A. Core Sequence  

1. Abuse and Proper Use of the Equal Sign -  Mistakes to Avoid
2. Real Number Properties, algebraically described -Real Numbers - It is not enough to state the properties of real numbers algebraically,  how to apply the algebraic description needs to be explained. This lesson raises the issues, but does not fully address it.
3. Square Root Simplification.  In symbolic arithmetic, decimal approximation are avoided and numbers are represented exactly with (i)  whole numbers and fractions, for example 18, ¼ , ½ and ¾, the fractions being in reduced form (a first cosmetic convention); with (ii) constants (special symbols) such as as π (pi) and e (the natural number); and with (iii) roots with the latter being expressed in a reduced form - an second cosmetic convention. 
4. Absolute Value |x| - To get the absolute value of a signed number, drop the sign prefix.  Or, the absolute value of positive number is the number itself; the absolute value of a negative number is the additive inverse of the number; and finally, the absolute value of zero is zero. 
5. Rename an inequality sign.  Here is a short essay on why the label greater than sign for the sign > should be replaced by the label more than sign.  The first label agrees with the latter in the case of unsigned numbers, but its common use (a comparison of magnitudes) differs in the case of signed numbers: Integers, Rationals and Reals.
6. The i - Explanation with algebraic derivation of the algebraically described properties.
7. Abs Value Eq'ns & Inequalities - Solving Equations of the form |x - a| - examples and a few exercises. 
8. Rectangular Coordinates in 1, 2 & 3 Dimensions- How to Locate points.
9. Distance Formulas in 1, 2 and 3D for  Points on the Line, in the Plane and in Space - Explanation that includes use of the absolute value sign and employ the Pythagorean Theorem - see  Chinese Square Dissection Proof
10. Shortest Path Problem -  Geometric Solution for a ball or beam of light reflecting off a wall. Why is the angle of incidence and reflection be equal?
11. Triangle Inequality.- Geometric explanations of the triangle inequality | x - y| < |x| + y| in the line and in the plane, if not in space. Algebraic explanations based on the properties of coordinates are not provided here.
12. Three Operations with Rectangular Coordinates: Addition, Subtraction & Real Multiples of Points in the Plane
13. Polar Coordinates and Scalar Multiplication (1) Polar coordinates (R, a) can be used to locate points [a, b]  in the plane. Polar coordinates (R, a) for points [a,b] in the plane can also be measured. So rectangular and polar coordinates are interchangeable at least through measurement. Trigonometry provides further  methods. (2) We observe for k > 0 that  points with rectangular coordinates [ka,kb] have polar coordinates  (kR, a) when the point with rectangular coordinates [a,b] has polar coordinates  (R, a).(3) Multiplication by k > 0 distributes over rectangular methods for adding points in the plane.  k ([a,b] + [c,d]) = k[ a+c, b+d].

14. Radian Measure of Anglese (appears in trig because of calculus) Radian Measure of angles is employed in Calculus as tje introduction of radian measure simplifies formulas in for derivatives of cosines and sine, and their approximations by sequences of polynomials.  more precisely power series expansions. Radian Measure is a proportionality constant given by  the ratio of arc-length to radius for arcs that span the same central angle in concentric circles. 

15. What Are Vectors. The assumption of coordinates and the properties of real numbers provide the simplest starting point for a thought-based development of vectors and their properties.

16. Arithmetic With Vectors. Addition, Subtraction and Scalar (Real) Multiples of Vectors based on Rectangular Coordinates.

17. What is a Dilation? Short Definition

18. What are Translations Short Definition

19. Navigation on Maps - Application of Vectors. The foregoing lesson showed how to add vectors with the aid of rectangular coordinates. This lesson shows how to add vectors geometrically.

20. Four (presumably  equivalent) ways to add vectors: Head-to-tail, tail-to-tail with a parallelogram, with components, with coordinates.  Plus discussion of how multiplying by positive numbers distributes over addition. 

21. Polar Coordinates and Rotation of points and sums of points in the plane. This lesson starts with the polar coordinate introduction of rotation in the plane and then shows how rotation of points distributes over addition. 

