Estimation

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.

So your male and female horseshoe crabs are spread over the floor, hopefully over a distance of at least 1000cm, you have one popsicle stick quadrant and one 50cm length of string per group, now what? Choose a different starting point for each group, so that you have random quadrants being counted. Tell the students to place their quadrant directly on top of the coins (horseshoe crabs) and to count how many crabs are in the quadrant. At this point, you will run into two problems: first, what counts as being "in" the quadrant? Is it half of the crab, all of the crab, any part of the crab in the quadrant? That decision is up to you, but here's an idea. For the first trial, have them only count those crabs that are entirely within the quadrant, nothing else counts. For the second trial you can change the criteria to include the ones that are at least half in the quadrant, and for the third trial you may include any crab that has any part of them inside of the quadrant. This way you will have three sets of data that can be compared within each group, and across the classroom. The other problem that you will run into is when the kids realize they do not have anywhere to record the data they are collecting. Instead of providing you with a template, I like for my students to create their own data tables for the different projects we are working on. Different projects require different data tables, so I like to see what they come up with own their own. What they will need for their data table however, is a column for males, females, and total crabs. They will need a separate row for each quadrant that they counted as well, with at least 10 rows for 10 quadrant counts. At the top of their data table I would have them include space for their name(s), the date, the time, and the starting position for their group. Remember that you will be starting each group at a different random place. For instance, if group 1 starts right at 0cm, they would mark that on their chart, but if group 2 starts at 11cm, they should mark that as their starting point.

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.

Based on the numbers that you gave to them, do they have the ability to estimate the total crabs found on each beach? Can they estimate the total number of crabs found on all of the beaches for one night's survey? They will need to add the total distance of all of the beaches to begin working on this problem. Once they determine these answers, have them find the average number of crabs per square meter, for all of the beaches combined, based on their answers. Then have them compare their answers to the data from the actual survey from those beaches on those dates. Are they close? What type of math did they do to determine their answers? This activity is again focusing on estimations, so remind the students that the leading digit of their answer should be enough, they should not have exact answers. An estimate to the leading digit will still be accurate enough to see if the students' answers are within two powers of ten, and therefore close enough to the actual survey results recorded in this data table from 2001. Presenting their answers to the class will be useful to stimulate group discussion about the problem, their methods, and their answers.