[Trois notions qui mènent à l'algèbre] [Logique basée sur des règles] [Deux définitions d'une variable] 

HOW-TOs and Leading Questions

Leading questions  will  become HOW-TOs as answers become available. Visit the webvideo section of this site for a growing number of audio-videos that will offer help on demand, as you need it, for learning or teaching key ideas in mathematics.

FOR

• logic   
• whole numbers and fractions
• arithmetic Leading Questions,
• algebra - words before symbols
• trigonometric  - complex #s before or after.
• complex numbers,
• geometry,
• calculus 
• real analysis.

The best source of information in a mathematics course should be a well-selected, precisely written textbook.

Advice

Logic HOW-TOs

From Volume 1A, Pattern Based Reason, and Volume 2, Three Skills for Algebra

1. How to use implication rules, one at a time, and one after another, to arrive at conclusions, one at a time or one after another. NB. the route, how it is covered, may be of more interest than the conclusion.

2. How two-way implication rules say more than one-way implications. 

3. How to read precisely what is written.  

4. How the ability to climb a ladder or not is related to mathematical induction.  

5. How different starting points may make a subject easier or harder to explore. 

6. How to justify entries in truth tables for one-way implications starting from the common notions of a rule being obeyed, disobeyed or not disobeyed.

7. How to use the contrapositive of a one-way implication to arrive at a conclusion (indirect reason I)

8. How proof by contradiction may be recast as using the contra-positive of a one-way implication.

Skill and confidence in logic (rule-based thought) comes from precisely applying implication rules, one at a time and one after another, to arrive at conclusions, one at a time and one after another, with the knowledge that any other thinking starting with the same circumstances and using same implication rules, may follow your chain of reasoning with the same result.  This makes logic (how to argue you must) repeatable and reproducible, independent of the thinker or initial discover of a chain of reason. But an a mistake in one step of a chain of reason makes all the rest suspect.

Whole Numbers and Fractions
Before and After Axioms for Real Numbers

1. Do you see how the figure 

   	3 4
   2 3

   suggests  (3/4)(2/3) = 6/12 = 1/2. Do you see the algebraic pattern  (a/b)(c/d) = (ac)/(cd) for multiplication and the algebraic pattern (ps)/(qs) = p/q for simplification or reduction?
   

2. Do you see how the figure 

   3
   	4				5

   suggests or illustrates the grouping  3 x 4 + 3 x 5 = 3 ( 4+ 5) ?

3. Do you see how the figure 

   a	Area ab	Area ac
   	b	c

   suggests or illustrates the grouping (distributive law)  ab + ac = a ( b+ c)?

In elementary school mathematics, operations on fractions and grouping may be introduce, illustrated, suggested or justified via diagrams or figures. With the introduction of axioms for real numbers (or complex numbers),  justification for operations on fractions and grouping may be derived from the axioms instead of from figures. But the two justifications need to coexist side by side as follows:

Teachers might say repeatedly, here is a message worth repeating, that you have met these diagrams or similar ones to justify arithmetic operations, but mathematics has a set-based codification in which the properties of real numbers are described by axioms (assumptions) about real numbers or obtained from these axioms. After this diagrams and figures may be kept to illustrate the axioms and as memory aids. Students are most at ease with formulas or equations with geometric interpretations in which the letters and computations represent a length or area, etc. The notion that a letter may stand for a number not yet given is harder. 

Course design for completeness of exposition should add to the axioms statements about the decimal representation of whole numbers by themselves or in the numerator a and denominators c of fractions, about the decimal representation of fractions a/c where  c is or is not a multiple of 2 and/or 5. (Is leads to finite decimal expansions, NOT leads to decimals expansions that repeat),  and about the existence and convergence of decimal expansion of irrationals. 

Arithmetic Leading Questions

The first sign of skill,  confidence and intelligence in mathematics comes from applying methods of arithmetic, one step at a time, and one step after another, to arrive at numerical results, with the knowledge that an mistake in one step makes all the rest wrong or suspect.

