Estimation

Objectives

Overview

I asked my professor if the math book I was using for research was "too old," implying that its contents might be inaccurate and obsolete. He emphatically replied, "Math is timeless!" Over the course of my seminar at Yale University with Professor Roger Howe, I realized the absolute truth in his statement. Math concepts survive over a long period of time only if they are widely applicable. Through my professor's lectures, I've learned that the use of estimation dates back to at least ancient attempts to measure land area and time. Ideas involving estimation have occupied many of the most famous mathematicians of all times. For instance, I learned that over two thousand years ago, Archimedes estimated pi as 223/71 Π 22/7, and in the seventeenth century, Newton developed a sophisticated method for approximating solutions to equations. Within the past hundred years the emergence of statistics has become an important discipline that studies populations through estimates. It was the latter fact that moved me to write a unit involving understanding estimation of large numbers. What besides math is timeless? Baseball! Napoleon Bonaparte once said, "The advancement and perfection of mathematics are intimately connected with the prosperity of the State." So too can be said with a great baseball team.

My unit will emphasize estimation by exploring relative order of magnitude and relative size as it relates to our decimal system. Too often students don't understand the difference between a thousand and a hundred thousand, or even a million and a billion. I want students to have a better understanding of the value of very large numbers! I will accomplish this by using baseball stadiums to investigate several quantitative aspects of this game. The first goal will be to investigate the economic impact, and perceived value on the part of the fans: attendance, concession sales, ticket prices, ballplayer salaries, and other related topics of student interest. (All issues students want to research for clarification should have a relevant purpose before they go looking for data.) Secondly, students will investigate physical characteristics of a baseball stadium: they will contrast distances of the outfield fences to home plate. All playing fields vary in size from one to the next, therefore, students can determine whether the standard for a home run is fairly constant. Distances will be compared with homeruns hit in various parks located around the world. A possible extension would be for students to determine the area of the playing field and compare it to the area of the stands for that stadium.

One reason I am using baseball as an analogy might be because my husband is a coach. Most importantly, baseball is an immensely popular sport where I live (North Carolina). And it is especially motivating to children. Many of my students (boys and girls) play on some sort of local little league team. According to the official web site of Little League Baseball, there are more than 2.3 million players in Little League Baseball, worldwide, as of 2007, including 400,000 girls registered in Girl's Softball ¹. A final reason I chose baseball is because professional baseball teams are located in other developed countries of the world. Baseball's popularity has been spreading in recent decades, but it already had spread to a number of countries in the 1860s and 70s. The game is followed with fervent interest in Japan, Taiwan, South Korea, Mexico, Venezuela, Cuba, the Dominican Republic, other Caribbean countries, and elsewhere. The International Baseball Federation (IBAF) was founded in 1938 and now has 112 member countries; it has organized the Baseball World Cup since it's start. These facts will allow me to offer a further exploration into relative sizes by researching ballparks in these developed countries across the globe. Students will be able to apply their knowledge at the end of the unit by completing a web-based research project. Requirements for this project will be differentiated to meet all students' needs. They will research a country that has a professional baseball team. Students will collect the same quantitative baseball data as before, but this time it will be for their chosen country. In addition, the students will collect quantitative data such as total population, annual population growth, population density, land area, and any other related topics as they become of interest during research. This final formative assessment project is intended for students to discover that there could be a proportional relationship between population growth and its impact on the land and the environment. Its anticipated purpose is for them to consider what the environmental impact might be of an overcrowded town or city.

Why am I making the connection between baseball and different countries around the world? I teach at an International Baccalaureate school. At the elementary school level, basically this means that internationalism should be reflected in almost everything we teach. One ultimate goal of the IB program is for students to build the desire to take action after they complete research and interpret their findings. Upon completion of this entire unit, students will become more conscious of not only just how large big numbers are, but also how numbers affect our lifestyles and our environment. My hope is for my students to be reflective and responsible citizens. I want them to ascertain reasonable ways to reduce the human and environmental impacts they've discovered from their research.

