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Geometry and Formula Evaluation

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.  

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: Recognition that multiplying by a half gives the same result as dividing by a half sets the stage for the introduction of algebraic identifies - 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. 

Geometry with Maps, Plans and Designs 
to Complex Numbers

Maps, plans, designs and drawings made to full, partial or oversized scale 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 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.
   
4. Rectangular Coordinates 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.  Absolute coordinates would use coordinates of the form [A, B] = [a units, b units] with ordered pairs of mixed number multiples of units (quantities). 
   
5. Rectangular Coordinates 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. 
   
6. 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 absolute for degree measure.  Absolute coordinates would use polar coordinates of the form (R, q) = (r units, q )  with R being the absolute quantity r units, and q (still) being the 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. 
    
7. 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.  
   
8. 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.
   
   

   (C) 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 ANDing or adding of displacements or arrows or vectors in a head to tail manner.  Diagrams imply that addition is clearly associative.

   
   

9. 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. 
   
10. Orienteering:  Walking through parks and bush with the aid of maps and compass may introduce map usage 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.
     
11. 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 or vectors 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.
    
    Technical Note: The Head to Tail Addition of a sequence of displacements, 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 vector, all without changing the net displacement from the initial point of the route to the terminal point of the route. 

Theory and Use of Scale Factors (Proportionality Constants) for 
Maps and Models in 2 and 3D

1. Measuring lengths and areas with relative and absolute measures: Suppose we take 1 meter to be the unit length for measurement of distance.  Then a  curve or length with actual or absolute length of 5 meters has a length of 5 relative to the unit length of one meter.  Further a region with actual or 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 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 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 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 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 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 absolute length = K * original length                    or  L_image  = K L_original
   image absolute area    = K² original absolute area            or  A_image  = K² A_original
   
   and 
   image absolute volume    =  K³ original 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. 
   

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 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. 
   

IV. More on Signed Numbers and Real Numbers

Extrinsic Development of arithmetic methods for signed numbers: Start with Multiplication of Displacements (arrows or vectors)  by whole numbers and fractions which have signs as prefixes.  Here multiplying by a negative number reverse the direction of an arrow.  Next consist multiples of a single nonlinear vector. The multipliers are then signed numbers.  The addition of those multipliers implies rules for the addition of the multipliers or signed numbers. The multiplication of a multiply implies rules for multiplication of the multipliers. Optional: The extrinsic development of arithmetic with whole numbers and fractions may be explained here for the sake of continuity or comparison.  

Material to be re-organized

1. Addition of Collinear Movements: Define, then show this Addition is commutative. Then show identify repeated Addition of a Single Collinear with multiplication by whole numbers.  Then defined Multiplication of  Collinear Movements by proper and improper fractions - whole numbers and mixed numbers included. Finally,  extend that that multiplication to included multiplication by signed numbers.  Observe resultant of a head-to-tail sum of pair of collinear arrows has length equals  the sum of their lengths when the vectors have the same direction, and length equal to the difference when the vectors have opposite direction. Observe addition of displacement vectors or movements is commutative and associative.  The zero movement gives the additive identity property.  Each displacement has an negative, its additive inverse, a vector with the same length and the opposite direction. 
   
2. On a finite or infinite straight line, choose an origin and then use it to define position vectors for points in the line.   The addition of position vectors (signed numbers) is then defined by the head to tail addition of those collinear position vectors - possible in any order since addition of collinear displacements is commutative. The head to tail addition of position vectors is also associative. 
   
3. On a finite or infinite straight line, choose an origin and then use a unit vector (displacement), to defined signed coordinate for points on the line relative to the unit vector.  Each point may be identified with a position vector, the vector from the origin to itself. Each position vector is a signed number multiple of the unit vector. The addition of signed coordinates (signed numbers) is then defined or implied by the head to tail addition of those collinear position vectors - possible in any order since addition of collinear displacements is commutative. The head to tail addition of displacement is also associative. The identification of signed numbers with collinear movements (displacements) along a straight line thus defines addition, implies the effect of adding a zero displacement or zero; and implies the existence of additive inverses.   The previous discussion of multiplication of collinear displacements by signed numbers suggests how to multiply signed coordinates. 
   
