Times

49 /www/nrich/html/content/98/08/bbprob2/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> I've been thinking about the time for some time now! What a funny thing to say. Let me explain. Often during the day I have to look at my watch to see what the time is. "Am I late?'' "Will I get there on time?'' "This is very boring, is that all the time is?'' "I can't wait, I wonder what the time is now?'' and so on. I have a digital watch on my wrist, one in the car and one by the side of my bed. For quite a while I've been fascinated by the way the digital numbers are displayed and the patterns they show. Some of my friends also get caught up in this. I'll explain more ... When it's:- <mdo:image height="100" width="218" alt="pic1" src="Picture%201.jpg"></mdo:image> I'm very likely to say, "Well, that's interesting, if you put a mirror down where the dots are you find you get symmetry; there's definitely a line of symmetry down the middle.'' So I'll then look for another, and here's one:- <div style="text-align: center;"><mdo:image height="100" width="262" alt="pic2" src="Picture%202.jpg"></mdo:image></div> Is it the next one or are there some in between? Whatever the next one is, I wonder how long it is between the two times? So there we go, the first challenge is to find times on a 24 hr digital watch/clock that have this line of symmetry down the middle. You might find some rather different ones in which you have to decide whether they are allowed or not, that's all part of doing investigations. But, you know, there are other sorts of shapes that fascinate me. What about this one for instance: <mdo:image height="100" width="199" alt="pic3" src="Picture%203.jpg"></mdo:image> This is just good because when you stand on your head, or show someone else your watch [with your watch upside down to them], you find it can say the same time. Maybe we have to start ignoring the dots in between the hours and minutes, so we take it to look like: <mdo:image height="100" width="200" alt="pic4" src="Picture%204.jpg"></mdo:image> As before there are others like:- <mdo:image height="100" width="200" alt="pic5" src="Picture%205.jpg"></mdo:image> which we may prefer to see as simply:- <mdo:image height="100" width="200" alt="pic6" src="Picture%206.jpg"></mdo:image> I wonder just how many there are? Can we use the seconds numbers as well? Does it help to always have your watch set to being a 24 hour display? Well all of that was something to do with shape. What about the actual numbers, or digits? Here's an idea which I've used when running a residential course for youngsters. I told them that we would have a break and the next time that we would meet would be when the digits added up to 10. [We did not include the seconds.] So after the lunch break we met at 1:54. Here are three times that add up to ten: <mdo:image height="100" width="191" alt="pic7" src="Picture%207.jpg"></mdo:image> <mdo:image height="100" width="270" alt="pic8" src="Picture%208.jpg"></mdo:image> <mdo:image height="100" width="232" alt="pic9" src="Picture%209.jpg"></mdo:image> Have a go yourself and see what others you can come up with; you may think about changing the rule later on. I've had a go at times that add up to 12! <mdo:image height="100" width="175" alt="pic10" src="Picture%2010.jpg"></mdo:image> <mdo:image height="100" width="209" alt="pic11" src="Picture%2011.jpg"></mdo:image> <mdo:image height="100" width="205" alt="pic12" src="Picture%2012.jpg"></mdo:image> <mdo:image height="100" width="210" alt="pic13" src="Picture%2013.jpg"></mdo:image><mdo:image height="100" width="206" alt="pic14" src="Picture%2014.jpg"></mdo:image> <mdo:image height="100" width="208" alt="pic15" src="Picture%2015.jpg"></mdo:image> Well, that's it. "I wonder what would happen if ...?'' Do enjoy this. Looking at the time may never be the same again! Remember to send in ideas, even if you do it on your own and not through a school. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Even though this challenge is about times on clocks, the display on a calculator will help you to explore the patterns more easily than trying to write the numbers on paper. So grab a calculator and punch in some "times". A little hand mirror can help check for symmetry and reflections too. <a href="mailto:nrich@damtp.cam.ac.uk" style="font-weight: bold;">Email</a> us to let us know what you find! Please don't worry that your solution is not "complete" - we'd like to hear about anything you have tried. Teachers - you might like to send in a summary of your children's work. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Times</h2> I've been thinking about the time for some time now! What a funny thing to say. Let me explain. Often during the day I have to look at my watch to see what the time is. "Am I late?'' "Will I get there on time?'' "This is very boring, is that all the time is?'' "I can't wait, I wonder what the time is now?'' and so on. I have a digital watch on my wrist, one in the car and one by the side of my bed. For quite a while I've been fascinated by the way the digital numbers are displayed and the patterns they show. Some of my friends also get caught up in this. I'll explain more ... When it's:- <mdo:image alt="pic1" height="100" src="Picture%201.jpg" width="218"></mdo:image> I'm very likely to say, "Well, that's interesting, if you put a mirror down where the dots are you find you get symmetry; there's definitely a line of symmetry down the middle.'' So I'll then look for another, and here's one:- <div style="text-align: center;"><mdo:image alt="pic2" height="100" src="Picture%202.jpg" width="262"></mdo:image></div> Is it the next one or are there some in between? Whatever the next one is, I wonder how long it is between the two times? So there we go, the first challenge is to find times on a 24 hr digital watch/clock that have this line of symmetry down the middle. You might find some rather different ones in which you have to decide whether they are allowed or not, that's all part of doing investigations. But, you know, there are other sorts of shapes that fascinate me. What about this one for instance: <mdo:image alt="pic3" height="100" src="Picture%203.jpg" width="199"></mdo:image> This is just good because when you stand on your head, or show someone else your watch [with your watch upside down to them], you find it can say the same time. Maybe we have to start ignoring the dots in between the hours and minutes, so we take it to look like: <mdo:image alt="pic4" height="100" src="Picture%204.jpg" width="200"></mdo:image> As before there are others like:- <mdo:image alt="pic5" height="100" src="Picture%205.jpg" width="200"></mdo:image> which we may prefer to see as simply:- <mdo:image alt="pic6" height="100" src="Picture%206.jpg" width="200"></mdo:image> I wonder just how many there are? Can we use the seconds numbers as well? Does it help to always have your watch set to being a 24 hour display? Well all of that was something to do with shape. What about the actual numbers, or digits? Here's an idea which I've used when running a residential course for youngsters. I told them that we would have a break and the next time that we would meet would be when the digits added up to 10. [We did not include the seconds.] So after the lunch break we met at 1:54. Here are three times that add up to ten: <mdo:image alt="pic7" height="100" src="Picture%207.jpg" width="191"></mdo:image> <mdo:image alt="pic8" height="100" src="Picture%208.jpg" width="270"></mdo:image> <mdo:image alt="pic9" height="100" src="Picture%209.jpg" width="232"></mdo:image> Have a go yourself and see what others you can come up with; you may think about changing the rule later on. I've had a go at times that add up to 12! <mdo:image alt="pic10" height="100" src="Picture%2010.jpg" width="175"></mdo:image> <mdo:image alt="pic11" height="100" src="Picture%2011.jpg" width="209"></mdo:image> <mdo:image alt="pic12" height="100" src="Picture%2012.jpg" width="205"></mdo:image> <mdo:image alt="pic13" height="100" src="Picture%2013.jpg" width="210"></mdo:image><mdo:image alt="pic14" height="100" src="Picture%2014.jpg" width="206"></mdo:image> <mdo:image alt="pic15" height="100" src="Picture%2015.jpg" width="208"></mdo:image> Well, that's it. "I wonder what would happen if ...?'' Do enjoy this. Looking at the time may never be the same again! Remember to send in ideas, even if you do it on your own and not through a school. </div> <h3>Why do this problem?</h3> <div>This <a href="http://nrich.maths.org/public/viewer.php?obj_id=49&part=">fun activity</a> can be used as a short or long activity as part of both spacial awareness (transformations such as line symmetry and rotation symmetry) and number recognition.</div> <h3>Possible approach</h3> <div>If pupils are experienced in telling the time in a digital way then a brief introduction will enable them to choose what they are going to explore.</div> <h3>Key questions</h3> <div>Tell me what's special about this time you've chosen.</div> <div>Can you find any more like that?</div> <h3>Possible extension</h3> <div>Learners could look at the number of 'special times' that occur between certain times. More advanced pupils may find systems to allow them to know how many there will be without actually writing them down. You could set a 'simmering activity', for example invite children to find times in which the digits add to $10$, or where there is a horizontal line of symmetry. They could work on this task whenever they had spare time, or at home and could contribute ideas to a designated area of the classroom wall.</div> <h3>Possible support</h3> <div>It would help to have mirrors available for those that would like to use them.</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> You may find it helpful to have a mirror handy. Don't be afraid to turn your paper! </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> </mdoxml> 2 3 0 1 0 0 0 Times Which times on a digital clock have a line of symmetry? Which look the same upside-down? You might like to try this investigation and find out! Mathematics Tools Clock Measures and Mensuration Time Transformations and their Properties Reflections Calculations and Numerical Methods Addition & subtraction Using, Applying and Reasoning about Mathematics Investigations