1. Can you evaluate arithmetic expressions carefully and precisely, and give answers with proper use of notation?

2. How do grouping and conversion of units of time explain carries and borrows when adding or subtracting lengths of time of the form  a days, b hours, c minutes and d seconds? The answer may help you appreciate metric measurement systems - this example should be preserved for the sake of that appreciation.

3. How do grouping and the addition, subtraction, multiplication and division of polynomials in powers of ten with coefficients limited to the digits 0 to 9 may first justifies carries and borrows in decimal arithmetic first with whole numbers and then finite decimal expansions.

4. How does multiplication with decimals or polynomials in powers of ten with coefficients limited to the digits 0 to 9 imply  the product of nonzero numbers is non-zero?

5. How do you use scientific notation to estimate the results of a division or multiplication.

6. How do to describe errors due measurement or due to round-off in calculations, and how these errors affect sums and products?

7. Do you know or  assume that infinite decimal expansions define and represent  rational or irrational numbers?

8. Do you know how to evaluate formula, given numbers or quantities to use in it?

Hint: Try these Arithmetic Review Problems with Hints of Algebra and have someone correct your notation. A stitch in time, saves nine.

Note: Decimal notation gives the first representation of whole numbers, numerators and denominators of fractions.  Decimal notation with finitely many decimal places after the decimal points provides a representation of a decimal fraction plus a whole number. Infinite decimal expansions follow for further fractions and for irrational numbers.  In all, real numbers with signs or not are first met in the form of decimal expansions.  Opinion: Introductions to the set-theoretic codification of mathematics could begin by assuming a set of real numbers given by decimal notation, finite or infinite, and by summarizing the properties of real numbers as assumptions about decimal arithmetic.

Algebra HOW-TOs and Questions

Mainly from Volume 2, Three Skills for Algebra

1. Do you know how to evaluate formula, given numbers or quantities to use in it?

2. How to talk about variables - words before symbols A

3. Independent versus Dependent Variables - often a matter of choice.

4. How to talk about numbers and quantities - words before symbols continued.

5. How to describe calculations with words or formulas

6. How to change a calculation into another which gives the same result

7. How to interpret axioms for numbers as rules for arithmetic  which say when two different calculations give the same result.

8. How Words are Not Enough - why algebraic shorthand is used..

9. How to use or introduce shorthand notation for numbers, amounts and quantities

10. Can you use the simple interest formulas directly or indirectly to find one quantity when the other numbers or quantities in these formulas are given.

11. How to use the compound interest formulas directly or indirectly to find one quantity when the other numbers or quantities in these formulas are given.

12. How to solve a linear equation or a system of linear equations numerically and algebraically.

13. Can you translate simple word problems into a single equation or system of equations in one or more unknowns.

14. Can you solve quadratic equations with quadratic formula and  factorization-by-inspection (or trial and error)  methods?

15. Can you solve some  cubic equations involving the difference of two cubes?

16. Can you solve a nonlinear system of equations involving a linear equation and a quadratic equation in two unknowns?

17. How theorems may be disguised as problems, and vice-versa (painless theorem proving).

18. How to justify the Pythagorean theorem by dissecting a square

19. Can you locate points in the line, the plane or space using rectangular coordinates?

20. Can you locate points in the plane using polar coordinates?

21. Can you (by measurement) find the rectangular coordinates of a point given its polar coordinates?

22. Can you (by measurement) find the polar coordinates of a point given its rectangular coordinates?

23. How to multiply points in the plane using polar coordinates

24. How to describe and visualize complex numbers geometrically

25. Can you define, recognize, add, multiply, divide,  subtract,  whole numbers, fractions, rational numbers, irrational numbers, real numbers, complex numbers, vectors and/or matrices? Your answer may yes and no if you have more to learn.

26. Can you add, multiply, divide,  subtract, graph, define or recognize  linear functions mx +b, quadratic functions ax²+bx +c, polynomials, ratios of polynomials (rational functions), the six trigonometric functions sine, cosine, tangent, cotangent, secant and cosecant, logarithms to base 10, the natural logarithm, the exponential function (as the inverse of the natural logarithm)?

