5 What is Similarity
1 Early Concept of Like or Similar Shapes
2 Similarity By Design
3 Similarity by Design with coordinates
4 Similarity - Definition with Coordinate
5 Similarity of Circles Squares and Rectangles
6 Geometric Diagrams in Class
7 Translations Rotations Reflections Dilatations
8 Similarity of Triangles and Polygons
9 Similarity of Triangles Usual Criteria
10 Similarity of Triangles - Equivalent of Two Criteria
11 Triangle Similarity Missing Side Problem
12 Triangles Similarity More Problems
13 Navigation Location from Angles to 2 Landmarks
This folder What is Similarity provides a general and unified
treatment of the likeness or similarity of squares, circles, triangles
and arbitary regions in the plane. The lessons here can be covered after
section 1 on maps, plans and measurement and after the introduction of
Cartesian coordinates. The objective here is to explain and reconcile
different characterizations of similarity.
The key question is how to recognized similarity. In modern life, objects
are similar by design if they stem from the same plans but are built to
different scales. That reflects a coordinate view point of similarity
objects. Two objects are similar if we can attach coordinate system to
each so that the set of coordinates for the points for one object
essentially provides the plan for the other as is or after the
application of a scale factor - a dilatation. This set of coordinate
development easily explains how and why circles, squares and rectangles -
those with a common aspect ratio - may be similar. The coordinate
perspective of similarity in the case of similar triangles and more
generally in the case of similar polygons implies corresponding angles
are equal and corresponding sides are proportional. A partial proof of
the converse is included in the section what is similarity - a full proof
is left to later as site development to do.
Preamble and Context
Similarity is an artifical concept. It main comes from the construction
of rectangular, circular and triangular or polygonal shapes for walls,
doors windows and layouts or floor designs of buildings large and small;
and from the construction of furniture tops, sides and legs. Where
construction is is based on plans and drawings, actual lengths and
surface areas, and volumes are by design proportional via a scale factor,
its square and its cube to corresponding parts on the plans and drawings.
In this corresponding angles are equal. Further circles and squares in
the drawings correspond to circles and square areas or objects in the
construction The proportionality still holds when the same plans or
drawings are implemented at different scales, albeit too great variation
in scale may lead to structural instability. In modern times with the
advent of digital plans and drawings, planned objects are described in
terms of coordinates - sets of coordinates or the data needed to
determine those sets. In this, the digitilized plans and drawings may be
displayed at different scales. In nature, similarity appears in the form
and inner components of larger living beings, but as beings grow, the
measures and dimensions of the "similar shapes" are not proportionality.
No doubt there will be a few exceptions.
Primary School Geometry - Like Shapes
Primary level instructions may ask students to identify like shapes,
years before the secondary level mention and introduction of similarity.
Indeed, young children may recognize like shapes from their senses of
vision and touch, and from the function of objects in the plane or space.
The ability to recognize like shapes allows children and people in
general to recognize others and navigate their way their local
environments, all without or all before any formal acquaintance with the
concepts of similarity.
This late primary or secondary level geometry may introduce maps and
diagrams drawn to scale. At home or in a library, students can be shown
maps and plans of their community and its surroundings. Maps may be
employed in the description of routes followed by family members, cars,
buses, trains, and planes, etc. School going children and teenagers will
likely see maps of school building and terrains. On these planar maps -
when drawn to the same scale horizontally and vertically, actual
rectangular, circular and triangular regions appear as rectangles,
circles and triangular regions respectively. The study of maps may be
employed to point and recognize similar shapes before any formal
discussion of similarity. That being said, this level may show and imply
that angles on these maps and drawings equal the corresponding angles in
the drawn objects. When a unit length on the map coresspond to a unit
length in actuality, this level may further slowly show that the number
of map unit lengths needed to cover a drawn length and map unit squares
needed to cover a drawn rectangular region equals the number of real unit
lengths and real unit squares needed to cover the actual lengths and
region. The use of the scale factor or it reciprocal as is or squared
then gives a proportionality constant between drawn and actual lengths
and areas. Most likely, the length case and the area cases should be
treated differently.
Late primary and early secondary quantitative skill development should
emphasize measuring skills with rulers, tape measures and protractors for
measuring lengths and angles in the environment and on maps and plans
drawn to scale. Tutors and teachers may observe that the map measurements
are often easier to make - require less movement. This level may show
students how to recognize different kinds of triangles and quadrilaterals
and connect the latter to parallel lines or line segments. All the
foregoing may be done on paper with maps, plans or drawings. Coordinates
signed and unsigned should be introduced along line segments and for maps
and plans. The game of Battleship may be adapted to test mastery of
coordinates. Students may be further given a sequence of coordinates for
points in the plane to join to test and reward coordinate mastery. The
joined points or dots may form a picture of objects or animals in the
local environment.
Familarity with maps and diagrams drawn to scale should make the
assumptions or axioms of coordinate free Euclidean more self-evident. An
operational mastery of maps and plans drawn to scale sets the stage for
site simplified treatment of Euclidean Geometry, one reserved for the
keener students in senior highschool, one sufficient to introduce
students to the use of deductive reason in mathematics.
Finally, secondary level geometry instruction may tell and show students
that all circes are similar and all squares are similar; it may provide
students criteria for the similarity of triangles, criteria based on
equality of corresponding angles, or proportionality of corresponding
sides; and it may criteria for the similarity of rectangles based on
equal aspect ratios or equivalently based on the on the proportionality
of corresponding sides. The foregoing rules and criteria are consistent
with each other. To the foregoing, I would have secondary school geometry
add the coordinate perspective: Two objects in the plane or space are
similar if coordinates systems can be choosen for both such that the set
of coordinates for one are proportional [scaled versions] of the other.
The latter criteria echo the modern day use of plans and drawings as
indicated above to digitize actual or concieved objects in two and three
dimensions, that is, in planes and in space. The coordinate perspective
is timely if or when secondary level geometry talks about the
proportionality of corresponding lengths, area and/or volumes in the
discussion of similar objects in a plane or in space.
Proportionality of Map and Actual Measurements Explained
A map unit length corresponds to a given length, a unit length, in the
real world. Here the number N of map unit lengths needed to cover a path
on the map is the same as the number N of given lengths needed to cover
the corresponding path in the real world. The ratio of the number N of
given lengths to the number N of map unit lengths equals the the ratio of
the given length to the map unit length, that is the map scale factor.
Likewise, the number M of map square units needed to cover a region in
the the map equals the number M of of squares with sides provided by the
given lengths needed to cover the corresponding region in the real world.
The ratio of the number M of given lengths squared to the number M of map
unit squares that equals the the ratio of the given length squared to the
map unit square, that is the map scale factor, squared.
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Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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