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For Montreal Students:  Head Start Math Tutoring  is available from the site author. 

YOU are better than YOU think. Show yourself  how:

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 Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Do not leave here without it -  Logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

After logic,   (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.

George Orwell:   Is it nonsense for arts and disciplines based on and respected for carefully  mastery of rules and methods, alone and combined, to face education reforms based on the supposition that mastery of rules and methods is not a sign of intelligence.  Would you like to rewrite 1984 to include that angle?

Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit twiddla.com to set up whiteboards to work with the webpage of your choice.

Precalculus sites mathsisfun & purplemath are visually more appealling than this one.  Do not go. 

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

Secondary I Mathematics
year of fractions and measurement,
skills and sense consolidation

First year high school mathematics for should review, consolidate and extend measurement and fraction skills and sense.  Part I below is cryptic. Part II expands on most details.  The following implies standards for instruction by identifying skills and topics whose mastery should be advocated, achieved or developed and then maintained in secondary I and later.

Students in the first year of high school may come with a weak to non-existence command of the times table (addition table too) and with a weak to non-existence fraction sense and abilities.  The most important service of first year mathematics in high school is to consolidate fraction sense and skills.

The ability to follow a multi-step process in a repeatable and reproducible manner, modulo some accidents, is a sign that the students master further multi-step operations in and outside of arithmetic. That is the skill or intelligence we seek. Start emphasizing in it in arithmetic. Calculators betray students by allowing them to skip a first example of a multi-step process in which accuracy is demanded at each and every step. The last topic, statistics,  should be exploited as much as possible to develop and reinforce fraction skills and sense. 

In place of a complete thought-based development of mathematics, each secondary course in mathematics should aim to show student how to use rules and patterns, one at a time and in combination, one after another another to arrive at numerical results or further rules and patterns in a repeatable and reproducible manner.  The ability to combine rules and patterns to arrive at or justify further ones should be presented in class even if not required of students to illustrate to the thought-based development and linkage of skills and concepts where some rules and patterns are assumed (learnt by rote if need-be) and others derived.

Part I: Aims and Methods in Brief
The Main Ideas

A  brief are skills and concepts that should be mastered before or in the first year of high school mathematics follows. Use the list to develop, verify or consolidate skills and concepts.

• Decimals: Place Value of Digits, individually and in groups of three (non-UK) or six (UK method), for Decimal Representation of Natural Numbers. Column Methods for addition, subtraction, multiplication and long division of whole numbers. Check these skills before covering fractions. And mention the rules for recognizing whole number multiples of 2, 3 and 5 from the decimal representation of whole numbers.
  
  Teach students how to read the decimal representation of whole numbers, for example 34, 567, 898, 789 backwards in groups of three. The latter example for instance can be read backwards as 789 ones, 898 thousands, 567 millions and 34 billions outside of the UK. People in the UK would read the latter backwards in groups of 6. Reading aloud a 24 or 27 digit number backwards in groups of three or six could be entertaining in a mathematics classroom. Reading backwards silently or alound helps with the recognition of place value for each group of there digits in the decimal. It further puts last the more important element of the number mentioned, so the largest element of the decimal is  more easily remembered. After reading backwards, Students and then teachers can than read the decimal value forward in groups of three or less.

  In the foregoing or before it, have students practice filling in tables of sums and products of all pair of numbers from 0 to 10 or 12.  The more students protest, the greater is their need to do these exercises one to three times.
  
  Rules for recognizing multiples of 2, 3, 5, 7, 9 are important aids to the simplification of fractions.
  

• Fractions: Fractions or ordered pairs of whole numbers numbers a and b written in form a/b)  Decimal Representation, Percentages.  Connection of ratios to fractions.  Efficient ways to add, subtract, multiply and simplify  fractions. Use of Prime Number Decomposition for gcd and lcd. Decimals and percentage and ratios as fractions.
  
