Mathematics Concept & Skill Development Lecture Series:
Webvideo consolidation of site
lessons and lesson ideas in preparation. Price to be determined.
Bright Students: Top universities
want you. While many have
high fees: many will lower them, many will provide funds, many
have more scholarships than students. Postage is cheap. Apply
and ask how much help is available.
Caution: some programs are rewarding. Others lead
nowhere. After acceptance, it may be easy or not
to switch.
For students of reason in society, science and technology:
Pattern Based Reason describes
origins, benefits and limits of rule- and pattern-based thought and
actions. Not all is certain. We may strive for objectivity, but not
reach it. Postscripts offer
a story-telling view of learning: [
A ] [
B ] [
C ] [
D ] to suggest how we share theories and practices.
These online chapters may amuse while leading 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. Site Theme: Different entry
points may be easier or harder for knowledge mastery.
Deciml Place Value - funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6, US-CDN, UK-German and Metric SI style.
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
fractions.
Arithmetic with units - Skills of value in daily life and in the
further study of rates, proportionality constants and computations in
science & technology.
What is
a Variable? - this entertaining oral & geometric view
may be before and besides more formal definitions - is the view mathematically
correct?
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.
Solve
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
equations algebraically.
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.
Using
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.
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www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Postscript-Unifying Theme - A Fourth Skill For Algebra Next: [ Chapter 14. Forward and Backward Use of a Formula.] Previous: [Chapter 13. Second Reading Guide.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17][18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]
A Fourth Skill For Algebra
Volume 2, Three Skills for Algebra
Direct and Indirect Use of Formulas, or Forwards and Backward Use of
Equations
Every formula met in mathematics, accounting, science, technology etc may
be used directly and indirectly, that is forwards and backwards.
The simple message that the forward and backward use of formulas
(direct and indirect use) is part of high school mathematics and
beyond names a required skill and allows us to recognize,
identify and thus emphasize the most frequent pattern in high school
mathematics and beyond.
This message needs to be given explicitly and early in secondary
mathematics. Otherwise the underlying skill become part of the hidden, or
silent and unspoken, agenda in mathematics courses.
Teachers: Consider combining the www.purplemath.com a two page lesson on
solving
literal equtions with the message above and the examples and
exercises indicated below. The page banner above was Forward and
Backward use of equations but it now reflects the purplemath
lesson, Solving Literal Equations.
First Site Example
Direct and Indirect Use of the Rectangle Area Computation Formula
Chapter 10 in discussing Direct use of A =WL assumes W and L are given.
Indirect use assumes A and one of W and L is given, and leads to the
calculation or formulas W = A/L or L = A/W. The explanation of
those formulas is a step towards algebraic reasoning - the direct and
indirect or forward and backward use of formulas.
More Examples: Formulas for perimeters and areas
of squares, circles, triangles, rectangles etc can be used
forwards and backwards. Finding the value of a proportionality constant
k say in an equation y = k x represents an indirect or backwards
use of an equation, a pre-requisite to further forward and backward use
of the equation y = kx. The calculation of parameters a and b in
y = ax + b (or y = mx +b) represents another backward use of a formula
or equation. Quebec students in secondary III have met the
forward and backward use of the Pythogorean equation
c2=a2+b2 where c is the length of the
hypotenuse and the two numbers a and b are the lengths of the other two
sides (legs) of a right triangle.
To Do: : Post some details and exercises
here to further illustrate and emphasize the forward and backward use
of common formulas.
Going Further (More on Substitution)
Chapter 10 before the forward and backward use of a formula goes further
in showing how to describe a the calculation of a box V = H(WL) and show
how to employ substitution (a new concept for students) to go between
this formula and V = HA where A = WL. Details are
given in the chapter. The details may be easier to grasp if
numerical examples are added to this exposition.
Seeing how a box volume formula V = hA and V
= h (WL) can be transformed into each other illustrates and may
introduce the notion of equivalent expressions. The law applied here is
A = WL is a geometric law rather than an algebraic law (like the
distributive law). None, the idea that an expression represents a
number or quantity and that there may be more than one ways to compute
the number or quantity is key to the notion of equivalence.
Students thus see how substitution in formulas leads to new
formulas, how arithmetic patterns may be used to use formulas
directly and indirectly, and how algebraic solutions may be more
general or powerful than arithmetic solutions.
Algebraic Exercises:
- Find a formula for the area of square in terms of its perimeter
(easy)
- Find a formula for the area of circle in terms of its perimeter
(easy)
- Find a formula for the perimeter of square in terms of its
areas (harder)
- Find a formula for the perimeter of circle in terms of its
areas (harder)
See www.purplemath.com two page lesson
on solving
literal equtions for hints or to learn more.
The exercises could be easier after reading the first sections of Chapter
15 and Chapter 14 here in Three Skills for Algebra. The chapter 15
material may be easier.
The first sections in Chapter 15, >Solving Linear
Equations derives an algebraic formula for the solution of equations of
the form ax + b = c, and so emphasize the use of algebraic shorthand
reasoning to imply solutions for many problems of a given form at once.
All the foregoing emphasizes the power of algebra, or the shorthand way
of writing and reasoning with letters in place of numbers. That being
said, numerical experience is still required with formulas and their
graphs, otherwise the connection between numbers and algebra may too
weak.
A Deeper Site Example
for now or later or never.
Chapter 14 introduces the direct and indirect use of
the compound interest formula A = P(1+i)n.
Chapter 14 presents algebraic and arithmetic solutions that may be used
to check the calculator skills of students while developing the
algebraic way of writing and reasoning. In the compound
interest formula A = P(1+i)n three of the four amounts
A, P and i and n are assumed known, and the problem is calculate or
find a formula for the missing fourth. The use of this formula is
indirect when the left hand side quantity A is given or known, and the
task is to find the value of the principal P, the interest rate i or
the number of compounding periods n. Add to chapter 14
coverage, numerical confirmation that the algebraic solution
works. The algebraic solutions for the indirect use of formulas involve
substitution and assumes the pattern (AB)/B = A.
Coverage of Chapter 14 is recommended as part of the next topic:
exponents and radicals.
www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Postscript-Unifying Theme - A Fourth Skill For Algebra Next: [ Chapter 14. Forward and Backward Use of a Formula.] Previous: [Chapter 13. Second Reading Guide.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17][18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]
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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?
Site Reviews
1996 - Magellan, the McKinley
Internet Directory:
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,
Number 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. ...
Maps + Plans Use - Measurement use maps, plans and diagrams drawn
to scale.
Euclidean Geometry - See how chains of reason appears in and
besides geometric constructions.
Coordinates - Use them not only for locating points in the plane
or space.
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
cross-products.
Lines-Slopes [I] - Take I & take II respectively assumes no
knowledge and some knowledge of the tangent function in
trigonometry.
What is Similarity - another view of using maps, plans and
diagrams drawn to scale in the plane and space. May buildings in
space are similar by design.
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 - this 1995-96 lesson introduces calculus
skills and concepts. It may also may be given to introduce further function maxima
and minima both inside and at the ends of closed intervals.
Check Arith. Skills - too many calculus and precalculus
students do not have strong arithmetic and computation skills. The
exercises here check them while numerically hinting at
equivalent computation rules.
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