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/www/nrich/html/content/98/11/bbprob2/
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2011-02-01T00:00:01
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<div style="float: left;"><mdo:image alt="" height="105" src="8.gif" width="116"></mdo:image></div>
<br></br>
When I went into a classroom earlier this week a child rushed up to tell me she was 8 that day!
<p>Well, Happy Birthday to everyone who has a birthday today!</p>
<p>If you are 8 then this could be for you, but if it is another number then you just change the 8 to whatever your age is today.</p>
<p>There is not a lot to say to introduce this challenge. It's really just to find a great variety of ways of asking questions which make $8$.</p>
<p>Things like $6 + 2$, $22 - 14$, <em>etc.</em></p>
<p>But you need to get examples that use all the different mathematical ideas that you know about.</p>
<p>$1$) So you could show some multiplications and some divisions.</p>
<p>$2$) If you know about fractions then you can add or subtract numbers involving fractions. You could also ask questions like "What is half of $16$?''; "What is four-fifths of 10?'' and so on.</p>
<p>$3$) If you've come across decimals then do a few of those also, perhaps using all the four rules [addition, subtraction, multiplication and division].</p>
<p>And so on.</p>
<p>Use whatever mathematics you know to find as many different ways of getting the answer $8$.</p>
<p>You may find some patterns that would go on for ever and ever. If you do, just put down a few, and then see if you can describe how the pattern works.</p>
<p>So if you're $8$ years old maybe you'll write something like this:</p>
<p>$16 \div 2$, $8 \div 1$, $4 + 4$, $2 + 6$, $9 - 1$, $12 - 4$</p>
<p>$1 + 1 + 1 + 1 + 1 + 1 + 1 + 1, 2 + 2 + 2 + 2$</p>
<p>$15 - 3 - 2 - 1 - 1, 5 + 3 + 6 - 3 - 3$</p>
<p>and so on.</p>
<p>But if you're much much older you may write something like:-</p>
<p>$4 \sin (\pi/2) + \sqrt{5^2 - 3^2}$</p>
<p>Whatever your age, and whatever ones you get caught up with, have a look at the ways that you can make new ones that have a similar pattern.</p>
<p>Your "What would happen if ...?'' questions may be a little different from our usual ones.</p>
<p>The 8 year old might ask "I wonder what would happen if I tried to use multiplication and addition to make 8?''</p>
<p>The much older person (17 years old perhaps) may well ask "I wonder what would happen if I used matrices?''</p>
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<p class="editorial">Rachel from West Flegg Middle School has made
decisions about things such as what numbers she wanted to use and
what sort of mathematics she used. And she look for patterns! She
says:</p>
<p>Hi, I'm Rachel. I am nearly eleven, so I thought I would write
about "I'm Eleven''. In my investigation of different ways to find
eleven, I will be using addition, subtraction, fractions, decimals
and timesing. Some sums will use all and some will use some, but
whatever, I will make eleven.</p>
<table border="0">
<tbody>
<tr>
<td>1.</td>
<td>(10 + 78) $\div$ 8 = 11</td>
<td></td>
<td>2.</td>
<td>(0.8 $\times$ 10) +3 = 11</td>
</tr>
<tr>
<td>3.</td>
<td>0.11 $\times$100 = 11</td>
<td></td>
<td>4.</td>
<td>((11/12 of 72) $\div$ 11) +5 = 11</td>
</tr>
<tr>
<td>5.</td>
<td>50 - 39 = 11</td>
<td></td>
<td>6.</td>
<td>(3 $\times$ 12) - (100 $\div$ 4)= 11</td>
</tr>
</tbody>
</table>
<mdo:image width="250" height="250" align="right" alt="" src="rachel_idea.gif"></mdo:image>
<p>I also, apart from these sums, found 2 sets of patterns. Here
they are:</p>
<p><strong>Pattern 1</strong></p>
<table border="2">
<tbody>
<tr>
<td>(4 $\times$3) - 1 = 11</td>
</tr>
<tr>
<td>(5 $\times$3) - 4 = 11</td>
</tr>
<tr>
<td>(6 $\times$3) - 7 = 11</td>
</tr>
<tr>
<td>(7 $\times$3) - 10 = 11</td>
</tr>
</tbody>
</table>
<br></br>
<p><br clear="all"></br>
<strong>Pattern 2</strong> <mdo:image width="250" height="250" align="right" alt="" src="rachel_idea2.gif"></mdo:image></p>
<table border="2">
<tbody>
<tr>
<td>(4 $\times$11)-(3 $\times$11) = 11</td>
</tr>
<tr>
<td>(5 $\times$11)-(4 $\times$11) = 11</td>
</tr>
<tr>
<td>(6 $\times$11)-(5 $\times$11) = 11</td>
</tr>
<tr>
<td>(7 $\times$11)-(6 $\times$11) = 11</td>
</tr>
</tbody>
</table>
<br></br>
<br clear="none" all=""></br>
<table border="0">
<tbody>
<tr>
<td>7.</td>
<td>(132 $\div$ 10) - 2.2 = 11</td>
<td></td>
<td>8.</td>
<td>52 - 41 = 11</td>
</tr>
