Developing Geometric Reasoning Part 2. Focus on Quadrilaterals
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1 Developing Geometric Reasoning Part 2 Focus on Quadrilaterals Math Teacher Leader Seminar January 8 & 11, 2007 Henry Kepner Kevin McLeod DeAnn Huinker Connie Laughlin Karen Corlyn Lee Ann Pruske Paige Richards Mary Mooney This material is based upon work supported by the National Science Foundation Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).
2 Session Goals Examine how students understanding of geometry develops through defined stages, the van Hiele Model. Consider implications for instruction to move students along in developing their geometric reasoning. Investigate the Wisconsin sub-skill area of Describing Figures. Compare, contrast, and connect properties of quadrilaterals. 2
3 Geometry Wisconsin Mathematics Standard Describing Figures Sub-skill Areas Coordinate Systems Spatial Relationships and Transformations 3
4 Tricky Triangles Revisited Please take out the samples of student work that you brought to the meeting. Share your student work with the others at your table. Explain why you chose each of the pieces of work. What have you learned about the reasoning of your students and its relationship to the van Hiele levels? What strategies have you tried or considered to move your students along the van Hiele continuum? 4
5 van Hiele Levels of Geometric Reasoning Level 0: Visualization Recognize figures as total entities, but do not recognize properties. Level 1: Analysis (Description) Identify properties of figures and see figures as a class of shapes. Level 2: Informal Deduction Formulate generalizations about relationships among properties of shapes; Develop informal explanations. 5
6 Instructions Van Hiele Test 1. Circle the correct letter. 2. When complete, separate the copies, keep the top copy. 3. Return the bottom copy, with the test. 4. Please do not write on the test paper. 6
7 Quadrilateral Sorting Activity Assign Roles Facilitator: Give all a voice. Recorder: Take notes on recording sheet. Directions 1. Remove a card from the envelope. 2. Sort the quadrilateral shapes by the directions on the card. 3. Record the sort and discussion on the recording sheet. 4. Push all of the shapes back into a pile. 5. Pass the envelope to the left and repeat steps 1 4 with the next card. 7
8 all right angles Property Criteria at least one right angle both pairs of opposite sides congruent both pairs of opposite sides parallel both pairs of opposite angles congruent congruent diagonals diagonals bisect each other at least one pair of parallel sides exactly one pair of parallel sides 8
9 Some authors choose to define trapezoid as a quadrilateral with at least one pair of parallel sides. That definition is more inclusive and leads to the conclusion that all parallelograms are trapezoids. The Navigation books adopt the classical definition that a trapezoid is a quadrilateral with exactly one pair of parallel sides." Navigating through Geometry in Grades 6-8, page 11. by Pugalee et al. (2002). Reston, VA: National Council of Teachers of Mathematics. 9
10 Trapezoid Definitions Everyday Math Expressions Scott Foresman that has exactly one pair of parallel sides. (K & 1 st ) with at least one pair of parallel sides. (5 th ) with one pair of parallel sides. (3 rd & 4 th ) with only one pair of parallel sides. (5 th ) that has exactly one pair of parallel sides. CMP Glencoe Holt with at least one pair of opposite sides parallel. (6 th ) with one pair of opposite sides parallel. (7 th ) with one pair of parallel sides. (8 th ) with exactly one pair of parallel opposite sides. with exactly one pair of parallel sides. 10
11 Parallelogram Definitions Everyday Math Expressions Scott Foresman with two pairs of parallel sides. Opposite sides of a parallelogram are congruent. Opposite angles in a parallelogram have the same measure. (K-2 nd ) in which both pairs of opposite sides are parallel and opposite angles are congruent. (3 rd -5 th ) with both pairs of opposite sides parallel. (3 rd & 4 th ) in which opposite sides are parallel. (5 th ) with both pairs of opposite sides parallel. CMP Glencoe Holt with opposite sides parallel. Both pairs of opposite angles are also equal. that has both pairs of opposite sides congruent and parallel. with two pairs of parallel sides. 11
12 True or False? A square is a special kind of rectangle. It is a rectangle in which all four sides are the same length. A parallelogram is a special kind of trapezoid. It is a trapezoid with two pairs of parallel sides. A rhombus is a special kind of kite. It is a kite in which all four sides are the same length. 12
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16 Assessing the van Hiele Test van Hiele Geometry Levels Answer Sheet Name or Employee ID # School Name 1. A B C D E 2. A B C D E 3. A B C D E 4. A B C D E 5. A B C D E 6. A B C D E Directions 1. Circle correct letter. 2. When complete, separate the copies, keep the top copy. 3. Return the bottom copy, with the test. 4. Please do not write on the test paper. 7. A B C D E 8. A B C D E 9. A B C D E 10. A B C D E 11. A B C D E 12. A B C D E 13. A B C D E 14. A B C D E 15. A B C D E 16
17 Big Ideas Two- and three-dimensional objects can be described, classified and analyzed by their attributes. Objects can be oriented in an infinite number of ways. The orientation of an object does not change the other attributes of the object. Some attributes of objects (e.g. area, volume, perimeter, surface area) are measurable and can be quantified using unit amounts. Attributes of objects are not necessarily independent: there can be logical relations and implications between them. 17
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