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YOU are better than YOU think. Show
yourself how:
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Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately
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What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
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First Year of Secondary School
See too the Logic and Algebra tips at this site
Skills that should be mastered before the age of 14
- Telling Time: Your charge should be or become familiar with
AM and PM - 12 hour clocks, with 24 hour clocks, with days of the
week, names of months, days in months in and out of leap years
(exceptions), history of calendars, different calendars used around
the world. What year is it anyway?
- Check that your child can count, add, multiply, subtract
and divide with fractions and decimals without a
calculator. Addition, multiplication, subtraction and division
are repeatable, reproducible and in the latter sense, verifiable
methods: two doing the same computation should arrive at the same
result. Check that your child can add, multiply, subtract and
divide with fractions. The may employ knowledge of prime numbers,
greatest common divisors and least common multiples.
The latter can be computed using the prime number decomposition of
whole numbers.
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More Skills for Ages 14 and UP
- Temperature: Celsius scale: 0 to 100 degrees -- from
Freezing point of water to its boiling point at sea level or one standard
atmosphere. Explain boiling point changes with height (altitude) due to
changing atmospheric pressure. Explanation of Mercury and Alcohol based
thermometers. Marking of class into 100 equal-length pieces. Fahrenheit
scale: O to 212 degrees from temperature of coldest basement to boiling
points of water. (Observe 32 degrees Fahrenheit is the freezing point of
water. The boiling point of water is 212 degrees Fahrenheit is 180 degrees
Fahrenheit higher. (Does this have any historical link to the division of
angles.) So a change of a change of 100 degrees Celsius = 180 degrees
Fahrenheit. From this, explain Conversion between Celsius and Fahrenheit.
E.g. 10 degrees Celsius = 18 degrees Fahrenheit above the freezing point of
water = 32+18 degrees Fahrenheit =50 degrees Fahrenheit.
- Units of Measure for Quantities, physical or monetary. Metric.
Imperial. Origins of Various Units (Consult Dictionary or Encyclopedia).
Conversion of physical units. Carrying units in calculation. A child
familiar with more than one system of units has a cultural advantage over a
child indoctrinated with only one system. From the cultural perspective,
describing the history of units of measurements implies there is more to
arithmetic and computation than following rules for this or that. Discussion
of units and their origins, and their different types, is part of the
history of mathematics, science, technology and society.
- Metric Regulations. Regulations requiring the use of metric (ok)
but also regulations forbidding use of imperial (bad). The olde addition of
English pounds, shilling and pence provided a non-metric & non-decimal
example of how to carry or convert units from one column into another.
Miles, Yards, Feet, inches provide another non-metric and non-decimal
example of how to carry. Familiarity with these non-decimal units may
give a better understanding of the carrying operations in decimal operations
and in metric operations. Operations with nonnumeric units can be employed
to explain the advantages of metric units (simplification in unit
carries/conversion) and to illustrate the carry/unit conversion process. See
the previous item.
- Simple Formulas. Show how to compute perimeters and areas of
squares, rectangles and circles. Explain these computations with words and
with formulas. Explain use of formulas for perimeters and/or areas for
triangles, squares, rectangles and circles. Justify the formulas for
squares, rectangles and triangles. Give formula for volume of a box, and
justify.
- Negative Numbers and their interpretations. In dealing with
temperature, instead of saying above and below zero degrees, introduce +ve
and -ve as an alternative. In dealing with assets and debts, instead of
saying amount due and amount owed, introduce +ve and -ve as an alternative.
Also mention the accountant habit of using parentheses around negative
numbers. In dealing with height above sea level, instead of saying above and
below, say +ve and negative. In dealing with a distance to the right or left
of a point on a line, introduce +ve and -ve as an alternative.
- Plot the amount of money in an compound interest savings account,
initially and at the end of each period for several periods. The formula for
this is A=P(1+i)n where n is the number of periods. If B is the
amount of money at the start of the present period (= the end of the
previous period), then B(1+i) is the amount of money at the end of the
present period = the start of the next. Join the plot points by straight
lines. For two different periods, compute the rise over run, that is (amount
at end of period - amount at start), divided by the amount at the start of
the period, and then subtract 1. (If you wish take P = 50 dollars, and i= 5%
= 5 times (1/100) = 0.05. Use a calculator or slide rule.)
