Here we look at quick ways of counting a collection of dots by putting dots into groups. The collections in the first two task don't have a particularly clear, overriding pattern so there can be lots of fairly efficient ways of grouping them! The aim is to encourage children to look for (or impose) structure, even though the structure might not be general. In the later tasks there is the potential to generalise, which some children might find both challenging and exciting, but the tasks can be worked on fruitfully at a non-general level.
Task 07A: The collection of dots here is not entirely random - it has rotational symmetry. However, initially, children might group the dots in fairly random ways, in which case it would help to show them groupings that contain more of a pattern. [It would also help to provide copies of the dot-collection - or squared or dotted paper for children to draw their own copies.]
Once children get the hang of it, they might enjoy the fact that one can group the dots in many, not-entirely-random, ways. Here are some examples.
Task 07B: Another collection with rotational symmetry. You might also want to create you own collection, or ask children to do so.
Some possible groupings:
Task 07C: This collection is already quite structured, with two lines of symmetry (and rotational symmetry). Some children might spot that we can think of the collection as showing two copies of the third triangle number, 6. If we push the triangles together horizontally we can form a rectangle containing 4 row of 3 dots, as hinted at in the given example.
Task 07D: Here we have moved entirely away from the idea of a semi-random collection of dots that we had in Tasks 07A and 07B. This collection is highly structured, which makes it mathematically more interesting, including in ways that are quite challenging.
A nice, but challenging, feature of these ways of structuring the dots is that they can all be generalised. Imagine a 'square' of dots with, say, 10 dots rather than 5 along each side. Then we would get these structurings corresponding to the ones in the figure below:
4×10 – 4, 4×(10–1), 4×(10–2) + 4, and 10×10 – (10–2)×(10–2).
You might want to see whether children can generalise any of their structurings in this way.
Task 07E: Here is another well-structured pattern, which can be generalised should you wish to do so.
The collection of dots might remind children of the way 5 dots are arranged on dice. One can think of the collection as three overlapping 5s, as in the first example, below: there are three 5s but with 2 lots of 2 overlapping dots. The third example includes a phantom dot.
As in Task 07D, these structurings can be generalised, although they are probably even more challenging. Consider, for example, a collection that looks like a row of 10 overlapping 5s. Then we would get these structurings correspoding to the three in the figure below:
10×5 – (10–1)×2, 5 + (10–1)×3, and 3×(10+1) – 1.
Again, you might want to see whether children can generalise any of their structurings in this way.
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