Mathematics and Logic - Skill and Concept Development

This folder What is Similarity provides a general and unified treatment of the likeness or similarity of squares, circles, triangles and arbitary regions in the plane. The lessons here can be covered after section 1 on maps, plans and measurement and after the introduction of Cartesian coordinates. The objective here is to explain and reconcile different characterizations of similarity.

The key question is how to recognized similarity. In modern life, objects are similar by design if they stem from the same plans but are built to different scales. That reflects a coordinate view point of similarity objects. Two objects are similar if we can attach coordinate system to each so that the set of coordinates for the points for one object essentially provides the plan for the other as is or after the application of a scale factor - a dilatation. This set of coordinate development easily explains how and why circles, squares and rectangles - those with a common aspect ratio - may be similar. The coordinate perspective of similarity in the case of similar triangles and more generally in the case of similar polygons implies corresponding angles are equal and corresponding sides are proportional. A partial proof of the converse is included in the section what is similarity - a full proof is left to later as site development to do.

Preamble and Context

Similarity is an artifical concept. It main comes from the construction of rectangular, circular and triangular or polygonal shapes for walls, doors windows and layouts or floor designs of buildings large and small; and from the construction of furniture tops, sides and legs. Where construction is is based on plans and drawings, actual lengths and surface areas, and volumes are by design proportional via a scale factor, its square and its cube to corresponding parts on the plans and drawings. In this corresponding angles are equal. Further circles and squares in the drawings correspond to circles and square areas or objects in the construction The proportionality still holds when the same plans or drawings are implemented at different scales, albeit too great variation in scale may lead to structural instability. In modern times with the advent of digital plans and drawings, planned objects are described in terms of coordinates - sets of coordinates or the data needed to determine those sets. In this, the digitilized plans and drawings may be displayed at different scales. In nature, similarity appears in the form and inner components of larger living beings, but as beings grow, the measures and dimensions of the "similar shapes" are not proportionality. No doubt there will be a few exceptions.

Primary School Geometry - Like Shapes

Primary level instructions may ask students to identify like shapes, years before the secondary level mention and introduction of similarity. Indeed, young children may recognize like shapes from their senses of vision and touch, and from the function of objects in the plane or space. The ability to recognize like shapes allows children and people in general to recognize others and navigate their way their local environments, all without or all before any formal acquaintance with the concepts of similarity.

This late primary or secondary level geometry may introduce maps and diagrams drawn to scale. At home or in a library, students can be shown maps and plans of their community and its surroundings. Maps may be employed in the description of routes followed by family members, cars, buses, trains, and planes, etc. School going children and teenagers will likely see maps of school building and terrains. On these planar maps - when drawn to the same scale horizontally and vertically, actual rectangular, circular and triangular regions appear as rectangles, circles and triangular regions respectively. The study of maps may be employed to point and recognize similar shapes before any formal discussion of similarity. That being said, this level may show and imply that angles on these maps and drawings equal the corresponding angles in the drawn objects. When a unit length on the map coresspond to a unit length in actuality, this level may further slowly show that the number of map unit lengths needed to cover a drawn length and map unit squares needed to cover a drawn rectangular region equals the number of real unit lengths and real unit squares needed to cover the actual lengths and region. The use of the scale factor or it reciprocal as is or squared then gives a proportionality constant between drawn and actual lengths and areas. Most likely, the length case and the area cases should be treated differently.

Late primary and early secondary quantitative skill development should emphasize measuring skills with rulers, tape measures and protractors for measuring lengths and angles in the environment and on maps and plans drawn to scale. Tutors and teachers may observe that the map measurements are often easier to make - require less movement. This level may show students how to recognize different kinds of triangles and quadrilaterals and connect the latter to parallel lines or line segments. All the foregoing may be done on paper with maps, plans or drawings. Coordinates signed and unsigned should be introduced along line segments and for maps and plans. The game of Battleship may be adapted to test mastery of coordinates. Students may be further given a sequence of coordinates for points in the plane to join to test and reward coordinate mastery. The joined points or dots may form a picture of objects or animals in the local environment.

Familarity with maps and diagrams drawn to scale should make the assumptions or axioms of coordinate free Euclidean more self-evident. An operational mastery of maps and plans drawn to scale sets the stage for site simplified treatment of Euclidean Geometry, one reserved for the keener students in senior highschool, one sufficient to introduce students to the use of deductive reason in mathematics.

Finally, secondary level geometry instruction may tell and show students that all circes are similar and all squares are similar; it may provide students criteria for the similarity of triangles, criteria based on equality of corresponding angles, or proportionality of corresponding sides; and it may criteria for the similarity of rectangles based on equal aspect ratios or equivalently based on the on the proportionality of corresponding sides. The foregoing rules and criteria are consistent with each other. To the foregoing, I would have secondary school geometry add the coordinate perspective: Two objects in the plane or space are similar if coordinates systems can be choosen for both such that the set of coordinates for one are proportional [scaled versions] of the other. The latter criteria echo the modern day use of plans and drawings as indicated above to digitize actual or concieved objects in two and three dimensions, that is, in planes and in space. The coordinate perspective is timely if or when secondary level geometry talks about the proportionality of corresponding lengths, area and/or volumes in the discussion of similar objects in a plane or in space.

Proportionality of Map and Actual Measurements Explained

A map unit length corresponds to a given length, a unit length, in the real world. Here the number N of map unit lengths needed to cover a path on the map is the same as the number N of given lengths needed to cover the corresponding path in the real world. The ratio of the number N of given lengths to the number N of map unit lengths equals the the ratio of the given length to the map unit length, that is the map scale factor. Likewise, the number M of map square units needed to cover a region in the the map equals the number M of of squares with sides provided by the given lengths needed to cover the corresponding region in the real world. The ratio of the number M of given lengths squared to the number M of map unit squares that equals the the ratio of the given length squared to the map unit square, that is the map scale factor, squared.

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