found that Maria does real well in this PEGGY: Too big. He accepts the information as important chair. WhM does it have to do with, making How do we know when your chair is just right? ("That's one part"), but challenges students the chair fit and you fit in the chair? to think further. This balance between SEAN: The body. PEGGY: When your knees are not sucking acceptance and questioning provides a up. scaffold for children's thinking without doing What do you mean about the body? [Pointing at Maria, whose legs are dangling the thinking for them. BARBARA: She's small. above the ground) Her knees are not sticking She's small. But I could say Ricardo's small. up. But that's one part.

64 Measuring: From Picea to Feet 116 .117 may be reiterated until the piece of writing is Revision in data analysis may include finding (k1TEACHER NOTE finished. new ways to organize and represent the data, developing better descriptions of the data, Phases of data analysis: Data analysis has four phases parallel to collecting additional data, or refining the re- Learning from the process those in the writing process; search questions and collecting a different kind of data. approach to writing Phase One: Brainstorming and planning. During this time, students discuss, debate, Phase Four: Publication or display. The The process of data analysis is similar to and think about their research question. In nature of "publishing" the results of data many other creative processes. Students do- some cases, defining and agreeing upon the analysis varies, Just as it does for a story or ing data analysis follow the same processes question may take a considerable amount of essay. Sometimes students develop a theory that adults do; the analyses may be less time. Having defined the question and agreed that is the basis for a report on a particular complex, but the procedures are the same. In upon terms, students consider possible topic; at other times they may develop a the- data analysis, as in writing or art, teachers sources of data, ways of recording them, and ory that inspires further investigation. A help children do real work rather than stilted how they might organize themselves to col- completed report of a data analysis investi- school assignments requiring fill-in-the- lect needed information. gaUon may involve a written description of blank responses. The teacher's role is rela- the study with conclusions and recommen- tively subtleshaping the process, asking Phase Two: Putting it on paper. For the dations, final presentation graphs of infor- questions that guide the students' progress collection and representation of data, stu- mation previously displayed in working toward their goals, hearing and responding dents develop their discovery draftswhat graphs, or a verbal or written presentation of to their ideas and theories. Students are ex- we call "sketch graphs"the first draft of the the report to an interested audience. pected to have something origthal and inter- information on which they base their esting to say, and the teacher provides an developing theories. Students represent the When teachers think about the writing pro- environment that enriches and supports stu- data in a variety of ways to help them cess, their role as facilitator and helper dents' self-expression. describe the important features. They use seems familiar and obvious. Of course stu- their first drafts as tools as they look for dents need time to think and revise their Data analysis has many similarities to the relationships and patterns in the data. work! Of course they need to be challenged process approach to writing, which typically and led, sensitively, to the next level of includes four phases. The process starts with Phase Three: Revision. Writers are encour- awareness. The writing p:ocess seems more a planning phase (often called pre-writing or aged to share their drafts with their peers in familiar to most of us than the mathematics brainstorming). This is followed by the writ- order to determine how an audience per- process because we, too, have done writing. ing phase, when a vely rough draft of ideas is ceives their work. Similarly, in the data anal- first put down on paper. The third phase is ysis process, the students often present their The process of data analysis needs the same the revision or rewriting phase when the sketch graphs, preliminary findings, and be- kind of teacher support. Students need to try writer elaborates, clarifies, restructures, and ginning theories to their working group in their ideas, to rough them out, to be chal- edits the piece. The final phase is the publi- order to see whether their interpretations lenged and encouraged to go further in their cation phase, when the writer's completed seem supported by the data, and whether thinking. It is important that they have time piece is shared with others. These processes others see things they haven't noticed. to think and consider optionsand vitally

