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byBrian P. Bell

Overview

Estimation is a vague topic in most Math classes, regardless of the grade level. In most of the workbooks that I use with my sixth grade students throughout the school year, we see the word estimation in many of the questions. What we don't see however, is an explanation as to what estimation is, what it is used for, or how to use it. Too often estimation is thought of as merely a guess and not much more. Estimation is in fact not a guess, nor an educated guess, but rather an effective way of calculating an approximate answer to a question. Estimation is based on Math, not on a guess, educated or otherwise. There are several math concepts that students must be familiar with in order to effectively utilize estimation in the classroom including, place value, expanded notation, very round numbers, and order of magnitude. A familiarity with the powers of 10 would be beneficial as well. Since these concepts are ones that we teach in the sixth grade, I have incorporated teaching them into this unit.

At the beginning of each school year, I go through the same ritual with my Math students. I ask how many of them love Math. As a Math teacher, you probably already know where this is heading, but I'll tell you how it usually goes for me. If I get four confidently raised hands out of the 32 in the class, it is a lot. The next question I ask catches them by surprise. I ask how many of them hate Math. There are always the mischievous smiles on many faces, combined with a glance around the room to the other students looking for reinforcements before they commit to raising their hand. I tell them that yes, it is OK to answer the question honestly because it helps me know how I need to approach the class. Then my third question - Why do you hate Math? I usually get the same three answers each year; it's boring, I didn't like my teacher, or simply put, Math sucks! We can even thank Jimmy Buffett for writing a song of the same title, Math Sucks. By asking these questions, I get a good idea of the overall feel of the students' attitude towards the subject. My goal? That by the end of the year I can change their perception of Math and their preconceived notion that they cannot do well in it because they never have in the past. As teachers, if we can make the classroom environment lively and entertaining then we can hold their attention long enough to teach them Math and get them engaged in our lessons. We must also make our lessons as interesting as we can, which is why I chose the horseshoe crab (limulus polyphemus) as the focus for this unit.

Objectives

A recent fossil find in Manitoba places the horseshoe crab's origins at least as far back as 450 million years ago! 1 It is a unique creature that inhabits the Delaware Bay, amongst other places, and most of my students have seen them at the beach. Our school is located within a half hour's drive to some of the most popular horseshoe crab spawning beaches on the planet. Each year, I assist one of our 8 th grade teachers, Garth Stubbolo, on his annual overnight "Green Eggs and Sand" fieldtrip to the beaches to count the spawning horseshoe crabs. Green Eggs and Sand is a multi-state coordinated environmental education program designed to teach students about the horseshoe crab/shorebird connection in the Delaware Bay. 2 On the fieldtrip, it is always incredible to see the excitement and focus on a student's face as he/she holds a horseshoe crab for the first time and they can actually tell you about the different body parts and behaviors it exhibits. Its importance to the migratory shorebirds, commercial fishermen, the biomedical field, the major controversy surrounding its use and management, plus the fact that it lives right in our "backyard," - all these things combine to make the horseshoe crab a promising subject for creating an entertaining and educational math unit on estimation.

In 1990, the first annual horseshoe crab census was organized by the Delaware Sea Grant to estimate the number of spawning horseshoe crabs in the Delaware Bay. 3 This annual census continues to this day. However, with the implementation of the Atlantic States Marine Fisheries Commision's (ASMFC) fisheries management plan for the horseshoe crab in 2001, a more statistically rigorous scientific spawning survey was developed and has been in use since. 4

Why the need to estimate the number of horseshoe crabs in the Delaware Bay? Many people are directly affected by the laws pertaining to the harvesting of the horseshoe crab, especially the eel and conch fishermen. They use the horseshoe crab to attract conch and eel to their traps. The male horseshoe crab does not attract the eel the way the female horseshoe crab does, therefore, the eel fishermen are mainly interested in harvesting the females. Since the females are the ones who lay the eggs, many environmentalists worry that the harvesting of the females could be, if it has not been already, detrimental to the future of this species. To ensure future generations of this ancient mariner, many environmentalists want a complete moratorium on horseshoe crab harvesting. This occurred in Delaware, though the decision was reversed in 2001, mainly due to the fact that the livelihood of the fishermen would be greatly affected, and the fact that the horseshoe crab estimates do not warrant a moratorium at this time. 5

