Chapter 11 Elementary Instruction

The Inductive Approach

There is no single route in primary school for the teaching or learning of arithmetic and geometry, the associations between them, and their uses. Any route followed will I suspect circle around the same or related concepts and in the act cover more detail with much redundancy.

This chapter constructs an image, approximately correct perhaps, of primary instruction, that is, the explanation or description of mathematics before deductive reasoning and algebraic thought are emphasized and relied upon. This image is intended not only for elementary school teachers but also for the teachers of intermediate and advanced mathematics. The image approximately represents [1] the background

and expectations of students at the finish of primary math instruction.

The initial aim of primary math instruction is descriptive and corroborative. It suggests patterns from experience. It gives rules for calculation and shows how to verify results. The rules and patterns may involve geometric or arithmetic ideas. The rules and patterns are not derived from first principles but, altogether, the rules and patterns set a stage. They introduce some rule- and pattern-based reason together with the observation that rules and patterns, if applied without error, lead to repeatable, reproducible and thus verifiable results. Such use of rule and pattern-based methods precedes deductive reasoning and is for many people a secure substitute. Deductive reasoning itself represents a refined attachment to repeatable, reproducible and thus verifiable results. Ideas on and for primary mathematics instruction follow.

Cue Cards, a Verbal Start

Primary school students, from five to ten years old say, may see learning as part of the process of becoming an adult. Why learn this or that need not be explained in the first instance – an instance that is all to fleeting. Some persuasion as to why learn will be eventually required as otherwise initial reasons and enthusiasm for school, if any, will be lost.

In teaching these students to read, we introduce them to the alphabet and possibly the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9. The mastery of arithmetic requires some minimal reading skills: at least recognition of the symbols 1, 2, 3, 4, 5, 6, 7, 8 and 9. In counting, numbers describe how many. Just as a child may learn to recognize the word cat with the aid of a live animal or a picture on a cue card. The child may similarly learn to recognize the digits 1 to 9, and possibly the digit zero 0, standing for nothing – the content of an empty bag. Imagine a cue card bearing the digit 4 or the word four along with the image of four like or unlike objects by themselves or in a container. Cue cards can also help children name and recognize common geometric shapes: squares, circles, disks, triangles, etc. And so, a mastery of some mathematics may begin.

Counting

Writing the numbers on paper introduces decimal notation, and the concept that the value of digit in the decimal representation of a number is determined by its position. Thus the student becomes familiar with the unit, tens, hundreds and thousands positions in the decimal coding and expression of whole numbers. Whole numbers can be used to count the number of objects in a group, for instance the number of feet (or meters) between two points along a measuring tape or the number of marbles in a bag. Numbers written on paper and the rules for arithmetic represent the first symbolic manipulations that appear in mathematics.

Multiplication and addition are present in the decimal system. In particular, to envision the number 34, we can envision three groups of 10 squares counted together with another 4 squares. The image of a rectangle covered by 4 rows, each consisting of ten squares, leads to the notion of area – exactly how many squares are needed to cover a region. The area of this rectangle is forty squares. Units can be introduced: square inches, square centimeters, etc. (How to compute the area of a rectangle is being introduced or hinted at in the latter example.)

To physically represent the notion of multiplication, cue cards or pictures of rectangles and squares with varying heights and widths, as indicated by the number of squares in a horizontal row, or vertical column can be employed. These images can illustrate or define the 10, 12 and 16 times table, etc. They can be used to observe that 4 times 6 gives the same result as 6 times 4 – the order of multiplication is not important. Further examples of a similar or different kind will be generated by a teacher. The object of each is to introduce a new idea or to widen and reinforce a previous one.

Before multiplication can be described further, addition needs to be discussed. Addition can be initially viewed physically as the combination of two or more objects, or groups of objects together. This is a physical definition that is easily understood by students before the addition of decimal numbers has been explained or even mentioned. The addition of various numbers of objects can be illustrated with small groups of marbles, dots, squares, etc. By this method, the addition of pairs and even triplets of the numbers 1 to 9 can be introduced and illustrated. For instance, in a repeatable and reproducible fashion, a child may see two plus three is five simply by combining a group of two marbles with a group of three. For a child and possibly some adults, such considerations inductively show why 1+1=2 or 2+2=4. No deep philosophy is required. The meaning, justification and interpretation here is a consequence of the adjectival role of numbers in counting how many and the conservation of objects, say marbles.

The grouping concept is further useful in developing or explaining the distributive law of multiplication over addition. For example, five bags of 4 marbles plus three bags of 4 marbles gives five plus three, that is, eight bags of 4 marbles. The physically observed conservation of marbles now suggest the distributive law

Decimals and Addition

For primary school students, combining two or more groups of objects together and counting the number of objects in the resulting combination, a single group, represents their first computational perspective of addition. Of course, saying how to obtain the result of an operation, defines that operation. For the students, this serves as a definition. Decimal methods for the addition of two numbers (counts), the calculation of the number that will be in the combined groups, represent a shortcut for the original addition process – computing the number of objects in the union of two or more (finite) groups. Here students learn how to add pairs of single digits before learning about carries and advancing to the addition of multi-digit numbers or counts. But this practice with decimal addition methods may leave the original perspective, counting the number of objects in the union or combination of two groups, as a secondary, physical interpretation of addition.