118 Classroom furniture: Doour chairs fit us? 65 119 important that they see their work as part of Each of the bears had a chair to sit in: a little a process. Data analysis, like writing, is not TEACHER NOTE chair for Baby Bear, a middle-sized chair for cut and dried. There are many ways to ap- Goldilocks and the three bears Mama Bear, and a great, huge chair for Papa proach a question and many conclusions to Bear, And they each had a bed to sleep in: a be drawn. Like writing, mathematical investi- little bed for Baby Bear, a middle-sized bed gation is a creative blend of precision and Here is a version of the familiar Goldilocks story for use as an introduction to the for Mama Bear, and a great, huge bed for imagination. IIII Papa Bear. investigation,Do our chairs fit us?* One day, after they had made the porridge for their breakfast and poured it into their porridge pots, they walked out into the wood while the porridge was cooling, so that they might not burn their mouths by beginning to eat it too soon. And while they were walking, a little girl called Goldilocks came to the house. First she looked in at the window, and then she peeped in at the keyhole; and, seeing nobody in the house, she opened the door. The door was not fastened, because the bears were good bears who never did anybody any harm, and never suspected that anybody would harm them. So Goldilocks opened the door and went in; and well pleased she was when she saw tae porridge ONCE UPON A TIME there were three bears on the table. If she had been a thoughtful who lived together in a house of their own in little girl, she would have waited till the a wood. Baby Bear was a little,small, wee bears came home and then, perhaps, they bear; Mama Bear was a middle-sized bear; would have asked her to breakfast; for they and Papa Bear was a great, huge bear. Each were good bearsa little rough, as the of the bears had a porridge pot: a little pot manner of bears is, but for all that very for Baby Bear, a middle-sized pot for Mama good-natv red and hospitable. The porridge Bear, and a great, huge pot for Papa Bear. looked tempting, though, and little Goldilocks set about helping herself.

'This version was adapted from The Green Flaky Book, edited by Andrew Lang (New York: McGraw-Hill. 1966. reprinted from First she tasted the porridge of Papa Bear, 1892). 1 21 66 Measuring: From Paces to Felt and that was too hot for her. And then she said Papa Bear in his great, rough, gruff And you know what Goldilocks had done to tasted the porridge of Mama Bear, and that voice. the third chair. was too cold for her. And then she went to the porridge of Baby Bear and tasted that; And when Mama Bear looked, she saw that 'SOMEBODY HAS BEEN SITTING IN MY CHAIR, AND HAS SAT and that was neither too hot nor too cold, the spoon was standing in hers, too. 'ME BOTTOM OUT OF 111" but just right, and she liked it so well that said Baby Bear in his little, small, wee voice. she ate it up. "SOMEBODY HAS BEEN AT MY PORRIDGE!" said Mama Bear in her middle voice. Then the three bears thought it necessary Then Goldilocks sat down in the chair of that they should make further search; so Papa Bear, and that was too tall for her. And Then Baby Bear looked at his, and there was they went upstairs into their bedchamber. then she sat down in the chair of Mama the spoon in the porridge pot, but the por- Now Goldilocks had pulled the pillow of Papa Bear, and that was too big for her. And then ridge was all gone. Bear's bed out of its place. she sat down in the chair of Baby Bear, and that was neither too tall nor too big, but just -SOMEBODY HAS BEEN AT MY PORRIDGE. AND HAS EATEN "SOMEBODY HAS BEEN LYING IN right. So she seated herself in it, and there IT ALL UK' MY BED!" she sat till the bottom of the chair fell out, and down she went, plop, onto the ground. said Baby Bear in his little, small, wee voice. said Papa Bear in his great, rough, gruff voice. Then Goldilocks went upstairs into the bed- At this, the three bears, seeing that someone chamber. And first she lay down upon the had entered their house and had eaten up And Goldilocks had pulled the bolster of bed of Papa Bear, but that was too high at Baby Bear's breakfast, began to look about Mama Bear's bed out of its place. the head for her. And next sae lay down the room. Now Goldilocks had forgotten to "SOMEBODY HAS BEEN LYING IN MY BED!" upon the bed of Mama Bear, but that was put the cushion straight when she rose from too high at the foot for her. And then she lay the chair of Papa Bear. said Mama Bear in her middle voice. down upon the bed of Baby Bear; and that was neither too high at the head nor atthe "SOMEBODY HAS BEEN SITTING IN And when Baby Bear came to look at his foot, but just right. So Goklilocks covered MY CHAIR!" bed, here was the bolster in its place; and herself up comfortably and lay there till she the pillow in its place upon the bolster; and fell fast asleep. said Papa Bear in his great, rough, gruff upon the pillow was the head of Goldilocks voice. which was not its place at all, for she had no By this time, the three bears thought their business there. porridge would be cool enough; so they had And Goldilocks had flattened down the come home to breakfast. NowGoldilocks had cushion of the chair of Mama Bear. "SOMEBODY HAS BEEN LYING IN MY BED-AND HERE SHE left the spoon of Papa Bear standing in his isr porridge. "SOMEBODY HAS BEEN SITTING IN MY CHAIR!" said Baby Bear in his little, small, wee voice. "SOMEBODY HAS BEEN AT MY PORRIDGE!" said Mania Bear in her middle voice. Goldilocks had heard in her sleep thegreat.