  For the sake of an operational command of fractions:  Students who have seen fractions before can be given an operation command of fractions through the following steps:  (i)  Learn how to simplify fractions by canceling common factors in enumerators and denominators; (ii) Learn how to multiply fractions but with an emphasis on postponing multiplication in favor of  factoring the numerator and denominators of products in order to cancel and simplify; (iii)  Learn how to divide fractions by turning divisions into multiplication by a reciprocal, and then applying the efficient product simplification methods in step; (iv) learn how to add and subtract fractions with like denominators and how to simplify the sum; (v) learn how to add and subtract fractions with unlike denominators and the role of least common denominators in reducing the amount of simplification needed in sums. 

  In the foregoing,  prime decompositions can be introduced to aid simplification and to aid the computation of least common denominators and greatest common divisors. Teach students to look for factors of whole number among those primes whose square is less than or equal to the whole number in question. If none those of those primes are factors, the whole number in question is prime. Calculators and knowledge of all primes less than 50 are sufficient to quickly generate the prime number decomposition of all numbers < 2500.
  
  Decimal Fractions: Decimal representing proper fraction, for example 0.567,782,780 outside of the UK can be read forward in groups of three. The example reads 567 thousandths, 782 millionths and 780 billionths. For mixed numbers with fraction part equal to a decimal, the whole number part can be read aloud backwards and then forwards in groups of three or six as you like, and the mixed number, whole number part and fraction part can be read forward in groups of three or six as you like.   For example the whole number part of 
  
  34, 567, 898, 789.567,782,780
  
  can be read backward as above, and then read forward as 34 billion, 567 million, 898 thousands, 789 ones and 567 thousandths, 782 millionths and 780 billionths.
  
  Remark on Developing Algebra and/or Fraction Sense and Skills: The development of an operational command of fractions needs to be accompanied by fraction sense - what is a fraction.  The site area Fractions,  Ratios, Rates, Proportions   & Units includes material to develop fraction sense and to give a thought based comprehension of all fractions operations from simplification to addition described above. But the Preparation for Algebra below in requiring fractional operations on line segments or stick diagrams may consolidate and if need be develop sufficient fraction sense for high school students while also developing algebra skills.
  

• Formula Evaluation and Evaluation Format. Student are given formulas for calculating areas and perimeters of figures.  To ensure proper use of the equal sign and to tell students how to show their work, how to reason and how to present or communicate in mathematics  have students employ the following steps in evaluating a formula with or a geometric meaning.  Require the same steps, except for the drawing of a diagram, for problems without a geometric meaning. 

• Preparation for Algebra:  Linear Equations and Stick Diagram, How to Solve Essentially One variable systems, Solving Word Problems. Proper Use of Equal Sign.
  
  Compare problem solving in mathematics with solution of jigsaw puzzles -- there are methods to speed the solution of jigsaw puzzles, but trial and error still required. 
  
• Statistics:  Collecting Data. Interpreting and Creating Diagrams, Charts and Tables of Data including line, bar and circle graphs.  Here is an opportunity to Check and emphasize fraction sense and operations while covering statistics.  Talk about critical thinking, how a single number (average, median, range) gives an ideas about data but does not describe it fully.  Talk about faulty impression left by different visual presentations of data. How to best present the data for one end or another.  The treatment of statistics is not essential. 
  
• Integers: Four Operations of addition, subtraction, division and multiplication. Law of signs, Geometric or Displacement Significance of Addition, Subtraction as adding the negative or additive inverse. Order of Operations.
  
• Measurement Skills and Concepts.  Students should be able to measure lengths and angles with the aid of rulers and protractors. Students should learn that the zero point on a ruler need not be an end of the ruler or tape measure. 

• Ruler and Compass Constructions: Students should learn the Side-Side-Side, Side-Angle-Side and Angle-Side-Angle  methods to construct triangles from given data and to duplicate other triangles.  They may see that the duplicated triangles are isometric to the original via a correspondence - a matching, pairing or mapping that associates vertices and hence measures in different triangles. Then corresponding sides have equal length measure and corresponding angles have equal angle measure. Following that they may see two triangles constructed from the same data with the Side-Angle-Side, Angle-Side-Angle or Side-Side-Side methods can be considered duplicates of each other, and so are isometric.

