88
/www/nrich/html/content/01/04/bbprob1/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
I was exploring a puzzle in which headless match sticks had to be
moved to make a different number of triangles.
<p>I made one small triangle</p>
<table cellpadding="10">
<tbody>
<tr>
<td style="text-align: center;"><mdo:image alt="Triangle from three matches." src="onetriangle.gif"></mdo:image></td>
<td>$3$ matches</td>
</tr>
</tbody>
</table>
<br></br>
I made it into $4$ small triangles by adding $6$ matches.<br></br>
<table cellpadding="10">
<tbody>
<tr>
<td style="text-align: center;"><mdo:image alt="Four triangles." src="fourtriangles.gif"></mdo:image></td>
<td>$9$ matches</td>
</tr>
</tbody>
</table>
<p>I added another row and counted the number of small triangles
and counted the matches.</p>
<p><mdo:image alt="9 triangles from matches." src="manytriangles.gif"></mdo:image></p>
<div style="clear: both;"><br></br>
<div style="clear: both;">I made a table of my results and
continued adding rows. I found many patterns.</div>
</div>
<br></br>
Have a go and see what patterns you can find. You do not have to
use match sticks (or cocktail sticks) - drawing lines will do just
as well.
<p>Find a good way to record your results. See if you can predict
the numbers for rows of triangles you have not drawn.</p>
<p>When you have done all you can with triangles, see if you get
the same sort of results with squares. Then think of other shapes
which might make number patterns as they grow.</p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
Tom is putting a great deal of effort and hard work into
investigations.
<p>The first thing that Tom did is an important, helpful strategy
whenever you are problem solving, he used a tool. In this case it
was matchsticks. Being able to 'build' the problem and see how it
'looks' and then being able to move pieces of the problem around is
one of the most useful and vital ideas in solving problems.</p>
<p>Organising your findings into an easy to read table is also
important. This way patterns get revealed making it easier to
predict what happens next.</p>
<p>Here's Tom's ideas for the <strong><em>Sticky
Triangles</em></strong> investigation.</p>
<p>I used matchsticks to work out the answers for the one, two,
three and four rows of triangles. I put my answers in a table and
saw there was a pattern and I worked out what I thought the answer
would be for five rows. I checked it with the matchsticks and it
was right.</p>
<table cellspacing="0" cellpadding="4" border="1" summary="Data table of Tom's solution">
<tbody>
<tr>
<th>Matches</th>
<th>Triangles</th>
<th>Rows</th>
</tr>
<tr>
<td>3</td>
<td>1</td>
<td>1</td>
</tr>
<tr>
<td>9</td>
<td>4</td>
<td>2</td>
</tr>
<tr>
<td>18</td>
<td>9</td>
<td>3</td>
</tr>
<tr>
<td>30</td>
<td>16</td>
<td>4</td>
</tr>
<tr>
<td>45</td>
<td>25</td>
<td>5</td>
</tr>
</tbody>
</table>
<p>This can be explained using equations. Letters can be used to
represent the different pieces of information. This saves writing
out the full words each time. Tom used:</p>
<p><strong>m = number of matches<br></br>
t = number of triangles and<br></br>
r = number of rows</strong></p>
<p>By looking at the table above, Tom was able to work out
that:</p>
<p><strong>t = r <sup>2</sup></strong></p>
<p>(which is the same as saying that the number of triangles is the
same as the number you get when you square number of rows -
meaning, you multiply by the number of the rows by the same
number)</p>
<p>and</p>
<table>
<tbody>
<tr>
<td>m = t + 2</td>
<td>for the first row</td>
</tr>
<tr>
<td>m = t + 2+3</td>
<td>for the second row</td>
</tr>
<tr>
<td>m = t + 2+3+4</td>
<td>for the third row</td>
</tr>
<tr>
<td>m = t + 2+3+4+5</td>
<td>for the fourth row</td>
</tr>
<tr>
<td>m = t + 2+3+4+5+6</td>
<td>for the fifth row</td>
</tr>
<tr>
<td>and so on.</td>
</tr>
</tbody>
</table>
<p>I didn't know how to write this as an equation so I went to
<strong>"Ask Nrich"</strong> and they told me it was:</p>
<p>m = t + r*(r+3)/2</p>
<p>Using <strong>"Ask Nrich"</strong> is using your initiative Tom.
