Beginner Geometry

Steps to develop practices, or skills and concepts, with minimal theory, or very short chains of reason.  That is, prerequisites for each practice are minimal.  Each practice may appear to be obvious to students, so that minimal justification is needed after their introduction. 

Steps or Topics

1. Geometry and Formula Evaluation
2. Geometry in or with maps, plans and designs
3. Length, Areas and/Or Volume Scale Factors Backwards and Forwards
    for Maps, Plans and Models in 2 and 3D
4. Right Triangle Trigonometry upto law of sines
5. Rectangular and Polar Coordinates with unsigned and then signed numbers -
6. More Geometry with maps - Navigation and Orienteering. 

Step 1: Geometry and Formula Evaluation,
a step common to both geometry & algebra

The algebraic description of length and areas of triangles, squares, rectangles, trapezoids, parallelograms, circles and fractions of circles provides formulas for student to evaluate.   Detail formatting rules for the evaluation of geometric formulas, diagram drawing and labeling included,  show students how to show work - how to communicate the setting, the steps in their reasoning and results in the evaluation of geometric formulas in an observable and correctable manner on paper.  That is a performance objective easily understood and met.. 

The use or role of letters or more generally symbols as placeholders in formulas and identifiers (labels) on diagrams provides a starting point for algebraic ways of writing and reasoning in general where letters or symbols or expression are placeholders for numbers and quantities without an immediate geometric significance.

Examples:

1. Give Formula Evaluation Exercises for areas of squares, rectangles,  triangles, parallelograms and circles with justification where possible of all except for the formula for the area of the circle. That latter requires calculus (or a numerical study of how the area of of circles is proportional to the square of the radius).
   
2. Give Formula Evaluation Exercises for perimeters of squares, rectangles, circles and semicircles, triangles, parallelograms, regular polygons.  justification where possible of all except for the formula for the area of the circle. The justification of the circle perimeter formula  requires calculus (or a numerical study of how the perimeter of a circle is proportional to its radius).

Teachable Moment (painful): Recognition that multiplying by a half gives the same result as dividing by a half is an example of the notion that different formulas when evaluated will give the same result, or in brief the notion that two different expression may be equal or have the same value.  The idea for this come from a student painful objection to my writing two formulas for the area of triangle- one using the factor one half and the other using division by two. I was not being consistent. Consistent use of one or the other formula might have avoided the issue.

Step 2: Geometry with Maps, Plans and Designs 

Maps, plans, designs and drawings made to scales less than, equal or greater than 1  may be used for locating objects and for describing movements along trails or paths, actual or intended.   

1. Maps: In maps drawn or redrawn, the image of a straight line segments and circular arcs are also straight line segments and circular arcs.   Whence the images of figures made of straight line segments and circular arcs are also made of straight line segments and circular arcs. Image element are seen to be proportional to their pre-images in the original figures.  All the foregoing can be shown or implied by many examples, and then assumed as a drawing and design shortcut or tool.
   
   There is an innate ability to recognize like shapes, close-up and far-way, within the level resolution capabilities of eyes - a level that may vary. The ability to read and write letters, digits and further symbols, and to recognize (read) and draw line segments,  squares, circles and semi-circles depends on that ability.  The abiltiy to recognize shapes and figures in pictures and diagrams  also depends on this ability.  Primary students and teachers learning to read and write, and learning geometry, may recognize like or similar shapes without any mention of the formal characterization of similarity that appears say in secondary school mathematics. Geometric optics suggest two figures, polygonal or not, in different maps have the same shape if one is the projection or scale drawing of the other  - undistorted.  Distortions would follow from different scales on different axes. The secondary level discussion and definition  of similarity of polygons and circles in a single plane or appearing on different maps characterizes and codifies similarity in terms of corresponding angles being equal and corresponding lengths being proportional formalizes or codifies that innate ability but not fully as the geometric optics projection, perspective geometry and/or scale drawing viewpoint.    The equivalence of the latter to the primary school identification of geometric figures and curves having like or same shapes is incomplete as the formal discussion only involves polygonal figures.  
   