Another group of people who are interested in the laws affecting the harvest of the horseshoe crab are those who are concerned about the plight of the different species of shorebirds. There are many species of shorebirds that briefly stop over in the Delaware Bay in May to gorge themselves on the eggs of the horseshoe crabs, which are laid in the sand of the beaches. The horseshoe crab nests are usually just a few inches deep and are often disturbed by other nesting horseshoe crabs, thereby scattering many eggs near or on the surface making it easy for the shorebirds to eat them. One female horseshoe crab can lay up to 100,000 eggs in one season, so there are billions of eggs on the beaches during the peak mating season. 6 Although this is a huge number of eggs, with 1,000,000 or so shorebirds arriving in the Delaware Bay to feed on the eggs, some people believe that these numbers need to be sustained for the survival of the birds. The shorebirds are on an annual migration from the southern tip of South America to their nesting grounds in the Arctic, a trip that covers 10,000 miles for many of them. Some people feel there is a direct correlation between the number of horseshoe crabs and the number of shorebirds. This would mean that a decline in the number of horseshoe crabs will result in a decline in the number of shorebirds. (One species, a subspecies of the red knot, the most long-distance migrant to visit the Delaware Bay, has declined dramatically in population over the last decade and is now being considered for addition to the federal government's Endangered Species List. 7) The shorebirds also bring in an estimated annual income between \$30 to \$70 million combined, for Delaware and New Jersey, due to eco-tourism. 8 There are many people who come to the Delaware Bay each May to see the shorebirds and their visits generate money for the local economies. This money would certainly be missed if the shorebirds numbers decline and they no longer offer the current "Wow" appeal based on their sheer numbers.

The third major group that is interested in the laws governing the horseshoe crab is the biomedical industry. The coastwide biomedical industry currently has a harvest of approximately 500,000 horseshoe crabs, which are bled, then returned to the wild. It is estimated that 10% to 15% of these crabs do not survive the bleeding process. 9 Why is the biomedical field interested in harvesting horseshoe crabs? Actually we are all benefited by the horseshoe crab's contributions to the medical field and to our own lives. Horseshoe crab blood is used to test that medicines (those that are injected) are free of contamination from bacterial endotoxins. 10 This is the main current biomedical field use of horseshoe crabs. However new research is suggesting promise of even more benefits to human health, including discoveries in anti-viral and anti-cancer activities in the proteins of the horseshoe crabs' blood. 11

Mathematical Background

Now that you have a better understanding as to why estimating the horseshoe crab population in the Delaware Bay is important to many different groups of people, let's teach the students estimation using the place value system, expanded notation, very round numbers, order of magnitude, and the powers of 10.

Each of the math concepts listed above is directly connected to the others when teaching estimation. If you don't teach your students place value, how can you expect them to understand the relative sizes of different numbers. To understand that one number is 10, 100, or 1000 times larger than the other, the student must have a familiarity with the names within the place value system. How do you teach place value without a discussion leading into expanded notation? Expanded notation often helps students make sense of the place value system by showing them how the large numbers are broken down into smaller numbers, based on the place value system. It is almost impossible to teach expanded notation without a discussion on the concept of base ten. All of the topics are intertwined and each of them must be taught. However, the order in which they are taught may be more of an individual decision. Each school district, and perhaps even schools within the district have their own way, and order, of teaching these concepts, so do not feel obligated to teach them in the order that I have set forth in this unit. As long as the topics are taught and the students are able to use those topics with little teacher assistance, then the students will be able to accurately answer estimation questions.

