Estimation

Classroom Activities

Overview

How will I develop sound estimating strategies in my classroom? I will use a variety of situations and real-life problem-solving experiences. According to Whitin and Wilde, my students will then become flexible thinkers (1992). In this unit, I will also use children's literature and Discovery Education videos to demonstrate why people use estimation in the first place. Students will learn the sizes of individual place values in the decimal system from the thousandths place up to the billions place. They will discover that each place is ten times bigger as they move to the left and ten times smaller as they move to the right. The notational system hides this fact so it is hard to realize how big a number can actually get (Howe 2008). Students will discover the importance of the leading digit in a given number and the increasing irrelevance of subsequent digits when dealing with extremely large numbers that often change; populations of countries or square surface area of an ocean. Each day, estimation problems taught in class will be based on quantitative inquiries related to the game of baseball; attendance, concession sales, ticket prices, area of the playing field, area of the stands, ballplayer salaries, and other related topics of student interest. We will take these original findings and then compare and contrast them with baseball stadiums located in other developed countries around the world. Students will discuss the class findings and defend any opinions they will have formed based on the results. For homework they will complete real-life estimation problems relevant to their own lives or the lives of their parents/guardians. Students will be able to apply their knowledge at the end of the unit by completing a web-based research project. It will be differentiated in its requirements however all students will have the same goal. 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 of their interest. This final formative assessment project is intended for to students discover that some countries are densely populated and therefore have much larger impacts on the land and the environment. They will again defend an opinion that will form based on the outcome of their research. For example, India is much more densely populated than most countries - approximately 10 times the density of the United States! It has a reputation of being overcrowded, dirty, and having poor environmental statistics. On the other hand, Canada has less than 1/10 ^th the density of the United States. It has a reputation of being a very clean, beautiful, and environmentally conscious country. They take pride in their high standards for air and water quality. Are these accusations due to the human impact based on their populations? Many opinions can be debated.

Day 1-2

Joe Torre has a lot in common with me in that we both have high expectations of our players. In baseball, because of experience, a hitter can reasonably predict what throw the pitcher will be throwing. In the classroom, because of experience, a teacher can reasonably predict which students will understand the lesson and which will need extra help. However, none of us are mind-readers so occasionally we strike out! Therefore, we need to set some benchmarks that our players can strive for. The word "about" is such a sophisticated idea and should not be used as sparsely as it is. If Joe Torre asked Chan-Ho Park "about how many strikeouts do you throw in a year? He most likely will get a plethora of answers depending on when he asked. However, if Joe Torre narrowed his field of choices by asking, "Would you say you throw between 100 and 200 strikeouts a year?" he would get a more accurate estimate. In my classroom, figuring out the order of magnitude of a quantity by comparing it with powers of ten will allow us to set initial ballparks for students to estimate between.

A good introductory book to read to the class is called Math Curse by Jon Scieszka and Lane Smith. There are also two short videos available on the Internet that could be shown before the unit begins ³. A large decimal place value chart will be constructed by students and hung from the ceiling in front of the room. Each place will be labeled with a single place component in standard and word form. Students will also bring in photos or examples of objects that visually display relatively the same size. For example, with regards to the hundred's place, students can paste 100 pennies on the chart paper, along with a photo of a building that is 100 stories tall, a copy of a $100 bill, and anything else the students can find. For fifth grade, the chart will encompass places from the thousandths up to the billions. The entire place value chart will be displayed throughout the unit so students can add to it as they become more knowledgeable about each place's size. There are many books available to help students visualize the relative size of some place values ⁴. There are also two Internet sites that offer interesting visuals for the larger place values ⁵.

Day 3-4

After benchmarks are set and easily accessible in the classroom, students will further explore the size of numbers by expanding each single place component to its digit and base ten unit. There is an Abbott and Costello video that is available on the Internet that would be a fun place to start off this lesson ⁶. The book, The Best of Times by Grag Tang is another tool you can use to get students motivated before they work collaboratively. With a partner students will break apart several single place numbers and simple multiplication problems to discover the value of each single place component. For example: 7,986 = 7 X 1000 + 9 X 100 + 8 X 10 + 6 X 1. They will further expand each unit as a power of ten. So: 7 X (10 X 10 X 10) + 9 X (10 X 10) + 8 X 10 + 6 X 1. They will be able to see that each place is simply ten times larger than the one before it.

Day 5-6

After a good week or so, students can begin to apply their place value knowledge with real-life baseball estimations. There is a short movie clip on the Internet from the movie Little Big League that will surely catch students' attention ⁷. The books, Betcha! by Stuart J. Murphy, Great Estimations and Greater Estimations by Bruce Goldsone can also be used to introduce easy estimating strategies. They will collect information off the Internet regarding several quantitative aspects of a single baseball game. Students will then have to determine this information for the entire season. For example: $3586 in hotdog sales for one game = $3600 possible estimate. 162 ballgames per season = 160 possible estimate. So, $3600 X 160 games = ($36 X 16) X (10 X 10 X 10) = $576,000 = $580,000 in hotdog sales per season.

Once students are comfortable with relative place value, they can explore much larger numbers. There is a short movie clip from Back to the Future Part III available on the Internet to use at the beginning of this lesson ⁸. The book, On Beyond A Million by David M. Schwartz can also be used. In the classroom, will collect the world's current population from as many sources as possible. The teacher should compute the relative difference between the maximum number found and the minimum number found. He/she will determine if these reported numbers are reasonable estimates or simply approximations. Remember if the percent difference is less than 10%, it is a reasonable estimate.

Possible Extension Activity: With a partner, students will then apply this new knowledge and compare values of data they have already collected from their baseball inquiries. They will find the percent difference for each inquiry. They will take turns with whose data will be the important value (V) and whose will be the estimated value (U). Students will be able to determine if their answers were reasonably close to other data in their small groups. Next, students will compare Charlotte's Knights Castle data to a professional baseball stadium in another country. They will use the internet to find the same inquiries as before. Students will conclude with the idea that only the first 2-3 digits really matter in large inexact global numbers. They will discuss their findings and formulate an opinion debating whether extremely large numbers of measurement are accurate or not.

Day 7-10

Students will begin working on their research project. A rubric with requirements and grading expectations will be handed out the first day. (See no.4 in Appendix A) They will choose 2 countries, one from a provided list, and one using a world atlas-to find one not found on the list. (See no.5 in Appendix A) They will collect the required data and create a chart to show all findings. Students will compare and contrast the two countries figures and reflect on whether there is a greater human impact on the environment in one country versus the other and why. They will think responsibly and write a proposal as to how the country could conserve their environment rather than continue to damage it. Several Internet sites can be used to help students find their information, especially www.wikipedia.org and www.answers.com. (Excellent literacy supplements can also be found at the same websites!)