There are to be 6 homes built on a new development site. They could
be semi-detached, detached or terraced houses. How many different
combinations of these can you find?
Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
How many triangles can you make using sticks that are 3cm, 4cm and 5cm long? Each side can only use one stick but a triangle can use more than one stick of the same length.
You could use the interactivity below to investigate the problem and try making some triangles. Choose three sticks by clicking on the red, green or blue buttons. You can then rotate each stick (turn it round) by clicking and dragging on its end. If you click and drag near the middle of a stick, you can move it.
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?