The Moons of Vuvv

1066 /www/nrich/html/content/01/07/penta3/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> The planet of Vuvv has $7$ moons which lie spread out on one plane in a great disc round it. These Vuvvian moons all have long and confusing names so scientists usually call them by their initials: $A, B, C, D, E, F$ and $G$ starting from the nearest one to the planet. <p style="text-align: center;"><mdo:image height="200" width="497" src="1066.gif" alt=""></mdo:image></p> <p>When two of these moons line up with the planet it is called a 'lunar eclipse'. When three line up with the planet it is called a 'double eclipse', when four do it is a 'triple eclipse' and so on. Once in a while all seven moons line up with the planet and this is called a 'super-eclipse'.</p> <p>Moon $A$ completes a cycle round the planet in one Vuvvian year, moon $B$ takes two years, moon $C$ takes three years, moon $D$ takes four years and so on.</p> <p>How long is it between each 'super-eclipse' on the planet of Vuvv?</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> "I enjoy your website", wrote <strong style="font-weight: 400;">Becky</strong> , from Carleton St Hilda's C. of E. Primary School. Becky sent her Vuvviaan moons solution to us in the form of a table. Becky's solution was one of three different suggestions for solutions to this problem sent in. The three relate to each other. See if you can find out how and why. Then have a look at the problem and see if you can decide which of the answers you agree with and why. This is how Becky began her solution search: <p><mdo:image src="solpic7.gif" alt="Becky's solution."></mdo:image></p> <p>Now, I wonder what Becky changed her search to? If Becky is going to change her search to try and arrive at an answer perhaps she wants to think about this idea.</p> <p><strong><span style="font-weight: 400;">Alex</span></strong> and her family from Leicester, England worked on this Vuvvian problem. Alex explains how they set about arriving at a solution:</p> <ul> <li>We started off by doing the seven times table, because that was how long the last moon took to go round Vuvv.</li> <li>Next, we checked if the multiples of seven were also in the 2x, 3x, 4x, 5x, 6x tables. This was so we'd know if they (Vuvv moons) would line up.</li> <li>We got fed up working out the multiples of seven, because they got way too big. So, we used a calculator! We pressed <strong>+7===</strong> to get the multiples of seven.</li> <li>We found out that it would take <strong style="font-weight: 400;">210</strong> Vuvvian years between each super eclipse.</li> </ul> <p>Now although 210 years is a long time, Anita and Jing Jing from Kilvington Girls' Grammar in Australia, think that's only the half of it...in fact, they think that it is 420 year wait between Super-eclipses. But Thomas, from Suffolk, thinks it's way longer than that - twice as long.</p> <p>He wrote: "There are 840 Vuvvianyears between the Super Eclipses. I worked it out by going through the 7 times table and writing it down seeing that this would be the hardest to divide by. I then realised that the number had to end in a zero because it had to be divisible by 5 and 2. That meant that all I had to do was find out if the multiples of 7 were divisible by 3, 4 and 6 and if they were multiply them by 10. In the end 84 was the lowest number divisible by 3, 4 and 6 so I multiplied it by 10 and got 840."</p> <p></p> <p>Franco and Jonny from Northamptonshire agree that is it 420. They say:</p> <p>We started off with 42. Every number goes into 42, except 5, so we multiplied it by 5.</p> <p>6 doesn't go into 210, so we went back to 42. We then multiplied 42 by 10, to get 420. We checked by dividing 420 by 1, 2, 3, 4, 5, 6 and 7. They are all factors of 420. So the overall answer is 420. </p> <p>So, who is correct?</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>The Moons of Vuvv</h2> <br></br> The planet of Vuvv has $7$ moons which lie spread out on one plane in a great disc round it. These Vuvvian moons all have long and confusing names so scientists usually call them by their initials: $A, B, C, D, E, F$ and $G$ starting from the nearest one to the planet. <p style="text-align: center;"><mdo:image alt="" height="200" src="1066.gif" width="497"></mdo:image></p> <p>When two of these moons line up with the planet it is called a 'lunar eclipse'. When three line up with the planet it is called a 'double eclipse', when four do it is a 'triple eclipse' and so on. Once in a while all seven moons line up with the planet and this is called a 'super-eclipse'.</p> <p>Moon $A$ completes a cycle round the planet in one Vuvvian year, moon $B$ takes two years, moon $C$ takes three years, moon $D$ takes four years and so on.</p> <p>How long is it between each 'super-eclipse' on the planet of Vuvv?</p> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=1066&part=index">This problem</a> offers opportunities for pupils to reinforce their understanding of factors and multiples, and, in a simple example, see an illustration of 'lowest common multiple'. It would fit in well when revising multiplication tables or working on multiples and factors.<br></br> <h3>Possible approach</h3> <div>You could start on this problem with a whole class activity counting in, for example, $2$s and $5$s. When do you say the same number in both? Try also two numbers which have a common factor, for example, $4$s and $6$s. When do you say the same number first in both?</div> <br></br> <div>After this you could introduce the problem either verbally, as a printed sheet or on an interactive whiteboard. Once the children have understood what they are to do, they could work on it in pairs. Some children might benefit from using a calculator for this activity both for multiplying by $7$, and for checking results. You may wish to stop the class part way through to share some of the different ways they are working and recording. Some may be drawing pictures, others may be listing numbers. You could talk about the benefits of the different ways and it may be that some children adopt other representations following this sharing process.</div> <br></br> <div>A discussion of methods and comparison of answers in a plenary may well bring up different results. This would be a good opportunity to discuss the meaning of lowest common multiple.</div> <br></br> <h3>Key questions</h3> <div>How many years does it take for these two moons to coincide?</div> Do these two moons coincide sooner than that?<br></br> <h3>Possible extension</h3> Tell the children that more moons have been discovered circling Vuvv. Get them to work out the length of time between the super-eclipses if there are also moons that cycle taking $8, 9, 10 \ldots$ years.<br></br> <h3>Possible support</h3> Suggest starting with just three or four moons and slowly adding the higher numbers.<br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">You could look at just two moons and work out how many years it takes for them to coincide.<br></br> Which moons might it be good to look at first?<br></br> It might help to use a calculator and to jot your ideas on paper.<br></br></mdoxml> 2 3 0 1 0 0 0 The Moons of Vuvv The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse? Using, Applying and Reasoning about Mathematics Working systematically Numbers and the Number System Factors and multiples Admin Upper primary mapping document