Formulas for simple interest and for compound interest or investment
accounts answers the questions of how money in a bank account or an
investment may grow. A sequel in Volume 2 (chapter 14) is given by the
discussion which illustrate the direct and indirect applications of the
compound interest formulas and also reinforces and displays further the
algebraic writing and thinking.
- Compute slopes (rise over run ratios) of roads and railway lines as
numbers and percentages. Say what is steep. Indicate the usual range or
maximum for roads, railways, downhill ski trails (novice, intermediate,
expert, racing,), stairs, etc. Note that units cancel when rise and run are
measured with the same units. What happens otherwise?
- Explain distance = (speed) x (time). Give examples. Use
distance/time to compute speed. Introduce units for time and distance, and
from them obtain units for speed, ratios of units of distance over
unit of time.
- Graph distance versus time for a constant speed journey. Observe
how the speed equals the graph slope, that is, its rise over run. Note that
units of speed equal the units of distance over those of time. Give examples
where different units are used to measure distance and time. Use both metric
and imperial units of distance. On a distance versus time graph, draw a
straight line through the origin. Measure its slope, and relate this slope
to the speed of travel.
- Graph the amount of simple interest earned versus time. Compute the
slope and explain how it is related to the interest rate (it is a constant
multiple). How is the slope affected if the amount of interest or money is
described in (i) dollars and (ii) pennies. (The amount of simple interest is
a quantity, a number times a unit of money.) How is the slope affected if
time is measured in years, months etc.
- Show how to measure angles with a protractor. By measurement, show
how the sum of angles in triangles in the plane add up to 180 degrees. By
measurement, show how the sum of angles in a rectangle add up to 360
degrees.
- Triangle Inequality: Observe the sum of lengths of
two sides of a triangle is greater than a third. Illustrate this by joining
two ends of a string together and then forming triangles with it. This
suggests the shortest distance between two points is a straight line. (The
word linear in mathematics comes from line. On a flat plane, a taught cord
or line between two points defines a straight path. On a curved surface, a
taught string between two points may provide the shortest path between those
points)
- The Pyramid or Tetrahedron Inequality. If you can
construct a Pyramid or tetrahedron from four triangular faces (for instance
a pyramid), observe that one faces serves as the base, its area is less than
the total area of the other three.
- The area of a right triangle
A
o
|
| x
| .
| c
|b x
|
| x
|90 deg
+-----------o
C a B
with perpendicular sides of length a and b, respectively is (ab)/2
Geometric Proof: Complete the rectangle determined by the
perpendicular sides of the right triangle. The hypotenuse of the original
triangle equals the hypotenuse of a rotated copy of the original --- a
triangle obtained by rotating the original 180 degrees. The area ab of the
rectangle is the sum of the areas of the triangles = twice the area of the
original right triangle.
A
+-----------o
| |
| x |
| . |
| |
|b x |
| |
| x |
|90 deg |
+-----------o
C a B
- State the Pythagorean Theorem. Next illustrate it with 3-4-5
right triangle and with the 5-12-13 right triangle. The Chinese square
dissection proof of the Pythagorean theorem requires a knowledge of
how to compare, if not compute, the areas of triangles and the areas of
squares. The proof given here is painless (and it includes a diagram
better than that shown below). The treatment of trig and complex
numbers at this website in webpages and web videos points to another way
to prove the Pythagorean theorem.
A
o
Theorem: If triangle ABC is a right
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triangle with hypotenuse AB of length c
|
x
and other sides of length a and b, then
|
.
|
c
a**2 + b**2 = c**2
|b
x
|
|
x (where
q**2 is shorthand for q times q)
|90
deg
+-----------o
C a B
- Maps -- basic concepts. Take a map of your home town or region.
Explain how letters and numbers are used to located grid squares/regions.
Include explanation of scaling. Identify North-South, East and West.
Standard Convention North at top -- explain exceptions are possible. (The
floor plan of a house for instance need not have North at the top.).
- Maps -- use of coordinates. Points in the plane can located by
identifying the square to which they belong, but coordinates provide the
location more precisely. Coordinates may be introduced in two steps.
- First Quadrant Coordinates. Positive
Rectangular Coordinates: locate origin (0,0) at bottom-left
corner and then use ordered pairs (a,b) of nonnegative numbers to locate
points in the plane. (Descartes when he introduced coordinates only
employed them in the first quadrant. Negative numbers were thus not
needed.) Also employ polar coordinates (r, theta)
where r is distance to the origin and theta is between 0 and 90 degrees,
to show a second way of locating points. The origin (0,0) is at distance
0 from itself, and traveling 0 units in any direction from the origin,
represents the zero displacement. By measurement, show how to go
back and forth between polar coordinates and rectangular coordinates.
