Light the Lights Again

7035 /www/nrich/html/content/id/7035/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <p><span style="font-style: italic;">This problem is similar to</span> <a href="http://nrich.maths.org/7044&part=" style="font-style: italic;">Light the Lights</a><span style="font-style: italic;">, but slightly more tricky. You may like to try that one first.</span></p><br></br> Here is a machine with four coloured lights. Each light responds to a rule. <br></br><br></br> When you type in a number, lights go on if their rule is satisfied. <br></br> If the number satisfies more than one rule, then more than one colour will light up.<br></br><br></br> Type in some numbers and see which lights you can switch on.<br></br> <br></br><br></br> <div style="background-color:#000;padding:20px;padding-bottom:20px;width:260px;height:280px"> <div style="position:relative;"> <label style="position:absolute;top:10;color:#fff;font-size:12px;font-family:helvetica,arial,sans-serif">Type in a number</label> <input id="numberText" type="text" value="0" name="number" tabindex="1" maxlength="8" size="8" style="position:absolute;top:-10px;left:130px;font-size:1.2em" /> <img id="red" src="red.png" style="position:absolute;left:20px;top:30px;opacity:0;z-index:1" /> <img id="redOff" src="redOff.png" style="position:absolute;left:20px;top:30px;opacity:1;z-index:0" /> <img id="yellow" src="yellow.png" style="position:absolute;left:120px;top:30px;opacity:0;z-index:1" /> <img id="yellowOff" src="yellowOff.png" style="position:absolute;left:120px;top:30px;opacity:1;z-index:0" /> <img id="green" src="green.png" style="position:absolute;left:20px;top:130px;opacity:0;z-index:1" /> <img id="greenOff" src="greenOff.png" style="position:absolute;left:20px;top:130px;opacity:1;z-index:0" /> <img id="blue" src="blue.png" style="position:absolute;left:120px;top:130px;opacity:0;z-index:1" /> <img id="blueOff" src="blueOff.png" style="position:absolute;left:120px;top:130px;opacity:1;z-index:0" /> <input id="restartButton" name="restartButton" type="button" value="Restart" style="position:absolute;top:260px" /> <br style="clear:both" /> </div> </div> <br></br><br></br><br></br> What is the smallest number which lights up all the lights?<br></br><br></br> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <span class="editorial">Many of you arrived at the same correct conclusion that $36$ is the smallest number which lights up all the lights.</span><br></br> <br></br> <span class="editorial">Some used trial and improvement, testing out all the numbers from $1$ until they reached $36$. This is one way to answer the question, but what we'd really like to know is: what are the rules behind the lights?</span><br></br> <br></br> <span class="editorial">Students in Mrs Keary's Maths Group of Bushloe High School summed this point up well:</span> We worked on this problem for a whole lesson and only just found all of the reasons right at the end, though we did find $36$ quite quickly!<br></br> <br></br> <span class="editorial">So what are the rules? Let's start with the Green Light.</span><br></br> <br></br> <span class="editorial">Class 4 of Grampian Primary School said:</span><br></br> <br></br> The first pattern we spotted was that all the even numbers would light the 'Green' light. We experimented with larger numbers to check that this still worked. We agreed that 'any multiple of $2$ would light the Green light'.<br></br> <br></br> The second pattern we spotted was that the 'Red' light would light up when you entered a number that ended in $6$ or $1$. We tested this with four digit numbers and it still worked. We agreed that 'any number ending in $1$ or $6$ would light up the Red light'.<br></br> <br></br> <span class="editorial">Year 6 of the Croft School had a good approach which made it easy to identify the rule for the Red light:</span><br></br> <br></br> <div>We started off using guess and check and wrote down what lights lit up. We found that $36$ lit all the lights but we did not know why. We decided to try the lights from $1$, and to colour in a hundred square to look for patterns for each colour. We noticed the red squares formed columns downwards, like the $5$ times table does, but they are not $5$ times table, you start with $1$ and add $5$ each time.</div> <div> </div> <div><span class="editorial">Class 4 continued:</span></div> <div> </div> <div>The Yellow and Blue lights were much more difficult to spot. We started to look at the number sequences and looked at what you had to do to one number to get to the next number. We spotted a pattern and could then say what the next number would be. We tested these in the problem. Here is a copy of our recordings.</div> <div><mdo:image width="552" height="387" src="Light%20the%20lights%20again%20recordings.PNG" alt=""></mdo:image></div> <div> </div> <div><span class="editorial">Many of you correctly identified the numbers in the Yellow column as the Triangular numbers and the numbers in the Blue column as the Square numbers.</span></div> <div> </div> <div><span class="editorial">Finlay and Ben of St Andrew's School wrote down all the formulae which lit up all the lights:</span></div> <div> </div> <div style="margin-left: 40px;">Green = Even number = $2n$</div> <div style="margin-left: 40px;">Yellow = Triangular numbers = $n(n+1)\over 2$</div> <div style="margin-left: 40px;">Blue = Square number = $n^2$</div> <div style="margin-left: 40px;">Red = Every five number = $5n+1$ </div> <div> </div> <div><span class="editorial">Here are some more pictures sent into us by Class 4 of Grampian Primary School investigating triangular numbers and square numbers with counters.</span></div> <div> </div> <div> <mdo:image width="348" height="261" src="Light%20the%20lights%20again%20counters%202a.PNG" alt=""></mdo:image><mdo:image width="369" height="275" alt="" src="light%20the%20lights%20again%20ish.jpg"></mdo:image></div> <div> </div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Light the Lights Again</h2> <p><span style="font-style: italic;">This problem is similar to</span> <a href="http://nrich.maths.org/7044&part=" style="font-style: italic;">Light the Lights</a><span style="font-style: italic;">, but slightly more tricky. You may like to try that one first.