Math in the Middle... of Oceans Objectives TURTLES Graph data on nest locations Use data on turtle nest locations to make recommendations on construction near beaches Compute growth rate of turtles Key Terms dimensional relationship loggerhead incubate hypothesize survey reference point Math Concepts Integers Percent Computation/Estimation Ratios/Rates Proportion Patterns Statistics/Data analysis Probability Measurement Surveying techniques (location given by angle and distance) Electronic Fieldtrips Loggerhead Sea Turtle Conservancy at Bald Head Island, North Carolina Math in the Middle... of Oceans TURTLES LESSON 10 Page 1
NCTM Standards Connections MATHEMATICS AS PROBLEM SOLVING Using data on turtle nest locations to make recommendations on construction near beaches MATHEMATICS AS COMMUNICATION Using data on turtle nest locations to make recommendations on construction near beaches Use of surveying techniques to record nest locations MATHEMATICS AS REASONING Using data on turtle nest locations to make recommendations on construction near beaches Relating stage of development of turtle eggs to potential reasons for unhatched eggs MATHEMATICS AS CONNECTIONS Using growth rate measurements to predict age of turtles Using data on turtle nest locations to make recommendations on construction near beaches Relating stage of development of turtle eggs to potential reasons for unhatched eggs NUMBER AND NUMBER RELATIONSHIPS/NUMBER SYSTEMS AND NUM- BER THEORY Measurement of turtle shells at various times to predict growth rate of tagged turtles Relation of temperature of nest sand to sex ratio of hatchlings Math in the Middle... of Oceans TURTLES LESSON 10 Page 2
COMPUTATION AND ESTIMATION Computing growth rate of turtles Computing the number of eggs produced in a lifetime by a turtle PATTERNS AND FUNCTIONS Graphing data on nest locations Relation of temperature of nest sand to sex ratio of hatchlings Relating stage of development of turtle eggs to potential reasons for unhatched eggs Predation patterns STATISTICS Graphing data on nest locations Using temperature data to predict sex of hatchlings Relating stage of development of turtle eggs to potential reasons for unhatched eggs Collecting and analyzing data on predators PROBABILITY Predictions on survival rate of hatchlings MEASUREMENT Measurement of turtle shell at various times to predict growth rate of tagged turtles Using measurement to record locations of turtle nests Math in the Middle... of Oceans TURTLES LESSON 10 Page 3
Getting Started - Discussion Topics Sea turtles are some of the most fascinating creatures on Earth. The main turtle we study in this lesson is the loggerhead. The loggerhead sea turtle can grow to weigh over 350 pounds and have a shell three feet long. The largest sea turtle in the world, the leatherback sea turtle, can have a shell nearly 6' long and weigh close to 2,000 pounds! Having survived since the time of the dinosaurs, sea turtles are connections to the ancient life of this planet. Thought they have survived for millions of years, while other large reptiles died off, they are all now threatened with extinction because of the carelessness of humans. There are seven species of sea turtles recognized in the world today. Five of these species regularly inhabit the Atlantic coast of the United States including the leatherback, green, hawksbill, Kemps s ridley and loggerhead. All of the species are listed as endangered and protected under the United States Endangered Species Act except for the loggerhead, which is listed as threatened. Due to the perilous status of these marine reptiles, a great deal of time and money has been spent worldwide to study the life and history of these animals. While many facets of the life of the sea turtle have been studied extensively, still other aspects need to be researched to help fill in the puzzle of sea turtle biology. The loggerhead is the most common inhabitant of the southeastern United States and thus the easiest to study. Most sea turtle research and conservation in the United States therefore focuses on the loggerhead sea turtle. The loggerhead is regularly seen in temperate, subtropical and tropical waters of the Atlantic, Pacific and Indian Oceans. Adults feed on mollusks, crabs and other bottom dwelling invertebrates during a life span many speculate to be sixty years or more. Age at maturity is not known, but has been estimated at six to thirty years of age and is believed to vary between populations. Most of this lesson is based on loggerhead turtle nesting because that is the best known part of these turtles' lives. How they live when they are in the ocean is not well known. Like other aquatic turtles, loggerheads must return to dry land to lay nests. Each spring in the U.S., adults mate in the shallow waters of the Atlantic near shore. During the summer, females crawl out of the water to dig nests in the sand where they will deposit an average of 120 eggs. Females can return and nest again up to six times in one summer, at intervals of about two weeks. No maternal care is provided for the eggs, which will incubate in the warm summer sand for around sixty days. Most nesting occurs on temperate and subtropical beaches. The southeastern United Stated represents a major portion of worldwide nesting with an estimated 14,000 females nesting there annually. High temperatures during the day normally keeps newly hatched turtles in the nest until nightfall. The benefits of this include predator avoidance and avoidance of lethal daytime temperatures. Once hatchlings emerge from the nest chamber, they quickly move down the beach towards the ocean. Once in the ocean, each turtle swims nearly continuously for twenty-four hours and nearly continuously during daylight each day following until it reaches the open ocean where it will spend the first years of its life. Math in the Middle... of Oceans TURTLES LESSON 10 Page 4
