5648
/www/nrich/html/content/id/5648/
1
2011-02-01T00:00:01
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Here is a four by four dotty grid:<br></br>
<br></br>
<mdo:image height="177" width="177" alt="four by four dotty grid" src="grid.gif"></mdo:image><br></br>
<br></br>
I have joined three dots on the grid to make a triangle which has
one dot inside it:<br></br>
<br></br>
<mdo:image height="177" width="177" src="grid1dot.gif" alt="triangle with one dot inside"></mdo:image><br></br>
<p>How many different triangles with one dot in the middle can you
draw?</p>
<p>How do you know have found them all?</p>
<p>You may like to experiment with the pegboard interactivity
below, or you could print off <a href="/content/id/5648/dottysheet.pdf">this page</a> of dotty grids
to work on.</p>
<p><a href="/content/id/5648/circleAngles.swf">Full Screen
Version</a></p>
<mdo:flash height="400" width="550"><param value="/content/id/5648/circleAngles.swf" name="movie" ></param><param value="7" name="flashplayerversion" ></param></mdo:flash><br></br></mdoxml>
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<br></br>
<p class="editorial">The Four Mathmateers at Brocks Hill Primary School said they drew lots of triangles and used trial and error (or trial and improvement) to answer the question. I think, though, that they might have counted the same triangle more than once.</p>
<p><span class="editorial">Starting off using trial and improvement is an excellent idea! How will you then decide whether you have found all the triangles? Try looking at the</span> <a href="http://nrich.maths.org/public/viewer.php?obj_id=5648&part=clue" class="editorial">hints</a> <span class="editorial">for some ideas.</span></p>
<p><span class="editorial">Neptune Class from Riverley Primary wrote to say:</span></p>
<p>We think that there are $6$ triangles in total.</p>
<p>We made sure that each triangle was a different type (scalene, isosceles, right-angled and equilateral) and we experimented with different shapes on the pin-board.<br></br>
<br></br>
We thought that there must also be a mathematical way to systematically calculate the solution to the problem, but we're still working on it.</p>
<p class="editorial">I think there are a few more than six triangles, but I like the way you thought that working systematically would help. That's exactly what Greg from Swanland County Primary School did. He said:</p>
<p>First I tried to just make triangles with a dot inside.</p>
<p><mdo:image src="Sol1.png"></mdo:image></p>
<p>I found these but I thought there might be more.</p>
<p>So I tried every one with a three dot base.</p>
<p><mdo:image src="sol2.png"></mdo:image> $4$</p>
<p>Then I tried ones with a four dot base.</p>
<p><mdo:image src="sol3.png"></mdo:image> $3$</p>
<p>But the purple one here is the same as the green one above. So only $2$.</p>
<p>Then I tried one with a two dot base.</p>
<p><mdo:image src="sol4.png"></mdo:image></p>
<p>I think there are $7$ different triangles.</p>
<p><span class="editorial">Swifts Class from Southill Lower School said that they found $9$ triangles with one dot but unfortunately they didn't send us a picture of their triangles.</span></p>
<p><span class="editorial">I think I agree with Swifts Class. Can you find the two triangles that Greg missed? <a href="mailto:enquiries.nrich@maths.org">Let us know</a> if you do.</span><br></br>
</p>
<br></br>
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<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><div class="embed">
<h2>Inside Triangles</h2>
<br></br>
Here is a four by four dotty grid:<br></br>
<br></br>
<mdo:image alt="four by four dotty grid" height="177" src="grid.gif" width="177"></mdo:image><br></br>
<br></br>
I have joined three dots on the grid to make a triangle which has one dot inside it:<br></br>
<br></br>
<mdo:image alt="triangle with one dot inside" height="177" src="grid1dot.gif" width="177"></mdo:image><br></br>
<p>How many different triangles with one dot in the middle can you draw?</p>
<p>How do you know have found them all?</p>
<p>You may like to experiment with the pegboard interactivity below, or you could print off <a href="/content/id/5648/dottysheet.pdf">this page</a> of dotty grids to work on.</p>
<p><a href="/content/id/5648/circleAngles.swf">Full Screen Version</a></p>
<mdo:flash height="400" id="/content/id/5648/circleAngles.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/5648/circleAngles.swf"></param>
<param name="flashplayerversion" value="7"></param>
</mdo:flash></div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/5648&part=">This activity</a> is accessible to all pupils but has the scope to be extended in many directions. In order to make a start, children will need to be familiar with properties of triangles but drawing triangles on the grid will help them to clarify for themselves what they understand by "triangle". In order to find all the possible
triangles, pupils will need to work in a systematic way. </div>
<br></br>
<h3>Possible approach</h3>
<div>You may want to begin this task with the whole class and, this way, the notion of "different" will come up quite quickly. How is the group going to define "different"? This is a great discussion point and one where there isn't a right or a wrong answer. You could decide to count triangles which could be picked up and placed exactly on top of another triangle as the same. Or, you could decide