2. Map Drawing or Construction:  In drawing maps of physical situations and objects or points there-in,, students may determine the image of an object or map in the map by using physical measurements to determine the location of the point relative the bottom-left corner of the map with the aid of real-life unsigned rectangular and/or polar coordinates. For example, students may be asked to draw or map to scale, their current classroom and the location of key objects there-in -  desk and chairs, blackboards, doors, windows, etc.  Line segments, squares and rectangles, and part of circles, may be used to depict the latter objects on the map.  Desk should be drawn in proportion - so that aspect ratio of their sides (top view) is maintained.  Teachers could introduce four objects  with a triangular top view in the classroom and get students to plot them in a room map or plan with the aid of (i) three vertex coordinates,  (ii)  the coordinates of the end points of one side (top view) and the use of  the SSS, SAS and ASA physical measures to draw the images of the objects (triangular top view) in the map.  Division of the room and map into corresponding grids may help.
   
3. Why Measure or Calculate Distances and areas with Maps and Plans.  Students may measure the drawn, on-map distance between two points on a map using a ruler or a tape measure, and then determine the pre-image points with by multiplying by a scale factor (proportionality constant).   Let the unit length in the map be the image of an actual or real-life unit length.  Then map unit square is the image of the actual or real-life unit square. Simple examples may imply that measure relative to the unit lengths and areas are invariant - that is the same in the map and in actuality. Whence lengths and areas of a figure or its map image can be measured or calculated relative to unit length and area on the map or in real life.  The advantage of maps, plans and drawing in calculating lengths and measures, and in route planning, appears when the actual or real life actual (absolute) measures are not feasible.  In other words, maps, drawing and plans provide a means for the indirect measurement as relative lengths and areas are invariant. Whence on-map (on drawing or on-plan) measurements provide an alternative to real or actual measurements.   For surveying and navigation, information that is sufficient to draw a length or figure to scale allows the missing dimensions and areas in the figure to be determined from the drawing.  
   
   The foregoing may be done before the use of coordinates and then after. See the introduction of coordinates below.
   

Steps 3: Scale Factors Backwards and Forwards for Maps and Models in 2 and 3D

1. Measuring lengths and areas with relative and actual (absolute) measures: Suppose we take 1 meter to be the unit length for measurement of distance.  Then a  curve or length with actual or actual (absolute) length of 5 meters has a length of 5 relative to the unit length of one meter.  Further a region with actual or actual (absolute) area of 14 square meters, that is, 14m² has area 14 relative to the unit area of m² or 1 square meter.  
   
2. Relative lengths are invariant: Suppose a line segment 8 m (8 meters) long is drawn on a map with a scale of 10 to 1. Then the drawing of the line segment will be 8 dm (8 decimeters) long.  The original unit length, that is one meter, is gives or corresponds to a 1  decimeter unit length.  The original line segment and its image both have relative length 8 with respect to the original unit length (1 meter) and the unit length (1 dm) on the map.  So relative lengths are unchanged.  Likewise, if a sequence of line segments forms a piecewise linear path in the original plane, then the images of the sequence drawn on the map forms the image path, piecewise linear too,  in the map.  Both paths will have different actual (absolute) lengths, but identical lengths relative to the unit length in the physical situation and in the map.  So again, relative lengths are invariant. 
   
   Finally and optionally, if the relative length of a curve in a physical plane is the limit of the lengths of sequence of piecewise linear approximation to it, the original curve,  then the image curve in a map will be the limit the lengths of sequence of corresponding piecewise linear approximation to IT, the image curve, and vice-versa. Whence the image curve and the original curve will have the same lengths.  The piecewise approximation can be taken with zero error (to be exact) on any portion of the curve which is linear. 
   
3. Relative areas are invariant: Suppose a rectangle with dimension 3 meters by 4 meters is drawn on a scale of 1 to 10 on a map. The image is a rectangle of dimensions 3 dm by 4 dm.  The actual (absolute) or actual area of the original square is 12 square meters or 12m²   while the area of the image is 12 square dm or 12 dm².  Observe that the image of a the unit of area, that is a square meter, is the unit of area, a square decimeter, in the map.  Here we see the area 12 of the original rectangle relative to the original unit area equals the area 12 of the its image rectangle on the map with respect to the map unit area = the image of the original unit area.  So area of the rectangular region and its image defined relative to the unit squares  and its image is the same. Likewise,  the areas of square and their images relative to the unit areas in the pre-image and image planes are equal.  The key word here is relative.  
   