</span></p> <br></br> Here is a machine with four coloured lights. Each light responds to a rule.<br></br> <br></br> When you type in a number, lights go on if their rule is satisfied.<br></br> If the number satisfies more than one rule, then more than one colour will light up.<br></br> <br></br> Type in some numbers and see which lights you can switch on.<br></br> <br></br> <br></br> <div style="background-color:#000;padding:20px;padding-bottom:20px;width:260px;height:280px"> <div style="position:relative;"><label style="position:absolute;top:10;color:#fff;font-size:12px;font-family:helvetica,arial,sans-serif">Type in a number</label> <input id="numberText" maxlength="8" name="number" size="8" style="position:absolute;top:-10px;left:130px;font-size:1.2em" tabindex="1" type="text" value="0"></input> <mdo:image id="red" src="red.png" style="position:absolute;left:20px;top:30px;opacity:0;z-index:1"></mdo:image> <mdo:image id="redOff" src="redOff.png" style="position:absolute;left:20px;top:30px;opacity:1;z-index:0"></mdo:image> <mdo:image id="yellow" src="yellow.png" style="position:absolute;left:120px;top:30px;opacity:0;z-index:1"></mdo:image> <mdo:image id="yellowOff" src="yellowOff.png" style="position:absolute;left:120px;top:30px;opacity:1;z-index:0"></mdo:image> <mdo:image id="green" src="green.png" style="position:absolute;left:20px;top:130px;opacity:0;z-index:1"></mdo:image> <mdo:image id="greenOff" src="greenOff.png" style="position:absolute;left:20px;top:130px;opacity:1;z-index:0"></mdo:image> <mdo:image id="blue" src="blue.png" style="position:absolute;left:120px;top:130px;opacity:0;z-index:1"></mdo:image> <mdo:image id="blueOff" src="blueOff.png" style="position:absolute;left:120px;top:130px;opacity:1;z-index:0"></mdo:image> <input id="restartButton" name="restartButton" style="position:absolute;top:260px" type="button" value="Restart"></input><br style="clear:both"></br> </div> </div> <br></br> <br></br> <br></br> What is the smallest number which lights up all the lights?<br></br> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div><a href="http://nrich.maths.org/7035&part=">This problem</a> requires children to make sense of information and work in a systematic way, thereby modelling the process of scientific enquiry. It is a motivating context in which use their knowledge of number properties.</div> <br></br> <h3>Possible approach</h3> <div>This activity will require children to have access to computers, ideally in pairs.</div> <br></br> <div>You could start by dividing the board into two columns, one headed with a tick and the other headed with a cross. Ask learners to suggest numbers, and write each suggestion in the appropriate column according to a rule of your own choice. Make it clear to the class that the activity is designed to model working like a scientist - there is no-one to say whether they have the right answer or not. So they can come up with a hypothesis for your rule, but you will not confirm their hypothesis, you will only place numbers in the appropriate column.</div> <br></br> <div>Here are some suggestions for rules:</div> <ul> <li>Multiples of $3$</li> <li>Prime numbers</li> <li>One less than a multiple of 10</li> </ul> <div>Once the class has tried the activity with a couple of rules and everyone is reasonably convinced their hypothesis holds, move on to the main task.</div> <br></br> <div>To introduce the main task, show the interactivity and demonstrate entering a couple of numbers to see what lights up. Make sure learners understand that more than one light can light up at once, and that each light is governed by its own simple rule. Learners should work in pairs at a computer, and their challenge is to try to light up all the lights.</div> <br></br> <div>While the class is working, note any particularly good ways of recording or working systematically, and highlight them to the rest of the class.</div> <br></br> <div>Towards the end of the time working on the problem, leave some time for the class to come together to share what they have done. As well as discussing the number that lit up all the lights, encourage children to explain how they approached the problem.</div> <br></br> <h3>Key questions</h3> <div>Which numbers have you tried? What happened?</div> <div>Which number would be good to try next? Why?</div> <div>How will you remember which lights each number lights up?</div> <br></br> <h3>Possible extension</h3> <div>Some children may want to try a harder version of this problem, <a href="http://nrich.maths.org/7024">Charlie's Delightful Machine</a>.</div> <br></br> <h3>Possible support</h3> <div>Some children might find it helpful to use a <a href="/content/id/7035/1-100_NumberGrid.pdf">hundred square</a> to record their findings. They also might like to try <a href="http://nrich.maths.org/7044">this version</a> of the problem which uses slightly lower-level mathematics.</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">How will you keep track of which lights light up for each number?<br></br> You may find <a href="/content/id/7035/1-100_NumberGrid.pdf">this hundred square</a> useful.<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Even, square, 5n+1, triangular<br></br></mdoxml> 2 3 0 1 0 0 0 Light the Lights Again Each light in this interactivity turns on according to a rule. What happens when you enter different numbers? Can you find the smallest number that lights up all four lights? Using, Applying and Reasoning about Mathematics Making and testing hypotheses Information and Communications Technology Interactivities Numbers and the Number System Properties of numbers Numbers and the Number System Factors and multiples Numbers and the Number System Square numbers Numbers and the Number System Triangle numbers Admin Upper primary mapping document