Once they have a basic familiarity with the number names, I like to break the numbers down for them into the expanded form of the number. This is critical for getting them to understand that the leading digit of a number is the largest and therefore the most important digit in the number. As we go through this, they learn that the digits to the right of the leading digit are smaller and therefore a less significant part of the number. This is an important concept when we discuss rounding. Because the digits to the right of the leading digit are smaller, when we round a number, we are only changing the overall number by a small amount. If I give them a number, let's say, 4,235,681, I like to break the number down into its base ten expansion (expanded notation):

4,235,681 = 4,000,000 - millions

200,000 - hundred thousands

30,000 - ten thousands

5,000 - thousands

600 - hundreds

80 - tens

1 - ones

When a number is written in expanded notation, I call the individual components very round numbers. By writing the numbers out in their expanded notation, it is easier for the kids to see that place value is simply a way of writing the sum of a group of numbers of a special sort. For example, 4,235,681 is really the sum

4,000,000 + 200,000 + 30,000 + 5,000 + 600 + 80 + 1.

If your students are ready for it, you can also use this opportunity to talk about how these special numbers are written in terms of powers of ten. For example, 4,000,000 is written as 4 x 10 6. I find that they quickly understand the concept of the powers of ten. I start by teaching them 10 can also be written as 10 1. The one tells them how many zeros to write after the one, or the number they are using for a particular problem. For 10 2 I tell them that it simply tells us how many times we multiply 10 to itself. Make certain they understand that they are multiplying 10 by itself, not multiplying 10 times 2. This is a common mistake they make as they are learning the powers of ten. To practice base ten expansion (expanded notation), give them different numbers and ask them to write the numbers out in their base ten expansion. I find that they catch on to this quickly with relatively little assistance.

This is also the time to discuss order of magnitude. As we increase by 10 we are increasing by one order of magnitude. Remember that to begin with order of magnitude, we must have a certain unit to work with for our problems. In this case, let's say a penny is our unit. Therefore, a penny has an order of magnitude of zero. Your base unit will always have zero magnitude, by definition. And remember that we only increase our order of magnitude when we multiply by 10. Therefore, any pennies from one penny to nine pennies will have an order of magnitude of zero. Ten pennies, or a dime, will have an order of magnitude of one, because a dime is 10x's larger than the penny. So, to get to an order of magnitude of three, we will have to multiply the dime by 10 to get 100 pennies, or one dollar. That means pennies 10 - 99 will have an order of magnitude of one. It is not an order of magnitude two until we reach 100. Again, this is just the definition. It allows us to talk about size in a qualitative and still precise way. As you see here, and hopefully your students will see, that the order of magnitude are bounded by successive powers of ten. An easy way to remember order of magnitude is to take the number of digits in your number and subtract one. 12 For example, 8,421, has an order of magnitude of three. There are four digits, so four minus one is three. Don't just give your students this shortcut without explaining the meaning, otherwise the shortcut will be useless because they won't know why they are getting the answer, or what it means.

In terms of horseshoe crabs on the beach, this means that, for the estimate on Slaughter Beach (discussed below), we should be more than happy to replace the reported figure of 23,800 with 24,000. For many purposes, it might be satisfactory to report just the first digit: 20,000. We lose some horseshoe crabs here, but if on another beach there are 26,000 and we report that at 30,000, we get them back. The figure 24,000 is so close to the reported estimate (less than 1% more), that we can't be sure that the actual number might not be 24,000, or even a little more.

Strategies

Now it's time to start applying some of the ideas expressed thus far in this unit. The states of Delaware and New Jersey conduct a horseshoe crab census every year to estimate the number of spawning horseshoe crabs in the Delaware Bay. The numbers collected by this census have been used to create laws, some of which have had a significant impact on many people's livelihoods. Some of these laws have even been overturned due to a lack of convincing evidence, including the use of these estimated population numbers.

