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Dynamic Geometry Activities for Grade 9 Math Club

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Descriptions

The content of these activities is appropriate to the Nova Scotia school curriculum for grades 6-9. Most of the material is review, although our experience is that it was all new to at least some students. Cabri Geometry was used for these activties.

Activity A

Activity A is a sequence of three activities concerning the construction of parallelograms. In A1 the pointer tool, the line menu (segment and polygon tools) and the construction menu (parallel line tool) are introduced. The Drag Test is also defined and used. In A2 the measure menu is introduced (angle and distance tools). In this activity the students look for properties of parallelograms, including equal opposite angle, supplementary adjacent angles and equal opposite sides, as well as diagonal properties (mutually bisecting). In A3 a number of tools are mentioned without describing them, and the focus is on constructing a parallelogram without using the parallel line tool.

Activity B

Activity B is a sequence of three activities concerning the angle sums of polygons. B1 introduces the line menu (triangle and polygon tools), the Label tool, and the measure menu (angle, distance, area and calculate tools). The constant angle sum of a triangle, the triangle inequality and the non-relatedness of the area and perimeter can be observed in this activity. B2 repeats the same questions for quadrilaterals, and analogous properties can be observed. In addition Question 9 invites a connection to be made between the angle sum of the quadrilateral and the angle sum of the triangle. B3 is less directive but again concerns angles in a general polygon. In addition the number of diagonals in a polygon is suggested as worth exploration, and question 9 invites a connection to be made between properties of polygons and properties of quadrilaterals and triangles.

Activity C

Activity C is a sequence of four activities concerned with locus definitions of circle, perpendicular bisector and angle bisector, and the circumcentre and incentre of a triangle. Unlike activities A and B, these activities have a “real world” story as a frame. C1 involves constructing the locus of points equidistant from a single point and observing that this is a circle. C2 involves constructing the locus of points equidistant from two points and observing that this is the perpendicular bisector of the segment between the points. C3 involves constructing the locus of points equidistant from two lines and observing that this is the angle bisector of the angle formed by the lines. C4 involves locating the point equidistant from the three vertices of a triangle, and the point equidistant from the three sides, and constructing these points exactly.

Activity D

Activity D involves the truth or falsity of seven statements and explanations. The first two statements concern perpendicular bisectors and the circumcentre of a triangle and are directly related to Activity C. The next three statements concern angle sums and diagonals of polygons and are directly related to Activity B. The last two statements are new, one being the triangle midpoint theorem and the other the Varignon parallelogram.

Activity E

Activity E is a problem solving activity concerning the quadrilateral formed by the intersections of the perpendicular bisectors of a quadrilateral.

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Activities

The activities are in a Word document. To download all the activities in a compressed zip file, click here.

Activity	Topic	Cabri tools introduced	Math concepts reviewed or introduced
A1	Constructing a parallelogram	Polygon, Pointer, Segment, Parallel Line, Point, Hide/Show	Definitions: Quadrilateral, parallelogram.
A2	Properties of parallelograms	Angle measure, Distance measure.	Properties of parallelograms: Opp. angles congruent, Adj. angles supp., Opp. Sides congruent, Diagonals bisect each other, etc.
A3	Redefining “parallelogram”	Rotate, Circle, Midpoint, Translate, Vector
B1	Angle sums of triangles	Triangle, Polygon, Label, Angle measure, Distance, Calculate…, Area.	Triangle properties, triangle inequality, area & perimeter, angle sum
B2	Angle sums of quadrilaterals		Quadrilateral properties, angle sum
B3	Angle sums of polygons		Angle sum in general, number of diagonals in general
C1	Equidistance: Point	Point, Label, Pointer, Distance, Trace, Circle	Circle as locus
C2	Equidistance: Two Points	Perpendicular Bisector	Perpendicular Bisector as locus
C3	Equidistance: Two Lines	Angle Bisector	Angle Bisector as locus
C4	Equidistance: Triangle	Segment, Triangle	Special lines in triangles: Perpendicular bisectors and angle bisectors.
D1	Review of established knowledge	None	Perpendicular Bisector as locus, circumcentre, Angle sum of polygons, number of diagonals of polygons, triangle midpoint theorem, Varignon parallelogram.
D2	Explanations: Perpendicular Bisector	None	Perpendicular Bisector as locus, circumcentre.
D3	Explanations: Polygons	None	Angle sum of polygons, number of diagonals of polygons
D4	Explanations: Varignon	None	Triangle midpoint theorem, Varignon parallelogram.
E	Perpendicular Bisectors of Quadrilaterals	None	None

Activty B is also avaible for Geometer's Sketchpad. Click here for the Word file.

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Comments

Activity A

Activity A1 seemed to be effective as an introduction to some basic tools. It was insufficient, however, as an introduction of the Drag Test, probably because there was no special reinforcement of this idea in later activities. A2 allowed the discovery of angle and side properties of parallelograms, but these discoveries were not consolidated in any way, so they did not influence the students work on activity A3 very strongly. Question 2 would be better if it required the students to record which angles are congruent and supplementary. The property that seems most present to the students was the supplementarity of adjacent angles, which is not useful for constructing a parallelogram given Cabri’s tools. In Activity A3 the rotational symmetry of the parallelogram, which had not been observed as a property, was used to construct a parallelogram by several groups. This seems to have been prompted by the suggestions to use the rotate and point symmetry tools.

Activity B

All the properties of polygons the might be observed in Activity B are included in the geometry curriculum of grades 6-8. The only one that the students recalled was the angle sum of the triangle. In one case a student recalled learning the formula for the number of diagonals of a polygon but could not remember it and took a long time to rediscover it.

Activity C

In activity C1 the students found it obvious that the locus was a circle. In activity C2 they could visualise that the locus was a line through the midpoint, but they did not associate this with the perpendicular bisector (in fact they seemed not to know this term). The angle bisector, in C3 was both more difficult to visualise and to describe. The term “angle bisector” was unknown to them. Activity C4 was much too difficult in this context. Although they could locate the point equidistant from the three vertices of the triangle by measuring, they made no connection to the previous activities.

Activity D

All the groups had difficulty interpreting the statements, especially statement a. Once they understood the statements they had little difficulty correctly determining their truth using empirical methods. Their explanations reveal both unstructured thinking and writing. The results of this activity make it clear that for work on reasoning, attention must be paid to communication first of all.

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Supported by a research grant from the Social Sciences and Humanities Research Council of Canada

Page last updated May 2005 by David Reid