4901
/www/nrich/html/content/id/4901/
1
2011-02-01T00:00:01
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You might like to try <a href="http://nrich.maths.org/public/viewer.php?obj_id=1813&part=index">
Transformations on a Pegboard</a> before looking at this problem.
Here are some ideas to take this activity some stages further. One
part of the problem was about Billy with a triangle and the other
was about Tanya with her enlarging rectangle.<br></br>
<br></br>
You may like to use this interactive pegboard to try out your
ideas. Read underneath the interactivity for the challenges! <br></br>
<br></br>
<a href="/content/id/4901/circleAngles.swf">Full Screen Version</a>
<br></br>
<mdo:flash height="400" width="550"><param value="/content/id/4901/circleAngles.swf" name="movie" ></param><param value="7" name="flashplayerversion" ></param><param value="400" name="height" ></param><param value="550" name="width" ></param></mdo:flash><br></br>
<br></br>
Billy asked, 'By moving just one peg can you make a right-angled
triangle?'. Well, I had a go starting the same way: <br></br>
<br></br>
<mdo:image height="295" width="379" alt="Billy" src="Billy1.jpg"></mdo:image><br></br>
<br></br>
Here are two I found moving one peg each time (from the
original):<br></br>
<br></br>
<mdo:image height="281" width="375" alt="billy 2" src="Billy2.jpg"></mdo:image><br></br>
<br></br>
<mdo:image height="291" width="383" alt="billy 3" src="Billy3.jpg"></mdo:image><br></br>
<br></br>
And so on and so on . . . . . How many different ones can you get
if you stick to a grid of $7$ rows of dots and $9$ columns of dots?
<br></br>
<br></br>
The triangle started off having a base of $6$ and was $3$ high.
Look at the other triangles you have found that have this same base
and height. <br></br>
What can you say about the areas of this set of triangles? (You
might like to draw some more triangles with a base of $6$ and a
height of $3$ which don't have right angles to test your
ideas.)<br></br>
<br></br>
<br></br>
<br></br>
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<h2>More Transformations on a Pegboard</h2>
<br></br>
You might like to try <a href="http://nrich.maths.org/public/viewer.php?obj_id=1813&part=index">Transformations on a Pegboard</a> before looking at this problem. Here are some ideas to take this activity some stages further. One part of the problem was about Billy with a triangle and the other was about Tanya with her enlarging rectangle.<br></br>
<br></br>
You may like to use this interactive pegboard to try out your ideas. Read underneath the interactivity for the challenges!<br></br>
<br></br>
<a href="/content/id/4901/circleAngles.swf">Full Screen Version</a><br></br>
<mdo:flash height="400" id="/content/id/4901/circleAngles.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/4901/circleAngles.swf"></param>
<param name="flashplayerversion" value="7"></param>
<param name="height" value="400"></param>
<param name="width" value="550"></param>
</mdo:flash><br></br>
<br></br>
Billy asked, 'By moving just one peg can you make a right-angled triangle?'. Well, I had a go starting the same way:<br></br>
<br></br>
<mdo:image alt="Billy" height="295" src="Billy1.jpg" width="379"></mdo:image><br></br>
<br></br>
Here are two I found moving one peg each time (from the original):<br></br>
<br></br>
<mdo:image alt="billy 2" height="281" src="Billy2.jpg" width="375"></mdo:image><br></br>
<br></br>
<mdo:image alt="billy 3" height="291" src="Billy3.jpg" width="383"></mdo:image><br></br>
<br></br>
And so on and so on . . . . . How many different ones can you get if you stick to a grid of $7$ rows of dots and $9$ columns of dots?<br></br>
<br></br>
The triangle started off having a base of $6$ and was $3$ high. Look at the other triangles you have found that have this same base and height.<br></br>
What can you say about the areas of this set of triangles? (You might like to draw some more triangles with a base of $6$ and a height of $3$ which don't have right angles to test your ideas.)<br></br>
<br></br>
<br></br>
<br></br>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>This <a href="http://nrich.maths.org/public/viewer.php?obj_id=4901&part=">activity</a> offers a good chance for pupils to develop concepts surrounding triangles, including their areas. It is an engaging activity which allows pupils to work in twos and have fruitful disussions.</div>
<br></br>
<h3>Possible approach</h3>
<div>As we all learn in different ways it would be good to allow learners to explore this activity with a number of aids at their disposal, for example:</div>
<div>- <a href="/content/id/4901/1cmdotty.doc">Spotted paper in squares</a></div>
<div>- Nail boards with elastic bands</div>
<div>- The on-screen interactive board</div>
<div>- Squared paper</div>
<br></br>
<div>Encourage them to start with the original shape and move one point at a time to see the results. They will need to do this in a systematic way if they are going to find all the possibilities and it is likely that the group will benefit from discussing some strategies for doing this as a whole. This might also require you to model the first few steps in a particular way of being systematic,