Classroom furniture: Do our chairs fit us? 67 122 123 rough, gruff voice of Papa Bear, and the middle voice of Mama Bear, but only as if she had heard someone speaking in a dream. But when she heard the little, small, wee voice of Baby Bear, it was so sharp and so shrill that it awakened her at once. Up she started; and when she saw the three bears on one side of the bed, she tumbled herself out at the other and ran out the window. Now the window was open, because the bears, like the good, tidy bears that they were, always opened their bedchamber window when they got up in the morning. Out Goldilocks jumped, and ran through the woods as fast as she could runnever looking behind her. What happened to her after- ward I cannot tell you. But the three bears never saw anything more of her again.

68 Measuring: From Paces to Feet 124 HOW CLOSE CAN YOU GET TO A PIGEON?

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( INVESTIGATION OVERVIEW

What happens VDecide whether students will collect ideas about how to measure the distance data during class or at another time. If a from themselves to a nervous pigeon, and to- Students experiinent to find out how close field trip to a nearby park is in order, gether the class will define and refine their pigeons (or some other wildlife common in make the necessary arrangements. methods until they have some specific proce- your area, such as squirrels, ducks, or gulls) Provide materials for presentation dures to follow. will let people approach them. First students V graphs and written work, including Experiencing all the phases of the data establish a method and formulate a plan for writing paper, graph paper of various data collection; then they collect the data; analysis process. Data collection, display, sizes, markers, stick-on dots, pencils, and analysis in a sustained investigation like finally they organize and analyze the data scissors, and construction paper. and try to draw conclusions about pigeons' this one is somewhat different from that which took place during the short-term "comfort distance" with people. In a final Important mathematical ideas report, students describe how they conduct- investigations earlier in this unit. See the ed the investigation and explain their Defining the question. 'I i':ust phase in Te:.,:her Note, Phases of data analysis: Learn- findings. data analysis is defining the question and ing from the process approach to writing (page deciding what data are needed to answer it. 65) for a description of a more extended in- The activities take three class sessions of Encourage discussion about what the ques- vestigation. Students can explore their ques- about 45 minutes each. tion means: Can we run up to the pigeons? tions further because of the additional time; Should we be allowed to feed them? you may find students inspired by this investigation to undertake others. Related What to plan ahead of time Finding a workable means of measure- issues in their own livesinstances of y Have rulers, yardsticks, tape measures, ment. The question of how to measure dis- crowding in the school, or preferences for adding machine tape, and pencils avail- tance when the target doesn't hold still is an personal spacemay take on importance for able. important one. Students will have lots of some or all of your students.

1 25 How close can you get to a pigeon? 69 126 Br ;:.ding theories based on data. When SESSION ACTIVITIES how close you can get to a pigeon. You will be- students understand their data and present come scientists and conduct an experiment to it clearly, they can then go on to build theo- find out whetiva there seem to be patterns in The timing of parts of this investigation de- ries and to develop stories based on their the comfort distances of pigeons. First, you pends largely on your class and your own data. The measuring task in this investiga- will decide what information you need and needs. Some teachers introduce the investi- tion presents a good opportunity for story- make decisions about how to collect data. gation and send students out to measure telling and theory-building. Your role is to After you go out to collect your data, you'll distances after practicing in the classroom. support and encourage this kind of expres- spend some time back in class putting They then split the next two sessions be- sion. everyone's data together. From all the data, tween recording and organizing data and we'll see whether pigeons seem to follow any making final presentations based on those Making a report. Many students find writing patterns about how close they let humans get. data. At the end of the third session, they about a mathematical experience to be very Maybe all pigeons have about the same call everyone together to report about their challenging and satisfying. Writing about comfort distance. Maybe they varywe'll find findings. mathematics needs to be encouraged, since out. it provides both an avenue for creative expression and a way to express mathematical Other teachers allow more time for establishing methods and measurement techniques at concepts in *plain language." Some students Deciding on a method of data collection: the start. Because of the need for a final re- find it easy to write about presentation Measuring to a moving target port, they spend one extra session on the graphs; others find it difficult. This should be final report itself. You will find natural no different from writing in English class; Ask questions that help students focus on breaks in the activities and can decide your only the content will be different. The reports the nature of the task and make decisions own best pace. or summaries can be fairly short, but should about how best to handle it. contain thoughtful treatment of both the How might you collect information about pi- mathematical content and the interpretation Considering the problem: How close geons comfort distance? What problems might of their data. can you get to a pigeon?* come up? What could you do about them? Many animals fly or run away when humans What tools would you need? Could you do the get too close. You've probably noticed that pi- task alone? How many pigeons should we try geons for whatever local animal will work for this with? your investigation] let us get a little bit close, but never very close. The distance they insist Pick some students to demonstrate the on keeping is called their "comfort distance." method the class finally settles on. For some ideas, see the Teacher Note, Measuring your Your challenge for this project is to find out distance from a pigeon: Some ideas (page 72). Remind students of the research question 'This problem was sugAested by Marilyn Burns in TheI Hate Mathematics! Book(Boston: Little, Brown, 1975), page 16. It they will be addressing with these data: How makes a good cooperative project in data collection and analysis. close can you get to a pigeon?

