6152 /www/nrich/html/content/id/6152/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <ul id="stemLinks"> <li><a href="http://nrich.maths.org/7275">Warm-up</a></li> <li><a href="http://nrich.maths.org/6140">Try this next</a></li> <li><a href="http://nrich.maths.org/6151">Think higher</a></li> <li><a href="http://plus.maths.org/content/reconstructing-tree-life">Read: mathematics</a></li> <li><a href="http://en.wikipedia.org/wiki/Evolution">Read: science</a></li> <li><a href="http://plus.maths.org/content/os/latestnews/may-aug10/gene_package/index">Explore further</a></li> </ul> <div> </div> <p>An evolutionist suggests that at some distant point 400 million years in the past one of his ancestors was an early form of fish. Try to estimate the number of generations that link the scientist to the fish. To make your estimation, you will need to fill in sensible values for the species in the table below (you will possibly need to research these numbers, and feel free to use a spreadsheet for the calculations. A good place to start is <a href="http://en.wikipedia.org/wiki/Timeline_of_human_evolution">http://en.wikipedia.org/wiki/Timeline_of_human_evolution</a>)</p> <ul class="noindent"></ul> <table style="" border="1"> <tbody> <tr> <th rowspan="2" scope="col" width="100">Category</th> <th rowspan="2" scope="col" width="75"> <div align="center">First appearance (million years ago)</div> </th> <th colspan="3" scope="col">Similar modern day example</th> </tr> <tr> <th scope="col" width="100">Similar modern day creature</th> <th scope="col" width="100">Time from birth to producing first offspring</th> <th scope="col" width="100">Lifespan (years)</th> </tr> <tr> <td>Bony fishes</td> <td style="text-align: center;">400</td> <td>Coelacanth</td> <td> <div align="center"> </div> </td> <td> <div align="center"> </div> </td> </tr> <tr> <td>Amphibians</td> <td> <div align="center">350</div> </td> <td>Lungfish</td> <td> <div align="center"> </div> </td> <td> <div align="center"> </div> </td> </tr> <tr> <td>Reptiles</td> <td> <div align="center">300</div> </td> <td>Lizards</td> <td> <div align="center"> </div> </td> <td> <div align="center"> </div> </td> </tr> <tr> <td>Early mammals</td> <td> <div align="center">200</div> </td> <td>Small dogs</td> <td> <div align="center"> </div> </td> <td> <div align="center"> </div> </td> </tr> <tr> <td>Mammals</td> <td> <div align="center">75</div> </td> <td>Lemurs</td> <td> <div align="center"> </div> </td> <td> <div align="center"> </div> </td> </tr> <tr> <td>Apes</td> <td style="text-align: center;">15</td> <td>Gorillas</td> <td> <div align="center"> </div> </td> <td> <div align="center"> </div> </td> </tr> <tr> <td>Humans</td> <td> <div align="center">1</div> </td> <td>Charles Darwin</td> <td style="text-align: center;">30</td> <td style="text-align: center;">73</td> </tr> </tbody> </table> <p>Can you produce a sensible estimate for a number that you are confident exceeds the actual number of generations?</p> <p>Can you produce a sensible estimate for a number that you are confident is definitely less than the actual number of generations?</p> <p>There will be various estimates and assumptions that you need to make in this question. Can you clearly state the most important factors?</p> <p> </p> <p> </p> <h6 style="font-style: normal;"> </h6> <br></br> <div class="framework"><br></br> NOTES AND BACKGROUND<br></br> <br></br> Suggested timelines for evolution form fascinating reading.<br></br> <br></br> Interestingly, the bony coelacanth fish was considered extinct by scientists until fishermen caught one alive in 1938. Now they occasionally turn up, caught in deep-water nets.<br></br> <br></br> <br></br>  </div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <span class="editorial">We had an interesting anonymous solution to this problem, giving rise to some sensible numbers. We think that similar investigations would make for interesting data-handling project work.</span><br></br> <br></br> We used the internet to research lifespans. We were surprised to see that fishes and reptiles live for a very long time. A modern day bony fish is the sturgeon, from which caviar is taken. Lizards seem to have quite a wide range of lifespans (between 10 and 50), so we picked 30 as an average. The numbers for dingos and lemurs seemed typical for these sorts of creatures. We also changed the human numbers because in the past both would be smaller. Our table became<br></br> <br></br> <a onclick="tablealter(1)" class="control">@</a> <table border="1"> <tbody> <tr> <td style="text-align: center;">Creature</td> <td style="text-align: center;">Timespan</td> <td style="text-align: center;">Time to maturity</td> <td style="text-align: center;">Lifespan</td> <td style="text-align: center;">Min number</td> <td style="text-align: center;">Max number</td> <td style="text-align: center;">Average number</td> </tr> <tr> <td>Fishes</td> <td style="text-align: center;">50,000,000</td> <td style="text-align: center;">20</td> <td style="text-align: center;">100</td> <td style="text-align: center;">500,000</td> <td style="text-align: center;">2,500,000</td> <td style="text-align: center;">833,333</td> </tr> <tr> <td>Amphibians</td> <td style="text-align: center;">50,000,000</td> <td style="text-align: center;">20</td> <td style="text-align: center;">100</td> <td style="text-align: center;">500,000</td> <td style="text-align: center;">2,500,000</td> <td style="text-align: center;">833,333</td> </tr> <tr> <td>Reptiles</td> <td style="text-align: center;">100,000,000</td> <td style="text-align: center;">2</td> <td style="text-align: center;">30</td> <td style="text-align: center;">3,333,333</td> <td style="text-align: center;">50,000,000</td> <td style="text-align: center;">6,250,000</td> </tr> <tr> <td>Early mammals</td> <td style="text-align: center;">125,000,000</td> <td style="text-align: center;">2</td> <td style="text-align: center;">10</td> <td style="text-align: center;">12,500,000</td> <td style="text-align: center;">62,500,000</td> <td style="text-align: center;">20,833,333</td> </tr> <tr> <td>Mammals</td> <td style="text-align: