6158 /www/nrich/html/content/id/6158/ 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/7418">Warm-up</a></li> <li><a href="http://nrich.maths.org/6308">Try this next</a></li> <li><a href="http://nrich.maths.org/7412">Think higher</a></li> <li><a href="http://en.wikipedia.org/wiki/Sampling_%28statistics%29">Read: mathematics</a></li> <li><a href="http://en.wikipedia.org/wiki/Haemocytometer">Read: science</a></li> <li><a href="https://motivate.maths.org/content/MathsHealth/PositiveTest">Explore further</a></li> </ul> <div> </div> <p>A biologist is interested in the number of cells in a sample of fluid. A small amount of this fluid is taken from the sample and put into a haemocytometer. The configuration details for the haemocytometer grid are given on the right hand side. By counting the cells and scaling up appropriately, work out an estimate for the number of cells in the original sample.<br></br> <mdo:flash height="550" width="550"><param name="allowfullscreen" value="true" ></param><param name="movie" value="/content/id/6158/haemocytometerv4.swf" ></param><param name="flashplayerversion" value="8" ></param></mdo:flash><br></br>  </p> <h6>Send <a href="mailto:post16.nrich@maths.org?subject=Feedback%20on%20NRICH%20problem%20id%206158">feedback</a> on this interactivity</h6> <br></br> <br></br> <div class="framework">NOTES AND BACKGROUND<br></br> The haemocytometer is a device which is used to count the number of cells in a sample of fluid, from which concentrations and other quantities might be calculated. A haemocytometer is made from a small glass chamber crossed with calibrated wires to aid counting. A small sample of the fluid is placed in the chamber and then examined under a microscope.They are commonly used in laboratories.</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> This problem illustrates calculations typical of those routinely encountered at University-level. Rather than memorising the steps explained below, it is far more beneficial to understand the process by which the answer is obtained.<br></br> <br></br> The problem can be tackled in a few easy steps, which are applicable regardless of the set-up of the Haemocytometer:<br></br> <br></br> 1) Count the number of cells you can see - although this seems like an easy task, it should be noted that in many configurations of the haemocytometer it may be difficult to distinguish between overlapping cells. This is particularly illustrative of real laboratory specimens, where cells will often significantly overlap in dense mediums.<br></br> <br></br> 2) Work out the volume of a the haemocytometer - this is calculated by multiplying together the length, width and depth of the given square-size (either large or small) and then multiplying this up to give the total volume. The volume of a small square should be multiplied by 144, and the volume of a large square by 9 to give the total volume of the haemocytometer. <br></br> <br></br> It is also important to note that since 1ml corresponds to 1 cm$^3$, converting the length, width and depth measurements into cm from mm will greatly ease the calculation.<br></br> <br></br> 3) Work out the number of cells per ml - We have already calculated the number of cells in the haemocytometer, and its volume. Dividing the number of cells by the volume will give the cells per ml (provided that the volume is in cm$^3$)<br></br> <br></br> 4) Scale up to the original sample - With the number of cells per ml already calculated, multiplying this number by the appropriate number of ml in the original sample will give the original cell number.<br></br> <br></br> <br></br> As an example, consider this particular configuration of the Haemocytometer:<br></br> <br></br> <mdo:image height="437" width="579" src="Haemocytometer%20example.png" alt=""></mdo:image><br></br> <br></br> 1) A quick count of the cells reveals that there are 39 present in the haemocytometer.<br></br> <br></br> 2) Since the haemocytometer is comprised of 9 large squares, the average number of cells per large square is $\frac{39}{9} = 4.33$<br></br> <br></br> 3) Each small square measures 0.25mm x 0.25mm, and so each large square must measure 1mm x 1mm. This is equivalent to 0.1cm x 0.1cm. The depth of a small square is the same as a large square and is given by 0.25mm, which is the same as 0.025cm. Therefore a large square measures 0.1 x 0.1 x 0.025 cm, giving a volume of $2.5 \times 10^{-4} cm^3$, which is the $2.5 \times 10^{-4}$ ml.<br></br> <br></br> 4) We have 4.33 cells per large square. Each large square has a volume of $2.5 \times 10^{-4}$ ml. Therefore the number of cells per ml is $\frac{4.33}{2.5 \times 10^{-4}} = 17333$.<br></br> <br></br> 5) 17333 cells per ml. Therefore in our sample of 100ml, we have $100 \times 17333 = 1733300 =1.73 \times 10^6$cells.<br></br> <br></br> <br></br> Can you work out a general formula which allows the calculation of the cells in the sample in a single step?<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <h3>Why do <a href="http://nrich.maths.org/public/viewer.php?obj_id=6158&amp;part="> this problem</a> ?</h3> The haemocytometer is an important piece of scientific apparatus. This interactivity will allow students to practice the relevant skills of proportional reasoning required for its use.The <a href="http://nrich.maths.org/admin/edit.php?obj_id=6158&amp;part=solution"> solution</a> to the problem provides lots of scientific insight into the real use of such a piece of equipment,<br></br> <h3>Possible approach</h3> <div>This resource could be used for whole class or individual use. The configuration for the question is randomly generated each time the refresh button is pressed. This will require students to stay on their toes whilst answering the questions.</div> <div>Once students have become skilled in the use of the interactivity, you might like to try a race: project a configuration onto the board and the first student to obtain the correct answers is the winner, allowing them to balance speed with accuracy.</div> <br></br> <div>There are two key issues involved with using a haemocytometer:</div> <br></br> <div>1) The sample chosen on the slide might not uniformly represent the density in the big sample of the medium</div> <div>2) The cell count might be made difficult due to overlapping cells on the microscope slide.</div> <br></br> <div>Students should realise that they always need to consider the questions</div> <div>1) Is my sample uniform?</div> <div>2) Are the obscured cells on the microscope slide.</div> <br></br> <div>You could discuss these issues with the help of multiple choices of cells and higher or lower density of medium (select these on the interactivitiy). Don't forget that you can zoom in and drag the grid around to focus on issues of overlapping cells.</div> <h3>Key questions</h3> <ul> <li>Have you understood the configuration of the grid? How big is the whole grid?</li> </ul> <ul> <li>Is there an obvious order in which to work out the various answers? Why?</li> </ul> <ul> <li>Given the details for a question, how might you jump straight from the count of the cells on the screen to the number of cells in the original sample?</li> </ul> <h3>Possible extension</h3> <div>Can students find algebraic answers to the questions for a general set of inputs?</div> <div>In what situations might the haemocytometer be more likely to give inaccurate results? How accurate would you expect the results to be in everyday usage?</div> <h3>Possible support</h3> At first, you might like to show the answers and ask students to replicate them.<br></br> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Be systematic in your counting.<br></br> <br></br> Break the calculation down into several small steps.<br></br> <br></br> Keep practising.<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><a href="/content/id/6158/haemocytometer.swf">Interactive 1</a><br></br> <br></br> <a href="/content/id/6158/haemocytometer2.swf">Interacitve 2</a><br></br> <br></br> <a href="/content/id/6158/haemocytometerv3.swf">Interactive 3</a><br></br> <br></br> <br></br> <br></br></mdoxml> 5 3 0 0 0 1 0 Practise your skills of proportional reasoning with this interactive haemocytometer. 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