www.whyslopes.com || Fit Browser Window

# 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.

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 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.
-
- 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 >> Volume 2 Three Skills For Algebra >> Appendix E. How To Study Mathematics and Why Next: [Postscript For Better Performance.] Previous: [Appendix D. What to do in School and Why.]   [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40][41] [42]

# Appendix E. How to Study Math and Why

Volume 2, Three Skills for Algebra

### A. Before You Stop

In school before you stop studying mathematics, please do the following.

1. Look at the chapters Implication Rules and Chains of Reason. These two chapters may help you to use and understand precisely rules, instructions, patterns, definitions and recipes in every subject and every area of skill or specialization including mathematics.

2. Read the first chapters on algebra. The description in them of the three key skills for algebra and the algebraic examples will, I hope, help you step from arithmetic to algebraic way of writing and thinking plus a little beyond. These chapters try to explain and describe with everyday words, how (algebraic) shorthand notation is used to describe and do mathematics after arithmetic. There is more to mathematics than just doing arithmetic well.

### B. Why Study

Mathematics courses are preparation for business calculations, for handling your money sensibly and for courses in sciences, engineering and technology. You should view mathematics as an opportunity to strengthen your thinking skills.

In mathematics courses you should not only meet calculations to do but also the chains and threads of reason and persuasion which justify them and links them together. Understanding and following the rules and patterns of mathematics, practices and nurtures an ability to think and reason well. Mathematics provides a neutral territory for the practice of rule and pattern-based reason and logic. The opinions and views you meet in daily life say and care little about what mathematical conclusions should be.

If you find yourself in a course which gives formulas and numbers to use in them but does not expect you to use algebra, you are wasting your time. Your time would be better spent studying algebra, and then taking a more advanced course that respects your intelligence. Similarly in college, if you find a course which gives you formulas and numbers to use in them and also talks at length about rates of change without expecting you to understand calculus,6 then a calculus course would be of better use of your time.

6 Calculus in the first instance provides formulas for the slopes of (nonlinear) curves and for the rates of changes of numbers or quantities.

### C. When to Study

Look at the description of courses you will take in and outside mathematics. From their description, see what mathematics course(s) you are expected to take with them (co-requisites) or before them (prerequisites). Methods taught in a co-requisite mathematics course are too often covered after they are required in another course. So take a mathematics course before there is any possibility the methods in it will be needed in another course. Then you may master the methods before they are required and not after.

If you follow the advice and the cautions below, you should have a mathematical foundation for any subject requiring calculation. In mathematics do the following.

1. Master arithmetic. Also master weights and measures. After you have mastered the rules of arithmetic, learn to use a calculator.

2. Master the algebraic way of writing and thinking. Also master the use of rules and patterns to arrive at conclusions. Mathematics after arithmetic builds on our abilities to talk about numbers and quantities, to describe calculations and to change the way calculations are done. Mathematics after arithmetic also depends on our ability to follow and understand rules and patterns. See the first chapters on reason and algebra in this book.

3. After algebra, take trigonometry and geometry.

4. Learn about money matters. Take a course on money calculations, preferably after a course in basic algebra. Most of us handle money for credit or investment. In your last year of studies before starting work, take a course on the mathematics and arithmetic of personal finances. This course should include balancing of budgets, description of typical household expenses for individuals or families in rented or mortgaged properties, problems involving simple and compound interest, and mortgage/pension calculations. Traces of such calculations appear in elementary and high school mathematics, but they are forgotten years before you need them. A course like the one described should be offered in schools and colleges for students in any art or science. Ask for one to be given, if it is not already offered. All this is practical mathematics. It should be more widely known.

5. If you go to college, take a year or two of the mathematics subject called calculus, a year of probability and statistics, and a year of matrix computations (or linear algebra). Calculus courses usually have trigonometry and algebra as prerequisites. Calculations in many trades, including business, engineering, computer technology, physics and health science, require calculus.

Mastering the rules and patterns of mathematics and reason (there is a connection) is good practice for mastering the rule and patterns of all disciplines. To master mathematics you need to read your course notes or course textbook carefully. Examples, solutions and proofs show you patterns to follow or imitate. Here every step not understood hides an idea from you.

The problems you find easy to solve should be done to restore and build your confidence and to reassure yourself that you have understood what they require. But after you have done a few such problems, you should look at the ones which appear harder. The problems which appear to be too hard should be noted and remembered. You can return to them later by yourself or with help from another. What is hard for you to solve may be easy for another, and vice-versa.

