LET’S MAKE SQUARES
Task 0:
Make (draw) figures on which appear 1, 2,3,... squares, using exactly 12 congruent segments,
without partial or complete overlapping, such that each endpoint of a segment must coincide with
the endpoint of another segment (there is no free endpoint).
We illustrate such a construction with n  5 squares and we give also some figures to illustrate the
unallowed construction.
Task 1:
Formulate questions, problems related to the previous activity!
Task 2:
Solve some of the formulated problems!
This activity can be used on different levels starting from 10-12 years old students to university
students and it has a good potential for PD courses, because participants can experience a lot of IBL
related processes. Depending on the age group the mathematical focus should be different. While
working with lower secondary school students the main focus should be on processes like
systematizing, visualizing, discovering relationships and connections. This focus moves to modeling
complex problems as the age of student increases. The strengths of the activity are that in most of
the cases the same questions arise during the activities: what is the maximal number of squares, is it
possible to construct a configuration for a given n, if yes, how many different configurations do exist,
when do we call two configurations different, etc. The answers to some of these questions can be
obtained intuitively, but for a rigorous proof you need a little bit more. In this way participants can
experience how mathematics works beyond the usual framework. The basic activity is in fact from
Spencer Kagan’s “Let’s make squares” project, but the main objectives here are different, we are
using the basic activity as an operational (experimental) base for the analysis, but the focus is more
on the formulated problems and their solution. During this activity students can exercise their
counting skills (counting squares on a grid or on an arbitrary configuration). From this viewpoint this
activity is a transformation of some
classical counting problems (counting the number of squares on a regular rectangular grid) into IBL
tasks. This material was developed for the Primas Project in collaboration with the SimpleX
Association.
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