Geometry is systematic. It is deductive. It begins with a set of postulates that explicitly formulate the assumptions on which it based (at least those of a geometrical nature), and from those it deductively derives its results.
I've given quite a bit of thought to my postulate set. When I first began to teach, I simply relied upon the postulate set provided by my text (a Glencoe monstrosity), but over time I became dissatisfied with it. It sometimes obscures issues of great importance (it makes a real mess of the Parallel Postulate), and it is quite bloated (each of triangle congruence principles SAS, SSS, ASA and AAS are treated as postulates).
I resolved to do better. A bit of research led me to the School Mathematics Study Group and its postulate set. (The SMSG postulate set was constructed expressly for use in the secondary classroom. But apparently textbook publishers didn't think it simple enough, for much that could be deduced from it they added as postulates.) Not content to simply take it over, I began to fiddle with it. Four years in the classroom have (I hope) given me some insight into the best way in which express our fundamental assumptions.
No doubt I'll continue to fiddle in the future, but I will share what I've done so far. Below is my (tentative) postulate set.
Postulate Set for Elementary Euclidean Geometry
P2 Ruler Postulate We may divide a line's length into a sequence of congruent segments. We may then assign integer values to the endpoints in such a way that, for a given choice of direction along the line, each next point is assigned the next integer. Each point between these endpoints is assigned a real in such a way that:
a. Distinct points are assigned distinct reals.
b. If, for our given choice of direction along the line, point Q comes after point P, then the real assigned to Q is greater than the real assigned to P.
On any such assignment, the real that corresponds to a point is its coordinate and the distance between points is the absolute value of the difference of their coordinates.
P3 Segment Addition Postulate Point T lies between points A and B just if the length of AB is the sum of the lengths AT and BT. P4 Shortest Path Postulate Of all possible paths that begin in one point and end in another, the shortest is the line segment.
P5 Point Existence Postulate Every line contains at least two points. Every plane contains at least three non-collinear points. Space contains at least four non-coplanar points.
P6 Line Containment Postulate If two points lie in a plane, then the line through those points is contained wholly in that plane.
P7 Plane Uniqueness Postulate Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane.
P8 Plane Intersection Postulate The intersection of two planes is one line.
P9 Plane Separation Postulate Given a line and a plane in which in lies, the points of the plane that do not lie on the line form two sets such that:
a. each of the sets is convex, and
b. if P is in one set and Q is in the other, then the segment PQ intersects the line.
P10 Space Separation Postulate The points of space that do not lie in a given plane form two sets such that:
a. each of the sets is convex, and
b. if P is in one set and Q is in the other, then the segment PQ intersects the plane.
P11 Protractor Postulate We may divide the circumference of a circle into 360 congruent arcs and assign to their endpoints in sequence the values from 0 to 359. We say that the direction along the circle is the direction of increase of these values. Points between these endpoints are assigned reals in such a way that:
a. Distinct points are assigned distinct reals between 0 and 360.
b. If point Q comes after point P for our direction along the circle, then the real assigned to Q is greater than the real assigned to P.
The measure of an arc is then the absolute value of the difference of the values assigned to its endpoints. Arc measure determines angle measure in this way:
d. We construct a circle whose center is the vertex of our angle.
e. We mark the points where the angle's sides intersect the circle. Call these S and T.
f. We then say that the measure of the angle equals the measure of the arc ST that lies in the angle's interior.
P12 Angle Addition Postulate If point T lies in the interior of ∠CAB , then m∠BAT + m∠CAT = m∠CAB . P13 Supplement Postulate If two angles form a linear pair, then they are supplementary.
P14 Side Angle Side Triangle Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent congruence.
P15 Parallel Postulate Through a point not on a given line, there is at most one line parallel to the given line.
P16 Similarity Postulate For any given figure and any real k, there exists a second figure similar to the first such that the scale factor from first to second is k.
P17 Area Existence To every finite planar region there corresponds a unique positive real number that we call its area.
P18 Area Equality If two closed figures are congruent, then the regions enclosed within them have the same area.
P19 Area Addition Suppose that the region R is the union of two regions R1 and R2. If R1 and R2 intersect at most in a finite number of segments and points, then the area of R is the sum of the areas of R1 and R2.
P20 Rectangle Area The area of a rectangle is the product of the length of its base and the length of its altitude.
P21 Volume Existence To every finite spatial region there corresponds a unique positive number that we call its volume.
P22 Prism Volume The volume of a right prism is equal to the product of the length of its altitude and the area of the base.
P23 Cavalieri’s Principle Given two solids and a plane, if for every plane that intersects the solids and is parallel to the given plane the two intersections determine regions that have the same area, then the two solids have the same volume.
No comments :
Post a Comment