Rationale

Approximately 35 people move to Charlotte every day. The district that I teach in, Charlotte-Mecklenburg Schools (CMS), grows by approximately 4,000 students per year! We are one of the largest school districts in the nation! My school is one of 173 in the CMS District that encompasses more than 133,000 students in grades K-12. They come from all different ethnic backgrounds, all different types of families, and are at all different levels of physical and academic ability. Nearly half of the students in the district qualify for free or reduced lunch, which is the federal standard for measuring poverty. We are home to students who are from 151 countries, and speak 120 different languages. According to my district's website, approximately 42% are African-American, 35% are Caucasian/White, 15% are Hispanic/Latino, 4% are Asian-American, and 4% are American Indian/Multiracial. These statistics only reflect the tip of the iceberg when it comes to diversity, as each of those groups contains many socioeconomic, cultural, ethnic, special-needs and religious subgroups ².

I teach at Lansdowne Elementary School in Charlotte, North Carolina. At Lansdowne we follow the Primary years Program (PYP) of the International Baccalaureate Organization (IBO). It is a transdisciplinary program of international education that draws from research and best practice from a range of national systems designed to foster the development of the whole child (IBO 2008). Every single student in my school receives the benefits of this program. My classroom is quite diverse not only because of my student's demographics, but also because the PYP brings the whole world into my room! Simply put, Lansdowne Elementary is an archetype of CMS.

Still, it is hard to imagine the growing impact of such a large district on any of our local schools. To the outsider, we are diverse, we get new students every couple weeks, and we work hard to make our national and state goals. But it seems the only thing people really notice is the new set of trailers (mobile classrooms) outside the schools each year! They really don't understand that student numbers are getting so large and are constantly changing. District leaders have no choice but to try and reasonably estimate this growth. How can they estimate such large numbers? Why is estimation significant to CMS, or to anyone?

Estimation is one of the most important strategies we use to compute everyday math problems and considered to be a significant topic in school mathematics, as confirmed in various curriculum documents (NCTM, 2000). Yet, the textbook we use in my classroom seems to give it little attention. In conjunction with this situation, I also confess that, in my classroom much more attention is paid to computation than to estimation! In my state's curriculum, estimation is treated as more of a process than as content knowledge. It is neglected, poorly motivated, frequently limited to rounding, and is not integrated into any other curriculum topics. There are several reasons for placing a greater emphasis on this strategy in our classrooms. Barbara Reys (1992) suggested that "over 80% of all mathematical applications call for estimation, rather than exact computation," and Northcote and McIntosh (1999) found that adults' everyday mathematics consists mostly of mental computations; and many of these involve estimation. Just think, how many times have we asked ourselves questions like: Do I have enough gas in the car to get home? What time do I need to leave to get to the 7:00 staff meeting? How many carbohydrates are in that cafeteria burger and fries? Can we afford a vacation during spring break this year? How much coffee should I make for my dinner party?

Thinking about these types of questions, I've discovered just how extensively we, as adults, use estimation in daily life. So I feel the absolute need to establish an estimation mindset in my students. What I've realized is this, much like professional baseball, estimating too, requires mental computation and thinking - neither process can rely solely on rules or mechanical procedures. My hope and main objective for this unit is to develop my student's number sense so that they will come to view estimation as a distinct way of thinking, rather than as a collection of unconnected rules; as now, I feel it has been in my class for years - strike one! Estimation is such a broad topic, therefore, I want my 5 ^th grade students to grasp only specific, important ideas that I feel have been lost in my classroom. In this unit I will focus on the importance of place values, as they are the building blocks of our decimal system, relative order of magnitude, and finally relative size of quantities.

Undoubtedly, the increasing emphasis on estimation in the elementary classroom may also be due to the constant advancements in technology. Especially with the advent of calculators in the classroom, in my opinion it is essential for students to not only derive an answer, but also, to be able to judge the reasonableness of what is displayed on their screen. As mathematician and professor at Yale University, Roger Howe states, "One of the worst features of calculators is that they make us stupid by providing too many decimal places" (2008)! Estimation is an important sense-making strategy and one goal of this unit is to show students that the first, or leftmost, digit matters the most.