One activity that would be a fun learning experience for the kids would be to replicate this estimation process on a small scale in the classroom. This was recommended to me by my seminar leader, Dr. Roger Howe. To recreate the meter quadrant I will use four popsicle sticks glued together in the shape of a square. Instead of a 20m distance between each counted quadrant, we will change it to 50cm. However, since this is a scale model, our 50cm will represent 20m. You will need to find something in your classroom that can represent the horseshoe crabs, which technically could be anything that can be counted. You are going to want as many as possible, the more you have the better the survey will be for the students. Whatever you choose, just remember one thing: you need to know the total number of them prior to beginning the assignment so the students can compare their estimates to the exact number - something that is not possible in the real world of counting horseshoe crabs. Do not give the students the accurate number of "horseshoe crabs" until after everyone has completed their estimates. Have the students find the average number of crabs per square meter, then multiply that number by the length of the "beach" in meters. Make certain your students remember to give their estimates as very round numbers. Have them present their answers, and the methods that led them to their answers, to the class. This way they can learn from each other, as well as from you. Once everyone has presented, give them the actual number of horseshoe crabs and discuss your findings.

Now that they have a good idea of how to get accurate estimates, here are some ideas for problems that you can give them.

1. If a female horseshoe crab can lay between 80,000 and 100,000 eggs in a year, 16 can you estimate the number of eggs that were laid on one beach in one night based on the census report for that night? Accuracy is going to be an issue here, so the students shouldn't be attempting to report any number past the leading digit. They will need to do a little research however, to find out how many nights per year a female may lay eggs on the beach. In Delaware a horseshoe crab egg has an average diameter of 0.7mm. 17
2. The Dover Air Force Base is home to a fleet of C-5 Galaxy cargo planes. Based on the dimensions of the cargo hold, estimate the number of male horseshoe crabs that will fit in the cargo hold. Do the same for the female horseshoe crab. The dimensions of the C-5's cargo hold are height 13.4ft, width 19ft, and the length is 143ft 9in. 18 The average size of the male horseshoe crab is 7-9in across, 2.5 inches high, and 13-16in long, the female is 9-12in wide, 3.5 inches high, and 16-20in long. 19 Keep in mind that one third of the horseshoe crab's length is its tail. Make certain to point out to the students that the cargo hold is a three dimensional object. Can they determine the number of C-5's it would take to transport the entire horseshoe crab population of a certain beach, or perhaps several beaches?
3. Let's say that Slaughter Beach has an area of 1.5 square miles shaped like a rectangle with a base of 3mi and a height of .5mi. Estimate the number of male horseshoe crabs that would fit on the beach. You cannot stand them on top of each other. Parts may not overlap each other. Do the same for females. Use the measurements of the horseshoe crabs from problem #2. Keep in mind when completing this problem that horseshoe crabs do not typically cover an entire beach when they spawn, we are only using that idea for this questions, for the sake of estimation.

Hopefully you can now see how the students will actually be using specific strategies to complete these estimation problems. These questions are not complete in the manner I have them written here. They are more for you to get idea of the types of questions you can ask your students. If you use these questions, then I would spend a few minutes asking the students what other information they feel they will need to know, or research on their own, to solve these problems. The students are not guessing the answers, they are not using "educated" guesses, they are using several mathematical applications to get as accurate an answer as is possible in the form of a very round number. There is more to estimation that most teachers realize and it is time we offer our students the necessary strategies to solve estimation problems with more than a guess.

Activities

The activities for this unit are not long, they are not in any way complicated, and they are certainly able to be interpreted or changed as you see appropriate for your students or for your teaching style. They are ideas that Dr. Howe and I discussed in class, though they are by no means the only activities you can do with this unit. At first, just read through them and see if they are appropriate for you. If they are not, feel free to alter them, or perhaps reading them gave you an idea for a better activity. No matter what you do, remember to make it enjoyable for the kids. If you change the activity, or create a new one, please let me know, I would love to hear from you and it would be beneficial for any teacher who is interested in teaching this unit. I may have written this unit, but I know that teachers are always able to improve upon lessons, so I look forward to hearing from you.

Activity 1

For this activity we will create a simulated horseshoe crab census for the classroom. Please keep in mind however, that the techniques described in this activity are not necessarily the way the horseshoe crab census is conducted in the real world. The way it is done now is a bit too complicated for the kids, and wouldn't serve us well for the sake of estimation. So, the main focus here is estimation, not a perfect simulation of the real world horseshoe crab census.

