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58 steps
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:
- 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).
- 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.
- 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.
- 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.
- 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.
- 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).
- 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.
- 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.
- 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.
- 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.
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(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. |
- 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.
- 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.
- 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
- 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.
- 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.
- 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.
- 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 Limage = K Loriginal
image absolute area = K2 original absolute area
or Aimage = K2 Aoriginal
and
image absolute volume = K3 original absolute
volume or Vimage = K3 Voriginal
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.
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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
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
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Correspondence
between triangles. Here is an explicit definition, not always seen in class.
- Isometry
of Triangles - Here is a definition.
- Side-Side-Side
method
- Side
Angle Side method
- Angle-Side-Angle
method
- Isoceles
and Equilateral Triangles
- Side-Side-Side
Failure
- SAS
Failure or Near Failure
- ASA
Failure - links with the parallel postulate
- Parallel
Lines - and angles associated with a transversal.
- Triangle
Angle Sum - from the parallel postulate
- Similarity
and Minimal Conditions for
Right
Angle Trig., from Similarity
Trig
& Similarity - More about the Connection
- Parallelograms
and their Properties
New:
- - //gm triangle construction method: Show how to construct a parallelogram
from a triangle
- 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.
- 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
- 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
- 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].
- 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.
- 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]
- 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
- 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
- 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.
- 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.
- 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.
- 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.
- 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.
- 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).
- 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.
- 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
- 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
- Construction, Dimensions, Areas and Perimeters of
Regular Polygons with the aid Roots of Unity and trigonometry
- 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.
- 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.
- 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]
- 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.
- 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. |
- 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)
- Geometric Applications of Cross-Products: Show how to
calculate area of a triangle and kites or parallelograms that may be
constructed using SAS data.
- 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.
- 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.
- 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.
| |
LAMP
(first
draft, June 2008) a program for adult
and teen mathematics education
Mathematics education standards implied by calculus should
be a factor, not the only one, yet not a forgotten nor hidden one in course design
Area Intro
Introduction
Arithmetic
Geometry
Algebra
Logic
Calculus
Musings - More Ideas
More About LAMP
Evaluation
Maths Cultural Origins
First Nation Education
Modern Mathematics
Before LAMP
Problem Solving Skills Routine to Non
Instructional Concepts
Student Cooperation
Maths Extrinsic Origins
Science Education
For further musings or thoughts see site books.
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