  The foregoing provides a first path to arrive at the Side-Side-Side, Side-Angle-Side and Angle-Side-Angle isometry or congruency properties or postulates (assumptions) of Euclidean Geometry.  Quebec students will see in secondary IV an alternative path based on the assumed properties of rotations, reflections and translations,

  Parallel Lines and Transversals: Comments in site pages about when SSS, ASA and SAS methods fail or work in unexpected ways point to a context for a later study of Euclidean geometry and a context for the discussion of when two lines will intersect or be parallel.

  Angle and Line Segment Bisection, etc. Students may also meet ruler and compass methods with justifications included (? ) for bisecting angles and line segments, and for dropping or drawing a perpendicular to a line from a point for (i) the point off line and for (ii)  point in line. Methods may given by rote - here are the constructions and apply them, or explanations of why the methods work may be based on the postulates.

  Students may met isosceles and equilateral triangles, and (enriched material)  see by the postulates that these triangles are isometric to their duplicates via two or three difference correspondences, and students may learn about the axes of symmetry for isosceles and equilateral triangles.   They may   see a proof that a triangle has two sides of equal length if and only if the triangle has two angles of equal measure. They may also learn (?) or see why a triangle is equilateral if and only if it is equiangular. The why here indicates enriched material.

  Remark A: The  correspondence between vertices of triangles here is a foretaste of the discussion of arrow diagrams and rules of correspondence (functions) in Quebec secondary IV mathematics.  The latter should be linked to the earlier experience while secondary I program could include a few remarks on mappings, pairings and arrow diagrams to introduce the notion of correspondence between the vertices and sides of a triangle employed in high school geometry, secondary I onwards.

• Translations, Rotations and Reflections (Transformation Geometry). With the aid of graph paper, if not coordinate systems in the plane, students may see how to translate, rotate and reflect points, triangles, circles and further figures in the plane. (i) The notion that two triangles are isometric if one is the image of the other under a translation, rotation or reflection may appear. (ii) The notion that two circles have the same radius if one is the image of the other under a translation, rotation or reflection may appear.  The two notions (i) and (ii), or (i) alone,  supports labelling translations, rotations and reflections being  as rigid body motions. 

Sequencing and Timing

 The sequencing and timing of topics, that is skills and concept,  is not important.  Skill development, perfection and retention are the aims.

In most tests and assignments include a few questions or problems to remind students that they are responsible for all or course material. Electronic calculators can be used to aid exact calculations with whole numbers and fractions but without lessening skills that would be required if no electronic calculators were allowed. 

In different school systems, what is taught in mathematics is more or less the same, but the order or arrangement may differ. So some items may not be present in your first year high school mathematics course, but those items are easily understood (we hope), their mastery may improve your study skills and mathematics comprehension, and sooner or later, students should meet them in the mathematics courses of your school system.  Sooner will do no harm, provided that does not distract you  and your students from other duties.

Mastery of prime numbers is not critical for the more important fraction skills and sense at the secondary I & II level where numerators and denominators are small, say less than 100 say in magnitude. That being said, the discussion of prime numbers and relatively prime numbers, mastery of arithmetic with fractions and mastery of long division provides a model for the later secondary IV to college  level  treatment of polynomials, their addition, multiplication and factorization; and the addition and simplification of their ratios- fractions with numerators and denominators given by polynomials. This treatment of polynomials in turn can be seen as preparation for calculus.  Mastery of prime numbers is an investment for further instruction in mathematics.

Present-day course design and delivery should be based on a knowledge of why topics appeared in earlier course designs. De-emphasizing topics by omitting them or introducing further ones may undermine mastery of key skills and concepts. .

