154 /www/nrich/html/content/99/07/letme1/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>Andrew decorated $20$ biscuits to take to a party.</p> <p>He lined them up and put icing on every second biscuit.</p> <p>Then he put a cherry on every third biscuit.</p> <p>Then he put a chocolate button on every fourth biscuit.</p> <p>So there was nothing on the first biscuit.</p> <p>How many other biscuits had no decoration? Did any biscuits get all three decorations?</p> <div style="text-align: center;"><mdo:image src="biscuit_i_che.gif" alt=""></mdo:image></div> <br></br> <p> <comment> End Biscuit Decorations </comment></p>  <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p><span class="editorial">We had over 80 solutions sent in for this challenge, thank you. Here are just a few showing the variety the pupils offer.</span></p> <p><span class="editorial">Francesca sent in a spreadsheet as follows;</span></p> <br></br> <mdo:image height="104" width="570" alt="Francesca" src="Francesca.jpg"></mdo:image><br></br> <div style="text-align: center;">--------------------</div> <br></br> <br></br> <p><span class="editorial">Jack sent in his result as pictures.</span></p> <br></br> <div style="text-align: center;"><mdo:image height="410" width="354" alt="pic2" src="Picture%202.jpg"></mdo:image></div> <div style="text-align: center;"><span class="editorial">-</span> -------------------</div> <p style="text-align: left;" class="editorial">Mia sent in a spreadsheet in this way.</p> <div style="text-align: left;"></div> <div style="text-align: center;"><mdo:image height="74" width="464" src="Picture%203.jpg" alt="pic3"></mdo:image></div> <div style="text-align: center;">--------------------</div> <br></br> <p><span class="editorial">Christabel wrote as follows;</span></p> <br></br> <br></br> What I did before: I first drew 20 cakes and then I put icing on the second one, cherries on the third one and then choco buttons on the next one and then Imissed one and Ikept on doing that until Igot to the end. <br></br> <br></br> What I'm doing now: I made cakes and then labeled them 1,2,3...Then I put icing on the ones that are in the 2 times table. I put cherries on the ones in 3 times table and choc buttons on the cakes if they're in the 4 times table.Then Iwould count how many toppings they have. <br></br> Q1. 1,5,7,11,13,17,19. <br></br> Q2.Number 12 did which had icing on cherries on and choc button's. Which means that there was only one.<br></br> <div style="text-align: center;">--------------------</div> <br></br> <br></br> <p><span class="editorial">Zahrah wrote it out in this way;</span></p> <br></br> <br></br> 1st biscuit= nothing <br></br> 2nd biscuit= Icing <br></br> 3rd biscuit= Cherry <br></br> 4th biscuit= Chocolate Button, Icing <br></br> 5th biscuit= Nothing <br></br> 6th biscuit= Cherry, Icing <br></br> 7th bicuit= Nothing <br></br> 8th biscuit= Chocolate Button, Icing <br></br> 9th biscuit= Cherry <br></br> 10th biscuit= Icing <br></br> 11th biscuit= Nothing <br></br> 12th biscuit= Chocolate Button, Icing, Cherry <br></br> 13th biscuit= Nothing <br></br> 14th biscuit= Icing <br></br> 15th biscuit= Cherry <br></br> 16th biscuit= Chocolate Button, Icing <br></br> 17th biscuit= Nothing <br></br> 18th biscuit= Cherry, Icing <br></br> 19th biscuit= Nothing <br></br> 20th biscuit= Chocolate Button, Icing <br></br> <br></br> 7 Biscuits had no decorations. <br></br> 1 got all three decorations.<br></br> <br></br> <div style="text-align: center;">--------------------</div> <br></br> <p><span class="editorial">Adam sent in this chart;</span></p> <br></br> <br></br> <mdo:image height="331" width="602" alt="Adam" src="Adam%205%20Biscuits.jpg"></mdo:image><br></br> <div style="text-align: center;"><span class="editorial">-</span> -------------------</div> <br></br> <p class="editorial">Eboselumen described her working as a set of instructions;</p> <br></br> <br></br> you label the biscuits 1-20 <br></br> you draw the icing on 2, 4 6,8,10,12,14,16,18,20 <br></br> the cherry on 4th, 8th, 12th, 16th, 20th <br></br> the chocolate button on 3, 6, 9 , 12, 15, 18<br></br> so 5 cakes did not get anything <br></br> nos 5,11,13, 17, 19 <br></br> its simple, the logic is the only prime nos can have nothing so all numbers that are a factor of either 2, 3, 4 will have something on it the only one with everything on it is 12 which is a factor of all three nos <br></br> <div style="text-align: center;">--------------------</div> <br></br> <br></br> <br></br> <p><span class="editorial">and finally Lorne used a spreadsheet to show her working;</span></p> <br></br> <br></br> <div style="text-align: center;"><mdo:image height="396" width="410" src="Lorne%2066%20Biscuits.jpg" alt="Lorne"></mdo:image></div> <div style="text-align: center;">--------------------</div> <br></br> <p><span class="editorial">Well done all of you! This may be a help for teachers to see the variety of approaches that we can start to expect from pupils.