First you are going to need some materials for this activity. You are going to need lots of pennies and nickels. The pennies will represent the male horseshoe crabs in our activity and the nickels will represent the females, therefore, you will need more pennies, than nickels. Why? Because there are more male horseshoe crabs on the spawning beaches than there are females, so we would like to keep this as realistic as possible. Now, before you spread out the coins, you need to know the exact number of them, prior to starting the activity. Write down the total number of pennies, the total number of nickels, and the overall total of all coins. Do not share this number with the students at this time. You will not give them these numbers until the end of the activity. It is the control that we will use to determine the students' estimation numbers. The number of coins you need depends on the location you have to work with. The larger the work area, the more pennies and nickels you will need. If you do an image search for horseshoe crab spawning, or horseshoe crab beach, you will see the way the horseshoe crabs come to the beach to spawn. This is what you are attempting to replicate in the classroom. Horseshoe crabs come up the beach, just like the water does, and they can be 30 crabs wide, sometimes more, and can extend for a kilometer or further, depending on the weather. A few hundred coins would be good - the more the better. But don't forget, you MUST know the exact number of pennies and nickels prior to starting the activity. Once you have all of your coins, spread them out in a long line across the classroom floor, at least 1000cm would be best, and perhaps 20cm wide. You may find it helpful to spread out all of the pennies first, then go back over them with the nickels, just make certain you spread them out a little more sparingly. It is perfectly fine for the coins to overlap each other, which is exactly what horseshoe crabs do when spawning on the beaches. This part of the activity can get messy, so I recommend spreading out the coins yourself, or having some type of frame so the students don't get the coins all over the classroom.

The other thing that you will need for this experiment is a scaled down model of the meter quadrant that is used in the horseshoe crab census. Four popsicle sticks are all that you need for this. Just glue the four of them into the shape of the square. This is what the students will use to count the number of horseshoe crabs. You will need one popsicle stick quadrant per group of kids. Most teachers have popsicle sticks in their rooms already, so it is usually an easy item to use for this activity. If you would like to substitute something else in place of them, feel free. Again, there is a lot of flexibility in these activities.

You will also need a length of string per group to measure the distance between each quadrant. The real census measures out a distance of 20 meters between each meter quadrant, however, to scale ours down, 50 centimeters may be the best length. You may increase or decrease the measurement, based upon the total length of your "beach" - the total distance from the beginning of your paperclips to the end of them. If you were able to spread out your paperclips over a total of at least 1000cm, then using the 50cm distance will ensure the kids are able to get at least 10 quadrant counts recorded. If you have a lot of students in your class, and if you have enough coins, you may find it beneficial to make two or three rows of horseshoe crabs. More than one group can work comfortably on one row; however, too many may make it difficult for the students to maneuver.

Now that you have solved their problems, after they count the crabs in their first quadrant, have them stretch out the 50cm string from the beginning of the first quadrant, until it is tight. This is the place they will lay down their quadrant for the second count. They will continue in this fashion until they have completed the number of trials that you request. It does not have to be 10, nor does it have to be the same number of trials per group. Change the number of trials for each group to see if that changes the estimates that the groups come up with. For each quadrant, the groups only need three pieces of information, the number of females, the number of males, and the total number of crabs for that quadrant. At the end of the data table, they should have a place for the total number of males, females, and all crabs for all of the quadrants combined.

Once they have all of the data collected, the group will have to come up with a way to determine the estimated number of crabs on their beach. They should be able to give you an estimate for the males, another estimate for the females, and if that went well they should easily have an estimate for the total number of crabs on their beach. Have each team also record the largest number of crabs they counted in one quadrant and the smallest number of crabs they counted in one quadrant. Then tell them to use the largest number to find the estimated number of total crabs, then the smallest number to estimate the total number of crabs. They can compare their estimates after using the largest number of crabs, the smallest number of crabs, and the average number of crabs. This part of the activity will provide the students the opportunity to see how uncertain this process can be for estimating populations. Answer questions as they arise, some groups may need you more than others. If necessary, have groups redo their counts if their numbers are not close to where they should be. If this happens, tell them they should not be discouraged since this happens in the real population studies as well.