   Finally and optionally, areas of regions in the original plane can be approximate relative to the unit square in the original plane by covering the region by small squares and finding the limit in relative or actual (absolute) terms. Do the same in for the image of the region and using the images of those small squares gives the same sequence of approximations for the relative area of the image and hence, in the limit, the image region has the same relative area as its pre-image - the original region. 
   
   Application to Note Taking:  A teacher draws a parallelogram on a board with height of 5 units and a base length of 4 units.  Each note taking student in the class draws a similar parallelogram with height 5 units and base 4 units, but the unit length used in all drawings of the students and their one teacher are not the same. None the less, the students and teacher all see that their version of the parallelogram, the original and all its images, have a common area of 20 = 5 x 4 relative to their unit of measure. In all calculations of area of a figures, figures whose corresponding dimensions relative to a unit length in the diagram or map containing the figure are identical, all have the same relative area.  Whence relative area calculation for a single figure - the original - may done with a figure that is similar to it. We may same for composite figures - figures that can be decomposed or split into smaller similar figures, so that the sum of the areas, actual or relative, equals that of the original composite.   TASK: Say or rewrite the foregoing in a clearer manner. 
   
   Extension: In a like manner, when 3 dimensional objects are designed or mapped, relative lengths, relative surface areas and relative volumes are invariant, that is, equal for each original object and any similar object that models it. 
   
4. Scale Factors K, K² and K³  for actual (absolute) Measures:  In mapping or modeling a 1, 2 or 3 D object or figure, the original unit length corresponds to an image unit length  = K times the original unit length. We take that image unit length to the unit length for the map or model, and thus for the calculation of unit area for 2D regions or surfaces, and for the calculation of unit volume for 3D models of 3D objects.  Whence 
   
   image unit length = K * original unit length 
   image unit area    = (K * original unit length)² =  K² (original unit length)² = K² original unit area
   
   and 
   image unit volume    = (K * original unit length)³ =  K³ (original unit length)³ = K³ original unit volume.
   
   That is
   
   image unit length      = K * original unit length 
   image unit area         = K² original unit area
   
   and 
   image unit volume    =  K³ original unit volume.

   For corresponding lengths, surface areas and/or volumes, the relative measures are equal by previous arguments.  Whence 
   

   image actual (absolute) length = K * original length                    or  L_image  = K L_original
   image actual (absolute) area    = K² original actual (absolute) area            or  A_image  = K² A_original
   
   and 
   image actual (absolute) volume    =  K³ original actual (absolute) volume  or  V_image  = K³ V_original

   Remark 1: The numbers are scale factors or proportionality constants for length, area and volume respectively. When one is calculated or obtainable, then so are the others via arithmetic operations of squaring, cubing, taking square roots and/or taking cube roots.  Moreover, if the ratio of a pair of image and original lengths, areas or volumes is known, then one of the scale factors K, K² and K³ and hence all may be calculated.  See site discussion of forwards and backwards use of formulas and proportionality relations to learn more.
   
   Remark 2:  Memorization of squares and cube roots of 1, 2, 3, 4 and 5 may help in the backward calculation of the scale factors in exercises that develop or encourage forward, backwards and sideways use of the proportionality between lengths, areas and volumes in maps and models, and in real life.
   
   Remark 3: The proportionality factors K, K² and K³ also apply to quantities that are proportional to length, area and volume in the building of models to part, full or oversized scale. Exercise: Explain why. 
   
   Remark 4:  Say the scale of a map or plan is 1 to 10000 then 1 cm on the map corresponds to 100 meters =10000 cm on the map. Now let d = distance measured on the map = n cm. Let D = the corresponding distance in practice (actuality) measure in meters. Then  we expect 1 to 10000 = d to D or  10000 to 1 = D to d. So 10000 = D/d or 10000 d = D.  Now d = n cm gives D = 10000 (n cm) = 100 x n 100 cm = n 100 meters.  Thus the "unit distance" 1 cm on the map corresponds to the unit distance of 100 meters in actuality. 
   