Getting Started - Discussion Topics continued Oceanic predation is largely a mystery to researchers. Many terrestrial predators of eggs and hatchlings have been identified including raccoons, ants, ghost crabs, foxes, coyotes, pigs, dogs, armadillos and birds. Oceanic predation is believed to be high but has not been looked at extensively, although sharks and large game fish such as dolphin, jacks and marlin are thought to be key predators. As you will see in the video, conservation efforts are in place to help the loggerhead sea turtle recover from its threatened status. Much of the energy is focused on protection of the nests by volunteers and paid sea turtle biologists. Some progress has been made through U.S. fishing laws as well. The National Marine Fisheries Service now requires all shrimping boats to install turtle excluder devices (TEDS) in their nets, which help sea turtles escape when caught in these nets. Before the widespread use of TEDS, hundreds of thousands of sea turtles died in shrimp nets each year. Since sea turtles are reptiles, they have to come to the surface to breath (they do not have gills). Shrimp trawlers will usually pull their nets for over an hour. Since the net is on the bottom of the water, any sea turtle caught in the net will often drown. They can usually only hold their breath for 30 minutes. Vehicles driving on beaches are another hazard to nesting sea turtles. The mother turtles and the baby hatchlings can be killed by vehicles. A less obvious hazard to the hatchlings is lights on beaches. When the hatchling first come out of their sandy nest they know their way to the ocean because of the reflection off the ocean. Even on a cloudy night, the reflection off the ocean is the brightest object on beaches without houses and hotels. When turtles nest on developed beaches, however, the hatchling will head towards whatever direction they view the brightest light. Artificial lights from buildings and parking lots have been to blame for many baby sea turtles killed on roads. Filters covering coastal lights can help solve this problem. To ensure the future of the loggerhead and other species of sea turtles, more advances must be made. The elimination of four-wheel drive traffic and lighting on beaches, the elimination of trash in ocean waters and the creation and preservation of reserves without beachfront development where turtles can nest undisturbed will go a long way towards helping the loggerhead make it into the 21st century. Math in the Middle... of Oceans TURTLES LESSON 10 Page 5
Activity - Population Projections We can use average egg and hatchlings success rates to figure out how many nesting mothers we may have in the future. Population estimates gained in this way are valuable tools for scientists trying to conserve sea turtles. This is how you go about estimating future additions to the nesting population: 1. If 150 eggs were laid in a nest and 90% of those eggs hatched and the hatchlings made their way to the sea, how many hatchlings would that be? ANSWER: 150 eggs x.90 hatch rate 135 hatchlings made it to the sea 2. Now if only one in every hundred hatchlings (1%) survives until adulthood, how many of the original 150 eggs will actually become adult loggerhead sea turtles? ANSWER: 135 hatchlings x.01 survival rate 1.35 turtles survive to adulthood from this nest 3. Obviously, you cannot have a part of a turtle swimming around, however, when you add several nest together, these extras become whole turtles. For example, if there were 200 nests on Bald Head Island one summer and 1.35 turtles survived to adulthood from each nest, how many turtles survived from Bald Head Island in total? ANSWER: 200 nests x 1.35 survivors per nest 270 turtles survived in total Math in the Middle... of Oceans TURTLES LESSON 10 Page 6
Activity - Population Projections continued 4. Sea turtle biologists also want to know how many hatchlings a female will produce in her lifetime and how many of these will survive to adulthood. To determine this, several factors must be taken into account. First, the average loggerhead will nest every three years over a period of about 32 years, or an average of ten different summers. During these ten summers, she will nest an average of four times with each nest having about 150 eggs. How many eggs will she produce? ANSWER: then: 150 eggs per nest x 4 nest per season 600 eggs 600 eggs x 10 seasons 6000 eggs 5. If 90% of these eggs survive to become hatchlings, how many is that? ANSWER: 6000 eggs x.90 hatch rate 5400 hatchlings 6. If one in one hundred of these survive to adulthood, how many adults has this mother turtle added to the loggerhead population? ANSWER: 5400 hatchlings x.01 survival rate 54 adult turtles added to population Math in the Middle... of Oceans TURTLES LESSON 10 Page 7