After all of the groups have reported to you that they are finished and happy with their answers, have them report their findings to the class. Have them show their data on the board, and have them show the math they used to determine their estimated number of crabs. This is why it is important not to share the total number of coins used at the beginning. The total number of coins used is the correct answer for this activity, however, we are only looking for estimates, and any number after the leading digit here may not be necessary. After you try the activity once, change the criteria, using the suggestions described above (5 th paragraph in this section), but if you are going to do that, then still do not share the correct answer with them. You don't want to influence their math.

Activity 2

First have the students choose the animal of their choice, in case they have had enough of the horseshoe crab. Once they have chosen a favorite animal, they need to do a little research and find a characteristic of that animal that can be compared to humans. If you have the time, you can work on the research in class (computer lab), or if necessary you can assign the research for homework. They should only need a one night, perhaps two, to complete the research. The animal and the characteristic they choose are really up to them, and don't worry about more than one student choosing the same animal, or characteristic. If they do, you have answers that you can compare in class, which can lead to some interesting math discussions if the answers are different for the same problem.

One example of something they could research could be the number of teeth an adult shark loses, compared to the number of teeth an adult human loses in a lifetime. If they can't decide on something that can be compared to humans, they can choose two animals and make the comparison. For example, they could compare the estimated number of sea turtle eggs to chicken eggs it will take to fill up a bathtub.

Another area that may be fun for the students to study is the population of humans compared to the population of their favorite animal. What is the estimated total weight of the human population compared to the estimated total weight of the animal that they chose? It would be interesting to see how much more humans weigh as a group, than say all of the blue whales remaining in the wild. It may surprise the students!

At this point in the unit, teacher intervention should be at a minimum, as the students' ability to estimate effectively should have improved to a level that allows them to work independently. You will need to give them some advanced guidance for their research. Things like the size of the above mentioned bathtub with have to be determined to answer that question. Also, after the students have chosen their characteristic, they need to decide what question it is they are trying to answer. This is the area that they will most likely need the most guidance from you.

Once everyone has finished this activity, it would be most beneficial to have them present their answers, and the math that got them their answers, on the board. This way the rest of the class can see how other kids are doing the math. This is good for students to learn new methods, and may help those students who are having difficulty grasping the idea of estimation. Remember that you want their explanations to include the concepts that you taught them in this unit. The more concepts they are able to utilize and discuss, the deeper their level of understanding.

Activity 3

For this activity you will need to access the following link, http://www.ocean.udel.edu/mas/bhall/hsccensus/2001%20season%20report.pdf

This is the 2001 horseshoe crab spawning survey report for the Delaware and New Jersey beaches on the Delaware Bay. Starting on page five you will find lists of beaches, the dates they were surveyed, and the average number of horseshoe crabs per quadrant for those beaches. We are going to use those numbers for this activity, but not all of them. We want to give the kids the total length of the beaches that were surveyed, but please do not feel as though you need to use all of them. I wouldn't use more than a page worth of this data with most of my sixth graders, so use what you feel is in accordance with your students' ability.

In the first column of the data, you will find the name of the surveyed beach and next to the name in parentheses you will find the total length of the beach that was surveyed. The length of the beach remains constant for each night the beach was surveyed. In the subsequent columns you will find the date of the survey, the number of horseshoe crabs per meter, and the estimated total of crabs for that particular beach for the dates listed. Give the students the list of beaches, including the beaches length and the average number of crabs found in each quadrant for that beach. That's all the information the students receive. Now the problem.

Teacher's Bibliography

Adam, John A. and Weinstein, Lawrence. Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin. Princeton, NJ: Princeton University Press, 2008.

The authors give an explanation of how to estimate the answers to problems with nothing more than basic math skills. The explanation only takes up the first section of the book. The rest of the book is dedicated to estimation problems, as well as their answers. This is a good starter reference for teachers who are new to teaching estimation.

