| Types of reasoning | Distribution (Percentage) | Explanation and Representative Examples from students’ questionnaires |
| Area calculations based on plane geometry proof | 15.8% (N = 9) | Explanation The students related to definitions, axioms, and proof in plane geometry and wrote the solution as a geometrical proof. A link was made between the problem and plane geometry. Examples “…we will mark the sides of the large square as X… [1A] we will place AB parallel to HN…” [24A] “…we will prove that MDNC is a rectangle” [26A] “…EF is parallel to AB…” [51A] *a fine example can be found in questionnaire [24A] |
| Area calculation | 40.4% (N = 23) | Explanation The students related to the geometrical figures as given figures, such as: rectangles or trapezoids as givens. They used known formulas to calculate areas. “… we will mark the sides of the large square…” [21D] “…we will calculate the area of the trapezoid…” [30B] “…we will calculate the area of the rectangle…” [32B] “…we will calculate the areas of the given rectangle…” [46B] |
| Verbal explanation | 28.0% (N = 16) | Explanation The students related to the problem as a collection of areas and given geometrical figures. Areas were described as equal and overlapping and were “cut out” and moved around. In order to find out who ate more, I tried to “put” the father’s pieces into the son’s…” [22C] |
| Other | 15.8% (N = 9) | “…I used the small square as a unit of measurement and reached the conclusion that the hearts were equal to 5 squares altogether…” [43D] |