<tr>
<td>9.</td>
<td>(77 $\div$ 11) + 4 = 11</td>
<td></td>
<td>10.</td>
<td>(11 $\times$ 11) $\div$ 11= 11</td>
</tr>
<tr>
<td>11.</td>
<td>249.15 $\div$ 22.65 = 11</td>
<td></td>
<td>12.</td>
<td>3$^2$ + 2 = 11</td>
</tr>
</tbody>
</table>
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<mdoxml version="1.0"><div class="embed">
<h2>I'm Eight</h2>
<br></br>
<div style="float: left;"><mdo:image alt="" height="105" src="8.gif" width="116"></mdo:image></div>
<br></br>
When I went into a classroom earlier this week a child rushed up to tell me she was 8 that day!
<p>Well, Happy Birthday to everyone who has a birthday today!</p>
<p>If you are 8 then this could be for you, but if it is another number then you just change the 8 to whatever your age is today.</p>
<p>There is not a lot to say to introduce this challenge. It's really just to find a great variety of ways of asking questions which make $8$.</p>
<p>Things like $6 + 2$, $22 - 14$, <em>etc.</em></p>
<p>But you need to get examples that use all the different mathematical ideas that you know about.</p>
<p>$1$) So you could show some multiplications and some divisions.</p>
<p>$2$) If you know about fractions then you can add or subtract numbers involving fractions. You could also ask questions like "What is half of $16$?''; "What is four-fifths of 10?'' and so on.</p>
<p>$3$) If you've come across decimals then do a few of those also, perhaps using all the four rules [addition, subtraction, multiplication and division].</p>
<p>And so on.</p>
<p>Use whatever mathematics you know to find as many different ways of getting the answer $8$.</p>
<p>You may find some patterns that would go on for ever and ever. If you do, just put down a few, and then see if you can describe how the pattern works.</p>
<p>So if you're $8$ years old maybe you'll write something like this:</p>
<p>$16 \div 2$, $8 \div 1$, $4 + 4$, $2 + 6$, $9 - 1$, $12 - 4$</p>
<p>$1 + 1 + 1 + 1 + 1 + 1 + 1 + 1, 2 + 2 + 2 + 2$</p>
<p>$15 - 3 - 2 - 1 - 1, 5 + 3 + 6 - 3 - 3$</p>
<p>and so on.</p>
<p>But if you're much much older you may write something like:-</p>
<p>$4 \sin (\pi/2) + \sqrt{5^2 - 3^2}$</p>
<p>Whatever your age, and whatever ones you get caught up with, have a look at the ways that you can make new ones that have a similar pattern.</p>
<p>Your "What would happen if ...?'' questions may be a little different from our usual ones.</p>
<p>The 8 year old might ask "I wonder what would happen if I tried to use multiplication and addition to make 8?''</p>
<p>The much older person (17 years old perhaps) may well ask "I wonder what would happen if I used matrices?''</p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>As a <a href="http://nrich.maths.org/public/viewer.php?obj_id=55&part=">number activity</a> I have found this to be one of the very best for both engaging pupils in thoughtful work and for getting them to push forward their own understanding of number.</div>
<br></br>
<h3>Possible approach</h3>
<div>Working at the front on some board to display the pupils' ideas I have started by vaguely grouping their suggestions into the separate four rules of number: addition, subtraction, multiplication and division. After a while I asked the youngsters if there are any more, but just wrote 'etc' to indicate that there were some more if they thought so.</div>
<br></br>
<div>Then I pointed out that they had used two numbers for each suggestion; so, could they use three or four numbers and, by adding, get to 8? After more examples I asked if they could start with a biggish number and then take some away and then have to take some more away in order to end up with only 8?</div>
<br></br>
<div>If and when slip-ups occur (suppose they have suggested 15 - 6 - 6), I would ask what has to happen to the answer so far, so that the answer can get to 8. I then talked with the children about the fact they they can use any mathematics that they understand as long as the answer is 8.</div>
<br></br>
<h3>Key questions</h3>
<div>(Not assuming that you follow their recording - which may have to be a bit unorthodox - it's good to pose questions that help you to know how the child was thinking.)</div>
<div>Tell me what you are doing here.</div>
<div>How have you got these?</div>
<div>Could you find any more like that?</div>
<br></br>
<br></br>
<h3>Possible extension</h3>
