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.
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 2way implication rules  A different starting point  Writing or introducting
the 1way implication rule IF B THEN A as A IF B may emphasize
the difference between it or the latter, and the 2way 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
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.
Early High School Geometry
Maps + Plans Use  Measurement use maps, plans and diagrams drawn
to scale. 

Coordinates 
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 humanmade objects
are similar by design.

7
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 >> More Algebra >> 3 Quadratics Geometrically >> 9 quadratics physical and further context Next: [10 quadratic exercises.] Previous: [8 quadratics  backward use of various formulas.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10][11] [12]
9. Quadratics: Applications in Geometry Physics Etc
Problems may come from several sources.
 Solving Systems of Equations  one quadratic, one linear.
 Examples from Physics.
 Constant Velocity Motion
 Quadratic in Time implies Constant Acceleration
 Constant speed and constant acceleration motion (enriched topic)
 Examples from Economics (do, but view with suspicion)
Mastery of quadratics is needed
for calculus and beyond in science, engineering, mathematics and other
quantitative disciplines based on calculus (or special functions such as
logarithms and exponentials.)
Remark: Applications in economics of quadratics
exist, and you may meet them, but those I have met seem more
unreal, contrived, or artificial than the physic applications.
Remark: The quadratic formula may be used to solve
ax^{2}+bx + c = 0 directly. Or, factoring by inspection and
factoring by completing the square and using the difference of two
squares can be use to say ax^{2}+bx + c = a(xr)(xs) for some
real numbers r and s
Problem Type: Intersection of a line and a parabola.
The intersection is found by solving a systems of Equations  one
quadratic, one linear.
The intersection of a line y = Ax + B and parabola y =
ax^{2}+bx + c may be found by solving Ax +B = ax^{2}+bx +
c. The latter yields ax^{2}+(bB)x + (cB) = 0 which can be
solved for x by the most convenient you see, say by inspection, by
completing the square or by the quadratic formula. The latter quadratic
in x may have two, one or no solutions. For each x solving the quadratic,
there is a y = Ax+B to be computed in order to obtain the coordinates
(x,y) of an intersection point.
Example: Find the intersection of the
straight line y = 3x3 and
the quadratic y = 3x^{2}6x+3
At the intersection points, if any, the right hand sides of the equations
must give the same value for y. Thus comparison of the two sides gives
the equation
3x3 = 3x^{2}6x+3
linear on one side and quadratic on the left. Add 3x + 3 to both
sides
3x 3 = 3x^{2}  6x + 3
3x + 3 = 3x +
3 +
0 = 3x^{2}  9x
+ 6
That implies that the first coordinate of any intersection point must
satisfy the quadratic equation:
0 = 3x^{2}  9x + 6
The latter can be solved with the aid of the quadratic formula or by
factorization. The latter route may give the least amount of work: Let us
try it.
0 = 3x^{2}  9x + 6
= 3(x^{2}  3x + 2)  take out common
factor 3
= 3(x2)(x1) since 2 has two possible
factorizations
2 = (2)(1) and 2 =(2)(1)
Here we are fortunate that 2  1 = 3.
That gives the factorization.
Now 0 = 3(x2)(x1) suggests x = 1 and x = 2 provide the
xcoordinates of the intersection points. Let's compute the
ycoordinates for each xvalue and verify that the two expressions y =
3x3 and y = 3x^{2}6x+3 give the same values for y.
x

1

2

3x 3

3 3 = 0

3(2)3 = 63
= 3

3x^{2}6x+3

36+3 = 0

3(2)^{2}6(2)+3 =
1212+3 =
3

Therefore

(x,y) = (1, 0) gives one
point of intersection, and

(x,y) = (2, 3) also gives an intersection point.

Remark: In my scratch work, I made an error in
the evaluation of at x=2 and had to reconsider my derivation of
the solution and after a short delay, saw my error. Knowledge of
how does not guarantee calculations are errorfree, but practice may
help you and I correct more quickly from errors or
inconsistencies in our solutions.
Remark: The above problem did not ask us to graph
the straight line and quadratic in the region about their intersection
points. However, a graph follows.
Problem Type: Projectile Motion
Let t denote time. Let y denote height. Then quadratics y =
at^{2}+bt+c may be used to describe or approximate the
height of thrown or freefalling projectiles such as bullets, rocks
and balls when air resistance is negligible or neglected. For such
projectiles, equations of the form x = pt+q may describe the projective
movement in a horizontal direction.
If we express t in terms of x, we see that time t is
given by a linear expression in x. That expression can be use to
eliminate t in y = at^{2}+bt+c to obtain a quadratic relation y
= Ax^{2}+Bx+C between the y and x coordinates of the
projectile. So we conclude, the projectile follows a quadratic path in
the xy plane. In the foregoing, the upper case letters A, B and C The
letters A, B and C depend on the coefficients p, q, a, b and c. The do
not have the same meaning or same value as the lower case letters a, b
and c unless x = t in a unitfree description of the physical
situation.
In practice, you may meet

Vertical projectile motion  the position of a falling object
subject to the constant pull of gravity at or near the earth's surface
can be described using quadratics expressions y = at^{2}+bt + c
with time t in place of horizontal coordinate x as the independent
variable. The direct use of this equation is to calculate
coordinate y given the value of time t. One indirect use of this
equation gives the value of y and asks for the value or possible values
of t. You will need to solve a quadratic equation for t and if
there are two numerical solutions, decide which one is required or
selected by the information at hand. Further indirect uses of the
formula may give you values of y and t, clearly or not, and ask you
find the values of the coefficients a, b and c, before using y =
at^{2}+bt + c directly, or indirectly again.