27. Have you met  the algebraic definitions of whole and rational powers, the algebraic definition of roots, and the general computation of real powers and roots via the natural logarithm and exponential functions? Have you met the definition of six hyperbolic trig functions and their inverse via the exponential function and the natural logarithm

28. How do the parameters a, b and c affect the graph of a curve y=a f(bx +c)?  (A parameter is a number or quantity which may change from example to example, but is constant in each example.)

29. When is a graph of y=f(x)  symmetic about an axis x =a?

30. When is a function y =f(x) odd?

31. When is a function y = f(x) even?

32. How a function may be defined by a rule or by a set of ordered pairs.

33. Can you define, recognize and compute the inverses to linear functions, quadratic functions, the six trigonometric functions, the natural logarithm  the exponential function, the six hyperbolic trig functions?   Inverse functions are useful in solving equations and useful in modeling situations, commercial or natural, albeit some models should not be taken seriously

34. How to compute arithmetic and geometric sums.

35. How to interpret shorthand notation for sum and products

36. How to do personal money computations

37. How to justify shortcut formulas for sums using mathematical induction.

38. Can you say why are Equalities important. For a first answer see Volume 3, Why Slopes and More Math, Chapters 1 to 18. For a second answer, learn about linear optimization problems.

39. Do you know how to compare numbers in  the line and plane by magnitude. See rename the greater than  sign

40. How to compare real numbers by more positive than. See rename the greater than sign

41. Did you know the following rule of thumb?  (i) letters at the start of the alphabet are used to represent numbers that may remain constants, (ii) letters at the end are used to represent numbers that may vary (or be unknown); and (iii) letters in the middle of the alphabet may represent parameters:  numbers which may change from example to example but are constant in each example. See the discussion of what is a variable.

42. Why do textbook employ the letters m and b as parameters in equations y=mx+b for straight line? Hint: Spend 5 seconds reading this question, and then forget it - do not waste your time in thinking further about it.

43. After you have learnt Gaussian Elimination,  return to the word problems that you began solving after learning to solve one linear equation in one unknown.  Is Gaussian elimination implicit in their solution?

44. Exercise: After you have learnt to solve two equations in two unknowns, algebraically, revisit the derivation of equations to straight lines for examples or special cases.

45. Exercise: Solve a system of equation by adding multiples of  equation to others to eliminate variables and arrive at a diagonal or scrambled triangular form.. Then rewrite the system and each intermediate system as an augmented matrices. Do you see how adding multiplies of equations corresponds to row manipulation on equivalent systems?

Shorthand notation in mathematics has three roles. The first role appears in recording and evaluating arithmetic expressions. Here no letters or symbols appear to represent numbers or quantities in calculations. Here order of operations may affect the result.  The second role of shorthand notation is given by formulas which describe calculations that can be done or given to another. Here letters and symbols appear to represent numbers or quantities in the calculation. The third role of shorthand notation lies in the statement of rules of arithmetic (properties of real or complex numbers) which generally say when two different calculations or formulas give the same result.

Slopes and Straight Lines

1. Do you know how similarity of right triangles implies the slope of straight line is determined by any pair of points of the line?

2. What is the point-slope equation for a straight line

3. What is the slope-intercept form of the equation of a straight line? How is it obtained from the point slope form?

4. What is the two point form of the equation of a straight line? How is it obtained from the point slope form?

5. What is the product of the slopes of a pair of slanted lines when one is perpendicular to another? Justify.

6. How do you find the intersection of two straight lines?

Quadratics

Logs and Exponentials

Set HOW-TOs and Questions

1. Can you find the intersection, union, complement of a set?

2. How does a set of order pairs in the plan which satisfies the vertical rule property define a rule for computing one number from another

3. How does a rule for computing one number from another define a set of order pairs which satisfies the vertical line property?

4. How does a set of order pairs in the plane which satisfies the horizontal rule property define a rule for computing one number from another?

5. How does a set of order pairs in the plane which satisfies the horizontal and vertical  rule property define two rules for computing one number from another, each of which inverts the other?

6. How or why sets appear in mathematics.

Remark: There are two rules for set definition in mathematics.

1. Set-builder notation { x in U | x has property p(x) } may be used define subsets of a set U?

2.  If A and B are sets, there exists a set C = A X B (called the crossed product) which consists of all ordered pairs (a,b) where the first coordinate a comes from A and the second coordinate b comes from B.