Part II: More Ideas

A. Proper Use of the Equal Sign:  

When we write an expression of the form a = b = c where a, b and c are numbers or completed expressions, the proper use of the equal sign is to say or imply that a, b and c all have the same value. Some students will write

• 3 x (4 x 5) = 20 = 60
  
  instead of 

• 3 x (4 x 5) = 3 x 20 = 60

The first expression is wrong. The students mean 4 x 5 = 20 and then 3 x 20 equal 60, but the student has written 20 = 60. While it is true that 4 x 5 = 20, the expression 3 x (4 x 6) fully evaluated  has the value 60 and not 20. Here is another abuse that may occur in learning and teaching algebra. 

B. Geometric Quantities from Measurement,  Calculation or both

• Arclength: Students should learn how to add lengths of line segments and arcs of circles to obtain formulas and numerical values for perimeters. 
• Formula Use: Student should be able to calculate the area of simple regions, say triangles, rectangles, circles, half-circles, quarter-circles and trapezoids alone. 
• Additive Property of Area: Students should be able to to add and subtract areas of simple regions to calculate the areas of more complicated regions - composite regions.

Questions requiring floor, wall and ceiling perimeters and areas directly or in cost computations for painting all or for floor covering may give a context for this. The foregoing provides a connection between geometry and arithmetic.

C. Whole Numbers and Prime Decomposition
Focus on Computational Skill

The list method for obtaining common denominators, that is listing multiples of both denominators until their product appears, provides a simple way to computationally introduce and thus define the concept of least common multiples for a pair of whole numbers. The further decomposition of whole numbers is a product of primes introduces exponents and their properties. This prime decomposition or prime factorization provides methods for calculating least common multiples and greatest common divisors for pairs of whole numbers, or even a triple or finite sequence of whole numbers.  The pattern that the product of the least common multiple and the greatest common divisor of a pair of whole numbers equals the product of the latter may be should numerically and perhaps through the addition of exponents.

Rules for recognizing multiples of 2, 3, 5, 7, 9 and 11 may included with the treatment of prime number decomposition, albeit the introduction of the rule for multiples of 11 should come after mastery of integers. Interested teachers and gifted students may see Number Theory. area for a treatment (end of second year high school level) for the justification or development of these rules.

D. Fraction Sense and Operations

Note the above paragraph on operational command of fractions followed by Solving Linear Equations with and then without Stick Diagrams   offers a short effectie way to develop and consolidate both fraction and algebra skills and sense for many students - not all. The one exception I met understood the stick diagram treatment and codification of linear equations but not their algebraic codification.

Two  site sections (i) Fractions,  Ratios, Rates, Proportions   & Units  and (ii) Number Theory illustrate and discuss the development of fraction skills and sense. The foregoing development of fraction sense and skills, being written later,  goes further in emphasizing the geometric or physical approach to a hands-on, thought-based,  understanding and explaining fractions. The treatment of prime number decomposition should be linked to doing fraction addition and multiplication efficiently with the aid of least common denominators and cancellation of any or greatest common factors. Arithmetic Videos (Realplayer format) cover most if not all of the fraction skills and prime number decomposition and usage skills mentioned here. They demonstrate efficient Exercises on Mostly Fractions will test fraction know-how. Students  should meet the physical interpretation of proper and improper fractions in measuring distances with rulers,  and how improper fractions can be expressed as mixed numbers.

Students should learn numerical methods for adding, subtracting, comparing, multiplying and dividing fractions directly and efficiently.  

• While students should learn that any common denominator suffices for the addition, subtraction and comparison of fractions, should should see that the use of a least common denominators results in intermediate calculations with smaller numbers in the denominators and numerators before the result of an addition or subtraction is simplified, that is before the greatest common factor in numerator and denominator cancelled. In summary,  work at the start of a fraction addition or subtraction in finding or recognizing a smaller or smallest common denominator leads to less work in simplification at the end of a calculation. 
• Students should also learn to multiply fractions by forming the products of denominators and numerators to obtain the result, and by forming such products after the cancellation of common factors.  The latter leads to small numbers in the product and no further cancellation if the original fraction factors are expressed in reduced form. In sum, work in identifying common factors of the numerators and denominators at the start of a fraction product calculation leads via cancellation to less work in simplification. 

