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Mathematics and Logic - Skill and Concept Development

Questions: Will these ends and values motivate? Will smaller & more steps in site lessons and lesson ideas build skills and confidence?
Should we emphasize how ideas & methods depend on earlier ones? Does concept & skill mastery need to be checked to be believed? What is a Variable?
30 pages en Francais || Parents - Help Your Child or Teen Learn
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.

For students of reason in society, science and technology: Pattern Based Reason describes origins, benefits and limits of rule- and pattern-based thought and actions. Not all is certain. We may strive for objectivity, but not reach it. Postscripts offer a story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theories and practices.

Site's Best Lessons

For Logic

These online chapters may amuse while leading 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. Site Theme: Different entry points may be easier or harder for knowledge mastery.

For Arithmetic

Deciml Place Value - funny ways to read multidigit decimals forwards and backwards in groups of 3 or 6, US-CDN, UK-German and Metric SI style.

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.

For 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.

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www.whyslopes.com >> Volume 3 Why Slopes - A Calculus Intro Etc >> Chapter 2. Slopes and Ski Trails Next: [Chapter 3. Slope Sign Analysis.] Previous: [Chapter 1.Introduction.]   [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 2 . Slopes and Ski Trails

Volume 3, Why Slopes and More Math.

  • [Play Video]80 seconds: Slope Sign Interpretation for Linear Functions. (Appeared earlier in Why Slopes Appetizer)

Slopes of Line Segments

A tour of calculus begins. Recall how the slope to a straight line is computed. The slope m of a straight line segment between two points ( x1, y1) and ( x2, y2) may be calculated as follows.


slope m = D y
D x 		= y2- y1
x2- x1 		= rise
run

The point-slope formula for a line
y = y1+ m·( x- x1)
implies
y- y1 = m·( x- x1)
The latter says that the change D y in y is proportional to the change in D x in x. For a straight line segment, the slope m is a constant of proportionality between D y = y- y1 and D x = x- x1.

Remark. A quantity Q2 is said to be proportional to a quantity Q1 when and only when there is a constant k such that Q2 = k· Q1. If a quantity Q2 is proportional to a quantity Q1 then the graph of Q2 versus Q1 is a straight line through the origin whose slope m = k is the constant of proportionality.

A Cross-Country Skier and Her Trail

[Play Video] 2¼ minutes: Slope Interpretation for a 2D ski hill y = f(x). (Appeared earlier in Why Slopes Appetizer)
Meet the cross-country skier, Barbara:

She has only one ski. Alternatively, you can imagine she always travels with both skis parallel. Travel with one ski was the way in which both alpine and cross-country skiing began. Also meet the Jack Rabbit ski trail y = h( x) (see below) which she skis, always in the direction -> from left to right.5 That is, she travels in the direction of increasing x.

5 Footnote: The slope to a curve at point can be approximated by taking the slope of a short line segment which has one end at the point and another end also on the curve. This approximation should get better as the line segment gets shorter. The finite limiting value of this approximation, should it exist, is taken to be the slope. Before discussing this approximation any further, we will make the improper assumption that the slope of a short ski placed on the graph of y = f( x), or the graph of one quantity versus another, is the slope to the graph. This will allow some exploration of why slopes are studied.

Imagine or suppose that the hill is smooth enough, so that, at most points, a ski can lie flat against the hill surface. The slope beneath a foot or ski gives what should be the slope of a tangent line to the hill. The slope of a ski can in principle be measured any time by freezing a skier in place, or equivalently taking a photograph (snapshot) and then measuring the slope from the photograph.6

6 footnote What happens if Barbara goes up and over a sharp peak? As she gets to the top and pivots from the uphill to downhill side, the slope of her ski goes from positive to negative.