4. Real Numbers: Signed numbers may be represented by proper and improper fractions, and by terminating or non-terminating decimals.  They may also be represented by square roots and arithmetic expressions.  And with coordinates on a straight line relative to an origin and a unit vector,  signed numbers may be identified with points on the line or their position vectors, and with a class of vectors of a given length and direction, equal modulo the location of their initial points. There-in an opportunity to introduce or name the real number line, and to identify key subsets of the real numbers: whole numbers, natural numbers, rational numbers and irrational numbers.
   
5. Signed Numbers: Arithmetic operations on signed numbers based on the addition and multiplication of collinear displacements: Signed numbers provide coordinates along the coordinate axes. They can be identified with displacements in the positive and negative direction along one of the axes - call it the horizontal axes.  Whence the rules for the addition, subtraction and multiplication of signed numbers follow from those for the addition, subtraction and multiplication of collinear vectors. The number of times one displacement is a multiple of another leads to the definition of division. 
   
   Remark: Now the commutative and associative property of addition of arrows implies the same for coordinates.  Teachers may tell students to assume them - give them as theorems with proofs available, but with proof mastery optional.  Formal discussion can be left to later. Focus on providing students with an operational command of arithmetic with signed numbers would be an option.  
   
6. Necessary Field Properties of Coordinates:  Commutative, Associative, Distributive Laws. Properties of 0 and 1. Use of Additive and Multiplicative Inverses in applying rules to subtraction and division. Products of non-zero factors are nonzero since product of nonzero unsigned numbers is nonzero, or since the area of a rectangle with sides > 0  is nonzero.  The distributive law is equivalent to a change of scale and direction of the unit vector for coordinates along an axis. 
   

On Euclidean Geometry 
(Geometry Before Coordinates) 

Euclid about 300 BC in his elements produced a codification of geometry before the invention of coordinates by Renes Descartes 1800 year later.  Knowledge of Geometry before coordinates is employed in the development of geometry with coordinates (analytic geometry, unit-circle trig, complex numbers, calculus, and so on).

This area on Euclidean Geometry  on geometry before coordinates offers thought-based explanation of the following.  Try to read them in sequence. There is more to Euclidean Geometry than this, but the following elements cover the least amount possible for the following site development of analytic geometry and trigonometry.

1. Correspondence between triangles. Here is an explicit definition, not always seen in class. 
2. Isometry of Triangles - Here is a definition.
3. Side-Side-Side method
4. Side Angle Side method
5. Angle-Side-Angle method
6. Isoceles  and Equilateral Triangles
7. Side-Side-Side Failure 
8. SAS Failure or Near Failure 
9. ASA Failure - links with the parallel postulate
10. Parallel Lines - and angles associated with a transversal.
11. Triangle Angle Sum - from the parallel postulate
12. Similarity and Minimal Conditions for
13. Right Angle Trig., from Similarity
14. Trig & Similarity - More about the Connection
15. Parallelograms and their Properties

New: 

16. - //gm triangle construction method: Show how to construct a parallelogram from a triangle
17. Show how to construct a parallelogram from two non-collinear vectors which share a common initial point

    Remark: The vector heads and their common tail give three of the four vertices of the //gm. The vector that goes from the the common tail to the fourth vertices is taken to be the sum of the two vectors.
    

18. Show how the SAS parallelogram construction methods commutes with scalar multiplication of the vectors where the tail location is fixed.
    Remark: This shows that common tail vector addition commutes with scalar multiplication
    
19. Show how the SAS parallelogram construction methods commutes with rotation of the vectors about a fixed point located at the tail.
     
    Remark: This shows that common tail vector addition commutes with rotation.
    

Arrows,  Vectors and Parallelograms

1. Addition of Order Pairs, Subtraction of ordered pairs,  and Signed Number Multiples: These may be defined as follows:
   
   [a,b] + [c,d] = [a + c, b + d]
   
   [a,b] - [c,d] = [a - c, b - d]
   
   k [a,b] = [ka, kb]
   
   Teachers may identify points in the plane with the heads (terminal ends) of displacements from the origin, associate points in the plane with position vectors (tails at the origin) and then give vectorial diagrams to illustrate the previous operations.
   