The site section  (i) Fractions,  Ratios, Rates, Proportions   & Units includes an explanation on how Division by a fractions can be expressed as a product by a reciprocal of the divisor.

Decimal Aside II:  Talking about the Place Value of Digits, individually and in groups of three or six, for Decimal Representation of Fractions or Irrational Numbers may set the stage for the properties of decimals and  column Methods for addition, subtraction, multiplication and long division. Check the mastery fo skills before covering fractions. The site coverage of Number Theory  (see decimal place value etc) may help - serve as online lessons for gifted students.

E. Algebra Skill and Concepts

The site area Solving Linear Equations helps students visualize fractions while giving geometric first steps in algebra. Our use of names (squares, cubes) for powers is an echo of the geometric origin of algebra. 

Pairs of stick will retained the same length if one what is done to one stick in terms of cutting (subtracting), multiplication and division is done to the other. There-in lies an informal visual introduction to the concept of maintaining equality. 

First year mathematics may introduce the stick and thne stick-free approaches to solving linear equations of the for m ax + b =  d and then  ax + b = cx + d. Selecting coefficients a, b, c and d so that  so that a > c > 0 and d > b > 0 implies the solution x is positive and avoids the appearance of negative numbers in the solution.  Thus the solution of algebraic reasoning can be introduced before or apart from negative number concepts and concepts. 

Verification of solution by checking is encouraged - strongly recommended.  If the check fails, or when it fails, students should be told that the error occurs between the start of their derivation of the answer, their solution, and the end of their check.  

Begin if you can with problems that have natural number solutions and then mix such problems with problems that have fractions, unsigned or negative, for their answers. Solving linear equations and checking them provide drill and practice in context for arithmetic operations with whole numbers, fractions and signed numbers. Judicious selection of coefficients (make them small whole numbers) may in the first instance minimized the appearance of fractions, yet fraction skills need to be emphasized and practiced sooner or latter. 

Going further in  introducing the Shorthand role of letters and symbols - verging on second year material.  The first pages of Chapter 15, Solving Linear Equations, in Volume 2, Three Skills for Algebra, offer arithmetic examples of solutions to the equation ax + b = c before introducing an algebraic solution of the formula x =  (c-b)/a for solving all equation of the type ax+b = c.  There-in lies an introduction to the algebraic way of writing and reasoning.  The previous chapter 14 in introducing the direct and indirect or forward and backward use of a formula,  a compound interest or growth formula A = P (1+i)ⁿ in particular, also gives arithmetic and algebraic solutions.   While the treatment here of the direct and indirect use of the compound interest or growth  formula  is not for first year year students, the treatment provides a model for the forward and backwards use of formulas in the second year and beyond of  high school mathematics.  That being said, time-permitting for sake of further introducing the algebraic role of letters and symbols, you could introduce the forward and backward use of equations for areas and perimeters of  rectangles, triangles, squares and circles. Chapter 10 in Volume includes the backwards and forward use of a rectangles area formula A = WL. All the foregoing may provide a context for algebra, a familiarity with the shorthand role of letters and symbols in the forward and backward use of formulas sufficient to understand algebraically described and stated axioms for algebra, more precisely,  algebraically described patterns for arithmetic with real numbers and subsets there-of: commutative law, associative law, distributive law, and so on. In sum, before introducing axioms for algebra or real numbers,  we can emphasize in instruction that each formula met in high school mathematics will be used directly and indirectly, that is forwards and backwards. 

F. Word Problems Made Easy

The site area Solving Linear Equations after introducing the stick and stick-free solution of equations of the form ax +b = cx + d introduces the solution of (i) systems of equations in essentially one unknown, and (ii) triangle systems of equations.