3. There exists an infinite set W whose first element corresponds to 1, whose second element corresponds to 2, whose third element corresponds to 3, whose n-th element corresponds to the whole number n.

The set-theoretic codification of mathematics follows from defining subsets of W and building Cartesian products A x B.

The set theoretic codification of mathematics begins with decimal-free assumptions about sets. That leads through a series of steps or constructions to sets that may be identified with the  real number line, and a derivation from assumptions about infinite sets of the properties of real numbers, including their decimal representation and the convergence properties of decimal expansions.  Modern mathematics instruction in the mid-950's started with the assumption that there was a set of real numbers, but not sanction nor admit the decimal representation in the further theoretical description or development of mathematics for people not specializing in the discipline.  That has impeded the development of the common knowledge and put aside what might have been a decimal context for the decimal-free development of the subject from arithmetic to calculus. See the chapter How or why sets appear in mathematics for the benefits of set concepts.

Trigonometric HOW-TOs and Questions

1. Have you met the right triangle method for computing sines, cosines and tangents of an angle.

2. How does similarity implies the right-triangle based computation of trig functions depends only on the angle and not on the size or scale of the right-triangles? 

3. How do you use trig functions to find angles and lengths of sides in right triangles?

4. How do you use or draw right triangles to compute distances in the plane for surveying and/or navigation problems?

5. How to explain [Complex Numbers] before and besides a trigonometry course - here motivation for the study of trig.

6. How does a knowledge of exponential & cis fns for angles and purely numbers turn proving or showing trig identities into simple algebraic exercises? College students in engineering and science use complex numbers and algebraic methods to avoid hard pre-complex number methods for solving or proving trig identities.

7. How to derive trig formulas for Dot & Cross Products  in the plane using the product of a complex number with the complex conjugate of another.

8. How to explain the Cosine Law with the aid of complex numbers

Complex Numbers HOW-TOs and Questions

1. How to add and multiply points in the plane with  knowledge of real numbers, rectangular coordinates and polar coordinates.

2. How to justify the distributive law for multiplication over addition with a knowledge of trigonometry.

3. How the distributive law for multiplication over addition gives two different ways for computing the product of a pair of points in the plane. See site area Complex No,s, Trig & Geometry

4. How to understand or explain complex numbers using the previous two examples. See site area Complex No,s, Trig & Geometry

5. How the law of signs for real numbers follows from the polar-coordinate method for multiplying points in the plane.

6. Why products of points in the plane,  nonzero ?  See site area Complex No,s, Trig & Geometry

7. How multiplication of a complex number by its conjugate implies a Pythagorean identity or theorem. See site area Complex No,s, Trig & Geometry

8. How multiplication of a complex number by the complex conjugate of another yields trig formulas for dot- and cross-products of points or vectors in the plane, and implies the cosine law for triangles. See site area Complex No,s, Trig & Geometry

9. How does the fundamental theorem of algebra imply polynomials of degree N in a single variable has N roots in the complex plane?

Teachers Note: If the distributive law for multiplication over addition is assumed, students who have learnt (1) arithmetic with unsigned rational and irrational numbers, (ii)  how to add real numbers (multiplication optional), (iii) how to locate points in the plane with rectangular coordinates (signed) and with polar coordinates (unsigned) may assume the distributive law instead of deriving it. In this case, unit circle definitions for sine and cosine provide a quick introduction to trigonometry in which the equality of two different ways for multiplying products gives most, if not all trig identities, and in which a discussion of similarity of right triangles or triangles in general provide a base for right triangle definition of trig functions using the ratio of sides of similar right right triangles, and thus a base for solving right triangles  - computing the measurements of a right triangle from knowledge of an angles and/or the lengths of  sides.  Here is a method for weaving a command of trig and complex numbers into an algebra course before or besides the explanation of real numbers.

Calculus HOW-TOs

Volume 3, Why Slopes and More Math,  points to a way through calculus and real analysis which may ease or avoid common difficulties

1. Why Slopes appear in high school mathematics - a calculus preview.

2. How can one avoid or ease algebraic shocks in a first and furthers courses in calculus. See Volume 3, Why Slopes and More Math, Chapters 1 to 12.