22. Complex No. Intro. Points in the plane added using rectangular coordinates and multiplied using polar coordinates provide the simplest way to defined complex numbers and to introduce complex number arithmetic. With real numbers identified with complex numbers with imaginary part zero, see how the law of signs follow from or agrees with the polar coordinate rule for product calculations; see how to calculate square roots of negative numbers,  learn about complex conjugates and reciprocals (multiplicative inverses) of complex numbers.  The polar coordinate rule for complex numbers (add angles, multiple lengths) is consistent with the product rule (multiply signs, multiple lengths) for real & rational numbers, and integers indicated the arithmetic video site folder. 

23. Properties: The foregoing lessons showed how multiplication by positive numbers and how rotation distribute over the addition of points or arrows in the plane. That implies the distributive law for complex numbers. All but the distributive law for complex numbers follow follow from corresponding properties of real numbers applied to polar and/or rectangular coordinates. The lesson describes a second, rectangular or real or imaginary part, method to form products of complex numbers. The fact that two different ways to multiply gives the same result has many consequences for unit-circle related trigonometry.

24. Pythagoras Thm, New Proof.  Suppose a point Polar coordinates (R, q) has rectangular coordinates [a, b] = a + ib  in the plane.  The product P of  (a+ib) = (R, q)  with its conjugate (a+ib) = (R, -q) can computed with polar coordinates to obtain P = R2 or it can be computed with rectangular coordinates to obtain  P = a2 + b2. That implies the Pythagorean identity  R2 = a2 + b2.  The foregoing represents another proof of the Pythagorean if in the preceding development of complex numbers and their properties, the distributive law for dilatations (positive real multiplication)  over point addition is implied by similarity arguments and not based on the Pythagorean theorem.  There-in the lies the first consequence of the equality of polar-coordinate and rectangular coordinate method to form complex number products in the plane. 

25. Trig for All Angles, Acute or Not, with aid of a  unit circle. In the earlier discussion of right triangle trigonometry in Euclidean Geometry without or before coordinates,  we saw how similarity implied the computation of sine and cosine for acute angles only depended on the angle and not on the scale or size of the right triangle drawn for the computation. All such triangles were similar. This lesson with the aid of similarity considerations,  shows how to compute sine and cosine for acute angles with the aid of a unit circle. Then it observes the computations work for angles that are not acute as well.  This leads to a unit circle based method for computation of sine and cosines for angles, acute or not, which is consistent with (agrees with)  the similarity-justified right triangle approach for acute angles. 
    
26. Trigonometry and Complex Numbers. In college engineering and physics, students are shown how to handle trigonometric calculations and how to derive trigonometric identities quickly with the aid of complex numbers, and the exponential or cis functions.  Trigonometry allows the rectangular coordinates (real and imaginary parts) to be expressed in terms of polar coordinates.  But there are two ways to multiply points in the plane, one with polar coordinates and another with rectangular coordinates.  The two different ways give different trigonometric expressions for the same product. Trig identities follow immediately from the equality of those expressions, or more precisely, their real and imaginary parts.  See B7 to B10 and Chapter 24 below.

Course Design Recommendation:  Classes which covers unit circle based Trigonometry courses should begin with a  development of vectors and  complex numbers, or the alternative development in the site folder Euclidean Geometry with Complex No.s, to provide students an easier and faster route for developing unit-circle relate trigonometric skills and knowledge required for calculus. The inclusion of complex numbers in this route would make discussion in calculus of partial fraction decomposition easier or more acceptable to students by removing the mystery that has surrounded the square root of negative numbers.  

B.  Straight Lines (9 lessons)

Prerequisite: a good command of Solving Linear Equations  

The following lessons here and elsewhere.

1. Numerical Introduction
2. Slopes and Lines - Deriving Equations
3. Perpendicular Lines. Understand why the slopes of perpendicular pairs of slanted lines (lines not vertical not horizontal) are negative multiplicative inverses (negative reciprocals) of each other. 
4. Three Equations Forms -  point-slope, slope intercept and two-point forms.  (Symmetric form not covered). 
5. From Equations to Lines Numerically - Here is the algebraic viewpoint of equations for straight lines.
6. Intersection point of lines- Two lines are parallel or they intersect. Learn how to recognize parallel lines from equations for lines, and learn solve systems of linear equations to find intersection points. 
7. Exercises - Here is an exercise set. 
8. Summary
9. Lessons Elsewhere. Do not put all eggs in one basket.  

give a thought-based development of basic skills and concepts.