In triangular systems, or equivalent systems, a first unknown can be obtained immediately from one of the equations. Then a second unknown can be found immediately from using the the value of the first in another equations. Finally, the values of each unknown, second onward, can be found one at a time and one after another from the values obtained for previous unknowns. That can be an exercise in arithmetic for students. 

Mastering the solution of systems of equations in essentially one unknown exploits the shorthand role of letters and symbols instead of avoiding it. Many of the word problems in first year high school mathematics can be posed as a single equation ax+b = cx + d appear to have several numbers in them which need to be expressed in terms of a single key quantity or unknown, here denoted by the letter x. Recognizing the key quantity can be difficult. On the other hand, first year high school courses often emphasis the translation of sentences or phrases into arithmetic or algebraic expressions and equations. Many of the aforementioned problems can be translated into systems of equations that have essentially one unknown, the key one, whose identify may be clear once the word clues have been written as a system of equations. The foregoing exploit algebra and turns the solution of word problems into a clearer and more accessible process.   

Remark: For further years of high school mathematics.  The solution here of systems of equations in essentially one unknown serves as stepping to the solution of systems of equations in two or more unknowns in higher level via substitution.  

G. Statistics

Calculation of average, median and extremes test student ability to follow and employ definitions.  Instruction and exercises for forming line, bar (histograms?) and circle graphs and diagrams provide an opportunity for students to graphically represent fractions as is or written as a percentage, and so may test or solidify fraction sense and skills.

Students should be able to construct graphs a and diagrams. They should also be able to interpret them.  Statistical data may be collected for items or topics of interest to students.  One reason for the inclusion of statistics and graph interpretation is the development of critical thinking skills, the ability to recognize the limitations of statistics and graphs met in daily life - when are they accurate, when are they misleading or not, and how the choice of scale and location (y-intercept) influence graphs and lead to impressions of great or small variation. 

Did I write the foregoing paragraph, or did I adapt or  copy it from somewhere.  I do not know. One possible source would be Quebec MEQ documentation for secondary mathematics.

Each statistic provides a window, a blinkered view of a set of statistical data. The question of which statistic, the average or median will give the best impression of salaries in a company or cost of houses in an area, points to the limitations of statistics - the blinkered view that statistic provide of data.  In repeated measurement of a single line segment, the average of a set of measurements may give a better estimate of the true value of a coordinate or quantity - that points to calibration methods and/or the scientific or technological use of statistics (averages) for the sake of greater accuracy or less probable error. That may be mentioned to students. 

Caution: But the presentation of statistics to develop critical thinking skills with numbers and their interpretation is some what absurd in classes where student command of arithmetic with and without calculators does not lead to repeatable and reproducible results. The prerequisite for critical thinking is the ability to follow multi-step methods, one step at a time, and one step after another, with care because of the knowledge that an error in one step leads to bad or incorrectly justified results. If a student lack precision in reading and writing mathematics, in doing calculations on paper, the development of critical thinking skills via the study of statistics is hopeless.

H. Integers {0, ⁺1,^-1, ⁺2, ^-2,⁺3,^-3, ....}

Signed Numbers and Coordinates

So far arithmetic with  natural numbers and fractions have been developed with no plus nor negative signs as prefixes, raised or not. These unsigned numbers may serve as coordinates on a half-line. By introducing raised prefixes + and - in front of unsigned numbers, coordinates for a line that extends in both directions from a point chosen to be the origin of the coordinate are obtained.  

Ordered pairs of natural numbers and fractions can locate points in the first quadrant of a coordinate plane. Simultaneously, the use of signs in order pairs gives coordinates for four quadrants in the plane.  

The role of signs in providing coordinates for a whole line and for the whole plane instead of a half-line or a quarter plane (the first quadrant) may gives a first geometric context for placing signs in front of natural numbers and fractions. 

 Integers and Directed Line Segments

Students may be taught by rote how to add, subtract and multiply integers, or they can be a offered a thought-based development.