center;">60,000,000</td> <td style="text-align: center;">2</td> <td style="text-align: center;">25</td> <td style="text-align: center;">2,400,000</td> <td style="text-align: center;">30,000,000</td> <td style="text-align: center;">4,444,444</td> </tr> <tr> <td>Apes</td> <td style="text-align: center;">14,000,000</td> <td style="text-align: center;">2</td> <td style="text-align: center;">35</td> <td style="text-align: center;">400,000</td> <td style="text-align: center;">7,000,000</td> <td style="text-align: center;">756,757</td> </tr> <tr> <td>Humans</td> <td style="text-align: center;">1,000,000</td> <td style="text-align: center;">13</td> <td style="text-align: center;">40</td> <td style="text-align: center;">25,000</td> <td style="text-align: center;">76,923</td> <td style="text-align: center;">37,736</td> </tr> </tbody> </table> <p>To get the minimum number of ancestors for each row we divided the time in years by the lifespan. To get the largest numbers of ancestors we divded the time in years by the time to maturity.</p> <p>This gave grand totals of between $19,658,333$ and $154,576,923$ ancestors.</p> <p>To get the average age of an ancestor when giving birth we found an average age = (Lifespan + time to maturity)/2 and timesed this by the timespan.</p> <p>Assuming that ancestors were born at an average point in their parents' lives, we get $33,988,937$ ancestors.</p> <p>Rounding up to sensible values, we think that there were at most $150$ million ancestors, at least $20$ million ancestors and probably about $35$ million ancestors.</p> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <h3>Why do this problem?</h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=6152&amp;part=">This task</a> will give practice in making sensible estimates, approximations and making a prediction based on a range of possible values. Although the content is relatively simple, answers could be given to this question at a range of levels and the problem would suit both Stage 3 and Stage 5 students as an exercise in thinking.</div> <h3>Possible approach</h3> <div>There are various ways in which approximations can be made and it might be good to let students attempt to make their own approximations before opening up to group discussion. As they work on the problem, various decisions will need to be made and various uncertainties will arise (for example: at what stage between maturity and death of a parent was a particular offspring in the chain of descendants produced?). Students should be encouraged to take notes of these points, as discussion of these forms a key part of the evaluation process.</div> <br></br> <div>It is interesting to ask students first to give a guess/estimate of the number of descendants without doing any calculation. Students could then work on the problem in pairs to come up with an estimate and then share their ideas with other groups. Whose estimations seem best? Can students justify clearly their estimations to each other? Do any students wish to revise their estimates in light of such interaction?</div> <br></br> <div>Once estimations have been constructed and discussed it would be worth relating the estimates back to the original question: how reliable might we consider our estimates of the number of generations to be? What questions would this raise? What information might lead to better estimates?</div> <br></br> <div>Finally, how did the students' initial guesses/estimates fit in with the calculated estimates? This allows students to reflect on the size of the final answers. Were they much larger or smaller than expected?</div> <h3>Key questions</h3> <div>Key questions should be used as appropriate to help students find their way into the question:</div> <ul> <li>Can you make a start to the question by giving a very rough estimate of the numbers that go into the table? How could you use these numbers to work out the number of generations?</li> <li>When you fill in the table, what sensible range of choices do you have for each entry?</li> <li>When you calculate the number of descendants from a filled-in table, what uncertainties are there?</li> <li>How could you use a filled-in table to calculate the maximum or minimum number of descendants?</li> </ul> <h3>Possible extension</h3> <div>The fullest solution to this task/main extension involves a careful investigation of the minimum and maximum possible number of descendants. For those with a knowledge of statistics, you might refer them to <a href="http://nrich.maths.org/public/viewer.php?obj_id=6354&amp;part=">Time to Evolve 2</a>.</div> <br></br> <div>As a very general extension discussion point, it is worth mentioning that the evolution from one species to another occurs through gene mutation. Can students get any sense of the rate at which the genes would have to mutate to account for the required genetic change over the given time period? This activitiy would, of course, be research intensive.</div> <h3>Possible support</h3> <div>Some students might feel uncomfortable that there are no 'right' numbers to fill into the table and no 'right' answer to the estimation question. Encourage them that it is OK to make a rough estimate to begin with and then to refine that estimate later. Give them some concrete estimates of time to maturity/death if necessary.</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Estimate lifespan and time to first offspring from those for the modern day equivalent creature.<br></br> <br></br> Think about the point in the lifespan of a descendant at which the next descendant might have been produced.<br></br> <br></br> Don't forget that this is an exercise in estimation and approximation: there is no 'right' answer, so try to put sensible guesses in the table.<br></br> <br></br> <br></br> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br></mdoxml> 2 3 0 0 0 1 0 How many generations would link an evolutionist to a very distant ancestor? Using, Applying and Reasoning about Mathematics Selecting and using information Calculations and Numerical Methods Estimating and approximating Applications STEM - living world Admin Discussion Admin Discussion