Apply the Pythagorean theorem, if it understood, to show r**2 = a**2 +
b**2
- Positive and Negative Rectangular Coordinates:
Locate origin (0,0) in the map interior. Use coordinates (a,b) to
indicate position relative to this origin. (Negative numbers need to be
understood first.) Also employ polar coordinates (r,
theta) where r is distance to the origin and theta is between 0 and 360
degrees. With polar coordinates, a comprehension of negative numbers is
not required. By measurement, show how to go back and forth
between polar coordinates and rectangular coordinates. Apply the
Pythagorean theorem, if it understood, to show r**2 = a**2 + b**2.
- Measuring the shortest distance between two points.
Take a chord or a piece of thread or string, and hold it taut
between the two points. Next measure the length of the string. A taut
string gives the shortest path between the two points.
- Map -- their use in navigation (describing journeys or
movements). The aim is to explain and describe Navigation in the Plane and
covertly introduce mathematical operations.
- Pirate Treasure stories may give directions to
buried treasure in terms of direction and steps. As a pirate treasure
alternative, in your kitchen, you may give your child a tape measure,
and then give directions (movements to do) in terms of distance and
angles, horizontal and vertical movements. E.g. From the door go 5 feet
south, then 3 feet north, then open the draw 24 inches off the ground to
find the buried treasure. The treasure could be some item of value to
the child, or another set of directions :)
- Rectangular Displacements and Movement. Examples:
from a location, illustrate movements 3 units rightward, 4 units upward;
or from another location, illustrate a movement, 12 units north and 5
units East. On the map or a piece of paper represent each sideward or
upward motion by an arrow. The head of each arrow or vector
should be at the destination (terminal point) of each motion. The tail
of each arrow should be at the initial point of each motion.
- As the Crow Flies: On a map pick an initial
point and a terminal point for a motion. Now draw an arrow, head at the
destination (terminal point) and tail at the starting point (initial
point). This arrow represents the straight or taut line motion
between the two points -- the path that a Crow could fly. (Land-based
animals may not be able to go in a straight line.) Observe that the
result of this single straight line motion can be given by a horizontal
motion followed by a vertical motion, or by a vertical motion plus a
horizontal motion. These motions are called the horizontal and vertical
components of the original motion. They too can be represented by
horizontal and vertical arrows.
- Basic Navigation Ideas. Find a real map of a
lake district, or draw on graph paper an imaginary lake with several
islands in it, and give a scale Water is crossed via a boat. Land is
crossed by carrying the boat over it -- a portage. Direction and
Distance Displacement/Movement -- From a location, specify a direction
(e.g. at angle 37 degrees above horizontal axis, 5 steps) or
(North-West, 13 steps), etc. Note how result of a rectangular
displacements is almost equivalent to a Direction-Distance movements,
and vice-versa, when they each movement has the same starting point and
finishing point. But direction-distance movement as the crow flies
is more efficient: the hypotenuse of a right triangle is shorter than
sum of the lengths of the other two sides. In consequence, may want to
replace a two-step rectangular displacement by more singe step direct
distance-direction movement. Again, the latter can represented by a
single arrow joining the initial point to them movement destination.
(Rectangular displacements can be represented by two arrows, each
parallel to the maps sides -- the map is assumed to be rectangular.)
- Arrow and Vector Operations: Coordinate Free
Perspective. Island hopping can be represented by a chain of arrows
joining the center of one island to the center of the next. Successive
straight-line motions in general can be represented by sequence of
direction distance movements, each represented by an arrow. (Pirate
buried or lost treasures provide an examples --- see above). Arrows or
displacement or movements can be added together graphically by placing
the tail of the first at the journey starting point, or at the head of
another. (For each arrow, tail = starting point, head = destination).
Alternatively, each arrow can be viewed as the sum of horizontal and
vertical displacement arrows, and the rectangular equivalent of the sum
of several displacement, can be obtained by their horizontal and
vertical displacements separately (several groupings of these
displacement is possible). The latter gives an arithmetic means to add
arrows together. See the next item. See Volume 3, Why Slopes and
More Math, as well.