Addition of Integers defined with the aid of Geometrical Displacements or Movements: Identify positive integers n with n steps to the right. So ⁺n = n R where R is one step to the right. Identify negative integers p = ^-m with  m  steps to the left. So p = m L where L is one step to the left.  The addition of integers is now identified or introduced as the addition of steps to the left or right.  Examples

• ⁺5 + ⁺9 = 5 R + 9 R = 14 R = ⁺14 
  (Steps have the same sign)
• ^-5 + ^-9 = 5 L + 9 L = 14 L = ^-14 
  (Steps have the same sign)
• ^-5 + ⁺9 = 5L + ⁺9 R  =  5 L + 5R + 4R = 0+4R = 4R = ⁺4 (steps to right more - they dominate)
• ⁺5 + ^-9 = 5R + 9L  =  5 R + 5L + 4L = 0+4L = 4L = ^-4 (steps to left more, they dominate)

When the steps have opposite directions, the larger number of steps equals the small number of steps plus a remainder - the difference.  The sign of the larger number dominates.  The subtraction of steps can be identified with addition of the additive inverse.  The additive inverse of 5R is 5L, and vice versa.

Optional: The addition of points in the plane using their coordinates takes the further viewpoint further and gives a coordinate perspective and/or definition of translations in the plane - a second year topic.

Multiplication of Integers with the aid of Plane Geometry or not.  Students can be given the law of signs

(+)(+) = (+)    (-)(-) = (+)
(+) (-) = (-)     (-)(+) = (-)

as part of the rule for calculating the product of positive and negative numbers. Opposite numbers or additive inverses ^-9 and  ⁺9 have a sign and a common magnitude or length  9 (an unsigned number) = |^-9| = |⁺9| obtained by omitting or dropping the raised positive or negative sign prefix.  The latter is also known as the absolute value. Saying how to compute it defines it. The definition here is quite simple and does not depend on knowledge of how to multiply signed numbers or on how to form the additive inverse -a of a signed number a.

Students know know to multiply unsigned numbers, and we have just given a rule for multiplying the signs. So the product ab of signed numbers a and b is given by the rule,

multiply the signs and multiply the magnitudes

Thus 

( ^-4)(⁺9) =  ^-36 since the product of signs (-)(+) = (-) and the product of the lengths or magnitudes (4)(9) = 36. 

Likewise

• ( ⁺4)(⁺9) =  ⁺36 since the product of signs (+)(+) = (+) and the product of the lengths or magnitudes (4)(9) = 36. 

• ( ^-4)(^-9) =  ⁺36 since the product of signs (-)(-) = (+) and the product of the lengths or magnitudes (4)(9) = 36. 

• ( ⁺4)(^-9) =  ^-36 since the product of signs (+)(-) = (-) and the product of the lengths or magnitudes (4)(9) = 36. 

Here again saying how to compute the product defines it. The algebraic properties or axioms of integers are not immediately by this definition or arithmetic operation rule. We are not justifying the rule. We are just giving it.

Radical Innovation: The site page on complex numbers (until now usually part of first year mathematics) introduces polar coordinates in the plane.  That is a small leap from the discussion and division of angles for pie charts in statistics. If one defines the product of points in the plane using the rule: add the angles and multiply the lengths,  the law of signs for the product of signed numbers, both integers and signed fractions follow immediately. Here positive numbers are identify with the angle 0 degrees, modulo 360 degrees, while negative numbers are identified by and with the angle 180 degrees, again modulo 360 degrees. We need only derive the  the law of signs for integers from the rule add the angles, and leave further discussion of addition and multiplication of points in the plane to later. 

Radical Innovation: The polar coordinate definition of products, multiplication by points on the unit circle, turns a rotation in clockwise or counter-wise angle into computational skill. The multiplication of points in the plane (order pairs) by signed fractions a  gives an analytic viewpoint of dilatations (x,y) --> (ax,ay) in the plane. Reflection of points of across the x-axis (the horizontal coordinate line) is associated with complex conjugation and an angle sign change. All the foregoing gives an initial coordinate-based viewpoint of translation, rotation and reflections which could be developed further in the following mathematics courses. 