The vertical line segment in the above graph represents a jump or cliff. The above diagram strictly speaking consists of the graph of a function y = h( x) plus a vertical portion to represent a ski jump.
  • Above the point x = a on the horizontal axis, the height above the x axis of her ski midpoint is y = h( a). We will call h( x), the height function.
  • The height function h( x) might be measured or computed from a formula, a map, or a graph, such as the one shown above. Or, in speaking of a height function h( x), we could leave its values unmeasured, not computed or unknown.
Barbara rides her ski both up and downhill. To go uphill, she may use her ski poles with great strength or skill. As she moves, the slope m of her one ski changes. It provides information about the hill.
The slope m is For Motion
> 0 (or positive) Uphill
< 0 (or negative) Downhill
= 0 (or zero) Horizontal

The shorthand for the word positive is +ve. The shorthand for negative is similarly -ve.

A Skier in Motion

Immediately below are a few snapshots of Barbara on another portion of the skill trails and hills y = h( x). In each snapshot, the slope of the ski is assumed to be equal to the slope of the hill at the ski midpoint.

In the diagram observe:

  • At x = c, the slope m > 0 and she is moving uphill: her height is increasing.
  • At x = b, the slope m < 0 and she is moving downhill: her height is decreasing.
  • At x = a, the skier is approaching the top of the hill. What is the sign of the ski slope before, at and after the top of this smooth hill?
Later, we will look at those points or intervals where the slope is increasing (becoming more positive or less negative), where the slope is decreasing (become less positive or more negative), and where the slope is greatest, least or zero. Ski trails in which the slope varies are of greater interest and possibly less boring than trails where the slope is constant. Again, the study and analysis of curves y = f( x) with varying slopes is one of the first subjects in a calculus course.

1. Local High-Points or Maximums

Load Flash Video
 3 minute summary of page topics.
High points of bumps and hills are called maximums.

As the skier Barbara moves up a hill (or a local bump), and then down the other side, the slope of her ski changes from positive on the uphill side to negative on the downhill side. At the very top of hill, her ski is horizontal and its slope is zero.7

7Footnote: What happens if Barbara goes up and over a sharp peak? As she gets to the top and pivots from the uphill to downhill side, the slope of her ski goes from positive to negative.

There can be several hills, with different and varying steepness on each side. From the slope of her ski, Barbara knows even without looking when she crosses over the top of a hill or a bump: the slope of her ski changes from positive to negative. The next diagram shows the sign of the slope (of the one ski) before, at and after a high point.

Knowledge of the sign of the slope does not provide all information about the ski trail y = h( x). In particular, unless she measures and records the height h( a) of each hilltop, she can not say which is the highest. Every hilltop gives a local maximum for her height. It has a height greater than or equal to ( ³ ) all nearby heights. A point with a height greater than or equal to ( ³ ) all other heights, is called an absolute maximum for the portion of the trail or curve y = h( x) being examined.

Remark (Advanced Material.) To be more precise, a point ( x0, h( x0)) is a local maximum for the portion of a curve y = f( x) where a < x < b if
  1. a < x0 < b. That is, x0 is in the interval being examined.
  2. There exists at least one interval ( x0- d, x0+ d) centered at x0 such that h( x) < h( x0) if a < x < b and x0- d < x < x0+ d.
These two conditions describe precisely what is meant in talking about all nearby heights. Note that in talking about numbers and quantities, a legalistic precision is required. Otherwise, tacit assumptions will be made differently by different writers and different readers.

Definition. (Highest of the High Points). A point with height greater than all other heights in a given portion of the trail, more precisely not less than all other heights in the portion, is called an absolute maximum for the portion or interval in question.

In the event of a tie, each of those points having or sharing the greatest height is called an absolute maximum.

2. Local Low-Points or Minimums

Low points of depressions are called minimums.

As our skier Barbara moves down the sides of a valley (or depression) and then up the other side, the slope of her ski changes from negative on the downhill side to positive on the uphill side.

As with high points (hill tops or maximums), several depressions or valley bottoms may be met and crossed in following a ski trail. From the slope of her ski, Barbara again knows even with her eyes closed when she crosses the valley bottom: the slope of her ski changes from negative to positive.

Note again that unless she measures and records the height h( a) of the low points in each depression or hollow, she can not say which is the lowest. The bottom of each depression or hollow gives a local minimum for her height. It has a height less than or equal to all nearby heights.