Howe, Roger. "Taking Place Value Seriously: Arithmetic, Estimation, and Algebra." (January 2008). http://www.maa.org/pmet/resources/PlaceValue_RV1.pdf (accessed July 09, 2008).

This paper was written by the Estimation seminar leader, Dr. Roger Howe, of Yale University. This paper provides the background of teaching estimation the way that I do in the unit. There is much more detail provided in this paper than was provided in my unit. This is a must read for any teacher looking to teach estimation.

Student Bibliography and Resources

Air Mobility Command. "C-5 Galaxy." (2008). http://www.af.mil/factsheets/factsheet.asp?id=84 (accessed July 12, 2008).

This is the United States Air Force website, which I used to get the measurements for the C-5 Galaxy cargo plane. The measurements for all different aircraft can be obtained through this site.

Delaware Department of Natural Resources and Environmental Control. "Green Eggs and Sand, Tri-State Horseshoe Crab/Shorebird Education Project." (2003). http://www.dnrec.state.de.us/fw/neware/Program%20Overview/Program%20Overview.htm (accessed July 11, 2008).

This is the State of Delaware's site for the programs provided through the Aquatic Education Resource Center. A listing and description of the programs is provided at this site.

Ecological Research & Development Group. "The Horseshoe Crab. In the News: Delaware Conch Fishermen Successfully Repeal Moratorium on Horseshoe Crab Harvesting." (2006). http://www.horseshoecrab.org/news/pdf/Pressrelease050807.doc (accessed July 11, 2008).

One of the most comprehensive horseshoe crab information websites available. This website provides all matters of things pertaining to horseshoe crabs. This particular link is to one of the articles that was found on the site, however, the main site, www.horseshoecrab.org is a great resource for any teacher looking to teach this unit.

Hall, William. "Horseshoe Crab Census Information." (2008). http://www.ocean.udel.edu/mas/bhall/hsccensus/index.html (accessed July 10, 2008).

This is a great place to find detailed descriptions of how the horseshoe crab census is conducted. Survey results from past years, a list of beaches, and a tally sheet can also be found on this site.

Howe, Roger. "Taking Place Value Seriously: Arithmetic, Estimation, and Algebra," http://www.maa.org/pmet/resources/PlaceValue_RV1.pdf (accessed July 9, 2008).

This paper was written by the Estimation seminar leader, Dr. Roger Howe, of Yale University. This paper provides the background of teaching estimation the way that I do in the unit. There is much more detail provided in this paper than was provided in my unit. This is a must read for any teacher looking to teach estimation.

Maryland Department of Natural Resources. "Horseshoe Crabs: A Living Fossil. Life History." (2008). http://www.dnr.maryland.gov/education/horseshoecrab/spawn.html (accessed July 09, 2008).

This site provides the reader with detailed information about the horseshoe crab. Similar to the site provided by EDRG, though has some information specific to the state of Maryland.

Mid-Atlantic Sea Grant Network, "Horseshoe Crab History and Biology." http://www.ocean.udel.edu/horseshoecrab/History/index.html (accessed July 11, 2008).

Detailed descriptions of the horseshoe crab are provided including, the history and biology, shorebird connection, human use, research, fisheries management, as well as a listing of prime horseshoe crab beaches that can be found on the east coast of the United States.

Mid-Atlantic Sea Grant Network, "Horseshoe Crab. Research. Biomedical. Eye Research." http://www.ocean.udel.edu/horseshoecrab/Research/eye.html http://www.ocean.udel.edu/horseshoecrab/History/index.html(accessed July 11, 2008).

Detailed descriptions of the horseshoe crab are provided including, the history and biology, shorebird connection, human use, research, fisheries management, as well as a listing of prime horseshoe crab beaches that can be found on the east coast of the United States.

