Teaching Tips - Secondary Maths from Fractions to Calculus
Lessons or lesson plans for secondary mathematics
follow.
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Year of Fractions -
Review and Extension of Primary School Material. The link here points
to a separate page. Next 5 links are internal to this page.
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Year of Algebra -
Formulas and proportionality relations forwards and backwards, see
and compare arithmetic & numerical solutions for questions.
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Year of 2D & 3D Geometry - Consolidate algebra and fraction
skills.
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Year of Proofs, Trig
& Functions - mastery of logic required here or before.
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Year of Analytic
Geometry - conic sections may appear here
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Year of
Calculus
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Year of Advance
Calculus (Real Analysis)]. First link is to another page.
Theories (skills and concepts) seen without examples give a vacuous
knowledge. Mathematics mastery in particular further requires numerical
and geometry drawing experience from examples and practice to put theory
in context. Plans for reform given without examples to show how are
vacuous in part and may be hazardous to education - a current
complaint. Reforms have to address why is mathematics and
select topics accordingly. Reform focused on delivery style, reforms
which ignores the role of long term objectives in mathematics education,
what should be taught and why, is bureaucratic. The content
issue is key. Calls to engage students with authentic, realistic
and genuine, relevant examples are vacuous when the callers or advocates
do not have a command of the content.
A six year mathematics
program: The first and second year of this program,
which may label secondary school mathematics, could or should
consolidate fraction skills and sense and introduce algebra, in
particular (i) the solution of linear
equations in one or essentially one unknown, and (ii) the direct
and indirect, forward and backward use of formulas, equations and
proportionality relations y = kx in arithmetic and then in algebraic
(literal) manners. While stick diagrams in item (i) are crutches to
develop equation balancing skills and sense, there cutting, duplication
of lengths in the approach develops fractions sense in the context of
line segments. So insist on stick diagrams. The third year could be the year
of examples - a reward for the first two years and preparation for the
next. Then fourth and
fifth years could emphasize logic and
proofs, the Euclidean
and Analytic Geometry of
straight lines (parallelograms & triangles included), similarity
and trigonometry, vectors and complex
numbers (geometric viewpoints), functions (various kinds) and
polynomials (some easily factored). A sixth year may introduce calculus. In the fifth year
or early, this lesson on complex
numbers (a not in the site area on complex
numbers) with field properties given or derived from 2D geometric
assumptions, yields an easier route to trig identities, and to further
material in science and engineering, and mathematics too.
The foregoing program would build a digital (decimal) and
geometric skills and sense which sanctions and extends the common
knowledge of arithmetic, coordinates and maps through explicit
assumptions given by interpolations and extrapolations of numerical
and geometric examples, with set notation and theory used to
facilitate and not dominate the development of mixed or impure
mathematics from numbers to calculus and real analysis. Pure
mathematics may build on the foregoing. The introduction of
mathematics needs to depend the assumption that points and
diagrams or sets of points in a 1D, 2D and 3D space are in one
to correspondence with coordinates (real numbers, ordered pairs,
triplets). Thus the introduction cannot be pure, and if it going to
be impure, it can serve and extend the common knowledge of decimals.
The program identifies a lean, fat-free, core sequence for high
school mathematics Fatty additions may include statistics, perspectives
drawing methods for art or construction, Euler Formulas relating
vertices, edges and faces; areas and volume formulas forwards and
backwards, and 2D or 3D geometric transformations, a coordinate
viewpoint after introducing functions and mappings in 1D. In this,
advances for instruction, how to understand and explain matters
in smaller more accessible steps, and emphasis of the verbal
description of numbers and quantities before and then beside symbols
could strengthen comprehension and give alternative routes for
instruction, repeatable and reproducible in the classroom with
fewer shortcomings. Students with learning difficulties should
focus on this lean sequence - the first three years might be
sufficient.
Volume 1, Elements of
Reason, introduces all site volumes.
[Online Books and More Site Areas] [Study Tips] [Directions for High School Mathematics -
Calculus Preparation] [Curriculum Shifts - Shorter,
Better, Stronger] [References]
Preparation for calculus provides the motivation for many skills and
topics in high school mathematics courses. Preparation for calculus
is good preparation for most, if not all, arts and subjects at work
and school that require some mathematics and logic.
Similar Directions: The earlier site preparation for calculus page (written earlier)
offers similar directions in three different ways - lean, wordy
and very wordy. The words comment on the development of ideas in the
classroom or historically.
Computer Games: If you play 3D computer games and want to write
your own, you will need a good command of logic, fractions, algebra and
geometry. The same advice applies if you want to enter a business,
trade or science.