70 Measuring: From Paces to Feet 1 27 128 Re liew with students the method they will be solve. Who has some ideas about we can go What do the data show? Are the results clus- usin, reminding them of the importance of about that? tered? What was the closest anyone got to the everyone's using the same method. Check to pigeon? What number shows a pigeon that flew see that they have the necessary materials in Students decide ways to organizing these away quickly? Look at the range of the data. Is their groups. Then send students out to do data so that they can put them all together there a lot of difference between pigeons? their measuring work or remind them to do to look for patterns. There may be some mea- What might account for some of those this as their mathematics homework surement problemssome groups may have differences? assignment. measured in inches, others in feet and inchesand some may not yet have For tips on guiding studeids through this measured their strips of adding machine discussion, review the Teacher Note, The Sharing stories about data collection: tape. Have calculators available so that they shape of the data: Clumps, bumps, and holes How did it go? can convert feet to inches or inches to feet if (page 22). they need to do so. This is likely to be time- What happened when you measured to see how consuming. You may want to allow a 10- or Presentation graphs: Building and close you could get to a pigeon? 15-minute work period and then resume illustrating theories This investigation leads to some interesting tomorrow when all the data are comparable. discussion and good stories. Since data col- What would you say if someone asked you how lection often produces surprises, let students Organizing and displaying data: What close you could get to a ptgeon? Can you now share their experiences. This will take some do our data look like? answer that original question? I. it possible to time. build theories from these data? Put students into their working teams to make quick sketch graphs that display allAsk each team to make a presentation graph Recording data: Getting it all together the pigeon data. As they make their sketch to illustrate their theory about pigeons' graphs, encourage them to talk with each comfort distance and show their results. Merging these data is important. Students other about what they see and to spend Have graph-making materials available. Ask need to put all their data together so that some time thinking about what the data each group to write the story of what they they can think about pigeons as a group. show. found out from these data, illustrating their They have enough experience now so that work with this final presentation graph. this will be a relatively easy problem to solve. Nevertheless, you need to be sure that all the Data analysis: Are there patterns data are available for everyone to see. in pigeons' comfort distances? Extensions How can we put all these data together so that Hold a conversation with the whole class In doing this investigation, your students we can see them all at once? Do you all have after the teams have made their preliminary may develop an interest in human comfor, the same kind of data? First let's simply list observations in small groups. Encourage distance. They might measure their own our data on the board so that you can all see students to make observations and comfort distance (the point at which they get whether we have any problems we have to comments. uncomfortable with the close presence of an-