</span></p> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Biscuit Decorations</h2> <br></br> <p>Andrew decorated $20$ biscuits to take to a party.</p> <p>He lined them up and put icing on every second biscuit.</p> <p>Then he put a cherry on every third biscuit.</p> <p>Then he put a chocolate button on every fourth biscuit.</p> <p>So there was nothing on the first biscuit.</p> <p>How many other biscuits had no decoration? Did any biscuits get all three decorations?</p> <div style="text-align: center;"><mdo:image alt="" src="biscuit_i_che.gif"></mdo:image></div> <br></br> <p><comment> End Biscuit Decorations </comment></p> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div>This <a href="http://nrich.maths.org/public/viewer.php?obj_id=154&amp;part=">problem</a> fits in well with counting and skip-counting (counting by twos etc.) and can be solved by physically modelling the biscuits and decorations with whatever objects are convenient. It is a good opportunity for children to choose the way they represent the problem in order to solve it. It may also be appropriate to introduce vocabulary such as "multiple".</div> <br></br> <h3>Possible approach</h3> <div>An important element in understanding the problem is the language of ordinal numbers, so &#39;warm-up&#39; activities which involve using the concepts of first, second, third and fourth would be worthwhile for young children.</div> <br></br> <div>Invite learners to work on the problem using whatever they find most helpful - have paper, pens, pencils, cubes, counters etc. easily available. You may like to stop them part way through to share some different representations with the whole group. Some children might have made models with differently-coloured cubes for the decorations, some may have drawn pictures, some may have used symbols. Invite the children to comment on the different ways of recording - what are the advantages of each way? You may find that some learners adopt a different representation following the discussion and it would be interesting to know why this was.</div> <br></br> <div>For those children who are more mathematically experienced, consider linking this problem with the idea of common multiples through the multiplication tables and the hundred square.</div> <br></br> <h3>Key questions</h3> <div>Which other biscuits have icing on?</div> <div>Which biscuits have a cherry on them as well as the third one?</div> <div>What about the biscuits with a chocolate button on them? Which ones are they?</div> <div>Tell me about the biscuits that have no decorations on them.</div> <br></br> <h3>Possible extension</h3> <div>Generate your own similar problems using a greater number of biscuits and different combinations of skip counting, or encourage investigation of the various possibilities. Can children find a combination of skip-counting that allows every biscuit to be decorated?</div> <br></br> <h3>Possible support</h3> <div>With practical equipment available to model the problem, it should be accessible to most learners.</div> <div> </div> <div> </div> <div>Handouts for teachers are available here (<a href="/content/99/07/letme1/Biscuit%20Decorations.doc">word document</a>, <a href="/content/99/07/letme1/Biscuit%20Decorations.pdf">pdf document</a>), with the problem on one side and the notes on the other. </div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Perhaps you could sketch the biscuits?<br></br> The second biscuit has icing on it. Which other biscuits have icing on?<br></br> Which biscuits have a cherry on them as well as the third one? <br></br> What about the biscuits with a chocolate button on them? Which ones are they? <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">Jack wrote to us and said:</p> The only numbers which are not divisible by any of 2, 3 or 4 would be the ones without any decorations. These biscuit numbers would be 1, 5, 7, 11, 13, 17, 19<br></br> The numbers which are divisible by all three - 2, 3 and 4 - would have maximum decoration on them. There is only one biscuit - Number 12.<br></br> <br></br> <p class="editorial">Thank you Jack!</p> <p class="editorial"> </p> <p class="editorial"> </p> <p class="editorial">Here are a set of answers (nos 1 - 10) for you to check yours with;</p> <div style="text-align: center;"><mdo:image width="400" height="22" alt="ans1" src="Bisc%20Ans%201.jpg"></mdo:image></div> <p class="editorial">and 11-20;</p> <div style="text-align: center;"><mdo:image width="400" height="22" src="Bisc%20ans%202.jpg" alt="ans2"></mdo:image></div> <div style="text-align: center;"></div> <br></br></mdoxml> 2 3 1 0 0 0 0 Andrew decorated 20 biscuits to take to a party. He lined them up and put icing on every second biscuit and different decorations on other biscuits. How many biscuits weren't decorated? Numbers and the Number System Common factors Numbers and the Number System Factors and multiples Admin Lower primary mapping document