Student Activity Sheet - Population Projections 1. If 150 eggs were laid in a nest and 90% of those eggs hatched and the hatchlings made their way to the sea, how many hatchlings would that be? 2. Now if only one in every hundred hatchlings (1%) survives until adulthood, how many of the original 150 eggs will actually become adult loggerhead sea turtles? 3. Obviously, you cannot have a part of a turtle swimming around, however, when you add several nest together, these extras become whole turtles. For example, if there were 200 nests on Bald Head Island one summer and 1.35 turtles survived to adulthood from each nest, how many turtles survived from Bald Head Island in total? 4. Sea turtle biologists also want to know how many hatchlings a female will produce in her lifetime and how many of these will survive to adulthood. To determine this, several factors must be taken into account. First, the average loggerhead will nest every three years over a period of about 32 years, or an average of ten different summers. During these ten summers, she will nest an average of four times with each nest having about 150 eggs. How many eggs will she produce? 5. If 90% of these eggs survive to become hatchlings, how many is that? 6. If one in one hundred of these survive to adulthood, how many adults has this other turtle added to the loggerhead population? Math in the Middle... of Oceans TURTLES LESSON 10 Page 8
Activity - Percentages It is also important to figure out what percentage of eggs were killed by a particular source. This helps sea turtle biologists know where to concentrate their protection efforts. Below is a list of how many eggs were lost to various sources in a nest. Change the numbers to percent lost from each source. There were 140 eggs in the original nest. Lost to raccoons 19 Lost to foxes 8 Lost to fire ants 5 Lost to water damage 3 What is the biggest source of loss in this nest? The least? Where should most of the protection efforts be placed - against high tides by moving the nest or by screening the nest for protection against hungry mammals? ANSWER: (Must first explain how to convert raw numbers into percentages if this hasn t been gone over in the school year already.) Number of loss Number in total = % loss Lost to raccoons 13.6% Lost to foxes 5.7% Lost to fire ants 3.6% Lost to water damage 2.1% Biggest source of loss: raccoons Least source of loss: water damage More screening of nest needed Female loggerhead sea turtles lay their eggs in sandy nests on the beach. They will dig a hole in the sand about two feet deep and lay up to 150 ping-pong ball size eggs in the nest. She will then cover the nest and return to the sea. Many biologists study what happens to loggerhead sea turtle eggs in the 60 or so days it takes them to incubate and hatch out. The location where the female turtle lays her eggs is very important for several reasons. If she buries her eggs too close to the ocean, a storm might uncover and sweep them away. If she lays them too close to the sand dunes, something very strange may happen. Math in the Middle... of Oceans TURTLES LESSON 10 Page 9
Activity - Percentages continued There are very few plants that live on the beachfront because it is a very harsh environment. The hills of sand, known as sand dunes support quite a few plants. Away from where the ocean water comes up, this area is better suited for plants than the beachfront is. However, the sand dunes are still a very harsh area for plants - this environment is much like a desert. The beach front is very much like a desert to plants. Rainfall is not absorbed very well by sand. The water seeps through the sand and into the groundwater. The sun is very intense on the beach because of reflection off of bright sand and water. On top of all of this, the salt that is blown off of the ocean by the wind sticks to the plants and can dry them out. The sand is very low in nutrients. The nutrients are drained from the soil with every rain. What do these plants do for nutrients? One important source is the sea itself. As the waves break, water is tossed in the air. The wind picks up the water molecules and carries them well inland. This is why cars rust so quickly at the beach. When you visit the beach on a windy day, your car windshield will be covered in spray. This same water spray also carries nutrients to plants on the beach front. On coastal islands, the further you move away from the shore line, the less the salt spray reaches the plants. The plants that are behind the sand dunes do not rely on the ocean spray for nutrients. Actually, the salt spray can damage these plants. The salt crystals settle on leaves and act like a magnifying glass, making the already intense beach sun hit the leaves even stronger. This causes a pruning effect and slows down the plant s vertical growth. So, what do beach plants have to do with turtle nests? Remember how we said the beachfront is a very desert-like environment? Moisture and nutrients are very valuable to these plants because there is very little of these requirements for life where they live. That is why, when a female turtle lays her eggs too close to these plants, their roots will penetrate the outer membrane of the turtle eggs to get to the moisture and nutrients inside. In the process, these plants are withdrawing essential fluids from these eggs and the embryos normally die. Knowing how many eggs are destroyed by various sources is very important to biologists who are trying to determine the best way to protect sea turtles. If you were a biologist studying sea turtle nesting, you would find there is an optimal range of the beach front that is best for sea turtles eggs to survive within. On the number line below, +10 marks the location of the ocean and -10 marks the location of the middle of the sand dune. Zero on the number line marks the optimal point on the beach for the turtles to make their nest. The numbers that appear on the line represent different nests. <...................... > -7-6 -5-4 -3 0 2 4 7 8 We find that we can follow the rule (n, n 2-11) to determine the egg loss compared to the distance of a nest from the optimal nesting area. n is the value on the number line marking the distance from the optimal nesting area (0). The y coordinate of the ordered pair is the Math in the Middle... of Oceans TURTLES LESSON 10 Page 10