Isn't it distressing to see upper-elementary students solve problems such as $2.75 + $1.25 by using a calculator? Children are so preoccupied with obtaining the exact right answer it often compels them to perform unnecessary calculations; and ultimately, it prevents them from gaining the experience needed to devise an appropriate estimate in a given situation! Therefore, we need to stress that estimation is so often used in real life as an important consumer tool. It helps people make choices about the level of accuracy required to solve real-life mathematical problems. Should I estimate or use mental computation? Should I use a calculator or paper and pencil? Comparison-shopping involves a lot of estimating. We (teachers especially) want to determine the best price for our purchases. We value that perfect price so much that large consumer stores such as Best Buy© have branded their name as such! Consumers are best served if they know a variety of estimation strategies.

Why would 5 ^th graders, in particular, need to estimate rather than just round numbers? First, they may need to estimate because an exact answer is simply unobtainable and they have no other choice (Usiskin 1986). This is especially true on state tests. In real life and example would be when our grade level goes on field trips. The cafeteria manager must estimate the amount of food to prepare for the children when we return for our late lunch. It is impossible to serve exact portions of food therefore our cafeteria manager must estimate what might be needed. It is her only choice!

A second reason students need to estimate is because estimating increases the ease of understanding (Usiskin 1986), especially when it comes to larger numbers of the world that continuously change, like the world's population. For example, in an environmental debate over the number of trees needed annually for the production of pencils, using an approximation of four million pencils rather than, lets say, the (possible) exact figure 3,958,241 makes it easier for students to understand the argument and remember the statistical reasoning. This approximation is easier for children to grasp than the more precise actual figure. Estimates used to increase clarity are almost always calculated by rounding, which underscores an interesting irony in mathematics: clarity and precision are often in conflict with one another (Whitin & Wilde 1992).

And lastly, students need to know that estimates are just easier to use (Usiskin 1986)! Estimating for a purpose focuses on operating with numbers rather than merely understanding them. Establishing reasonable benchmarks will be one of the first tasks students will accomplish as they begin this unit. Tom Parker (1983; 1987) describes establishing estimation benchmarks as his "rules of thumb" or personal estimates. Some of his examples are; one acre will park a hundred cars; it takes about forty gallons of maple sap to make one gallon of maple syrup; a farmer needs 6 tons of hay per cow per year. Students can use these estimates to solve a problem or make a decision much more efficiently than say computing an exact answer and then trying to round that number.

Mathematical Objectives

Tommy Lasorda once said about Fernando Valenzuela, "All last year we tried to teach him English, and the only word he learned was million!" I'd argue that Valenzuela knew more than his coach thought! The first stage of estimation is developing the idea of relative place value and that the biggest decimal place value matters the most. Place values are the building blocks of our decimal system. So at the very beginning of the unit students will explore the relative orders of magnitude (sizes of individual place values) in the decimal system. I will begin by having the students set reasonable benchmarks for each order of magnitude from the thousandths place to the billions place in our decimal system. I set this range according to the North Carolina Standard Course of Study. (See no.1 in Appendix A)

Students will become good estimators when they have certain benchmarks or reference points upon which to base their estimates. For example, they should discover the orders of magnitudes of familiar things: the width of their hand is about 10 centimeters (much more than 1 centimeter, and much less than a meter); the tile on their classroom floor is a foot square, or (about) 30 centimeters wide; the height from the floor to the door handle is 1 meter; the front chalkboard is 10 meters long; and the back side of the building is 100 meters long: and so forth. Classroom and body benchmarks should be set based on powers of ten, it's just simpler! The metric system is based on powers of ten so it should be utilized as much as possible when establishing these ballpark estimates. Students will then estimate with much more reasonableness throughout the year because they have these concrete benchmarks to work with.

Once the students establish a set of reasonable benchmarks based on the powers of ten, they will further explore the decimal place value system. Students will also establish how many decimal places are in a number. According to Roger Howe, Often, we want to

do more than say whether one number is larger than another, we want to say how much larger it is, or that it is very much larger. We also want to say when two numbers are close to one another. Paying attention to place value components also makes these things easy to do. The main point here is that multiplying by 10 just increases order of magnitude by 1. This is true no matter what decimal place we are talking about. Thus, given any decimal place, the place just to the left represents numbers 10 times as large as the given place, and the place just to the right represents numbers only 1/10 the size of the given place. Two places to the left, you find numbers 100 times as large as where you are, and two places to the right, the numbers are only 1/100 of the place where you are. For facility with estimation, it is important that students understand not only the values of each place, but also these relative values of the places (2008).