Projectile Motion in the Plane: Here y =
at^{2}+bt + c and x = Bt + C describes a falling body in the
vertical xy plane near the earths surface. You may be asked to analyze
these equations forwards and backwards.

Free Sliding Object on a slanted plane. y = at^{2}+bt +
c and x = At^{2}+ Bt + C but simplifications may follow, will
follow, from using a slanted coordinate system with x or y coordinate
in the plane. The equations for situation B or C may reappear.
This might be an enriched problem in a senior high school physics
course.
Problems of type B: A cannon ball leaves the mouth of a cannon
with an initial horizontal velocity of 800 meters per second in a
direction (say the xdirection) and an initial vertical velocity of
600 meters per second (say the ydirection). (i) How will the
ball be when the ball is 4000 meters horizontally from the initial
position. (ii) When and where will the ball hit the ground?
(iii) What is the maximum height of the ball? Assume the
position of the ball as it leaves the cannon mouth is almost ground
level, say y = 2 meters.
Solution: We assume air resistance is negligible for this
cannon ball projectile, that it flies in a vertical plane with
upward direction is positive. Then the x and y coordinates of the
projectile are given by two formulas from physics, namely
x = x(t) = x_{0} + v_{x}t and
y = y(t) = y_{0} + v_{y}t  ½gt^{2}
where t = elapse time since the projectile was in its initial
position, where g = 9.8 meters per second square = acceleration of
a freefalling object due to gravity at the earth's surface, where
(x_{0} , y_{0}) give the initial position of the
projectile; and where (v_{x} , v_{y}) = the
initial velocity of the projectile. The latter means v_{x}
= initial horizontal velocity and v_{y} =
initial vertical velocity
Substitution (Use) of data in equations.
We will take the initial position x_{0} = 0 and
use v_{x} = initial horizontal velocity = 800 meters
per second. So
x = x(t) = x_{0} + v_{x}t = 0 +

800 m
sec

t =

800 m
sec

t

Now (½)(9.8) = 4.7 gives
y = y(t) = y_{0} + v_{y}t  ½gt^{2} =
2 m +

600 m
sec

t

 4.7

m
sec^{2}

t^{2}

(i) Now we want to find the value of y when x = 4000 meter. The
latter condition implies
800 m
sec

t

= 4000m


t =


sec
800 m

4000 m

Therefore t = 5 seconds when x = 4000 meters. That
implies the value of y is given by the formula
y = y(t) = 2 m +

600 m
sec

t

 4.7

m
sec^{2}

t^{2}

evaluated at t = 5 seconds. That yields
y = [2 + (600)(5)  4.7 (5)^{2}] m =
[3002  4.7 (25)] m
= 2884.5 meters when x = 4000 meters.
That completes the solution to part (i).
In part (ii), the question is when will y(t) = 0 after t = 0. That
requires the solution of the equation
0 = y = 2 m +

600 m
sec

t

 4.7

m
sec^{2}

t^{2}

or equivalently
0 = 2 +

600[

t
sec

]  4.7[

t
sec

] ^{2}

The positive solution T_{+} =
t/_{sec} of this equation follows from the quadratic
with the aid of a calculator. That completes part (ii).
Calculate the negative solution T_{} as well for
use in part (iii).
Exercise: Compute T_{+} ,
T_{} and then the tcoordinate h below of the
maximum height with the aid of a calculator.
For part (iii), the high point of the trajectory
y = 2 m +

600 m
sec

t

 4.7

m
sec^{2}

t^{2}

occurs on the axis of symmetry t = h = b/2a. The latter can be
computed directly. The latter can be compute directly, or you
can use symmetry to observe h = (T_{+} +
T_{}) is halfway between the two zeroes of y =
y(t). From the value t = h, the maximum value of y =
y(t) can be obtained, again with the aid of a calculator.
Remark 1: In mathematics courses, I would advise students
to, try to delay or postpone the use of calculators and hence the
appearance of approximate calculations in a solution as much as
possible. The objective is to obtain an exact solution  one in
which there are no approximations.

Remark 2: The description of projectile motion
is provides a warlike qualitative idea of the flight of
projectiles. Suffice it to say, I do not like the connection
of mathematics to the arts of war, past and present. Mathematics skills
and concepts have been driven by various motivations in consumer life,
business, construction, science (planetary movements included),
technology and war. Projectile motion provides the application of
quadratics most easily visualized and so most useful for the development
of mathematical skills  ouch.
www.whyslopes.com >> More Algebra >> 3 Quadratics Geometrically >> 9 quadratics physical and further context Next: [10 quadratic exercises.] Previous: [8 quadratics  backward use of various formulas.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10][11] [12]
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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 headtotail
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 patternbased 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 storytelling view of learning: [
A ] [
B ] [
C ] [
D ] to suggest how we share theory and practice in many fields of knowledge.
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; patternbased 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
crossproducts, 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 howtos 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. ...

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 trigformulas for dot and
crossproducts.
LinesSlopes [I]  Take I & take II respectively assume no
knowledge and some knowledge of the tangent function in
trigonometry.
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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