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. Early High School Arithmetic
Deciml Place Value  funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6. Early High School Algebra
What is
a Variable?  this entertaining oral & geometric view
may be before and besides more formal definitions  is the view mathematically
correct? Early High School GeometryMaps + 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 
www.whyslopes.com >> More Algebra >> 2 NaturalLogarithms Exponentials Powers Roots >> 10 Exponential Growth and Decay Models Next: [11 Growth and Decay in Biology.] Previous: [9 Formulas for Real Exponents with Logarithms.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10][11] Exponential Growth and Decay Models  Algebraic ViewThis lesson on exponential growth and decay models provides an unified algebraic and view of discrete and continuous growth and decay models  discrete compound, continous compound, halflife and doubling time models. All are or can be expressed in terms of the natural logarithm and its inverse, the exponential. But halflife and doublingtime models may be expressed or analysed with the aid logarithms to base 2 and the 2^{x} exponential function. Which logarithms and exponentials are employed in the forward and backward use of these models is a matter of taste, guided by ideas of what is simplest. The general form is \[A(t)= A_0 e^{kt} \] for some constant k where t usually denotes time  a continuous variable. All the growth and decay formulas can be expressed in this form, and vice versa. If k is positive, the formula represents exponential growth in time. If k is negative, the formula represents exponential decay in time. Solving for kThe natural logarithm of $A(t)$ \[ y = \ln(A(t)) = \ln(A_0) + k \cdot t \] is a linear function of time t. So its graph is a straight with slope or rate of change \begin{eqnarray*} k&=&\frac{y_2y_1}{t_2t_1} \\ &=& \frac{\ln(A(t_2)) \ln(A(t_1)) }{t_2t_1} \\ &=& \frac 1{t_2t_1} \ln \left(\frac{A(t_2)}{A(t_1)} \right) \end{eqnarray*} The special case $t_2=t$ and $t_1=0$ gives \[ k = \frac 1t \ln \left( \frac{A(t )}{A_0} \right) \]Model Recognition
Exponential model recognition provides one reason of the use of logarithmic graph paper. In 1976, in a digression from mathematics studies, I developed an exponential decay model for the action of an ore grinding machine, and then asked for data to help get the model parameters  a grinding constant k. The engineer I was helping then provided a log paper of data points falling on a straight line. The development was my independent rediscovery of the work of others. Examples subject to limitations
Compound Growth and Decay ModelsThe discrete compound growth or decay model has the form \[A = P(1+r)^N \] with $A_0=$ where N is a number of periods, and r is the change period. Here r > 1 may be given by a decimal, a fraction or a percentage. The value of N may be determined from the values of A, P and r by taking the natural logarithm of both sides Here \[A = A_0(1+r)^N \] can be written in the form $A = A_0e^{kN}$ with $k =\ln(1+r)$ because \[A = A_0(1+r)^N = A_0 \exp( n \ln(1+r))\] Here $1+r = \exp(k)$ allows r to be computed from the calculation of k above, or in a similar, more direct manner. In the case of compound growth and decay, the graph of \[ y = \ln(A(N)) = \ln(P) + N \cdot \ln(1+r) \] versus the discrete variable n consists of discrete points on a straight line. with slope $\ln(1+r)$. The same equation may be used to approximate N when A, P and r are given.Practical Question: What do you do when the approximation does not give a whole number, but is close to a whole number? In practice the formula $A = P(1+r)^N $ may represent
Exercise 1 : $A = P(1+r)^N $ in the form $A = P 2^{\frac NT}$ Hint: Observe $ y = 2^{log_c(y)}$ and let $y=\frac AP$ Exercise 2 : Given r > 0, how may years T is needed for the amount A to double $A = P(1+r)^N $ ? Give an exact formula for T. Continous Doubling Time Models\[A(t) = A_0 2^{\frac tT} \] The property $A(t+T) = 2 A(t)$ (why?) implies T is the doubling time. Use a whole number approximation for T.Here \[A(t) = A_0 2^{\frac tT} \] can be written in the form $A = A_0e^kt$ because \[A(t) = A_0 2^{\frac tT} = A_0 \exp( \frac tT \ln(2))\] The graph of $y=\log_2(A)$ versus time t has slope $m = \frac1T$ Continous HalfLife Models\[A(t) = A_0 2^{\frac tT} \] The property $A(t+T) = \frac12 A(t)$ (why?) implies T is a halflife  in fact the halflife.Here \[A(t) = A_0 2^{\frac tT} \] can be written in the form $A = A_0e^kt$ because \[A(t) = A_0 2^{\frac tT} = A_0 \exp( \frac tT \ln(2))\] The graph of $y=\log_2(A)$ versus time t has slope $m = \frac1T$ with a negative value. Carbon Dating Before Atomic Bomb Testing 1950s onwardIn the upper atmostsphere, beyond the reach of human activities, the sun radiation in the form of high energy particles, gamma rays [?], hits air molecules, including those of CarbonDioxide [$CO_2$] The radiation converts some of the Carbon12 in the atmosphere into the radioactive form Carbon14. The latter has a halflife of 5,730 years.The density of Carbon14 in the atmosphere stabilises when the decay of Carbon14 present balances the generation of Carbon 14. In the last several thousand years before the industrialization and associated burning of fossil fuels released Carbon Dioxide into the atmosphere, we assume that the density of Carbon 14 in the atmosphere, its proportion to normal Carbon12, was constant. A living plant on land absorbs atmospheric carbon while growing, with the absorption stopping at the time of death. At the time of death, the fraction $f(t) = A(t)$ of Carbon14 to Carbon12 in the plant tissue equals that of the atmosphere. But over time, the Carbon 14 and hence the ratio decays. Here \[f(t) \approx A_0 2^{\frac tT} \] where T = 5,730 years, t = time since death, and $A_0$ = the [average] ratio of Carbon14 to Carbon 12 during the life of the plant. We assume that ratio was constant before, with regrets, the advent of humankind's atomic age. In this model, the time since death of the plant is unknown, but T, $A_0$ and the current value $f(t)$ are known or measureable. Whence \[ \log_2(\frac{f(t)}{A_0}) = \frac t{T}=\frac t{5,730 \hbox{ years}} \] provides an linear equation in t to solve for the latter's value. The Carbon14 dating is reliable for upto 60000 years according to the reference. That being said, in my youth, I heard this methods was reliable for less than 15000 years. The technology may have advanced while I was not looking. REFERENCE: How Stuff Works  Carbon Dating www.whyslopes.com >> More Algebra >> 2 NaturalLogarithms Exponentials Powers Roots >> 10 Exponential Growth and Decay Models Next: [11 Growth and Decay in Biology.] Previous: [9 Formulas for Real Exponents with Logarithms.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10][11] 
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 ReasonOnline 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 Reviews1996  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. ...
Senior High School Geometry

Euclidean Geometry  See how chains of reason appears in and
besides geometric constructions. Calculus Starter Lessons
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. 