Look at the expanded form of the base 10 number below and ask yourself, where is MOST of the number?

When we write in expanded form we do not need to write the 000 (hundreds) and the 0 (ones) place values. But, when writing a single place number, 2090, we need to write the zeros to understand the correct magnitude represented by each digit of that number. A basic fact is that order of magnitude sorts numbers according to size (Howe, 2008). Order of magnitude of a base ten number is defined as the order of magnitude of its largest (non-zero) place value component. So, 9,436 has order of magnitude 3, the same as its largest decimal component, 9000; and the 400 has order of magnitude 2, the 30 has order of magnitude 1, and the 6 has order of magnitude 0. The main point is that no two place value components have the same order of magnitude.

We call a digit times a power of ten a single place number, or a very round number. After dissecting (using the associative and commutative properties) single place numbers students will complete several calculations such as 7000 X 500.

Upon completion of the first stage of estimation, students should discover that, as noted in the quotation above, there is a fixed relationship between the places such that each decimal place to the left is ten times bigger and each place to the right is ten times smaller. Babe Ruth certainly understood this general idea when he said, "I'll promise to go easier on the drinkin' and get to bed earlier, but not for fifty thousand dollars, or two-hundred fifty thousand dollars will I give up women! They're too much fun."

The second stage of estimation is developing the importance of the leading digit, meaning the leftmost digit, or the digit of the leading (i.e., largest,) single place component! The decimal system is highly compatible with the ordering of the whole numbers, and makes it easy to compare numbers. For any number of any magnitude, the leading single place component tells you most (at least half) of the number. And each successive digit in a number is giving you a smaller and smaller "piece" of the number. Therefore, one can get a good idea of the value of a number just by its first digit!

I was a geologist before I became a teacher, and often the data I used for certain calculations included large measurements. For example, the height and length of mountain ranges and the radii of different planets were often used. Take the radius of the Earth. One can often find so-called "accurate" measurements such as 3,928 miles. How can one possibly measure the Earth's radius with all its mountain peaks, gorges and deep ocean crevasses to a fourth significant figure? It does not make sense. Therefore, we approximate it to 4,000 miles and can be defined to three significant figures. Roger Howe states that, "for many of our purposes, it suffices to know a number to one significant figure. For most purposes, two significant figures are enough and it's rare to know a real life number to more than three significant figures. All we need to know, or can expect to know, is an approximate value, and the largest decimal components supply this with great efficiency (2008)."

Students will duplicate this process by using hands-on manipulatives. Number cubes will help them visually dissect different single place numbers into their components and then further in to their expanded units. Again I will direct students to notice which place value holds MOST of the number. (See no.2 in Appendix A)

The crucial idea is that of emphasizing relative place value. As seen above, each successive single place component is giving you a smaller and smaller "piece" of the entire number! What about really large numbers like the population of the earth, which is constantly changing? One might ask if this same idea will still hold water. For larger numbers it is harder to estimate reasonably therefore, we must approximate. What is the difference between estimation and approximation you ask? We will say that estimation is getting within 10% error and approximation is anything better than 10%. For my purposes in 5 ^th grade, I will demonstrate the following percent difference formula for the whole class to help show the leading components of a single place number are the most important.

If possible, higher-level students can calculate the relative difference between a single place number and its minimum estimation:

The corresponding idea for approximation is relative error. Both relative and absolute error can be controlled in terms of decimal expansions. A key concept is significant digit. It is pointed out that relative accuracy of approximation improves rapidly with the number of significant digits. It is usually unreasonable to expect to know a "real-life" number (meaning the result of a measurement) to more than three or four significant digits, and often one must settle for, and can live with, much less. Failure to appreciate the limits of accuracy seems to be one of the most pervasive forms of innumeracy: it affects many people who are for the most part quite comfortable with numbers. Scientific notation, which focuses attention on the size of numbers and the accuracy to which they are known, is also discussed (Howe 2008).