C. Polynomial Addition, Subtraction, Multiplication & Division (5 lessons)

1. Distributive Laws and Product Calculation - a simple and effective approach to understanding and explaining distributive laws for multiplication, or methods for expanding products in which the factors are sums. 
   

2. Column Methods for Products This lesson continues the earlier one to  column method for their expansion or calculation of products of sums.
   

3. Area Development of Multiplication Rule for Polynomials This lesson proceeds from the area view of products of sums and the distributive law for their expansion to area view of products of polynomials and how column method for their expansion or calculation. Then it introduces column methods for addition and subtraction of polynomials.  
   
   Learn More: The site folder Number Theory. includes a similar development (html-based)
   
   Multipling Wholes
   Distributive Law  Preamble
   Distributive Law for Wholes
   Consequences 
   Multiplication Methods
   
   of column, place-value methods for decimals. The site folder Arithmetic Videos  offers 11 lessons on  multiplication methods and theory for decimals with theory and exercises.
   

4. Long Division with linear divisors - 3 animated examples, with checks based on polynomial addition and multiplication. Long division itself requires mastery of multiplication and subtraction of polynomials. Theorem: A polynomial p(x) has the form q(x) (x- a) for another polynomial q(x) if and only if p(a) = 0. 
   
   Learn More:  The site folder    Number Theory.also offers html-based page on Division Methods for Decimals .  The folder Arithmetic Videos  offers (12 lessons) long division methods and theory for Decimals -  whole numbers only,  
   

5. Long Division of  with quadratic factors - Here is an example with an exercise to try.

Remark: For pedagogical ease of development, the derivation of  multiplication methods for polynomials is valid only when the polynomials have non-negative coefficients and depend on a non-negative variable x > 0. A fuller, more rigorous treatment would require mastery of mathematical induction. Moreover in the above and most likely in all high school developments of addition, subtraction, long division and multiplication,  generalized associative laws for addition and multiplication are more likely tacitly employed and not explicitly mentioned. That flaw is continued in the above lessons. 

D. Quadratics in 10 Steps  

1. Graphing Quadratics from Standard form: Numerical Examples and Exercises
2. Discussion of the standard form y = a[(x-h)²+k] and location of zeroes 
3. Factoring Quadratics -(i)  Geometric Demonstration of 
     (x+A)(x+B) = x²+(A+B)x + AB with
   
   (ii) Factoring quadratics of the form x²+qx + c "by Inspection" when q and c are integers, using the prime decomposition of | c| to generate all integer factorizations of the form c = AB in the hope that for some pair of integer factors  A+B = q.
4. Difference of Two Squares etc
   .  
   • Column Multiplication Method yields (C+A)(C-A) =  C² - A²  
   • Zero Product Rule:  If a product equals zero then at least one of its factors equals zero - that is equivalent too: If all the factors in a product are non-zero, then the product is non-zero

   • Two ways to solve x² - A²  = 0 - using square route or by factoring & applying the zero product rule.

5. Completing the Square:  (i) How to convert a quadratic a x²+bx + c into standard form a[(x-h)²+k] with examples.  Note: Completing the square may lead to a difference of squares (the case k < 0); or a sum of squares - the case k > 0; or a perfect square - the case k = 0.  In the first case, how to factor the difference of squares leads to the factorization of quadratic expressions and to the solution of quadratic equations. If k > 0 then k = A² where A = sqrt(k) and sum of two squares result.  If -k > 0 then -k = A² where A = sqrt(-k).

6. Factorization, Arithmetic Approach. The quadratic formula for finding roots of expressions  comes follow from (i) completing the square and then (ii) factoring if (i) results in the difference of two squares. Examples follow to numerically illustrate completing the square and, if possible, factoring the difference of two squares. The quadratic formula itself (the algebraic shorthand description of all the numerical examples here)  is derived in the next lesson.