5. Similarity by Design: If two artificial bodies S and S' appear to have the same shape, then it likely but not guaranteed that there is a common plan
   

6. Preparation for Trig and Alternative to Trig:   Suppose S is a 2D or 3D figure that is similar to another figure S'. Then similarity or proportionality implies the ratio of any two sides, areas or volumes in the figure equals the ratio of the corresponding sides, areas or volumes in any similar figure S'.  If one aspect of  figure S is too large to measure, the construction of a similar figure may make that aspect measurable with the aid of a proportionality constant. 

Step 4. Right Triangle Trigonometry

A Message to Repeat in Class: All trig formulas may be used forwards and backwards alone or in combination with other trig formulas and the Pythagorean theorem to find missing angles and sides in triangles that are alone, adjacent to further triangles, or implicit in a diagram.

The latter  implies 

Triangle Area S  =  ½ c h  = ½ c a sin b 

By putting sin b   = sin  q  = sin (180^o - b)  when b  is between 90 and 180 degrees., the previous formula triangle area  = The latter half the product of two sides times the sine of the included angle.  is extended to  the case where the included angle is obtuse.  

Note:  A triangle is half a kite or parallelogram. So the foregoing also implies SAS formulas for the areas of kites and parallelograms which can be developed in class notes or in exercises. 

Derivation of The sine law:

From the SAS triangle area formula, the area of  triangle ABC 

is given by  S  = ½ ac sin B where B in an overuse of the symbol B, denotes the angle at vertex B.  In general, there are three formulas

• S  = ½ bc sin A
• S  = ½ ac sin B
• S  = ½ ab sin C

Equality of the first two formulas ½ bc sin A = ½ ac sin B gives   bc sin A =   ac sin B and hence   b  sin A =   a  sin B. The latter in turn (divide both sides by ab) gives 

 Equality of the last formulas ½ ac sin B =  ½ ab sin C likewise gives 

Likewise equality of the first and last formulas give

Now instead of writting (I), (II) and (III) separately, we write the simultaneous equalities

The latter expression is called the sine law, and its a brief form of stating (I), (II) and (III).

Remark: Formula (I) involves four quantities, namely two lengths and two angles.  The backward use of (I) as is, or rewritten as an equality of reciprocals, which ever is most convenient, means that whenever three of the four quantities is given, the fourth can be found.  Formulas (II) and (III) likewise involve four quantities.  They too can be used in a backward manner.

Sine Law Application Ideas:   Use the unit circle introduction of trig functions and reflection about vertical axis to show that sin(180 - A) = sin (A) for acute and obtuse angles A. Next prove the sine law for triangles, scalene or not.  Interpret the sine law in the case of right triangles and imply it works, with some overlap or redundancy [to do:  redundancy to be spelled out.]  Next apply the sine law forwards, backwards and side ways to solve right triangles.  Point out the option of drawing similar triangles and measuring in each way the law is used. 

Step 5: Rectangular and Polar Coordinates

Preparing for the Use of Coordinates and Arrows (to Represent Displacements) on Maps and Plans
Easily Covered Background Information. 

The introduction of rectangular and polar coordinates may be done here, in the development of algebra or with the further discussion of advanced geometry

1. Rectangular Coordinates for Maps with unsigned numbers:  Ordered Pairs [a,b] of Mixed numbers, proper and improper fractions and decimals with square brackets may be introduced as coordinates to locate points on rectangular maps when the origin of this unsigned coordinate system is place at say the bottom-left corner of each map. The introduction of coordinates is based on the introduction of unit lengths - keep it the same for horizontal and vertical directions - and based on the introduction of a square grid covering the map. Each square in the grid can itself by covered by a grid of smaller squares, and so on, ad infinitum.
   
   Note: the foregoing coordinates [a, b] are relative to the choice of unit length.  actual (absolute) coordinates would use coordinates of the form [A, B] = [a units, b units] with ordered pairs of mixed number multiples of units (quantities). 
   
