Mathematics

and what part it plays in the covering. For example, one student here wrongly
thought that “With the small discs you get better covering, on the other hand
with the big ones you get bigger gaps between them.” The second conjecture
was about which of the two orderings they found was the best. Here, we did
not ask for any justification on part of the students, as we only wanted to use
their intuition.
In order for the students to encounter the remaining task of the activity, they
were instructed to work in groups of five.
3.2.3.2 Task 2
“ i) Focus on the centre of the coins for each one of the two orderings, draw
repeated polygons in order to cover the entire surface.
ii) For each ordering find the smallest repeated polygon you can draw based
on the sub-question (i).”
Through this task we anticipated from the students to design various patterns
which would be generated by the repletion of a polygon. Furthermore, subquestion
ii) was added in order to motivate students to investigate and
discover the smallest geometrical unit. In the first ordering the smallest is a
right-angle isosceles and in the second an equilateral triangle. This subquestion
was added in order to facilitate the transition to the remaining tasks.
As far as the findings concerned, many students encountered difficulties to
generalize to infinity. Here is a quote from a student that demonstrates this
difficulty: “So now we need to make triangles across the whole area? Are they
crazy? I feel tired already.” Finally, some students managed to generalize to
the infinity and characteristically commented: “Okay, let's do the first 2 lines
anyway because the same pattern is repeated then!” and referring to the first
ordering: “Triangles. Every three is a triangle. And so the entire surface is
covered.”
The following pictures are examples of the successful answers of some
students. In other words, a group drew hexagons, focusing on the center of the
coins, and found the smallest geometrical unit, which in this case is an
equilateral triangle (Figure 4). Moreover, another group of students drew
squares and concluded that the smallest geometrical unit is a right-angle
isosceles (Figure 5).
Figure 4 Figure 5
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