3. How does slope sign analysis provide a context for the study of inequalties and factorization before calculus, and a simple preview of calculus. See Volume 3, Why Slopes and More Math, Chapters 1 to 6.

4. How to compute (in calculus) the derivatives (the slope) of  linear functions, quadratic functions, polynomials, rational functions, the six trigonometric functions, the natural logarithm  the exponential function, the six hyperbolic trig functions, and the inverses to all these functions. See a calculus text.  Hyperbolic functions re often met in the second term of calculus.

5. How to reverse the computation of derivatives or slopes for the foregoing functions. See a calculus text. 

6. How does an understanding of decimals and error control in decimal based computation provides a context for the discussion of limits. See Volume 3, Why Slopes and More Math, Chapters 14

7. How to evaluate limits algebraically. See Volume 3, Why Slopes and More Math, Chapters 15.

8. How are slopes or derivatives approximated and defined in the limit by approximations. See Chapter 15 in Volume 3

9. How is velocity approximated and defined in the limit by approximations. See Chapter 16 in Volume 3

10. How is area approximated and defined in the limit by approximations. See Chapter 17 in Volume 3.

11. How do the parameters a and b affect the computation of areas under a curve y=a f(bx).

12. How does anti-differentiation (the reversal of slope computation or differentiation) yield formulas for area under curves. See Chapter 18 Integration in Volume 3

13. How are logarithms defined by the area under a curve. See Chapter 19. Logs and Powers in Volume 3

14. How are exponentials may be defined via the inverse to the natural logarithm. See Chapter 19. Logs and Powers in Volume 3

Real Analysis HOW-TOs

See Volume 3, Why Slopes and More Math, Appendices  A to G for  details for almost all the following.

1. How is continuity related to unlimited error control.  See Volume 3, Why Slopes and More Math, Chapters 14

2. How is error control related to the study of inequalities. See Volume 3,  Chapters 14 & 15 and  Appendices  A to E

3. How an understanding of decimals and error control in decimal based computation leads to proofs for theorem normally stated without proof in calculus and to a starting point for real analysis. See Volume 3,   Appendices  A to E

Geometry HOW-TOs

Modern mathematics avoids a dependence on diagrams through analytic geometry: the use of coordinates and sets of ordered pairs to represent or mathematical model geometric objects and through the use of algebra  to arrive at conclusions. 

The drawing or careful construction of general diagrams in geometry to arrive at conclusions is suspect due to subtle errors that may occur and mislead.  Yet the drawing or construction of figures or general diagrams is second-to-none in giving geometric context & representation for real and complex number numbers & their arithmetic, and is needed or is  convenient for the application of mathematics.

From Triangles to Parallelograms

1. How do you  construct small  triangles in the plane and on the sphere using the SSS, SAS and ASA methods? Hints: Use taut strings to measure lengths and to provide straightest possible path. Use a protractor to measure angles. Observe the sume of angles in a triangle with base on the equator and vertex at the North Pole is greater than 180 degrees.

2. How  and when do SSS, SAS and ASA triangle construction methods or measurement lead to isometric or congruent triangles on the plane or sphere.

3. When does the ASA triangular construction method fails in the plane but not on the sphere? Hint: when the sum of the angles exceeds two right angles (180 degrees)

4. How does the failure of the ASA method in the plane (sum of angles = two right angles) leads to a criteria for and characterization of parallel lines.

5. Why are two lines in the plane are parallel if and only if alternate angles for a transversal are equal, if and only if corresponding angles are equal; if and only if the sum of interior angles on side of a transversal equals two right angles?

6. Can you explain or suggest why with the aid of a diagram, if two distinct  lines are both parallel to a third parallel then they are parallel to each other?

7. Can you explain or suggest why with the aid of a diagram, if two distinct  taut line segment are both parallel to a third line segment, parallel then they are parallel to each other?  Line segments are considered to be parallel if they are segments of parallels or segments of a single taut line.

8. For a plane quadrilateral with opposites non-intersecting, does equality of length of opposite sides  imply opposite sides are part of parallel lines, that is parallel?