2. Rectangular Coordinates for maps with Signs:  Ordered Pairs [a,b] of Mixed numbers, proper and improper fractions and decimals with plus and minus signs as prefixes may be introduced as coordinates to locate points on rectangular maps when the origin of this unsigned coordinate system is not placed at the bottom-left corner of each map. As before, the introduction of coordinates is based on the introduction of unit lengths - keep it the same for horizontal and vertical directions - and based on the introduction of a grid of unit squares covering the map. Each square in the grid can itself by covered by a grid of smaller squares, and so on, ad infinitum. The boundaries of the map need not be aligned with grid elements. Make sure that students are aware that the coordinates of a point are relative to the length of unit vectors. 
   
3. Polar Coordinates with unsigned numbers:  Ordered Pairs (r, q)  of Mixed numbers, proper and improper fractions and decimals with round brackets may be introduced as coordinates to locate points P on rectangular maps when the origin of this unsigned coordinate system is place at say the bottom-left corner of each map. The introduction of coordinates is based on the introduction of unit lengths - keep it the same for horizontal and vertical directions. Here r  = the distance of the point P from the origin while q = angle of the ray from the origin to the point P. The angle would be between 0 and 90 degrees for points in the first quadrant, and between 90 and 360 degrees for points in other quadrants.
   
   Note: the foregoing coordinates (r, q) are partially relative to the choice of unit length for distance and actual (absolute) for degree measure.  actual (absolute) coordinates would use polar coordinates of the form (R, q) = (r units, q )  with R being the actual (absolute) quantity r units, and q (still) being the actual (absolute) degree measure of angle.
   
   Note:  The angle q of a point is determined modulo 360 degrees.  One might speak of the angle, modulo 360 degrees, for the sake of having a "unique angle".  That angle might be identified with a point on a unit circle. 
   
   Remark for Items 1, 2 and 3 above:  It can be implied or suggested in Euclidean Geometry that rectangular coordinates [a,b] of point in a rectangular region of the plane which includes the origin [0,0]  uniquely determine a right triangle with one leg on the horizontal axis and a hypotenuse given by the straight line segment between the origin [0,0] and point [a,b]. Then the length of the hypotenuse gives R = r units while the standard angle it makes with the horizontal axis gives angle q.   Conversely, the polar coordinates of a point determines the legs of a right triangle with one leg on the horizontal axis.  The lengths of those legs and the identification of the quadrant in which point lies determines the points rectangular coordinates [a,b]. 
   
4. Map Mastery Exercises: Student mastery of rectangular and polar  coordinates may be developed and verified by exercises which require them to locate and plot individual points (dots) from point coordinates.  Student comprehension of rectangular coordinates may be further developed and verified by exercises which require students to join the points or dots that form the figure of a person, object cute animal or form a trail or path in the map with some amusing significance - path out of a maze, path between two cities following a road network,  path to buried treasure, etc, etc - where the etc, etc means I have run out of imagination. The introduction of coordinates is based on the introduction of unit lengths - keep it the same for horizontal and vertical directions.  
   
5. Map Usage:  From measurement and scaling of map coordinates, students may find the physical location of a point, or its image on another map.  Maps may also be used to draw and plan routes.  From measurement and scaling of map lengths with rulers, threads and measuring wheels (official name?), students may obtain the physical length of routes.  Bearing (angles) of a distance object and the endpoints of the line segment joining two bearings would allow students to locate on the map the distant object using the ASA method. The foregoing may be combined with more map mastery exercises.

   ---

Step 6: More on Geometry with Maps, Plans and Designs

Running Out of Steam: When this site author is sufficiently energized, the following will be reconsidered.
Question: Running out of Steam refers to what means of locomotion?

1. Arrows and Navigation: Actual or potential path (trips, voyages, routes) may shown on maps by curves - smooth or piecewise linear. The net result of a trip is a movement or displacement from the initial point (origin of the path) to the terminal point that can represented (drawn) as an arrow or vector with tail at the initial point and head at the terminal point. Paths that involve a sequence of net movements from one point to a next can be represent by piecewise linear curves in which linear part, an actual or net linear displacement,  is represented by an arrow.  The observation of a first displacement and then a second displacement and then a third displacement, etc, leads to the adding of displacements or arrows or vectors in a head to tail manner.  Diagrams imply that addition is clearly associative. 
   