### E. Cautions

When I taught a remedial algebra course, one of my students was a high school gym teacher. One of his past assignments was to teach algebra.7

7This should not occur, but in many school systems it does. When it does, it shows a lack of respect to students and a lack of purpose for education. It also suggests circumstances beyond the control of students and teachers.

In some schools due to circumstance beyond their immediate control, some instructors are required to explain ideas outside their own specialties. When or if you meet such an instructor, be polite and do not become a troublemaker. If a teacher sees you as a threat or troublemaker, you may suffer. When you meet a misplaced instructor, politely and diplomatically try to transfer to another class in the same subject or read the course textbook yourself and get a tutor.

### F. More Keys to Better Learning

Here are several more comments on learning mathematics or another subject.
1. How you find a solution to a problem is not important provided you understand fully the solution. (Some teachers may disagree.)

2. If you have to copy solutions blindly then you will not understand ideas well enough to pass tests and the final examination.

3. You should ask another to check that your written responses or solutions are both understandable and well-written. Mistakes brought to your attention in any manner improve your understanding. If such checking improves your ability to avoid mistakes in the future, then such checking should I believe be encouraged. Again, some teachers may disagree.

4. Students who know and identify in their solution those steps which are doubtful deserve more respect than students who don't. Knowing exactly where one is sure and where one is not is the sign of an alert mind.

5. Correct answers obtained accidentally, for instance by canceling errors in a solution should not be given full marks. Errors in a solution show that the subject is not carefully mastered.

6. Learning is better done in a cooperative atmosphere where students help each other to understand instead of a competitive one, where the success of one student is at the expense of others. (But you can not always choose your environment.)

7. Seeing two or more approaches to a subject can be better than one. What appears hard in one approach may appear easier in another.

### G. Calculators and Computers

Calculators lessen the need for us to do arithmetic but, in using them, mistakes can be made. Here you need to know in advance what kind of answer a computation will yield. If you think you have made a mistake in entering numbers or instructions, you need to reenter them again. If a different result appears from before, at least one of your efforts, the original or the check, will be in error. (Logic Question: What can you say for sure if the results agree?) Suggestion: remember or learn how to do arithmetic by hand and how to estimate the expected size of results for addition, subtraction, multiplication and division.

Computer programs can perform arithmetic and algebraic or symbolic operations. They can also draw graphs and solve some equations rapidly. These programs do not provide solutions to all possible problems. For the solutions they can provide, you have to understand the statement of the initial problem. Beyond this, a computer (or another student) cannot understand the chains of reasoning for you. Understanding is an personal affair. No computer and no other person can do this for you. But if you know what to expect from a calculation, calculators and computer programs can help you check your expectations and explore mathematical ideas. Here you can learn from your mistakes. In some cases, computer software can tutor you. They can tell what to expect in various circumstances. Today there are computer programs and on-line computer books which may help you master mathematics and other subjects. More are appearing everyday. I know of them, but I have no experience with them.

### August 2011 Postscript - Context

In mathematics programs for ages 4 to adult, the main objective appears to be the preparation of students for college level studies and careers based on calculus or statistics. But many or most students do not complete secondary school. While mathematics programs for ages 5 to 12 may develop numerical and geometric skills and practices of actual or potential value for daily and adult life, programs for ages 13 to 17 focus on a preparation for calculus- or statistic-based college studies, and they do so imperfectly.

Most secondary school mathematics - in North America at least - is taught by instructors who have not seen how or why calculus nor statistics is used in college studies and beyond. Most instructors started to teach mathematics not because they were trained in it, but because no one else was available to teach it. Besides that, year after year, the only immediate reason for understanding or explaining a skill or practice is its likely appearance on the next final examination. In a discipline whose mastery is taken to be a sign of good figuring and reasoning abilities, how or why the skill or practice will be needed in college will not be clear to learners and their teachers. Learners taking advanced courses covering biology, chemistry, physics and money computations will see high school mathematics in use - better late than never. Students in trade oriented courses may also see some applications. But almost everyone else will not. While mathematical preparation for college studies may be wanted and respected for its long term value, for most, the short-term value of doing well on test and finals is not inclusive and not appealing.

Present day students and teachers will have to do their best with local course design and materials. In site material, online books and further site material will help. See the advice in the other column. Here the description of alternative ends, values and paths for mathematics and logic-language skill and practice development may provide some immediate context and motivation.