Follow the steps below alone or with help. The review of fractions etc in
step 4 should come after steps 2 or 3. Other than that, which step to put
first appears to be a matter of taste. Site areas which do not appear in
these steps contain further material - optional reading. On first
reading, focus on learning how, and leave explanations why for later.
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Put logic First (if possible). Read the first logic chapters in Volume
2. Logic mastery will, we hope, ease fears and
difficulties, or if you have none, enrich skills and
knowledge.
Master logic
carefully to develop precision reading and writings.
Skills and knowledge are easier to obtain when you are able to read
precisely what is written, and do not assume too much. Marks in
all subjects are base on your written work. Precision reading will
help you recognize errors in your written work through the question:
does it, your written work, say precisely what you meant?
Secondary I and II
Material
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Meet the role of
fractions in algebra: Explore the site area Solving Linear Equation with stick
diagrams to further develop your algebra skills - those
needed for solving problems in one or essentially one unknown, and
see how fractions of line segments, the sticks, are combined (added,
subtracted, multiplied and divided) exactly in the solution of linear
equations.
Next read the Chapter 15, solving linear equations,
in Three Skills for Algebra, alone or with help. The discussion of
general systems is optional for junior high school students.
Test your algebra
skills and linear equation
problem solving skills.
Remark: Steps 1 to 4 may be covered in junior or
senior high school, the sooner the better. The following steps are
for senior high school students and older students in college or
adult education.
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Review or Develop Algebra and Fraction Sense and Skills.
Read (i) the algebra chapters 8 to 14
Volume 2, Three Skills for Algebra.
The shorthand role of letters and symbols is meaningless for many
people in school and out. But the shorthand role is
easier to grasp when we first learn to talk about numbers and
quantities, and how they
may vary, before the use of letters and symbols. Doing that
would make algebraic ways of writing and reasoning clearer in
calculus and all of high school mathematics.
Chapter 14, Compound
Interest, in Three Skills for Algebra, develops algebraic skills
with the aid of a calculator. Calculators are useful but success and
precision in mathematics requires efficiency with fractions without
one. --- Beside talking about numbers and quantities, there is
a fourth skill for algebra in Three Skills for
Algebra, namely a development of the ability to talk about or
describe the numerical and algebraic use of formulas and equations
with short descriptive phrases: (i) forward and backward use (or
direct and indirect use) and (ii) algebraic and arithmetic
(numerical) solutions. These phrases appear in Chapter 14. can be used
through out high school mathematics to identify recurring themes -
key objectives - and to provide another fresh perspective on the
algebraic way of writing and reasoning.
Alternate Between Steps 3 and 4 if you wish. Each one has a
different taste. The addition of animated graphic make Solving Linear Equation with stick
diagrams easier than before.
If you spend grades 1 to 11 or 12 in mathematics classes without
mastering fractions sense and skills properly and efficiently, you
have been cheated - several hundred or thousand hours of your time
has been wasted.
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Optional but Recommended: (i) Visit the fraction pages in the
site area, Fractions,
Ratios, Rates, Proportions & Units, to check your fraction
sense (step 4 could have helped in here) and to see the justification
of methods for adding, subtracting, dividing, multiplying and
comparing fractions. (ii) Develop an algebraic view of problem
solving with units and with rates and proportions, binary or
multiple, direct, joint or inverse. (iii) learn how to carry units
through solutions in a way that relies more on mechanical skill in
algebra than on thought. Here is an algebraic perspective and
clarification of skills and concepts in junior high school
mathematics, which may be read after steps 1 to 4 above.
The site area Fractions,
Ratios, Rates, Proportions & Units view of junior high school
concepts may help teachers & tutors develop skills and concepts.
Senior high school students may explore this area to review and
reform their understanding. Area material needs to be rewritten to
make it readable for junior high school students. Writing is an
iterative process in which the first draft is not always best.
Fractions are needed for algebra and beyond. In modern times, that is
today, we see and will see more and more cognitive
experts and curriculum advisors suggest the replacement of
fractions and algebra skills and sense development with
calculator push-button exercises in which the
intellectual component of mathematics instruction is eliminated
to provide a child- and technology- centered learning
environment. Yet arithmetic mastery was and remains a sign of
intelligence in work and study.
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Check & Consolidate your Arithmetic Skills. Do asap, the
first set of
arithmetic problems, chapter 7 of Volume 2, Three Skills for Algebra, See too
Simplification of square
roots. Logic
mastery asap is recommended for greatest benefit from site pages.
In doing exact arithmetic, if your result is not the same as that
of another, one of you has made an error. Learning how to
follow methods so that you obtain repeatable, reproducible and thus
verifiable results is a must, not always emphasized, for work, school
and home.