Part III. Still More Ideas

A. Modern Mathematics and Manipulatives

Schools of education which call for manipulative to be used in developing numerical sense and skills are providing a physical base and context for numbers and arithmetic in primary school.  In contrast, modern mathematics provides a context-free exposition or codification of numbers and arithmetic operation. While the modern mathematics based curricula of the late 1950's and 1960's and beyond, introduced the context-free view of number, the same curricula or allied courses introduced trigonometry with geometric drawings and diagrams (virtual manipulative) and employed that impure trigonometry and impure unit-circled based trig in the further development of high school and college mathematics. Thus the modern mathematics curricula had some context, accidental and not deliberate despite the emphasis on context-free thought-based development of skills and concepts.  Prior to modern mathematics codification or axiomatization of mathematics in a context-free fashion,  the development of geometry and number skills and sense developed from an ad hoc dependence on geometry, physical concepts, manipulatives of a sort, and from formal or uncodified use of letters and symbols in mathematics, science and technology. In other words, many of our numerical concepts and skills must or should of had a physical or geometric origion. Site material is pointing to a coherent, self-consistent, thought-based derivation of mathematics from geometric or physical context, manipulative actual or depicted, in a way that is full and complete or sufficient or adequate for most purposes, including applications in science, technology and society (daily life in particular);  in a way that leads to the algebraic-deductive skills necessary to mastery modern mathematics (if so desired) and in a way that provides a context for the latter.

B: Fractions and Whole Numbers Arithmetic, Geometrically Revisited

Length Arithmetic and Comparison

Before or besides four arithmetic operations on fractions and whole numbers, the physical addition and subtraction of line segments can be illustrated with the aid of rulers. Differences or subtractions that would result in negative numbers or fractions are avoided as shorter lengths can be physically subtracted from longer lengths, but not vice-versa.  Repeated addition of the same line segment leads to multiples of the same segment and shows how lengths can be multiplied by small numbers. Proper and improper fractions of lengths can be introduced and drawn as well. The number of whole times one length goes into another physically  reviews or introduces the concept of division with remainder.

All the foregoing gives a physical view and definition of the addition, subtraction, multiplication and division of lengths by proper fractions, whole numbers, improper fractions and mixed numbers. The concept of equivalent fraction or measures can be illustrated here with line segments multiples. 

D. Conversion of Length Arithmetic and Length Comparison into numerical operations

Remember saying how to do an operation defines it, albeit two how-to need to be consistent. Inconsistent how-to's for the same operation need to be avoided. Compare or combine the ideas here with the site Number Theory.  development of number skills and sense.

With the aid of rulers and tape measures, and in particular the use of unit distance for a divisor, lengths or  line segments can described as proper fractions, whole numbers, improper fractions and mixed numbers multiples of the chosen or implied unit length, say 1 cm (one centimeter). 

All lengths can be described as multiples of the unit length - an assumption with consequences.

Thus length comparison, which is longer or shorter, or implies numerical coefficient comparison in the description of lengths by numbers or numerical coefficients of the chosen unit length. This gives a physical base for the comparison of fractions. Moreover,  the physical or geometric addition, subtraction, multiplication and division of lengths implies and defines operations on the measures or numerical coefficients associated with the unit length.  That is,  numerical methods for addition and subtraction of fractions can thus be introduced or reviewed as means to compute the length of the products apart from physical measurement. The issue of irrational lengths is postponed. The foregoing gives a physical base for arithmetic with whole numbers and  fractions before the introduction of signed numbers. 

The foregoing gives a physical definition of addition, subtraction, multiplication and division of fractions, derived from their role as measures or coefficients of the unit length.   It gives a thought-based physical or geometric development of fractions. The tacit assumption that lengths are unique multiples of the unit length implies the physical viewpoint is well-defined. 

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