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Geometry
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
A quadrilateral is a trapezoid if two sides are parallel. In the notation of the figure on the right we have A+D=B+C=180∘,area=12(AB+CD)h. $$ A + D = B + C = 180^\circ ,\,\,\,\,\,{\text{area}}\, = \frac{1}{2}(AB + CD)h. $$
Examining pre-service mathematics teachers’ argumentation-supported lesson plans and their noticing during planning
Published in International Journal of Mathematical Education in Science and Technology, 2022
Berna Tataroğlu Taşdan, Ayşe Tekin Dede, Melike Yiğit Koyunkaya
When the PMTs’ statements were examined in detail, we identified that teaching strategies and teacher actions they responded to based on the situation they attended were varied. For instance, the PMTs preferred teaching strategies to construct the hierarchical relationships while teaching the quadrilaterals. Considering the importance of constructing relationships among geometric shapes in developing the ability of inquiry and reasoning (Clements, 2003; Fujita & Jones, 2007), it is deduced that it was important for the PMTs to notice the order of teaching of the quadrilaterals. Most of the PMTs decided to teach the trapezoid first and then planned to construct the relationship by adding properties for each quadrilateral and teach the parallelogram, rhombus, rectangle and square, respectively. It is thought that this situation is related to the nature of the selected objective. As Usiskin et al. (2008) mentioned, the PMTs preferred to add a specific property of a quadrilateral each time by considering whether the quadrilateral includes others or not to make a generalization and construct the hierarchical relationships among them. This strategy is more useful for the PMTs in the process of constructing the hierarchical relationships among quadrilaterals (Fujita & Jones, 2007).
Comparing the concept images and hierarchical classification skills of students at different educational levels regarding parallelograms: a cross-sectional study
Published in International Journal of Mathematical Education in Science and Technology, 2022
One aspect of geometric thinking development is linked to understanding inclusive relationships between the classification of quadrilaterals. The classification process depends on the ability to identify similarities and differences between figures and to explain why a figure is an example of a class (Walcott et al., 2009). Usiskin et al. (2008) state that based on the acceptance of the definition of the trapezoid, namely inclusive and exclusive, quadrilaterals are classified into two different categories. According to the exclusive definition, if a trapezoid is defined as ‘a quadrilateral with only one pair of parallel sides.’, then parallelograms and trapezoids are categorized as disjointed subgroups of quadrilaterals. Whereas if a trapezoid is defined inclusively as ‘a quadrilateral with at least one pair of parallel sides.’ in the inclusive definition, then all parallelograms (i.e. square, rectangle, rhombus, and parallelogram) are categorized as special types of trapezoids (Figure 1).
Conserved properties in polygons obtained by a point reflecting process
Published in International Journal of Mathematical Education in Science and Technology, 2021
First of all the participants discovered, by using the applets, that the reflecting process on an equilateral triangle creates an equilateral triangle that is congruent to the original one and whose sides are parallel to the sides of the original triangles respectively. Then the participants were asked to consider an arbitrary triangle. They found that also in this case a triangle is obtained whose sides are parallel to the sides of the original, and the two triangles are congruent. After that, the participants examined the process of reflection on a square and found that the resulting shape is a square of areas which is twice larger than that of the original. Then they were asked to guess without using the applets what will be obtained when the reflecting process is applied to the following shapes: rectangle, parallelogram, rhombus, kite, trapezoid, arbitrary quadrilateral. Each participant had to choose two shapes and was given 20 min to determine the hypothesis about the shapes obtained by the reflecting process. Below is a summary of the received responses.