Mathematics Concept & Skill Development Lecture Series:
Webvideo consolidation of site
lessons and lesson ideas in preparation. Price to be determined.
Bright Students: Top universities
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Are you a careful reader, writer and thinker?
Five logic chapters lead to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics.
1 versus 2-way implication rules - A different starting point - Writing or introducting
the 1-way implication rule IF B THEN A as A IF B may emphasize
the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
Deductive Chains of Reason - See which implications can and cannot be used together
to arrive at more implications or conclusions,
Mathematical Induction - a light romantic view that becomes serious.
Responsibility Arguments - his, hers or no one's
Islands and Divisions of Knowledge - a model for many arts and
disciplines including mathematics course design: Different entry
points may make learning and teaching easier. Are you ready for them?
Deciml Place Value - funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6.
Decimals for Tutors - lean how to explain or justify operations.
Long division of polynomials is easier for student who master long
division with decimals.
Primes Factors - Efficient fraction skills and later studies of
polynomials depend on this.
Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for
addition, comparison, subtraction, multiplication and division of
Arithmetic with units - Skills of value in daily life and in the
further study of rates, proportionality constants and computations in
science & technology.
a Variable? - this entertaining oral & geometric view
may be before and besides more formal definitions - is the view mathematically
Formula Evaluation - Seeing and showing how to do and
record steps or intermediate results of multistep methods allows the
steps or results to be seen and checked as done or later; and will
improve both marks and skill. The format here
allows the domino effects of care and the domino effects of mistakes
to be seen. It also emphasizes a proper use of the equal sign.
Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to
present do and record steps in a way that demonstrate skill; learn
how to check answers, set the stage for solving word problems by
by learning how to solve systems of equations in essentially one
unknown, set the stage for solving triangular and general systems of
Function notation for Computation Rules - another way of looking
at formulas. Does a computation rule, and any rule equivalent to it, define a function?
Axioms [some] as equivalent Computation Rule view - another way for understanding
and explaining axioms.
Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards.
Talking about it should lead everyone
to expect a backward use alone or plural, after mastery of forward use. Proportionality
relations may be use backward first to find a proportionality constant before being
used forwards and backwards to solve a problem.
Early High School Geometry
Maps + Plans Use - Measurement use maps, plans and diagrams drawn
Use them not only for locating points but also for rotating and translating in the plane.
What is Similarity - another view of using maps, plans and
diagrams drawn to scale in the plane and space. Many human-made objects
are similar by design.
Complex Numbers Appetizer. What is or where is
the square root of -1. With rectangular and polar coordinates, see how to
add, multiply and reflect points or arrows in the plane. The visual or geometric approach here
known in various forms since the 1840s, demystifies the square root of -1 and the associated concept of
"imaginary" numbers. Here complex number multiplication illustrates rotation and dilation operations in the plane.
- Geometric Notions with Ruler & Compass Constructions :
1 Initial Concepts & Terms
2 Angle, Vertex & Side Correspondence in Triangles
3 Triangle Isometry/Congruence
4 Side Side Side Method
5 Side Angle Side Method
6 Angle Bisection
7 Angle Side Angle Method
8 Isoceles Triangles
9 Line Segment Bisection
10 From point to line, Drop Perpendicular
11 How Side Side Side Fails
12 How Side Angle Side Fails
13 How Angle Side Angle Fails
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www.whyslopes.com >> Geometry - maps plans trigonometry vectors >> 5 What is Similarity
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
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
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.
www.whyslopes.com >> Geometry - maps plans trigonometry vectors >> 5 What is Similarity
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Road Safety Messages
for All: When walking on a road, when is it safer to be on
the side allowing one to see oncoming traffic?
Play with this [unsigned]
Complex Number Java Applet
to visually do complex number arithmetic with polar and Cartesian coordinates and with the head-to-tail
addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.
Pattern Based Reason
Online Volume 1A,
Pattern Based Reason, describes
origins, benefits and limits of rule- and pattern-based reason and decisions
in society, science, technology, engineering and mathematics. Not all is certain. We may strive for objectivity, but not
reach it. Online postscripts offer
a story-telling view of learning: [
A ] [
B ] [
C ] [
D ] to suggest how we share theory and practice in many fields of knowledge.
1996 - Magellan, the McKinley
Mathphobics, this site may ease your fears of the subject, perhaps even
help you enjoy it. The tone of the little lessons and "appetizers" on
math and logic is unintimidating, sometimes funny and very clear. There
are a number of different angles offered, and you do not need to follow
any linear lesson plan. Just pick and peck. The site also offers some
reflections on teaching, so that teachers can not only use the site as
part of their lesson, but also learn from it.
2000 - Waterboro Public Library, home schooling section:
CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as
how (and why) to learn mathematics in school; pattern-based reason;
finding a number; solving linear equations; painless theorem proving;
algebra and beyond; and complex numbers, trigonometry, and vectors. Also
section on helping your child learn ... . Lots more!
2001 - Math Forum News Letter 14,
... new sections on Complex Numbers and the Distributive Law
for Complex Numbers offer a short way to reach and explain:
trigonometry, the Pythagorean theorem,trig formulas for dot- and
cross-products, the cosine law,a converse to the Pythagorean Theorem
2002 - NSDL Scout Report for Mathematics, Engineering, Technology
-- Volume 1, Number 8
Math resources for both students and teachers are given on this site,
spanning the general topics of arithmetic, logic, algebra, calculus,
complex numbers, and Euclidean geometry. Lessons and how-tos with clear
descriptions of many important concepts provide a good foundation for
high school and college level mathematics. There are sample problems that
can help students prepare for exams, or teachers can make their own
assignments based on the problems. Everything presented on the site is
not only educational, but interesting as well. There is certainly plenty
of material; however, it is somewhat poorly organized. This does not take
away from the quality of the information, though.
2005 - The
NSDL Scout Report for Mathematics Engineering and Technology -- Volume 4,
... section Solving Linear Equations ... offers lesson ideas for
teaching linear equations in high school or college. The approach uses
stick diagrams to solve linear equations because they "provide a concrete
or visual context for many of the rules or patterns for solving
equations, a context that may develop equation solving skills and
confidence." The idea is to build up student confidence in problem
solving before presenting any formal algebraic statement of the rule and
patterns for solving equations. ...
Euclidean Geometry - See how chains of reason appears in and
besides geometric constructions.
Complex Numbers - Learn how rectangular and polar coordinates may
be used for adding, multiplying and reflecting points in the plane,
in a manner known since the 1840s for representing and demystifying
"imaginary" numbers, and in a manner that provides a quicker,
mathematically correct, path for defining "circular" trigonometric
functions for all angles, not just acute ones, and easily obtaining
their properties. Students of vectors in the plane may appreciate the
complex number development of trig-formulas for dot- and
Lines-Slopes [I] - Take I & take II respectively assume no
knowledge and some knowledge of the tangent function in
Why study slopes - this fall 1983 calculus appetizer shone in many
classes at the start of calculus. It could also be given after the intro of slopes
to introduce function maxima and minima at the ends of closed intervals.
Why Factor Polynomials - Online Chapter 2 to 7 offer a light introduction function maxima
and minima while indicating why we calculate derivatives or slopes to linear and nonlinear
curves y =f(x)
Arithmetic Exercises with hints of algebra. - Answers are given. If there are many
differences between your answers and those online, hire a tutor, one
has done very well in a full year of calculus to correct your work. You may be worse than you think.
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