7. Quadratic Formula - a full development.: Deriving the Quadratic Formula and factoring quadratics  in three steps with extras: (i) The Reducible or Irreducibility question; (ii) using the discriminant to count roots.  

8. Finding Coefficients - The quadratic formula and all associated formulas for locating maxima and minima of quadratics; and expressions for quadratics - standard or not, will be used forwards and backwards.  Your reference for the forward and backward use of formulas, algebraically and numerically,  is chapter 14 in site volume Three Skills for Algebra.

9.  Problems with Quadratics - Hunt for examples of the following.

   • Solving Systems of Equations - one quadratic, one linear.
   • Examples from Physics.
   • Constant Velocity Motion
   • Quadratic in Time implies Constant Acceleration
   • Constant speed and constant acceleration motion (enriched topic)
   • Examples from Economics (do, but view with suspicion)
     
10. Exercises with Quadratics - 8 little problems to keep you busy for a little while. These problems not cover all types likely to be met in mathematics or science (as in 9 above)

The Big Summary Page provided the initial introduction to quadratics.  It may still be too long for an introduction, but it could be used for as a checklist to review and consolidate the 10 steps below. 

Links to lessons elsewhere

• www.mathisfun.com offers the following webpages: Quadratic Equation Solver,  Completing the Square and Derivation of Quadratic Formula. 
• www.purplemath.com offers a multi page lesson on  Solving Quadratic Equations

E:  Monoticity & Sign Analysis. 

The question of where is a function y = f(x) increasing, decreasing, positive and negative answered in the case of functions f(x) that can be factored  - an intro .  (Prep for calculus)

• Sign, Zero and Monoticity Analysis
• For a Curve or function
• For A Quadratic (factored)

• For a Cubic Factored
• Uni-Sign Functions etc. (Functions which do not change sign over an interval)

Part I
Functions Before Sets
(Cover First)

Part  II
Functions with a
Set-Theoretic Focus

Methods to Define Computation and Assignment Rules :

1. Using Formulas (with use of function notation to indicate dependence of one number or quantity on several others. (math 436)
   
2. Using Arrow Diagrams, Tables and Sets of Ordered Pairs (listed or plotted) - functions with finite domains. (math 436)
   
3. Using Curves and Infinite Sets of Points in the Plane - When the vertical rule holds, a set of points or curve in the plane can be used to define a function f(x) via the vertical line method.  Note:  Graphing a function f gives a set of points or curve in the plane for which the vertical line method for computing a function yields the same function f. (math 436)
   
4. Functions with Infinite Domains - a few exercises (math 436)
   
5. Properties of Functions: or Definitions & Examples to introduce and describe:  Domains, Ranges, Injectivity (1 to 1) or not (many to 1), Onto or Surjectivity, Monoticity - where are real value functions of a single real variable increasing, decreasing or constant. Tools: Interval notation and symbols for there exists and for all. More examples given by calculus preview - geometric & algebraic (Material for calculus, if not an enriched 436 or 536)
   
6. Sign, Zero and Monoticity Analysis - Four Geometric Starter Lesson  or Exercises builds algebraic thinking skills.

Curve or  Set Viewpoint of Functions and Relations

In the foregoing examples, you have seen sets appear in the description of the domains and ranges of functions, and in the definition of function using sets of ordered pairs. The latter implies or suggest the Set Based View and Codification of what is a function in site pages with the following ideas. (Here are more ideas for math 436).

1. Set Existence and Construction (technical starting point)
   
2. Interval Notation. Next (?) see Domains and ranges for a zoo of functions using interval notation.
   
3. Assignment and Computation Rules without & then with ordered pairs.
   
4. Concept of a Relation, a Set-Based Codification and Generalization.
   
5. Why call a set of ordered pairs a relation? Numerical Exercise Included.
   
6. Source, Target, Domain and Range Set for functions and relations - plus Definition of subjection, injections and  bijections - set viewpoint
   
7. Injectivity of Real Valued Functions - injectivity, one-to-one, two-to-one, many-to- one, or not one- to-one.
   
8. Sign Analysis, Zero Analysis, Where are functions positive, negative or zero?
   
9. Monotonicity Analysis: Where are functions increasing, decreasing etc.Why strictlyincreasing and strictly decreasing functions are one to one, that is, injective.
   .
10. Extrema or Max-Min Analysis Where do they have their greatest and least values. What are minima and maxima.
    