9. For a plane quadrilateral with opposites non-intersecting, why does opposite sides parallel imply opposite are equal

10. For a plane quadrilateral with opposites non-intersecting, how does ne pair of opposite sides equal in length and parallel imply the other pair of opposite sides are equal in length and parallel?

Vectors in the plane - head to tail addition

A vector in the plane or sphere is given by a taut line which begins at one spot, called the tail, and ends at another spot, called the head. Visit the site area Complex No,s, Trig & Geometry for answers to some leading questions in this and the following sections on Geometry.

1. How is a succession of movements, one at a time, and one after another, on a plane,  sphere or map summarized or approximated by a sequence of vectors (movements) with the terminal end of one at the initial end of another?

2. How does the  head-to-tail addition of vectors (or movements) linked to the taut line approximation of movements on a map, plane or sphere?

3. Why is the head-to-tail addition of vectors associative?

Vectors and Parallelograms in the plane

1. Equality of Vectors in Plane or Along a Line: Two vectors in the plane are said to be equal when and only when the quadrilateral formed by joining the tails together by a taut string, the heads together by a taut string, has opposite sides non-intersecting and of equal length. Explain why the resulting quadrilateral is a parallelogram when  the  two vectors are not collinear.

2. Parallelogram method for Vector Addition: Two non-collinear vectors which share a common tail (origin or initial point) on a plane or sphere may be added together by taking a taught string whose length is sum of their lengths, and using it to construct a quadrilateral with opposite sides non-intersecting and of equal length. Explain why in the plane case, the resulting quadrilateral in the plane is a parallelogram and explain how opposite sides may be identified with pairs of vectors that are equal

3. Show the Parallelogram method for vector addition is commutative.

4. Explain how the addition of collinear vectors which share a common tail may be viewed as a limiting case for the non-collinear situation.

5. How is the parallelogram method for Vector Addition related to the Head-to-Tail Method.

6. Show if vector A shares a tail with B, vector C shares a tail with D, vector A=C and vector B = D then the parallelogram sum of A and B equals the parallelogram sum of C and D.

7. Show that the parallelogram method for vector addition is associative.

Vectors, Scaling and Coordinates

1. Scalar Multiplication: Say how a vector may be multiplied by an unsigned whole number, by a proper or improper fraction and in the limit, by a irrational number.

2. Introduce assumptions about similar triangles in the plane, to explain why scalar multiplication distributes over vector addition in the plane.

3. Introduce a right-hand coordinate system in the plane, show how vectors with tails at the origin may be viewed as sum of components parallel to the coordinate axises.

4. Introduce polar coordinates for the right-hand coordinate system in the plane, and explain how a vectors with tails at the origin may be identified with polar coordinates of its head.  Polar coordinates should have non-negative angles  less than 360 degrees.

5. Explain how scalar multiplication of a vector may be defined or represented by an action on polar coordinates of its head.

6. Explain how  rotating a vector by through an angle is defined or represented by an action on the polar coordinates of its head.

7. Explain how rotation through an angle distributes over the parallelogram method for the addition of vectors with tails at the origin of a right-hand coordinate system.

Complex Numbers in the Plane

1. Identify each point in the plane with its position vector - the vector with tail at the origin and head at the point.

2. Choose a unit of length in the plane.

3. Multiply a pair of points in the plane through  the rule: add the angles of the position vectors to get the angle of the position vector of the  product, multiply the number of unit lengths in their length to get the number of units in the length of the position vector of the product.  The latter gives the polar coordinates of the product and hence defines the product.

4. Explain why addition of angles is commutative.

5. Assume multiplication of lengths (number of units in the length) is commutative.

6. Show that a product of one point by another may be equals a rotation followed by a scalar multiplication.

7. Show that multiplication by a point or position vector in the plane distributes over the addition of position vectors in the plane.

8. Introduce complex notation for points in the plane, and derive the properties of complex numbers and real ones too from the previous considerations and assumptions about arithmetic with unsigned numbers.

9. What are the arithmetic properties of complex numbers?

10. How does multiplying a complex number by its conjugate imply the Pythagorean Theorem (provided of course, the Pythagorean Theorem has not been used to develop the properties of complex number)

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