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We consider groups defined by non-empty balanced presentations with the property that each relator is of the form, where x and y are distinct generators and is determined by some fixed cyclically reduced word that involves both a and b. To every such presentation we associate a directed graph whose vertices correspond to the generators and whose arcs correspond to the relators. Under the hypothesis that the girth of the underlying undirected graph is at least 4, we show that the resulting groups are non-trivial and cannot be finite of rank 3 or higher. Without the hypothesis on the girth it is well known that both the trivial group and finite groups of rank 3 can arise.
A crucial aspect of learning about (linear) functions is being able to change between graph and equation. Common German and Slovak textbooks propose different procedures for these representational changes. Within a sample of 49 German and 56 Slovak teachers, we analyzed if these different procedures can also be observed in the teachers’ corresponding knowledge of content and students, i. e. if the teachers expected different student strategies and errors. The results confirm this assumption and emphasize the importance to consider this teacher's knowledge in a country-specific way and being careful when comparing such knowledge of teachers from different countries.