   The Pythagorean Theorem (see Chinese Square Dissection Proof) implies the length of the position vector of  k [a,b] = [ka, kb] is k times the length of the position vector of [a,b].
   
2. Coordinate Description of Arrows:  Arrows drawn in the plane are characterized by their initial and terminal points which in turn are characterized in a coordinate systems,  a pairs of order pairs [a,b] and [c,d].  That being said, drawn arrows may be characterized by providing the initial and tail coordinates [a,b] and describing the displacement via the difference [c-a, d - b] = [dx, dy] = head coordinates - tail coordinates.
   
3. Position Vector:   The position vector of  point [a,b] in the plane is the arrow which terminates at [a,b] and which has initial point at the origin [0,0]
   
4. Equality of Arrows Modulo Initial Position: Two drawn vectors are said to be identical, modulo tail position, when and only when the differences  of head and tail coordinates result in the same ordered pair [dx, dy].  It follows that vectors identical modulo tail position have equal lengths, are parallel and have the same direction. With the aid of coordinates, the converse can be implied.
   
   Comparison of Arrows:  Two arrows with the same length and same direction are said to be equal or identical modulo the location of their tails (initial points).  That situation occurs when the difference
   
   head coordinates - tail coordinates = [dx, dy] 
   
   is the same for both arrows. The position vector of the point with [dx, dy] determines an arrow.  All the arrows equal to this arrow, modulo position of initial points, have the same same length and direction as this position vector. 
   
   Note:  head coordinates = terminal point coordinates and tail coordinates = initial point coordinates
   
5. What is a Vector:   Let [dx, dy] be a point in the plane.  The set of all arrows equal to the position vector of [dx, dy], modulo the location of initial point,  is said to be a vector.  Each element, an arrow, in the set (an equivalence class) is said to be an instance of the vector (equivalence class). All elements or arrows in the vector have same length and direction as position vector of [dx, dy]

Complex Numbers 

1. Distance r of a point [a,b] to the origin:   The Pythagorean Theorem (Chinese square proof) implies the formula  r = sqrt (a²+b²) for the distance r of the point [a,b] to the origin.
   
   The Pythagorean Formula also implies the formula d = sqrt ( [c- a]²+[d-b]²) for the distance d between two points [a,b] and [c, d] in the plane.  That formula implies the arrow from [a,b] to [c, d] and its additive inverse also have the length d.
   
2. Introduction of Dilatations that Fix the Origin:  The multiplication of points [a, b] by a number k > 0 gives  the image point k [a,b] = [ka, kb].  If the point [a, b] has distant r from the origin then the point [ka, kb] has distant kr from the origin.  
   
   Problems: Show if k is allowed to be a negative, the point [ka, kb] has distance |k|r from the origin.  Also show if d = sqrt ( [c- a]²+[d-b]²) gives the distance d between two points [a,b] and [c, d] in the plane, then the distance between image points [ka, kb] and [kc,kd] is |k|d.
    
3. Dilatations that fix the origin distribute over Point addition: The distributive law A(B+C) = AB +AC implies multiplying a points [a, b] by k > 0 and so multiply distances to the origin by k without changing direction distributes over addition of points in the plane: 
   
   k( [a,b] + [c,d])  = k [a + c, b + d] = [k(a+c), k(b+d)]  = [ka + kc, kb +kd] 
                             = [ka,kb] + [kc, kd]  = k [a,b] + k [c, d]
   
   Remark:  With the aid of parallelogram construction, triangle similarity may be used to show an equivalent result.  
   
4. Appearance of a Parallelogram: Points with coordinates [a, b] and [c, d] may be identified with position vectors - tails at the origin. Then   
   
   [a,b] + [c,d] = [a + c, b + d]
   
   can be identified with the head to tail addition of the vector drawn from [a,b] to [a+c, b +d ]. The latter vector, modulo tail position, is identical with the vector [c, d]. Likewise the position vector associated with [a + c, b +d] is given by the head to tail sum of a vector from [c,d] to [a+c, b+d] with the position vector of [c, d] since the addition of coordinates is commutative. The vector [c,d] to [a+c, b+d] is equivalent, equal, identical to the position vector of [a,b], modulo position of initial points. 
   