   Technical Remark: The Head to Tail Addition of a sequence of displacements (arrows), where subsequences are replaced by a net result (resultant vector) is  associative. [insert picture to demonstrate] Moreover, adjacent element of the sequence (subsequences) can be grouped and replaced by a single resultant arrow, all without changing the net displacement from the initial point of the route to the terminal point of the route. 
   
2. Orienteering:  Walking through parks and bush with the aid of maps and compass may introduce veering-off strategy of aiming for one side of a desired destination instead of heading directly for it.  For example if the destination is on a stream and there is some uncertainty or impossibility of heading directly for that destination, the orienteer would set a course or direction that guarantees hitting the stream above (or below) the destination, and then plan to walk downstream (respectively upstream) to the desired destination.  The alternative might lead the orienteer to the stream without being certain of being upstream or downstream of the target location.
   
   
3. Plans and Maps in Design and Navigation:  Students may be shown how to solve geometric design and navigation problems by drawing objects and paths to scale and then measuring lengths on the drawing to compute actual lengths and from them compute areas, and other quantities proportional to length and area. Here is a context for the introduction and study of similarity since  corresponding Angles are preserved and corresponding lengths are proportional in the drawing,  design and use of shapes and routes on maps and plans.  The study of similarity may focus on the use of scale drawing to draw conclusion about real or imagine world situations and applications with the aid of geometric formulas and real or on-map measurements. Applications may appear in interior design - the painting and design of rooms in homes and offices, and the calculation of quantities based on length, area and even volume. Application may also appear in surveying - determining heights of building from angle measurements and horizontal distances.  Application may also appear in navigation - route planning for vehicles on land, on water, or over and under land and water.  The common theme may drawing to scale on map or plans, and then rescaling map measurements to get or estimate real world measures. 
   
4. Navigation and Movements with arrows and vectors on Maps.    The planning of routes at sea and in the air may involve straight line segments with an initial and terminal point.  Each segment may be depicted by an arrow or vector with a tail starting at the initial point and a head ending at the terminal point. A piecewise linear route (top view) in the plane may be represented by a sequence of arrows, with the tail of the first arrow at the initial point of the route, and the tail at each further arrow at the head of is predecessor.  Each arrow represents a displacement from its initial point or tail to its terminal point or head. The arrow points in the direction of movement. So an arrow in depicting a displacement has a length (magnitude) and direction. The arrow also has an initial position (the tip of its tail) and a terminal position - the tip of its head. The head to tail placement of arrows represent a pair or sequence of displacements, a route in which the end of one displacement is the start of the next. The net displacement of a such a sequence is the arrow from the initial point of the route to the terminal point.  That arrow represents the net displacement of two to several displacements. In particular, the net result (sum or resultant) of two adjacent arrows or displacements in the route may be represented by a third arrow or displacement that starts at the initial point of the first arrow and ends at the terminal point of the second arrow. 
   
   A sequence of movements (displacements) plotted as arrows may give or approximate an actual or planned route. The map location of heads and tails may correspond to points on the route where bearing were taken to determine location on the map, or those map location may represent the intended location.  Maps and charts may show the intended and actual route of a vessel or vehicle across the sea or land.

Remark on Parallelism:   On maps and plans, the practice of indicating true North or Magnetic North provides a reference for describing the direction of line segments and movement. Two movement are declared to be parallel if they make the same bearing or angle with respect to true North or magnetic North.  Parallel movements are thus indicated by compass readings. Implicit here is theorem that could be stated and shown in particular cases, if not in general,  in Euclidean Geometry:  Namely, if at two distinct  points are located on parallel lines, and rays issuing from the points make congruent angles with the parallel lines, then the two rays and lines extending them are parallel. Likewise, the practice of covering maps and plans with grid also provides a reference for describing the direction of line segments and movement. How these practices are possible might be a subject of discussion in Euclidean Geometry - the applied math view of it.