Well done!<br></br>
Do you understand the equation our mathematical helpers gave you,
Tom?</p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><div class="embed">
<h2>Sticky Triangles</h2>
<br></br>
I was exploring a puzzle in which headless match sticks had to be moved to make a different number of triangles.
<p>I made one small triangle</p>
<table style="" border="">
<tbody>
<tr>
<td style="text-align: center;padding:10px;"><mdo:image alt="Triangle from three matches." src="onetriangle.gif"></mdo:image></td>
<td>$3$ matches</td>
</tr>
</tbody>
</table>
<br></br>
I made it into $4$ small triangles by adding $6$ matches.<br></br>
<table style="" border="">
<tbody>
<tr>
<td style="text-align: center;padding:10px;"><mdo:image alt="Four triangles." src="fourtriangles.gif"></mdo:image></td>
<td>$9$ matches</td>
</tr>
</tbody>
</table>
<p>I added another row and counted the number of small triangles and counted the matches.</p>
<p><mdo:image alt="9 triangles from matches." src="manytriangles.gif"></mdo:image></p>
<div style="clear: both;"><br></br>
<div style="clear: both;">I made a table of my results and continued adding rows. I found many patterns.</div>
</div>
<br></br>
Have a go and see what patterns you can find. You do not have to use match sticks (or cocktail sticks) - drawing lines will do just as well.
<p>Find a good way to record your results. See if you can predict the numbers for rows of triangles you have not drawn.</p>
<p>When you have done all you can with triangles, see if you get the same sort of results with squares. Then think of other shapes which might make number patterns as they grow.</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=88&part=index">This investigation</a> starts in a very practical way so that all learners can take part. It can lead to several interesting number patterns and is a good context in which pupils can begin to generalise.<br></br>
<h3>Possible approach</h3>
<div>You could draw one triangle on the board and indicate that it is made from three matches (or lines or lolly sticks etc). Next to this triangle, draw another identical triangle but this time say that you are going to extend the drawing with another row of triangles. Ask the children how many more matches you have used and therefore how many matches are now used altogether. Invite children to
predict how many more matches will be needed for another row in the pattern. Can they make a prediction without drawing? Take some suggestions with reasons for choosing that number, then check how many are needed by drawing the arrangement. Focusing on the different ways the children explain how they visualised the arrangement will help them to build up a pattern of what is happening.</div>
<br></br>
<div>Set the group off on the challenge. You could leave it very open-ended or you could say, for example, that you want them to be able to work out the total number of matches for ten rows of triangles. As they work, stop them at various intervals to share effective ways of recording results, for example in a table. Learners might find it helpful to have matches or sticks, and isometric lined or
dotty paper for recording the actual triangles.</div>
<br></br>
<div>To encourage them to look more carefully at the number patterns involved, rather than simply counting matches each time, suggest that you would like them, for example, to be able to work out the total number of matches for $100$ rows of triangles, which of course would take far too long to draw and count.</div>
<br></br>
<h3>Key questions</h3>
<div>How will you record what you have done?</div>
<div>Can you see why the number of matches increased by that amount when you added that row?</div>
Can you predict how many matches the next row will need? Why?<br></br>
Can you see a link between the number of rows and the total number of matches?<br></br>
Can you see a link between the number of small triangles and the total number of matches?<br></br>
<h3>Possible extension</h3>
<div>Some learners will be able to express the patterns they have found in terms of words, some might use a letter to stand for the number of rows, for example. In addition, the real challenge here is to explain the patterns found in the numbers.</div>
<br></br>
<div>This investigation can be continued with squares (see <a href="http://nrich.maths.org/public/viewer.php?obj_id=2290&part=index">Seven Squares</a> ) and even hexagons. Decisions have to be made on how these are to grow which means that variations on the numbers may be found.</div>
<br></br>
<h3>Possible support</h3>
<div>Some pupils will be able to organise their results without help but others might need the guidance of a table like this:</div>
<br></br>
<div style="text-align: center;"><mdo:image alt="table" height="115" src="table.gif" width="284"></mdo:image></div>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">You could start by making or drawing the first two rows.<br></br></mdoxml>
2
5
0
1
0
0
0
Sticky Triangles
Can you continue this pattern of triangles and begin to predict how
many sticks are used for each new "layer"?
Mathematics Tools
Matchsticks
Sequences, Functions and Graphs
Sequences
Using, Applying and Reasoning about Mathematics
Generalising
Using, Applying and Reasoning about Mathematics
Investigations
Admin
Upper primary mapping document