   The parallelogram will be squashed (flattened) when the origin, [a,b] and [c,d] provide coordinates of collinear points.  The foregoing implies the sum of two points can be obtained by drawing a parallelogram, that obtained by taking the position vectors of points with coordinates  [a,b] & [c,d] as adjacent sides.
   
5. Rotation Distributes over Point Addition:  The origin [0, 0], and three points (i) [a,b], (ii) [c, d] and (iii) their sum [a+c, b+d] are vertices of a parallelogram.  Euclidean geometry may be use to show that the construction of a parallelogram from two non-collinear arrows drawn in standard position commutes with the rotation of the arrows about the origin. 
   
6. Multiplication of Points by relative Polar Coordinates (r, q).  Let r be an unsigned number and q is an angle of rotation.  Multiplication by (r, q) multiples the length of a position vector by r and rotates it clockwise (top view of plane) through an angle q.  This multiplication commutes over point addition since multiplication by (r, q) may be regarded as dilatation which fixes the origin followed by a rotation which fixes the origin, and both operations commute over point addition. 
   
   Remark: In the modern mathematics curricula I saw as a student, basic trig identities were established using the properties of real numbers and and geometric assumptions about rotations about the origin of a unit circle. The above explanation of how and why origin-fixing dilatations and rotations distribute over point addition uses properties of real numbers and like or equivalent geometric assumption.  The use of geometric assumptions about coordinates departs from the instrinsic viewpoint of pure mathematics to ease comprehension and involves an extrinsic or operational viewpoint of mathematics.   The instrinsic viewpoint can be developed in advanced college level courses that develop mathematics from axioms about sets (or other objects).
   
7. Introduce Trig functions using the    The point coordinates (1,A) determine a point on the unit circle with angle A with respect to the horizontal (real) axis.  That point has rectangular coordinates [x,y] that depend on A.  It is clear that cos(A) and sin (A) are periodic functions with period 360 degrees. We write cos(A) = x and sin(A) = y.  Whence many many identities in the trig functions cos (A) and sin(A) follow from a comparison of polar-coordinate rule (add angles, multiple lengths) obtained expressions and rectangular coordinate (real and imaginary parts) expressions for products of complex numbers. The tangent function is then given by tan(A) = sin(A)/cos(A).  Analysis of the unit circle implies cos(A) is an even function and sin(A).  Find sine and cosine of angle A when A is a zero or a whole multiple of 45 degrees between 0 and 360 degrees. 
   
8. Complex Numbers: See this site introduction of complex numbers, the easy consequences, and the connection to complex number, algebraic approach and derivation trig identities.  For now or later, easy consequences include the cosine law and a converse to the Pythagorean theorem: If the Pythagorean identity a²+b²=c² for three side lengths a, b and c of triangle then the triangle is a right triangle with hypotenuse of length c. 

Trigonometry

1. Introduce Trig functions using the Unit Circle:   The point coordinates (1,A) determine a point on the unit circle with angle A with respect to the horizontal (real) axis.  That point has rectangular coordinates [x,y] that depend on A.  It is clear that cos(A) and sin (A) are periodic functions with period 360 degrees. We write cos(A) = x and sin(A) = y.  Whence many many identities in the trig functions cos (A) and sin(A) follow from a comparison of polar-coordinate rule (add angles, multiple lengths) obtained expressions and rectangular coordinate (real and imaginary parts) expressions for products of complex numbers. The tangent function is then given by tan(A) = sin(A)/cos(A).  Analysis of the unit circle implies cos(A) is an even function and sin(A).  
   
   Remark: Reflections about the horizontal and vertical axises, and the 45 degree line y = x  implies further algebraic properties of cosine and sine functions for presentation in all or part in this step or later, possibly with repetition.
   