How close can you get to a pigeon? 71 2 9 130 other person) and analyze those data. Mak- and, when the pigeon took off, came up to ing classroom furniture arrangements based TEACHER NOTE that spot and marked it with a finger. The on these measurements can be a good deal Measuring your distance from third was the measurer, who used adding of fun, with some students squished together machine tape to mark off the distance from or sharing desks while others are floating a pigeon: Some ideas the stalker to the pointer. A fourth student like islands by themselves. could be a recorder if you need a group of The comfort distance of an animal depends four. Analyzing the school for crowding can also on the circumstances. Of course, pigeons be useful. Perhaps students can look at the have more tolerance for t;ther pigeons than Your students may come up with a very suc- places and times in the school where and they do for people; they also have more toler- cessful method quite different from the one when they feel crowded and make recom- ance for quiet people than running people. Just described. In any case, you will want to mendations. Cafeteria tables and benches, People, too, have comfort distances that vary have students act out their measurement auditorium seats, and school bus seats could with the cimumstances. Students may want strategy in class so that they all do the same all be sources of data. How closely does their to talk about their own preferences for per- thing in the field. spacing match your students' preferences? sonal space as a way of understanding the problem of a pigeon's comfort distance. Some fast-food restaurants have tables with stools or chairs attached. Are those distances To measure the tolerance or comfort distance comfortable for your students? Do they re- of a pigeon (or another common animal), stu- flect a comfortable distance for conversation dents have to develop procedures for getting while eating? Perhaps your class would like a measurement that is as accurate as possi- to write to the restaurant to find out how ble. One image that sometimes helps in dis- those designs were decided on. A local archi- cussing the problem is the shot putter in the tect might be able to talk with the class Olympics. Because the shot rolls when it is about how human measurement data are thrown, judges on the field watch for the useful in his or her work. shot to hit the ground and immediately run over to the place it landed in order to mark it. Your students may have seen this on television. Using small groups or research teams of three or four students works well for this investigation. In one class, students worked together in threes. One was the pigeon stalker, who walked quietly toward the pigeon and froze when the pigeon took off. Another was the pigeon pointer, who stared at the pigeon,

72 Measuring: From Paces to Feet 132 131 7**,-

STUDENT GIANT STEPS AND BABY STEPS: MEASURING AT HOME SHEET I

Are you using giant steps or baby steps?

Count the steps:

From the stove to the front door.

From the kitchen sink to the living room couch.

From your bed to the bathroom.

From the bathroom to the stove.

Measure other distances at home. Write the results here.

From to is

Be sure you label to show giant steps or baby steps!

Make an estimate of the number of paces to your target. Then count how many paces the pacer actually takes.

Caller's name:

Pacer's name:

Estimate: Estimate:

Actual paces: Actual paces:

Estimate: Estimate:

Actual paces: Actual paces: V Ss%5

5. , 4 'A 1 s' Estimate: Estimate: s .5 5 5 55

Actual paces: Actual paces:

Estimate: Estimate:

Actual paces: Actual paces:

Write directions for your robot to get to each target.

First target Second target Third target

Forward paces Forward paces Forward paces Turn Turn Turn

Forward paces Forward paces Forward paces

Fourth target Fifth target Sixth target

Forward paces Forward paces Forward paces Turn Turn Turn Forward paces Forward paces Forward paces

......

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Start at:

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End at:

Total number of paces:

.0 Start at:7 Start at: Start at:

End at: End at: End at:

Total number of paces: Total number of paces: Total number of paces:

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Write the number of paces and turns to get to these places in the school.

From our classroom to: From our classroom to: From our classroom to:

Total number of paces: Total number of paces: Total number of paces:

With your foot ruler, measure the feet of everyone at home. If someone's foot is shorter than the ruler, write the person's name and age in the "shorter" column. If it is longerthan the ruler, write the name and age in the "longer" column.

If someone's foot is EXACTLY one foot long, write the name and age in the "equal" column.

Shorter than the foot ruler Equal to the foot ruler Longer than the foot ruler

Name Age Name Age Name Age

© Dale Seymour Publications asur ng: FtO m aces tO ee 145 146 Footstick pattern A footstick is a foot-ruler, 12 inches long, with no smaller units marked along its length. It is very useful for making comparisons of length and finding out whether something is exactly the same length as a foot, or shorter, or longer. Durable plastic footsticks can be purchased, or you can make your own out or tagboard and laminate them for extra strength.

8 117 Inchstick pattern An inchstick is a foot-ruler, 12 inches long, with inch lengths marked off in alternating dark and light squares. There are no numbers on an inchstick. This means that students have to count the inches to make measurements of length, rather than read off a number that they may not understand. Durable plastic inchsticks can be purchased, or you can make your own out of tagboard and laminate them for extra strength. Inch- square graph paper can be used to save time in measuring and marking the inch lengths.

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