11. Exercises with Formulas and Graphs - Numerical Experience (!)
    
12. Domains and ranges for a zoo of functions using interval notation.
    
13. The absolute Value Function (Qc math 536)
    
14. Functions Revisited (for teachers, if not students)

Functions - Part III

Or, Inverse Functions and Their Definition 

The set or curve in the plane viewpoint (Route 2)  has advantages in discussing the backward use of formulas y = f(x) where instead of calculating or obtaining  y from x as in the forward use, we try to obtain x from y.  Remember that when you meet the discussion of inverse functions.

A curve in the plane may be regarded as a set of points or ordered pairs.  The graph of a function f even when the function f or f(x) is introduced by other means  may be used for calculation of y = f(x), that is the forward use of the function,  and for the backward question of how x depends on y when y = f(x), y is given and x is to be computed.  This backward question provides a context for the following.

1. Using the Horizontal Line Method - Part  I . Just as there is a vertical line method for defining and calculating functions, there is also a horizontal line method.  Here if S is a set of points for which the horizontal line method can be used to compute a function y = f(x) then there is a twist, the graph of the function f
   
   graph (f) = { (a,b) | (b,a) belongs to S}
   
   is equal to a "transpose" of the set S in which the first and second coordinates are swapped. (Secondary V Subject).  Note: In earlier courses,  reflection of a point (a,b) across the line y  = x gives the point (b,a) with coordinates interchanged or swapped.
   
2. Using the Horizontal Line Method Part  II. If we apply the horizontal method to all or part of the graph of a function y = f(x) we may obtain another function h such that z = h(y) implies y = f(z), and perhaps, vice-versa. (Needed for discussion of inverse trig functions in calculus or secondary V mathematics)
   
3. Several more ways to define Functions - A brief glimpse of the future if you are in secondary IV or V, and glimpse of the present or past if you are studying calculus. 
   
4. Algebraic Calculation of Inverse Functions: .  Suppose y = f(x) where f(x) is a function given by a formula of some type.  The inverse function
   
    f^--1(x) = g(x) 
   
   if it exist, should have the property that g(f(x)) = x for each x in domain of f and also  f(g(w)) = w for each w in the range of the original function f. Now  f(y) = x may imply y = h(x) for some unique function h(x) or it may give more than one formula or solution  h(x) for y. In the latter case, the function f is not one to one.  In the former case, f is one to one,  f^--1 exists, and f^--1(x) = h(x). 
   
   Proof that f^--1(x) = h(x). :  If  x = f(h(x))   then by substitution  f^--1(x) = f^--1( f( h(x)) = f^--1(f(y)) = y = h(x) 
   
   Remark: If f(y) = x implies an equation linear in y (with the y coefficient nonzero) then y will be uniquely determined. If f(y) = x implies an equation quadratic in y (or more generally with a polynomial dependence on y) then their could 2 or more formulas h(x) for y, one formula per real root of a quadratic or more general polynomial in y. 

         Remark: The treatment of item 4 consists of its treatment here. There is as yet no webpage for it. 

The foregoing lessons  1 to 4 provide a basis for defining inverse trigonometry using parts of the graphs of trig functions - the restriction of the latter to intervals to obtain functions that are one-to-one (invective).  The twist, reflection across the line y = x in the Cartesian plane, connects the graph of a function and the graph of its inverse.  And in calculus, the area under the curve definition of the natural logarithm leads to a one-to-one function. Its inverse is the exponential function.

A. Core Material
B. Straight Lines
C. Polynomials
D. Quadratics
E. Zeroes & Monotocity
F. Functions

Extras

www.whyslopes.com   [ Next ] [Top of this Page]   

Road Safety Message  Do not walk on a road with your back to the traffic - rule of thumb
Please report by email,  errors in mathematics or grammar or terminology to site author
If an arithmetic topic you need is not covered in site pages,  report that as well. Topics in most demand
will be covered first in site growth.