Estimation

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.

So how do they do it? What are the procedures that are in place for determining these population estimates? The main tool used for these estimates is a one meter quadrant. That means it is a large square, made out of pvc pipe that is one meter in length on each side. This is the device that is used to actually count the crabs in a given area. Volunteers are given a tally sheet to assist them with their count and to help keep the data organized. Without getting too involved in the process (we will get to that later) the overall idea is this; volunteers place the meter quadrant in the sand and count the number of horseshoe crabs that are within the quadrant. They have specific guidelines as to what constitutes a crab that is considered within the quadrant. After one quadrant has been counted, the volunteers move 20 meters down (or up) the beach, lay the quadrant down again and count the crabs at this location. They continue in this fashion until they have covered at least 1km. These surveys can only be conducted during the new and full moon phases, and perhaps a day or two before or after, so it is important that the volunteers follow the procedures correctly the first time. Due to the limited amount of time available, and sometimes unpleasant weather, there aren't many opportunities for second chances. So, now let's say the volunteers have their data table loaded with information. It lists how many males and females were found within each grid. Those numbers are then averaged for the total number of grids that were counted. Let's say that for Slaughter Beach, DE, one of the most popular horseshoe crab spawning sites, a group of volunteers counted an average of 7.96 horseshoe crabs per square meter. If they covered a total distance of 3 km during that count that they would find their total of horseshoe crabs for that beach by multiplying the average number of horseshoe crabs by the total length of beach surveyed. So in this case they would multiply 7.96 horseshoe crabs by 3 km to get a total of 23,880 for Slaughter Beach for that one night. These are the actual numbers taken from survey results from Slaughter Beach on May 5 ^th of 2001. ¹⁵ After going over all of this with my students, I would take a few minutes to ask them what they think about the techniques that are being used as well as the numbers that are being reported. Remember the further to the right that we move down a number, the less and less accurate the digits are likely to be. Therefore, I would hope that the students would be able to tell me that they believe these numbers are accurate to the leading digit of the number, 20,000, and perhaps even to the second digit, the 3,000. However, make certain your students think about the 880. It's supposed to be an estimate and we like to see estimates to the powers of ten, preferably to the leading digit for accuracy. Whether the survey reports the number of horseshoe crabs to be 23,000 or 24,000 is not realistically going to change the count by a significant margin. But if we think about 24,000 horseshoe crabs crawling on a beach, can anyone really think they can estimate the number down to the tens place? I couldn't, not with confidence. The number is much too large to attempt to estimate the number to the tens place. In fact, 23,880 is only .5% less than 24,000.

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.