Miller, Mark. "The Issue: Too Close for Horseshoes." (2007). http://www.delaware-bride.com/ME2/dirmod.asp?sid=&nm=Archives&type=Publishing&mod=Publications%3A%3AArticle&mid=8F3A7027421841978F18BE895F87F791&tier=4&id=2DCD13F1B22344D39C6D309AC17D8DE5 (accessed July 10, 2008).

This article was found in the Delaware Today online magazine. It is a discussion on the effects of a harvesting moratorium. Topics include the effects on the eel and conch fishermen, as well as the effect on the migrating shorebird populations, and the resulting effect on the local tourism economies.

Norris, Scott. "Oldest Horseshoe Crab Fossils Found in Canada." (2008). http://news.nationalgeographic.com/news/2008/01/080131-oldest-crab.html (accessed July 17, 2008).

This article was just written this year and has changed the original estimate of the horseshoe crab's age - 300 million years old. Due this fossil find, the age of the horseshoe crab is now dated back to 450 million years ago, and some think it may be older than that.

Swan, Benji, and Hall, William, and Shuster, Carl Jr. "The Delaware Bay Horseshoe Crab Spawning Survey 2001 Season." http://www.ocean.udel.edu/mas/bhall/hsccensus/2001%20season%20report.pdf (accessed July 10, 2008).

This article provides detailed survey reports, beach by beach, for the 2001 season. This is the data that was used for the activities in this unit.

U.S. Fish and Wildlife Service. "News Release: Red Knot named candidate for Endangered Species Act protection." (2006). http://www.fws.gov/news/NewsReleases/showNews.cfm?newsId=A26DAA75-DFC1-18FC-1DF52CD3E63D886F (accessed July 09, 2008).

A news release describing the issues surrounding the red knot and its potential listing on the Endangered Species List.

United States Geological Survey. "Population Studies of Horseshoe Crab (Limulus polyphemus) in Delaware Bay." (2007). http://www.lsc.usgs.gov/aeb/2065 (accessed July 18, 2008).

Updated detailed information is provided on this site including population studies of the horseshoe crab. Projects, papers, reports, presentations, research, and a history of the horseshoe crab are also included. There are links to other horseshoe crab websites.

Wikipedia the Free Encyclopedia. "Horseshoe Crab." (2008). http://en.wikipedia.org/wiki/Horseshoe_crabs (accessed July 09, 2008).

A brief description of the horseshoe crab is provided. Some of the topics include, physical description, life cycle and behavior, evolution, and conservation.

Endnotes

1. Norris, Scott. "Oldest Horseshoe Crab Fossil Found in Canada."

2. Delaware Department of Natural Resources and Environmental Control. "Green Eggs and Sand, Tri-State Horseshoe Crab/Shorebird Education Project."

3. Hall, William. "Horseshoe Crab Census Information."

4. USGS. "Population Studies of Horseshoe Crab in Delaware Bay."

5. Ecological Research & Development Group. "The Horseshoe Crab. In the News: Delaware Conch Fishermen Successfully Repeal Moratorium on Horseshoe Crab Harvesting."

6. MDNR. "Horseshoe Crabs: A Living Fossil."

7. U.S. Fish & Wildlife. "News Release: Red Knot named candidate for Endangered Specied Act protection."

8. Miller, Mark. "Too Close for Horseshoe Crabs."

9. ERDG. www.horseshoecrab.org

10. ERDG. www.horseshoecrab.org

11. ERDG. www.horseshoecrab.org

12. Howe, Roger. "Taking Place Value Seriously: Arithmetic, Estimation, and Algebra."

13. Howe, Roger. "Taking Place Value Seriously: Arithmetic, Estimation, and Algebra."

14. Howe, Roger. "Taking Place Value Seriously: Arithmetic, Estimation, and Algebra."

15. Swan, Hall, and Shuster. "The Delaware Bay Horseshoe Crab Spawning Survey 2001 Season."

16. MDNR. "Horseshoe Crabs: A Living Fossil."

17. ERDG. www.horseshoecrab.org

18. Air Mobility Command. "C-5 Galaxy."

19. Wikipedia. "Horseshoe Crab."