that they are different if they are in a different orientation on the grid. The former suggestion makes a more manageable number to count!</div>
<div> </div>
<div>It might help to suggest working in pairs on this activity so that children are checking they haven't duplicated triangles. Learners might find it helpful to use a pegboard and/or to draw their triangles on <a href="/content/id/5648/dottysheet.pdf">this sheet</a> of grids. </div>
<div> </div>
<div>The interactivity, if projected or used on an IWB, allows findings to be shared easily. In addition to checking that the triangles are indeed all different, a plenary could focus on how the children know that they have found them all, which is quite a challenge. Listen out for learners who have a 'system' of some description which they follow to make sure they don't miss
any out. Alternatively, you could ask children to draw each triangle on a different grid and try to group the triangles that have been found. That way, the imposed method of grouping will help to identify any that have been omitted.</div>
<br></br>
<h3>Key questions</h3>
<div>Tell me about the way you're working.</div>
<div>How will you remember which triangles you've found?</div>
<div>How do you know that your triangles are all different from each other?</div>
<div>How do you know that you have found them all?</div>
<br></br>
<h3>Possible extension</h3>
<div>In order to extend the problem, pupils could be asked to find triangles with three spots inside them or no spots inside ... Differently sized grids could be drawn and compared. You could sort the triangles across differently sized grids, for example all right-angled triangles together, or all triangles which are the same shape but different sizes together.</div>
<br></br>
<h3>Possible support</h3>
<div>Having a range of different equipment available for children to use to tackle this problem (e.g. pegboards, <a href="/content/id/5648/dottysheet.pdf">grids on paper</a>, the interactivity) will help everyone get started.</div>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">You could try with a three by three dotty grid first to get the
idea of the problem.<br></br>
You might want to work with a partner so you have someone to check
that your triangles are all different.<br></br>
Have you got all the triangles which have a corner at the corner of
the grid?<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
Considering triangles drawn from corner dot first:<br></br>
<br></br>
<mdo:image height="196" width="200" src="sol1.gif" alt=""></mdo:image><mdo:image height="202" width="200" src="sol2.gif" alt=""></mdo:image><mdo:image height="197" width="200" src="sol3.gif" alt=""></mdo:image><mdo:image height="191" width="200" src="sol4.gif" alt=""></mdo:image><br></br>
<br></br>
<br></br>
<mdo:image height="203" width="200" src="sol5.gif" alt=""></mdo:image><br></br>
<br></br>
There are no other different ones (where different means could not
be placed on top of each other to "match") if you systematically
draw all triangles from the side dot and the middle dot. Therefore
total of 9.<br></br>
<br></br>
This activity is accessible to all pupils but has the scope to be
extended in may directions. In order to make a start, children will
need to be familiar with properties of triangles but drawing
triangles on the grid will help them to clarify for themselves what
they understand by "triangle". If you are not able to use the
interactivity, children might find this sheet of dotty grids
useful. <br></br>
<br></br>
You may want to begin this task with the whole class and this way,
the notion of "different" will come up quite quickly. How is the
group going to define "different"? This is a great discussion point
and one where there isn't a right or a wrong answer. You could
decide to count triangles which could be picked up and placed
exactly on top of another triangle as the same. Or, you could
decide that they are different if they are in a different
orientation on the grid. The former suggestion makes a more
manageable number to count! It might help to suggest working in
pairs on this activity so that children are checking they haven't
duplicated triangles. The interactivity, if projected or used on an
IWB, allows findings to be shared easily. <br></br>
<br></br>
In order to extend the problem, pupils could be asked to find
triangles with three spots inside them or no spots inside ...
Differently sized grids could be drawn and compared - for example,
you could sort the triangles across differently sized grids eg all
right-angled triangles together, or all triangles which are the
same shape but different sizes together. At a much higher level,
some children could be challenged to explain how they know they
have all the triangles each time which could lead into the
usefulness of having a system (e.g. drawing all those which have a
corner in the corner of the grid first, then all those that have a
corner on a "side dot" etc). <br></br></mdoxml>
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Inside Triangles
How many different triangles can you draw on the dotty grid which each have one dot in the middle?
Mathematics Tools
Pinboard/geoboard
Information and Communications Technology
Interactivities
2D Geometry, Shape and Space
Mixed triangles
Using, Applying and Reasoning about Mathematics
Working systematically
Admin
Lower primary mapping document