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2011-02-01T00:00:01
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<p>While I'm sitting in front of the computer I also have a dice and it is showing a $5$. Rather like this:-</p>
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This is quite a popular arrangement for this number.<br></br>
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Going around I see that some people wear ties or dresses that are very colourful and full of lovely patterns. The patterns may be made up of teddy bears, rockets, pink-panthers, flowers and all sorts of other things. However there arrangement - if I represent each item just as a black dot then the pattern is usually :-<br></br>
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<p>The arrangements usually sit quite well into a rectangular arrangement. So we might call this last one a $3$ by $5$ or a $5$ by $3$ arrangement. Squares are just special rectangles and so we could have a $4$ by $4$ :-</p>
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<p>When we have the $3$ by $5$ we can count the dots and see that we have $23$ altogether. Now $3$ times $5$ IS NOT $23$. So we cannot write it as:- $3 x 5= 23$</p>
<p>So we (You) will have to invent another name instead of times and a sign to go with it. Perhaps (because it is convenient on the computer) you may choose "?" BUT you can choose anything that you like!</p>
<p>Now I can write:-</p>
<p>$3 ? 5 = 23$</p>
<p>and when I look at the $2$ by $2$ and the $4$ by $4$ I can write them as:-</p>
<p>$2 ? 2 = 5$ and $4 ? 4 = 25$</p>
<p>If you now look at a bigger arrangement that follows the same kind of pattern you'll find various ways of counting up the number of dots needed.</p>
<p>So let's look at this one $8 ? 4$</p>
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<p>Count them up, calculate, or do whatever so that you find out how many there are altogether and we have:- $8 ? 4 = 53$. Invent some more questions of your own and work out the answers.</p>
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Now if this were an ordinary thing like multiplication, which you'd use in an ordinary rectangle like:-
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<p>You probably would know the answer straight away or you'd be able to work it out from things that you knew before. When people learn multiplication tables they often write them out in a big table. So you could explore what happens if you try a similar idea with doing "?" instead of multipliation and make a ?Table instead of a TimesTable.</p>
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There are lots of answers to this problem, depending on what
questions you choose to ask. <br></br>
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Have a go yourself, and if you discover anything interesting,
<a href="mailto:%20nrich@damtp.cam.ac.uk">e-mail</a> us to tell us
what you've done! 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.<br></br></mdoxml>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=66&part=">This activity</a> is extremely open and invites exploration at whatever the level of pupils' understanding. It can lead to a pupil having a wider understanding of what it is to multiply. Many opportunities are built in for exploring number patterns.</div>
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<h3>Possible approach</h3>
<div>I have found it good to lay out some counters or cubes in the initial pattern and invite pupils to talk about what they notice, with the simple prompt, "Tell me what you see!".</div>
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<h3>Key questions</h3>
<div>What do you notice?</div>
<div>Can you describe what you see for everyone so that they might see it too?</div>
<div>When appropriate - How did you work that out? What adding did you actually do?</div>
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<h3>Possible extension</h3>
<div>Get the pupls to imagine that the pattern you've presented to them at the start is just one item in a sequence. Ask them to create/talk about what the previous/next ones might be.</div>
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<h3>For the Most Able</h3>
When this acticity has satisfied the pupil then go to <a href="http://nrich.maths.org/7473&part=">3D Stacks</a> for a much marger 3D exploration giving numbers that have very many properties and relationships.<br></br>
<h3>Possible support</h3>
<div>Joining in with pupils so that they are very involved with the talk would be a big assett for many children.</div>
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This is an investigation that can lead to pupils thinking quite
hard. I've found that they really enjoyed this "Alternative"
multiplication activity.<br></br>
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For pupils who are confident and you're looking for some lengthy
extension work that allows them to think and discover then this may
be just what you want.<br></br>
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I found it extra helpful if they know about digital roots. See the
teachers notes for EWWNP for the work on Number Patterns if you are
unsure about this work.<br></br>
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The ?Table would start off as:-<br></br>
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2
3
0
1
0
0
0
Doplication
We can arrange dots in a similar way to the 5 on a dice and they
usually sit quite well into a rectangular shape. How many
altogether in this 3 by 5? What happens for other sizes?
Using, Applying and Reasoning about Mathematics
Investigations
Using, Applying and Reasoning about Mathematics
Generalising
Calculations and Numerical Methods
Multiplication & division
Calculations and Numerical Methods
Addition & subtraction
Sequences, Functions and Graphs
Arithmetic sequence