   Connect with Right Triangle Trigonometry via Similarity of Right Triangles in the first quadrant - next
   
31. Construction,  Dimensions, Areas and Perimeters of Regular Polygons with the aid Roots of Unity and trigonometry
    
32. Connect to Right Triangle Trigonometry.   
    . Similarity implies the ratio of any two sides in a figure equals the ratio of the corresponding sides in any similar figure.  Hence for acute angles A,  cos(A) and sin(A) are given by the ratios of adjacent and opposite sides for the angle A to the unit length hypotenuse of a right triangle determined by angle A in the first quadrant.  That implies standard right triangle trig formulas for cos (A) = adjacent/hypotnuse and sin(A) = opposite/hypotenuse.  Likewise, tan (A) = opposite/adjacent. Now the isoceles right triangle with legs of length 1 and hypotenuse of length sqrt(2) can be use to calculate cos(A), sin(A) and  tan(A)  for A  = 45 degrees.  Further the equilateral triangle of sides 2, bisected in two by the right bisector of one of its 60 degree angles, an altitude, can be used to calculate cos(A),  sin (A) and tan(A) for A = 30 degrees and A = 60
     degrees.|
    
    Tabulating trig functions provides an alternative to drawing diagrams and solving problems through the use of similarity - next. 
    
33. Trig function Values on the Unit Circle: For all multiples of 30 and 45 degrees in the range 0 to 360 degrees, and determine corresponding values of trig functions from their reflection induced algebraic properties.
    
34. Solving Triangles Using Similarity or Trig Functions:  Show how right triangle trig provides an alternative to similarity analysis in solving triangles. Given a large right triangle with determined (explain why) by SAS with the aid acute angle A and the measure of one its sides (opposite, adjacent or hypotenuse),  a triangle with sides that cannot be measured directly, we can find the lengths of the remaining sides by (i) drawing a similar right triangle and measuring (similarity implies the ratio of any two sides in a figure equals the ratio of the corresponding sides in any similar figure) or (ii) using tabulated or electronic calculator given values of right triangle trigonometry functions cos(A), sin (A) or tan(A) found without immediately drawing a similar right triangle, but found in principle from drawing many right triangles and tabulating the results.  In other words, the use of trig functions hides or buries the use of similarity in solving right triangles with missing lengths in the earlier link of  trig functions values for acute angles to  the ratios of adjacent sides of right triangles. [To do: rewrite]
    
35. Sine Law:   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. 
    
36. Cosine Law: Give or derive this law as an easy consequence of the properties of complex numbers - two ways or multiply. Next apply the cosine 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. |
    
37. Trigonometric Identities: Use properties of complex numbers (two ways to multiply) to algebraically derive and verify trigonometric identities - the engineering way.  That aids and speeds the coverage of this topic.  In sum, start with complex number viewpoint - real and imaginary parts of exp(iq) and  show algebraic development and verification of trig identities using exp(iq)
    
38. Geometric Applications of Cross-Products:  Show how to calculate area of a triangle and kites or parallelograms that may be constructed using SAS data.
    
39. Geometric Applications of Inner-Products:  Show how to compute components of a vector - horizontal, vertical and in any direction.  
    
    Optional: Connect to force analysis in physic and phasor analysis in electricity.

39. More Unit Circle Trig: Similar Sectors and Switch to Radians:  On a circle of radius r, the length of an arc subtended by a central angle A = n degrees is given by s = kn  where  k is a proportionality constant, and a backward use of the proportionality relation  s = kA when A = 360 degrees, implies 2pr = k 360 and hence k = pr/180. Now  
    
    s = k n = pr n/180  or s/r = pr n/180. 
    
    Whence the arclength the arc of a circle relative to its radius, in other words, the radian measure of the arc, is proportional to the central angle measure relative to degrees, and so is independent of the radius of the similar sectors of a circle - two sectors of different circles determined by a central sector being similar when and only when the central angles are equal. [Rewrite or clarify if need-be].  Give the radian measures exactly for all multiples of 15 degrees in the range 0 to 360 degrees. 
    
40. Trig function Values on the Unit Circle: Give the radian measures exactly for all multiples of 30 and 45 degrees in the range 0 to 360 degrees, and determine corresponding values of trig functions from their reflection induced algebraic properties.
    

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science