168 UNCOVERING STUDENT THINKING IN MATHEMATICS GRADES K S rectangle has four str
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168 UNCOVERING STUDENT THINKING IN MATHEMATICS GRADES K S rectangle has four stratght sides and four square corner (National Couucil of Teachers of Mathematics [NCIM), 2000, p 161) Students advance through levels of thought in geonnetry. Van EHlele bas characterized them as visual, descriptive, abstract/relatonal, and formal deduction (Van Hiele, 1986; Clements & Battista, 1992). At the first level, students identify shapes and flgures according to their concrete examples. At the second level, students Identify shapes according to their properties, and here a student might think of a rhombus as a figure with four equal sides. At the third level, students can identify relationships between classes of figures (e.g., a square is a rectangle) and can discover properties of classes of igures by simple logical deduction. At the fourth level, students can produce a short sequence of statements to logically justify a conclusion and can understand that deduction is the method of establishing geometric truth. (American Association for the Advancement of Science [AAAS]. 1993. p. 352) No single test exists to pigeonhole students at a certain level [see Van Hiele levels above]. At the upper elementary grades students should be pushed from level 1 to level 2. If students are not able to follow logical arguments or are not comfortable with conjectures and if-then reasoning, these students are likely still at level 1 or below. (Van de Walle, 2007, p. 414) Students should carefully examine the feature of shapes in order to define and describe fundamental shapes, such as special types of quadrilaterals, and to identify relationships among the types of shapes. (NCTM, 2000, P. 233) Teaching Implications In order to support a deeper understanding for students in elementary school in regard to rectangles, the following are ideas and questions to consider in con- junction with the research. Focus Through Instruction Students should be provided with materials and structured opportunities to explore shapes and their attributes. Students should ahalyze characteristics and properties of two- and three- dimensional shapes. . Students should sort quadrilaterals by looking at examples and nonexam- ples of special types. Students should engage in mathematical conjectures about geometric relationships, such as why a square is a rectangle, but a rectangle is not always a square. o Students need to develop more-precise ways to deserthe shapes using mathematies vocabulary associated with trapezoids, parallelograms, rec tangles. rhomb and squares ?rse interactive technology to have students sort shapes by various attributes.Explanation / Answer
Tom is correct since a square is always a rectangle, in fact a square is a special case of a recntagle where the length and breadth are equal to each other
Jack is correct, a rectangle can sometimes be a rhombus when its length is equal to its breadth, in which case it is a square and a square is also a rhombus, but a rhombus NEED NOT be a square always
Pam is incorrect as a rhombus has all sides equal but these sides need not be at right angle to each other, so angle between sides can be anything in a rhombus, hence a rhombus is not always a square
Sue is correct since a rectangle can be a square when its length is equal to its breadth;
Mike is incorrect as a paralleleogram need not always be a rhobmus. When the length and breadth of a parallelogram are not equal, it is not a rhombus.
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