Activity - Percentages continued number of eggs lost per nest due either to roots penetrating the eggs or to the ocean washing away part of the nest. Complete these ordered pairs using the above rule. (8, ), (7, ), (4, 5), (2, -7), (0, -11), (-3, -2), (-4, 5), (-5, 14), (-6, ), (-7, ). We find that the average number of eggs that were laid in these nests was 90. What is the total number of eggs that were lost out of these 10 nests? How many hatched? Take as a given that if y equals a negative number then the entire nest hatched successfully. ANSWER: Pattern: (8, 53), (7, 38), (4, 5), (2, -7), (0, -11), (-3, -2),(-4, 5), (-5, 14), (-6, 25), (-7, 48) Total number of eggs that were loss: 188 Total number of eggs that hatched: 712 Turtle eggs face other hazards as they warm in the summer sun, waiting to hatch. Foxes, raccoons and dogs will dig up the turtle nests and eat the eggs. It is important to understand the toll taken on sea turtles by predators so we can keep track of how many eggs are being laid and how many of these eggs are actually hatching. If a raccoon raids a turtle nest, a biologist would not be able to tell how many eggs were laid in the nest by taking a count. One way that the number of eggs laid could be estimated is by calculating the volume of the nest. The area of the nest where the eggs are deposited is shaped like a sphere. Therefore, by using the formula for the volume of a sphere, we can estimate how many eggs were taken from this sphere by studying patterns. The following measurements were taken from turtle nests, some of which had been raided by raccoons and foxes while others were intact. Nest: A B C D E Diameter of nest in decimeters: 3.0 3.25 3.4 3.5 3.75 Volume of nest in cubic dm.: 14.13 17.97 22.45 # of eggs recovered from nest: 71 90 112 Math in the Middle... of Oceans TURTLES LESSON 10 Page 11
Activity - Percentages continued Nests C and E were raided by raccoons and foxes so that a large number of the eggs that were originally laid in the nest had been eaten. How many eggs were originally in these nests? Follow the pattern of the other nests to find your answer. ANSWER: Students should have derived a pattern of 20 eggs / 4 cubic decimeters from the statistics on the intact nests. They should then solve for the volume of the nests that had been raided. Having found volume, they could apply the 20:4 ratio to derive the answer. The number of eggs originally laid in nest C was 103 eggs; in nest E there were originally 138 eggs laid before the raccoon raided the nest. Math in the Middle... of Oceans TURTLES LESSON 10 Page 12
Activity - Using Statistics Sea turtle biologists also use statistics of many kinds to create a kind of picture of what a population of turtles looks like. A mathematical picture, that is. Characteristics such as turtle size or number of eggs laid are combined together to tell scientists what the average turtle is like. Below is information from ten different turtles who have nested on Bald Head Island. If we average together all the numbers, we can create a picture of the average turtle from this group. Length Width # eggs in nests (cm) (cm) 1st nest per season Turtle ABC 110 94 146 2 Turtle BCD 105 85 166 3 Turtle CDE 112 100 133 4 Turtle DEF 108 97 151 4 Turtle EFG 107 97 88 3 Turtle FGH 111 99 127 5 Turtle GHI 100 84 101 2 Turtle HIJ 105 95 110 6 Turtle IJK 111 98 155 4 Turtle JKL 109 99 134 4 What is the average length? The average width? The average number of eggs in the first nest of the season? The average number of nests in a season? ANSWER: (Must explain averaging if not already discussed previously in the school year.) Add all values in a category and then divide by the number of values in that category (10 in our case) to get the average. Average length 107.8 cm Average width 94.8 cm Average # of eggs in 1st nest 131.1 Average # of nests per season 3.7 The mathematical picture of our average loggerhead from this group is as follows: She is 107.8 cm long, 94.8 cm wide, had 131.1 eggs in her first nest and nests an average of 3.7 each season. Math in the Middle... of Oceans TURTLES LESSON 10 Page 13
Video Challenge The future of turtles may well depend on females producing a tremendous amount of eggs. Very few of these eggs survive, though. It is estimated that only 1 per 1,000 of these hatchling may survive to adulthood, and the actual number could be as low as 1 per 10,000. Our best means of helping them is to use the knowledge we have gained from studying sea turtles to give them a fighting chance against the odds. Find five different plants and five different animals that produce high numbers of offspring. Then find five different plants and five different animals that produce a low number of offspring. See what traits these different species share. How are they alike? How are they different? See how math can be used to help us understand their differences and similarities. How has a species pattern of reproduction worked for their particular ecological niche? Math in the Middle... of Oceans TURTLES LESSON 10 Page 14