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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 >> Volume 3 Why Slopes - A Calculus Intro Etc >> Chapter 6. Slopes and Vertical Shifts Next: [Chapter 7 Slopes and Velocity.] Previous: [Chapter 5. Slope Sign Tests.]   [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]

## Chapter 6. Slopes and Vertical Shifts

Volume 3, Why Slopes and More Math.

The vertical shift or motion of a curve y = f(x) adds a constant d (the displacement) to each point. It yields a new curve y = f(x)+d. The next diagram indicates an example.

A simple vertical shift or motion may change the height of a skier, but not the slope of his or her skis.

 Load Flash Video On the Constant Difference Theorem 4:32 minutes 404 by 408

## Constant Difference Theorem

What can be said for sure about two functions when they have the same slope (or derivative) everywhere? One response is given by the following assertion.

[Constant Difference Theorem] If the functions f1(x) and f2(x) have the same non-infinite slope (that is, derivative) m = f¢1(x) = f¢2(x) at every x in an interval (a,b) then the difference

 f2(x)-f1(x) = d
is constant for a < x < b. That is, there is a constant d such that
 f2(x) = f1(x) +d
for every x in the interval (a,b), This number d does not depend on x. The proof of this assertion is given in the appendices.

Note the assertion says that there is a constant d. It does not say how to find it. Here is an analogy: Saying there is a needle in a haystack, does not say how to find it. Note also the word every. If there is a point x1 in the interval (a,b) where the slope is not defined then no conclusions can be drawn from the Constant Difference theorem.

Remark. If f1(x) and f2(x) satisfy the conditions (hypotheses) in the Constant Difference theorem with f2(c) = A and f1(c) = B at some point c in the interval (a,b) then at x = c,

 d = f2(x)-f1(x) = f2(c)-f1(c) = A-B
This says how to compute the value of d.

Remark. Mathematical assertions and theorems which say that a number (or limit) is not defined or does not exist actually mean that a finite number (or finite limit) does not exist. The vertical motion theorem given above applies only when the common slope is finite in the interval (a,b) of interest.

 Load Flash Video When the Constant Difference Theorem fails 404 by 408 3:36 minutes

Different vertical displacements over different portions of the trail are possible. The next diagram gives examples of this. Upshifts have been made in the topmost curve of the previous diagram.

Between a and b, between b and c and between c and d, the two trails y = f1(x) and y = f2(x) shown have the same slope but not the same heights. At the ski jump and cliffs in the upper trail y = f1(x), the slope is not defined.

## Where Slopes are Not Defined

On a ski trail y = h(x), there may be a few places where the slope is undefined or a single slope to the graph or trail does not exist. In the following diagram, the slope is undefined at the ski jump above the point x = a. At sharp peaks and kinks, a short ski may pivot or rotate while keeping in contacting with one point, the kink or peak on the trail. So a single slope to the trail cannot be defined there. Where the trail has some vertical jumps, the graph ceases to be the graph of a function. The slope is said to be undefined or not to exist here, even though we might say it is infinite, +¥ or -.
 Load Flash Video 2 minute explanation of the following 1:50 minutes

Consider the next diagram. At the points a,b,c,d and e, the slope function or derivative m = h'( x ) cannot be given a single value. In general, a single value cannot be assigned to slopes at sharp changes in the direction of a curve y = h(x).

The study of generalized slopes or gradients replaces the discussion of a single slope to a point on a curve by the discussion the set of slopes to a point on a curve. This set based discussion is too complicated to be examined further here.

But a single slope may sometimes be defined before and after such kinks or sharp turns in the graph of a function.

The ski at the sharp peak is shown pivoting, that is rotating, as the ski passes over.1 In pivoting at the peak, a ski can have many orientations or slopes without intersecting the curve y = h(x) on either side.

1The slope values during this rotation form a set, the slope set. The value 0 belongs to this set.

Problem for Advanced Students. A graph with vertical segments is not the graph of a function, but it may be the image of a parameterized curve $(x(t),y(t))$ where $t$ belongs to some interval. Show that if $(x_j(t),y_j(t))$ for j = 1 and 2 are two continuous curves parameterized by $t \in [a,b]$, then \( (x_1'(t),y_1'(t))=(x_2'(t),y_2'(t)) $for all$t\in (a,b)$implies there are constants$d_1$and$d_2$such that$(x_j(t),y_j(t))=(x_j(t),y_j(t))+(d_1,d_2)\$.

This problem is both a consequence and an extension, a generalization, of the constant difference theorem just stated. It applies to some graphs with vertical segments.

www.whyslopes.com >> Volume 3 Why Slopes - A Calculus Intro Etc >> Chapter 6. Slopes and Vertical Shifts Next: [Chapter 7 Slopes and Velocity.] Previous: [Chapter 5. Slope Sign Tests.]   [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]

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.