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# Mathematics and Logic - Skill and Concept Development

with lessons and lesson ideas at many levels. If one site element is not to your liking, try another. Each one is different.

Online Volumes: 1 Elements of Reason || 2 Three Skills For Algebra || 3 Why Slopes Light Calculus Preview or Intro plus Hard Calculus Proofs, decimal-based.
More Lessons &Lesson Ideas: Arithmetic & No. Theory || Time & Date Matters || Algebra Starter Lessons || Geometry - maps, plans, diagrams, complex numbers, trig., & vectors || More Algebra || More Calculus || DC Electric Circuits || 1995-2011 Site Title: Appetizers and Lessons for Mathematics and Reason

Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined.

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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 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: Different entry points may make learning and teaching easier. Are you ready for them?

#### Early High School Arithmetic

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.

#### 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?
- 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.
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- 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 human-made 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 Natural-Logarithms 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 View

This 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, half-life and doubling time models. All are or can be expressed in terms of the natural logarithm and its inverse, the exponential. But half-life and doubling-time models may be expressed or analysed with the aid logarithms to base 2 and the 2x 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.

In the event time is given in discrete units, say years or months, a letter N favoured for the representation of discrete variable may be used. In this case $A(n)= A_0 e^{kN}$ where N denotes the whole number of elapsed units of time.

#### Solving for k

The 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_2-y_1}{t_2-t_1} \\ &=& \frac{\ln(A(t_2)) -\ln(A(t_1)) }{t_2-t_1} \\ &=& \frac 1{t_2-t_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

• If the graph of $y = \ln(A)$ versus t lies on a straight line y = kt + b then $A =\exp(kt+b ) = e^b e^{kt}= A_0e^{kt}$ where $A_0 = e^b$.

• If the graph of $y = \log_c(A)$ versus t lies on a straight line y = kt + b then $A =c^{kt+b } = c^b c^{kt}= A_0c^{kt}$ where $A_0 = c^b.$ The cases where c is 2, e or 10 are of the most interest.

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

1. Initial Population growth where food is plentiful. The model fails when the limit of food or resourse is met. The net result may be population stability - no growth - if mortality rates equal birth rates; or a population collapse due to a shortage of food or resources. For further thought, investigate bacteria growth in a petri dish, human population growth of your region or country; population growth of bacteria and animal life in the oceans.

2. The continuous growth an investment over time in which growth or interested in compounded continously.

3. Decay and half-life of radio active elements.

4. The amount of particles with diameter large than a given size d in a grinding machine.

5. Economic growth models of government planners whose optimism knows no bounds. In Quebec, the Mirabel white elephant Airport was justified by predictions of continued economic growth, exponential style, without bounds.

### Compound Growth and Decay Models

The 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

1. a compound interest bank account with interest rate r given as a number, a fraction or most often as a percentage. In years gone by, Swiss bank accounts use to charge a negative rate of interest for the storage of large sums of money in private. Whence $A = P(1+r)^N$ represents a compound decay or shrinkage model. I have with regrets never had such an account, albeit with bank charges, the effective interest rate on small bank accounts may be zero or negative.

2. Bird or whale or other plant or animal populations with growth rate r usually expressed as a percentage. In the case of Beluga whales, the percentage r appears to be a negative 1 tor 2 percent per year. Whence $A = P(1+r)^N$ represents a compound decay or shrinkage model.

All good things come to an end. Typically, the formula $A = P(1+r)^N$ represents growth or decay over a short period of time because the compound growth rate may vary.

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 Half-Life Models

$A(t) = A_0 2^{-\frac tT}$ The property $A(t+T) = \frac12 A(t)$ (why?) implies T is a half-life - in fact the half-life.

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 onward

In 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 Carbon-Dioxide [$CO_2$] The radiation converts some of the Carbon-12 in the atmosphere into the radio-active form Carbon-14. The latter has a half-life of 5,730 years.

The density of Carbon-14 in the atmosphere stabilises when the decay of Carbon-14 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 Carbon-12, 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 Carbon-14 to Carbon-12 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 Carbon-14 to Carbon 12 during the life of the plant. We assume that ratio was constant before, with regrets, the advent of human-kind'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 Carbon-14 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.

www.whyslopes.com >> More Algebra >> 2 Natural-Logarithms 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 head-to-tail 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 pattern-based 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 story-telling 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; 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.
... 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.
- 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 assume no knowledge and some knowledge of the tangent function in trigonometry.

#### 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.
- 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.