See too these Real Player arithmetic webvideos - a few a
day, not all at once.
Aim for a logic-based mastery of mathematics after
arithmetic. That being said, arithmetic can be learnt by rote,
know-how without the know-why, provided you put aside your
calculator and learn the times and addition tables and learn to
do arithmetic with fractions and decmals (add,
multiply, divide and subtract) in an objective, efficient and
automatic manner - arithmetic results should be repeatable
and reproducible, and you should know that an error in one step
makes all the rest wrong. Once you have a logic-based
mastery of mathematics after arithmetic, you can if you want
retreat to develop a deeper, logic-based understanding of
arithmetic, a retreat that could become easier, and a retreat that
can be woven in to the explanation of further mathematics for skill
perfection and enrichment.
Secondary IV and V
Material
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Master Geometry without and with coordinates: Site areas on
Euclidean
Geometry and Analytic
Geometry offer senior high school students and teachers
lean logic-based development and connections of plane
geometry, plane trigonometry and functions of one variable. The
site coverage of Analytic
Geometry does not include all that calculus requires, but is a
start, and the missing material can be found elsewhere.)
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Remark A: The treatment of Euclidean Geometry is not
full, but it is enough to provide a logic-based consolidation of
the skills and concepts seen in junior and high school
mathematics, those needed to develop analytic geometry and
calculus. The treatment of Analytic
Geometry assumes results of the site treatment Euclidean
Geometry with the assumption that real numbers alone or in
ordered pairs may provide coordinates for lines and planes in
space. The result is a logical, coordinate based, development of
the key skills and concepts in analytic geometry, plane
trigonometry and functions. The reliance seen here on geometric
diagrams can be replaced and will be in studies of modern pure
mathematics. Or, we could use the alternate route in Remark
B.
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Remark B: Step 6 follows the traditional path of defining
trigonometric functions for acute angles with the aid of
similarity postulates before defining them for all angles.
This complex numbers
introduction leads to trigonometry in general for all angles,
with right-angle triangle, similarity based, trigonometry coming
last. For the brave, that gives faster route for
developing the senior high school mathematics which calculus and
electrical studies requires. This route is leaner in that
its reduces the need for Euclidean
Geometry to a discussion of similarity
principles.
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Remark C: In the modern mathematics curricula of the late
1950s and 1960s, sputnik inspired, there is a fuller treatment of
coordinate-free Euclidean geometry along side a general emphasis
on logic. Geometric proofs were challenging - not student
friendly. So Geometry was eliminated. But Euclidean Geometry was
the traditional place for the emphasis of logic and Euclidean
model for reason. Site logic and Pattern Based Reason
chapters present the Euclidean model in a math-free way and do so
to develop better study skills - or the precision reading and
writing better work and study skills demand.
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Test your arithmetic and Algebraic Skills: Try the remaining problem sets in
Chapter 7 of Volume 2. Get someone to identify all errors in your
answers in notation and comprehension, so you can learn from your
mistakes.
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Optional: Explore the Number
Theory Site Area. Here is a mix of easy and challenging lessons,
some in sequence. If one lesson or sequence is not to your liking,
try another.
Secondary VI & VII
Material
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Meet or Revisit Calculus: Begins with the why slopes geometric preview
before the more algebraic
why slopes preview chapters in Volume 3. Then explore more of the
site Calculus
Introduction.
Remark: The introduction points to simpler ways to cover the
first steps in calculus. Those simpler ways are for all. The
algebraic way of writing and reasoning is usually required suddenly
in calculus. The previews here and the latter decimal view of limits,
continuity and convergence provides a more accessible and less
algebraic demanding or shocking approach to calculus.Then the
introduction includes enriched material - the proofs that are often
omitted in first courses. Innovations here make the proofs easier to
understand, but not simple. The enriched material is for people who
do not like to accept mathematical methods without proof. The site
area Real-Analysis-Decimal-View (advance
calculus) and the calculus introduction at this site emphasize an
error-control decimal view of limits, continuity, convergence.
Remark The Modern Mathematics movement of the 1950s and 60s
made calculus algebraically hard or inaccessible need-be by following
a decimal-free view prevalent in pure mathematics. Here is a
correction sufficient for students outside of pure mathematics that
may provide a stepping stone and context for the decimal-free,
epsilon-delta view of pure mathematics.
Remark: Steps 5 onward can be followed or explored in any order
you like.
Learners at all levels need someone to review their written work for
mistakes in notation and comprehension in order to learn from their
mistakes. Every time someone (on your side) identifies a mistake, say
thank you because now you know not to make that mistake again. Do
not worry, your helper will be employed in identifying further
mistakes. It is a win-win situation.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
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:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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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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