for example by moving the "top" corner of the triangle one peg up, then two pegs up etc. You can ask the children what to do next at each stage and talk about what might be "sensible" so that you don't miss any out. They may notice a pattern in the way the possible positions for this point lie so that it might not be necessary to actually test every point on the grid. Depending on the class'
experience, you could use a number of ways of checking whether an angle is a right-angle, for example by using a protractor or simply a corner of paper.</div>
<br></br>
When it comes to looking at the areas of sets of triangles (i.e. triangles which have the same base and height), the pupils will have different ways of calculating these areas and these should be acknowledged and shared. Many children will be able to make the full generalisation about areas of triangles with identical base and height through this investigation.<br></br>
<h3>Key questions</h3>
<div>How are you exploring where the elastic band can go?</div>
<div>How will you make sure you don't miss any triangles out?</div>
<div>How are you finding the areas of the triangles?</div>
<div>What do you notice about the triangles' areas?</div>
<br></br>
<h3>Possible extension</h3>
<div>In the original <a href="http://nrich.maths.org/public/viewer.php?obj_id=1813&part=">Transformations on a Pegboard</a> problem, Tanya asked about changing a square into a rectangle so that the area doubles by moving two pegs. You could go on to investigate this further with your class, but rather than find a rectangle, look for any shape which can be made by moving two pegs which has
double the area of the square. How many ways can they do this? Sometimes when you've exhausted a lot of ideas you have to change the rules again so here's an idea - keep the same starting square but have a larger grid, so it looks like this:</div>
<div style="text-align: center;"><mdo:image alt="Tanya2" height="185" src="Tanya2.jpg" width="210"></mdo:image></div>
<div style="text-align: left;">Can the group find some new shapes now that by moving two pegs will double the original area? (No matter what the final shape is!) What would happen if you could only move one peg to double the area?</div>
<br></br>
<h3>Possible support</h3>
<div>Using nail/peg boards or the interactivity on screen will help all children access this problem, but those with poorly developed motor skills may need help from an adult or fellow pupil.</div>
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You might find it useful to use one or more of the following:<br></br>
- <a href="/content/id/4901/1cmdotty.doc">Spotted paper in
squares</a><br></br>
- Nail boards with elastic bands<br></br>
- The on-screen interactive board<br></br>
- Squared paper.<br></br>
<br></br>
You could try finding all the different right-angled triangles by
moving one point first, then moving a second point, then the
third.<br></br>
Why not try with the top left corner to start with? Where could
this point move to? <br></br>
How will you kow that you have them all?<br></br>
Have you decided how to record your work?<br></br></mdoxml>
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There are 20 right-angled triangles in total: <br></br>
<br></br>
12 made by moving the "top"vertex - the 6 below and then each one
reflected in a vertical mirror line half way along the base:<br></br>
<br></br>
<mdo:image width="298" height="267" alt="Solution 1" src="cansol1.gif"></mdo:image><br></br>
<br></br>
4 made by moving the "left" vertex:<br></br>
<br></br>
<mdo:image width="270" height="256" src="cansol2.gif" alt="Solution 2"></mdo:image><br></br>
<br></br>
And 4 made by moving the "right" vertex:<br></br>
<br></br>
<mdo:image width="294" height="260" alt="Solution 3" src="cansol3.gif"></mdo:image><br></br>
<br></br>
The area of a right-angled triangle of base 6 and height 3 =
½ x 6 x 3 = 9 . This is because a right-angled triangle
has thesame area as half of a rectangle with base 6 and height
3.<br></br>
<br></br>
Other triangles with base 6 and height 3 have the same area
regardless of whether or not they are right-angled.You can
demonstrate this by dividing each of them into two right-angled
triangles and adding the areas together. For example let's look at
the original triangle in this problem:<br></br>
<br></br>
<mdo:image width="379" height="295" alt="" src="Triangle%20divided%20into%20two.gif"></mdo:image><br></br>
<br></br>
The triangle on the left has area = ½ x 2 x 3 = 3<br></br>
The triangle on the right has area = ½ x 4 x 3 = 6<br></br>
Total area = 9<br></br>
<br></br>
This always works, so it means that you can calculate the area of
any triangle in the same way that you would calculate it for
right-angled triangles, that is½ x base x height.<br></br>
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More Transformations on a Pegboard
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
triangle.
Using, Applying and Reasoning about Mathematics
Working systematically
Using, Applying and Reasoning about Mathematics
Investigations
Information and Communications Technology
Interactivities
Measures and Mensuration
Area
2D Geometry, Shape and Space
Right angled triangles
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Upper primary mapping document