### Doing Your Best with local course design and materials

If you find studying hard or not, then you should try to master logic - see site logic chapters - and you should be aware of and try to avoid the domino effects of errors in multi-methods at home in cooking, at school in mathematics etc, and at work in following instructions. Logic mastery with the greater precision in reading and writing it provides is one way to ease or avoid troubles in following instruction at school today and in work tomorrow.

For reading and writing, all the letters of your alphabet have to be met and remembered. Anything less will lead to difficulties in spelling and understanding words. The child who complains there are too many letters too remember will soon be corrected. For mathematics mastery, skill in counting, measuring and calculating with decimals, fractions, percentages and signs is a must. But too many students are not shown fully how to do so in the last years of primary school and the first years of secondary school. But such skill is a must for all further mathematics in secondary school and college. Site arithmetic steps will help. Use them to check for gaps in your command of fractions, primes and arithmetic with units of measure.

In my school days, the introduction and use of algebra included steps too large for most to follow. Site algebra chapters and steps provide starter and advanced lessons to give a remedy. In it, smaller and extra steps provide a more gradual, less steep, paths to try. See what works. Looking for a remedy began in the author's high school days. The first break-through came with the fall 1983 invention of three lessons: Three skills for algebra, why slopes and two logic puzzles. Talking about the three skills adds words to the exposition of mathematics. The lesson on why slopes appear in secondary mathematics lesson was an algebraically light calculus preview. Calculus is the subject of slope related computations forwards and backward for lines or linear functions y = ax+b and more generally for curves or nonlinear functions y = f(x). Preparation for calculus is the main reason for earlier mathematics in college or secondary school. Calculus and many other subjects require precision reading and writing. That was one reason for the two logic puzzles. While lower level mathematics may be taught by rote in a practice first, theory second or not all at all manner, that is easier, students heading for calculus-based college programs should meet and be able to master some of the theory - too much would be overwhelming. Site logic chapters provide a worky introduction, almost math-free, to the kind of logic used in advanced mathematics, logic whose mastery in all or part has take home value in sharpening skills and in easing or avoiding learning difficulties at home, work and school.

In the bottom margin of each page there are links to key lessons. Explore them. If one or a few are not to your liking, try the others. Site content indicated above should fill a gap or two in your education. That should be reason to explore more site material. Good luck.

### Alternative Paths for Mathematics Education

Imagine mathematic and logic education may come in overlapping layers or levels. The first level, usually for ages 3 to 14, would cover all the counting, measuring and figuring skills and practices that employ numbers and/or maps-plans-diagrams drawn to scale with actual or potential take-home value for daily- and adult-life. In that time, date and calendar matters; money matters; logic and decision making matters; and elementary knowledge of chance, probability and odds may be useful. The first level would describe some calculations with words - adding by adding subtotals would be one example; and other calculations with formulas. Calculation methods would be given for routine or common situations in daily or adult life, all in a scout-like, be prepared for what is likely, context. Instructors would show how to do and record work in steps that can be seen and checked as done or later. Avoiding the domino effect of mistakes in counting, measuring, figuring and in general would be emphasized as an end, value and tool for skill development and mastery. This first level might cover methods that are easily understood and verified in class until just before doing so becomes too repetitive. And if mathematics education was to stop or not be appreciated beyond this first level, at least the first level would provide skills and concepts that would or could be useful sooner or later.

The second level of mathematics would introduce the use of algebra for solving for unknowns or getting formulas for them, and the use of algebra to say when different computations give the same result. In the past, the algebraic shorthand role of letters and symbols has not been clearly introduced. I saw that in my school days. In my school days, I was able to rationalize the shorthand roles of letters in finding formulas for solutions, in solving equations and in describing when different calculations would give the same result. But many other students, more gifted than I in their reading and writing abilities could not. And one of my physic teachers did not understand algebra. The foregoing combination of students and teachers in my physic course made my instruction slower than need-be. Site algebra steps, smaller and extra, provide a more gradual path, less steep, for developing algebra skills and practices. A natural context for commutative, associated and distributive laws in arithmetic is provided by the new concept of equivalent computation rules. The latter has little or no take-home value for daily or adult life, but it makes preparation for mathematics-based college programs more accessible. Learning how to use formulas forwards and backward allows several related formulas with take-home value given earlier to be replaced by one. The foregoing may lead to a fuller and stronger mastery of money related calculations, there be take-home value in that for daily or adult life. Stopping here might leave a favourable impression of mathematics,

Three further topics, study in any order, in mathematics have some actual or potential take-home or intellectual value for daily and adult life.