According to this definition, all the points on a horizontal straight lines are local minimums. Excluding the word equal here would yield a strict local minimum: a height less than nearby heights. See the previous discussion of nearby heights.

Definition. A point with a height less than or equal to ( < ) all other heights in a portion of the trail is called an absolute minimum for the portion of the trail or curve y = h( x) in question.

In the event of a tie, each of those points having or sharing the least height is called an absolute minimum.

x-Dependence of Slope

[Play Video] 1¾ minutes: Along a 2D ski trail, see how height y = f(x) and slope m = f'(x) both depend on the horizontal coordinate x.

As Barbara, the one-ski skier, moves along a trail y = h( x), both the height of the ski midpoint y and the slope m of the ski depend on its x coordinate. The slope of her ski at x = a, x = b, x = c and x = 2 in the following diagram are all different. The slope of her ski midpoint depends on her location.

  • At each point, the slope m of the ski is determined by x and the shape of the trail y = h( x). That is, the slope depends on x and the hill y = h( x). To signal this dependence, we write m = g( x) = h'( x). As said before, the slope m = g( x) could be measured from a snapshot - freeze Barbara and her ski in place. The mathematical definition of \( m = h'(x) $ appears later.
  • The slope m = g( x) depends on the shape of the hills y = h( x). The slope when x = a is m = g( a) = h¢( a). The slope when x = b is m = g( b) = h¢( b), and so on.
  • In most problems that you will meet or be shown, formulas for the slope m = g( x) can usually be obtained or derived from formulas for the height y = h( x). For each new type of function added to your knowledge, there will be a differentiation rule to be learnt.
  • For a given formula or function y = h( x), rules of differentiation say how to obtain or derive a formula or function g( x) = h¢( x) from a formula for height h( x). Presumably, because of these rules for deriving or obtaining the slope m = g( x) = h¢( x) from formulas for h( x), the function g( x) = h¢( x) is also called the derivative, the first derivative. Note that the use of the word obtainable for slopes is not an accepted alternative to the term derivative.
  • There are several simple rules for calculation slope functions or derivatives. These differentiation rules are given in the first instance for the cases where an expression for the height function h( x) involves polynomials, logarithms, exponentials, sines or cosines. Here for each operation (addition, subtraction, multiplication, division and composition) involving functions and yielding a new function or formula, there are additional rules, all of which appear to be very simple after, but not before, they have been mastered. Meeting and mastering these rules requires or instills an understanding of the algebraic way of writing and thinking. The algebraic way of writing and thinking is seen or required here in full strength.

Notations for Slopes and Derivatives

Calculus has several expressions for the slope m of the ski or hill function y = h( x). Some follow. \[m = g(x)=h'(x)=\frac{dy}{dx} = lim_{\Delta x \to 0} \frac{\Delta x}{\Delta y} \] Still more may be seen. Different notations exist because calculus was discovered and employed by different people in the past four or five centuries.
www.whyslopes.com >> Volume 3 Why Slopes - A Calculus Intro Etc >> Chapter 2. Slopes and Ski Trails Next: [Chapter 3. Slope Sign Analysis.] Previous: [Chapter 1.Introduction.]   [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]

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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?

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.

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. ...
For Geometry

Maps + Plans Use - Measurement use maps, plans and diagrams drawn to scale.

Euclidean Geometry - See how chains of reason appears in and besides geometric constructions.

Coordinates - Use them not only for locating points in the plane or space.

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 assumes no knowledge and some knowledge of the tangent function in trigonometry.

What is Similarity - another view of using maps, plans and diagrams drawn to scale in the plane and space. May buildings in space are similar by design.

For Calculus

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 - this 1995-96 lesson introduces calculus skills and concepts. It may also may be given to introduce further function maxima and minima both inside and at the ends of closed intervals.

Check Arith. Skills - too many calculus and precalculus students do not have strong arithmetic and computation skills. The exercises here check them while numerically hinting at equivalent computation rules.

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