- Arrow and Vector Operations: Rectangular and Polar
Coordinate Perspective. Each arrow is the sum of a rectangular
displacements, one in the up and down directions -- the vertical (or
North-South) component, and one in the left and right directions -- the
horizontal (or East-West)component. Each arrow may be represented,
recorded or written down as an ordered pairs (a,b). Here a and b are
signed numbers or distances. Adding arrows can done nongraphically by
adding the ordered pairs together. The equivalence or interchangeability
of the graphical and arithmetic methods for adding vectors and their
ordered pairs representation should be illustrated via examples. (Their
magnitudes |a| and |b| can be obtained by dropping the signs or
replacing a negative sign, if present, by a positive one. The
Pythagorean theorem r =sqrt (|a|***2 + |b|**2) gives a means to compute
the distance r in the polar coordinate or distance-angle (r, theta)
description of the equivalent distance-angle description. The angle
theta can be measured (or obtained from trigonometry).
- Navigation games are possible. Offer rewards
for following a sequence of displacements (distance-direction,
rectangular or a mixture of both), for computing or measuring the
one-step distance-direction between the origin and finish of all the
displacements, and also for giving the equivalent two-step rectangular
representation. Rewards might be points, cookies, pennies, privileges,
or just a smile.
- The Simple Mathematics of Money Computations. There are few concepts to
be mastered here, but without a mastery of the algebraic way of writing
and thinking, these concepts appear complicated.
- Simple and compound interest formulas -- use and
justification.
- Geometric sums, their recognition, shortcuts (the
geometric sum formula) for their addition, and justification of
shortcuts.
- Geometric sums in money computations.
- Comparison of different investments and different
income streams via present value or future value -- the value of an
investment and the amount in a saving account changes with time, so one
has to talk about the value or amount now or at some future time.
- Effects of Inflation,
- Equivalent Interest Rates.
- Economic Uncertainties.
- Sets, Venn Diagrams, Functions. Modern mathematics depends on
deductive reason and a mastery of the algebraic way of writing and
thinking. Set concepts (or what I mischievous call bag theory) are
employed in the set-theoretic codification of arithmetic based
mathematics. Set concepts and functions are also useful in counting the
number of ways that this or that may be done. The latter is useful in the
calculation of chance or probability. In school, Venn diagrams can be used
to introduce ideas of set intersection, union and complements. Venn
diagrams can also be used to introduce the idea of a universal set. (Talk
about a universal set may seem unnecessary even to bright students, but
its mention is an echo of the absurdity-avoiding principle that we should
only talk about subsets of a set known to exist.)
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www.whyslopes.com
Help your Child or Teen Learn:
Area Intro
Up
1. Speaking Skills
2. Reading & Writing
3. Preparing for Science
4. Learning Takes Time and Effort
5. Math Books: kids & teens
6. Math Books: teens & adults
7. Readings for Parents
8. Patience Please
9. Who is in Charge
10. Motivation
11. Will to Learn
13. Links For Parents
14. JumpMath WorkBooks
15. Discipline in Schools
Maths for Ages 5+
Ages 5 or 6
Ages 6 or 7
Ages 7 or 8
Ages 8 to 9
Ages 10 to 13
Age 14
Where is it going
D
What to do in School & Why
E.How to Study Mathematics
To read, write and spell, your children need to
learn and memorize the alphabet. Anything less would be absurd. That being
said, learning and using mathematics demands that your children meet key
skills and concepts, and not skip any. Where local schools do not provide the
latter, you need to provide remedies.
Care and Precision: If your child can learn
to follow multi-step methods carefully and precisely in arithmetic, he or she
may do so in other subjects, as well. Get your child or teen, if you
can, to sit down and study. Suggest he or she aim for skill and concept
development and perfection for their own sake, not that of their teachers.
The will to learn is the key to success in
school. Parents do have to be educated to support or guide their
children and teens. What matters more is support for the will to learn, for
children and teens to be told to try to learn and to ask teachers, their
schools or classmates for help and more help, as needed. Teachers and parents
need to push students, help them find the will to learn, teamwork helps.
The main reason and focus for high school
mathematics is or should be preparation for calculus. That requires skill and
knowledge perfection with fractions, algebra, geometry, trig and functions.
Many high school programs do not provide this. Make sure alone or with
help that your children and teens have a good command of
fractions.
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