2105 /www/nrich/html/content/03/04/six5/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>Investigate the successive areas of light blue.</p> <p>What fraction of the total area of the square does the area of light blue take up in each case? </p> <p class="c1"><mdo:image width="593" height="162" align="middle" alt="Picture patterns based on squares" src="Light-Blue.gif"></mdo:image></p> Can you work out what the next two diagrams would look like?<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <span class="editorial">Francesca investigated this problem. She imagined that each time the big square was split up into little blocks that looked like the light blue ones. Then she counted how many light blue ones there were, and how many overall. This is what she got</span> :<br></br> <br></br> 1, $\frac{2}{3}$, $\frac{4}{9}$, $\frac{8}{27}$, $\frac{16}{81}$, $\ldots$ <br></br> <br></br> <div><br></br> <div style="clear: both;"><span class="editorial">She noticed that the number on top got multiplied by 2 each time, and the number on the bottom got multiplied by 3 each time.</span></div> </div> <div><span class="editorial">Some of our more advanced readers might know that we could write this as</span><br></br> <br></br> $\frac{2^n}{3^n}$.</div> <div><br></br> <div style="clear: both;"><span class="editorial">Francesca also noticed that the amount of light blue got smaller and smaller each time. She thinks that if we could do this forever, in the end the whole square would be dark blue.</span></div> </div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Light Blue - Dark Blue</h2> <br></br> <p>Investigate the successive areas of light blue.</p> <p>What fraction of the total area of the square does the area of light blue take up in each case?</p> <p class="c1"><mdo:image align="middle" alt="Picture patterns based on squares" height="162" src="Light-Blue.gif" width="593"></mdo:image></p> Can you work out what the next two diagrams would look like?</div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div><a href="http://nrich.maths.org/2105&amp;part=">This problem</a> gives children the opportunity to explore fractions in a practical context. They will be identifying and explaining patterns, and justifying their ideas.</div> <br></br> <h3>Possible approach</h3> <div>Show the images to the group, perhaps one at a time, inviting them to look carefully at each. Then, hide the images and ask learners to talk to each other about what they saw. After some time, bring them together to discuss as a whole group and at this stage you might encourage them to think about the fractions of each colour if it doesn&#39;t come up naturally. It will be necessary to reveal the images again at certain moments, in order to check ideas and facilitate the discussion.</div> <br></br> <div>Then ask children to work in pairs to recreate the images and extend the sequence. Squared paper and squares of coloured paper might be useful, but encourage each pair to make their own decisions about the materials they will use. This will be an ideal time for you to wander around the room listening to the conversations. Listen out for those learners who are challenging their partner&#39;s assumptions and justifying their answers.</div> <br></br> <br></br> <h3>Key questions</h3> <div>What fractions can you see?</div> <div>Is there a pattern to the fractions of area in each image?</div> <div>Can you convince us that your fraction is correct?</div> <br></br> <h3>Possible extension</h3> <div>Learners could also make up other patterns of their own for partners to extend.</div> <br></br> <h3>Possible support</h3> <div>Some children might benefit from having copies of the images available on a sheet to which they could refer when necessary.</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">It might help to try and re-create the images using squared paper and coloured pens/pencils or squares of coloured paper.<br></br></mdoxml> 2 4 0 1 0 0 0 Investigate the successive areas of light blue in these diagrams. Fractions, Decimals, Percentages, Ratio and Proportion Fractions Sequences, Functions and Graphs Limits of Sequences Sequences, Functions and Graphs Sequences