<div>1) You might find that, for example, a pupil continues with loads and loads of subtractions raising the starting number by just one each time. My own feeling is that I'll allow that to happen for the first two or three lessons in which I use this starting point. If they carry on in the next lesson I would encourage them to venture further. Usually there is no need, they have already changed
things. Maybe the pupil just had to work at something they felt very confident with, or maybe they just liked the patterns that came from the work.</div>
<div>2) Sometimes when children have written something very confidently you can 'dangle a carrot' in front of them and ask them if they know anything about halves or quarters, and if so they could use them also. Very often pupils have done so when they have received no formal teaching of that subject yet.</div>
<br></br>
<h3>For the exceptionally mathematically able</h3>
The pupil in this category will presumably have many more arithmetic and geometric skills and knowledge of more sophisticated processes. Then the pupil can be expected to obtain the number $8$ using their knowledge and experience.<br></br>
<h3>Possible support</h3>
<div>If a pupil has eight objects then they can access this activity by just putting the eight into a number of groups and say "this, plus this, plus this, makes 8". In this way, the eight objects can be put together and set out into different groups.</div>
<br></br>
<br></br></mdoxml>
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<mdoxml version="1.0">Try to use as many number operations as you can.<br></br>
Can you change the starting point of one of your solutions and then
tweak it a bit to find another way of making 8?<br></br></mdoxml>
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This lesson is the best number investigation that I have ever done
in mixed ability classes ranging from children aged 7 years to 11
years. When using it in a class situation I have had the children
brainstorming ideas and what I have done, for what it's worth, has
been as follows:- <br></br>
<br></br>
1) Vaguely grouped the suggestions into the separate 4 rules of
number, + - $\times$ $\div$. <br></br>
<br></br>
2) Asked the youngsters if there are any more, and just inserted
etc. if they thought so. <br></br>
<br></br>
3) Pointed out that they had used 2 numbers for each suggestion;
could they use 3 or 4 numbers and by adding get to 8? <br></br>
<br></br>
4) Asked if they could start with a biggish number and then take
some away and then have to take some more away in order to end up
with only 8. <br></br>
<br></br>
5) If and when slip ups occur (suppose they have suggested 15 - 6 -
6) I would ask what has to happen to the answer so far so that the
answer can get to 8.<br></br>
<br></br>
6) Talk with the children about the fact they they can use any
mathematics that they understand as long as the answer is 8. <br></br>
<br></br>
7) Then they're off on their own to work on this. <br></br>
<br></br>
Sometimes youngsters do things that surprise their teachers, e.g.
<br></br>
<br></br>
1) A very bright pupil continues with loads and loads of
subtractions raising the starting number by just one each time. My
own feeling is that I'll allow that to happen for the first two or
three lessons in which I use this starting point. If they carry on
in the next lesson I would encourage them to venture further.
Usually there is no need, they have already changed things. Maybe
the pupil just had to work at something they felt very confident
with, or maybe they just liked the patterns that came from the
work. <br></br>
<br></br>
2) Sometimes when children have written something very confidently
you can ``Dangle a carrot'' in front of them and ask them if they
know anything about halves or quarters, and if so they could use
them also. Very often pupils have done so when they have received
no formal teaching of that subject yet. <br></br>
<br></br>
Well, good luck with this, and make it an enjoyable experience for
the pupils giving them the understanding that they have the right
to make so many choices for themselves.<br></br></mdoxml>
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I'm Eight
Find a great variety of ways of asking questions which make 8.
Calculations and Numerical Methods
Multiplication & division
Calculations and Numerical Methods
Addition & subtraction
Admin
Lower primary mapping document
Secondary Mapping Document
Number operations and calculation methods
Admin
Upper primary mapping document