1. The Euclidean model for reason is introduced by site logic chapters in a math-free way. Those chapters may improve reading and writing skills, and help people see the difference between one- and two-way implications. Not seeing this easy difference is a source of confusion in following and giving instructions, in digesting information in and outside of mathematics; and in agreements. The latter and their small-print need to be read and fully understood to avoid surprises. Remember in making an agreement, all parties need an acceptable exit clause - an option to use if things do not turn out as wanted. Good luck. The site simplified coverage of Euclidean Geometry, error-free we hope, shows how implication rules can be used in sequence to arrive at conclusions - often further implication rules. Euclidean geometry provides a neutral territory for illustrating the deductive use of implication rules, alone or in sequence. All is simpler than you think.

2. Not all is certain. Earlier mathematics and life may provide examples of chance, likelihood, probability and odds. Probability theory in mathematics provides ways to estimate what is likely to happend when not is certain. That help people lessen or avoid risk. Mastery of equivalent computation rules, how to use formulas forwards and backwards, and simple operation with sets, should allow students to understand the first elements of this theory and its notation. Learning about probability, and above expected value of possible outcomes in daily life or in playing games will help decision making in matters of chance, when not all is certain, including the methods for arriving at conclusions. The elementary study of probability theory provides one model, more algebraic than geometric, of mathematical reason.

3. The natural logarithm appears in the backward use of the compound interest or growth formula. Secondary mathematics is not the place to describe the origins of this logarithm function and the anti-log or inverse exponential function. But a short, full theory of logarithms, roots and powers may be develop from the algebraic description of properties of the natural logarithm and its inverse. The elementary study of the latter theory provides another model of mathematical reason. Here it might include the forward and backward use of exponential growth and decay models.

4. Site geometry steps cover and employ rectangular and polar coordinates in the plane. The development of complex numbers from the properties of these coordinates and their interaction might provide another easy topic. In the prepartion of students for geometry-based careers and for calculus-based college programs, this topic would set the stage for the mastery of trigonometry. Many geometric problems can be solved by drawing to scale and then measuring. Some of these problems can be solved by sketching and using trigonometric calculations in place of drawing to scale precisely and measuring missing angles or lengths. Trigonometry in the first instances allows calculation guided by a sketch to be used in place of drawing to scale. There-in lies a context for trigonometry.

Mathematics education may continue with further topics - see site steps. The further ones also required by mathematics-based college programs. But those further topics would not have any actual or potential take-home value for daily or adult life like the three or four above. For students not heading for mathematics and statistic based college studies, the three or four topics described above might leave a favourable impression. Leaving a favourable impression, one that includes multiple skills and practices with actual or potential take-home value, and perhaps a thirst to learn would be better than covering too much, and leaving a bad impression or an alienated view of mathematics in the process. To avoid the latter, less done to perfection would be better. Mathematics education should continue beyond the first level only while the underlying topics are easy for students to master, and have some relevance for future studies or for life in the street.

2011-08 Remark 1: Statistics appears too much in the mathematics programs in UK, USA and New Zealand. The use of averages has some practical value. Pie charts may be use to illustrate proper fractions and percentages. Beyond that, the study of statistic in high school appears to have little or not take-home value for the daily and adult life. The year after year placement of statistics in late primary and in secondary school mathematics is mostly a distraction from the development of skills and practices with actual or take-home value, and from the preparation of students for calculus-based college studies. Less would be best.

2011-08 Remark 2: Teachers, if course design was to treat each year of instruction as if it was the last chance to provide students skills and practices with take-home value, the direction and priorities of course design would be different. They would, if I am not mistaken, after coverage of the easier skills and practices with actual or potential take-home value for daily- or adult life, identify and put those skills and practices in or helpful to calculus preparation with the greatest and than faintest take-home value first or as early as possible. That would be subject to the inclusiong of further skills and practices that make skill development less steep and more gradual.

2011-08 Remark 3: Teachers, In computer programming, subprograms or code that has inputs but not outputs is redundant - can be removed or archived. In mathematics, skills and practices whose practical or intellectual value to students and society is unclear or confused - due to years and years of committee based course design - can likewise be eliminated or archived. Critical path analysis - careful thought - is needed in course design to choose goals and objectives, clear and concrete, and identify what is needed. Read about which way to go below.

www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Appendix E. How To Study Mathematics and Why Next: [Postscript For Better Performance.] Previous: [Appendix D. What to do in School and Why.]   [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40][41] [42]

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.