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This problem lends itself very well to practical exploration using sticks or cuisenaire rods. Having this "hands-on" experience will help children to realise the need to be systematic and develop their own recording system. The interactivity will enable the whole class to share their findings after the initial practical exploration. The task offers children the chance to explore all the ways in which they can choose three different sticks from sets of sticks.
Ideally allow the children to explore making triangles using strips of paper of lengths $3$, $4$ and $5$ cm (or scaled up if you think these are too fiddly for young children). You could use Cuisenaire rods or construction toy strips instead. Ask the children to make triangles and see how many different ones they can find.
Once they have made some, bring the class together to share their results. Present them on the IWB or by creating a display. Encourage the children to use descriptive language that is accurate mathematically to tell you about the triangles they have made.
How many have you made?
How do you know they are different?
Have you found them all?
Are there any that are similar in any way? How?
Try introducing an additional length such as $6$ units. This will mean that there may be some triangles that can't be made. Why?
How many extra possiblilities would this create?
This approach lays the foundations for systematic working that is so important in mathematics. You could also introduce the idea of perimeter - the distance around the edge of the shape - using questions such as:
How could you make the triangle with the smallest possible distance around the outside (smallest perimeter)?
How about the triangle with the next smallest perimeter?