Practice with addition introduces students to a process with repeatable and reproducible results, independent of the person or machine performing the operation. The results are thus verifiable. Here addition with decimal numbers is (or was in the classroom) one of the first processes with repeatable, reproducible and thus verifiable results taught in elementary mathematics. Addition methods for decimal numbers give the first mathematical examples of a rule or pattern based reasoning process before and besides deductive chains of reason and mathematical induction

The notion of a process with repeatable, reproducible and therefore verifiable results or conclusions is fundamental in mathematics, in thought and processes of all kinds, in daily life and technology/science. We put confidence in such processes. All depends on such processes. Even the deductive reasoning employed to obtain conclusions in higher mathematics including geometry illustrates our reliance on results or processes which are repeatable, reproducible and thus verifiable.

The foregoing discussion is too complex for students as they are learning about counting and arithmetic operations. Students may be content with the message that results should not depend on who obtains them; and they should only depend on the numbers involved and the carefully followed methods of arithmetic. Beyond this, they should be alerted that one false step in addition cast doubt on all the following steps. This is of course true in any chain of reason.

Decimals and Multiplication

The foregoing sets a stage for the explanation of decimal methods of multiplication. The latter depends on the decimal methods for addition – previously well-practiced. Here there is a mixture of justification, along with some confidence-building reliance on repeatable, reproducible and therefore verifiable products. More verification follows by counting the number of squares in rectangles with small numbers of rows or columns. Rectangles with some agile wording (yours) can also be used to show how to multiply a single digit by a single digit multiply of ten, hundred or thousand.

In the foregoing, decimal methods are being described and motivated and extracted inductively from examples, but not deductively derived.

Decimals and Subtraction

Given a bag of 12 marbles, 5 can be taken out. This leaves a remainder of seven. This subtraction can be done physically. We note that adding those taken to those leftover yields the original number 12. This gives the first physical ideas of subtraction. Besides, we note that it is not possible to take more than 12 marbles from a bag. This physical subtraction has its limitations.

Now with decimal numbers, rules for the subtraction of a smaller number, the subtrahend, from a large number can be given and practiced. The results again are repeatable, reproducible and thus verifiable. Beyond this the result of the subtraction can be added to the subtrahend, and the sum should be the original number. This situation provides a corroboration of the marble example above. The operation of addition and subtraction can be used to undo each other: subtracting a group of marbles from a set, and then returning the group to the set leaves the original count unchanged; adding a group of marbles to another set and subtracting the same also leaves the original count unchanged.

With whole numbers, the comparison of numbers, what is smaller or greater, depends on the decimal representation. A whole number whose decimal representation contains more digits than another is larger than the other. And when two, decimal representations have the same number of digits, we compare the leading digits (lexicographically) to identify the greater or smaller. The justification comes from the positional basis of decimal notation. Leading digits are assumed to be nonzero.

The subtraction of a larger number from a smaller number appears impossible. More precisely, it cannot be understood via the bag of marbles analogy in a simple fashion. Other physical interpretations of subtraction consistent with the marble interpretation are possible. For instance, the partial payment of a debt of ten marbles with a partial payment of seven leaves a debt or residue of three to be paid. As with some merchants of the middle ages, signs can be used to record credits and/or debts. With these, it is possible to visualize subtraction of any whole number from another larger one. This is the first context for subtraction.⁴²

Fractions and Divisions

Fractions appear via the concept say of division, the division say of a pie, an amount or a group of objects into separate but equal portions. This leads to the concept of fractions: enjoying a fifth of a pie, consuming two quarters of pie, or being sick on three quarters of a pie. And when there are several pies, one can introduce improper fractions, for instance eating 7 quarters of more than two pies, the latter having being physically divided into quarters. Here fractions can be added or grouped together in a physical sense. Here the mathematical operation of division is linked to the semantic and physical concept or root.

In discussing pies, we observe eating [6/18]th or [4/12]ths of a pie is almost the same as eating a third, only the number of divisions may be different. Like wise, eating 15 out of 20 portions of a pie is the same as each 3 quarters of the pie. Pie based examples like this lead to the idea of physically equivalent fractions and the simplest form of a fraction - the form that requires that the least number of divisions in a pie.

Examples can be generated using marbles, distances, and measures of mass, area and so on. Further, some improper fractions, like [10/2] can be identified with whole numbers. That is, 10 halves of a pie (more precisely of several pies) is equivalent to 5 pies in total. The significance of [11/2] can also be explained. These observations when the numerator and denominator are small, can be made with the aid of simple examples.

A fraction is said to be simplest form when the numerator and denominator have no common divisors. The fraction [60/100] can be reduced to [6/10] and then to [3/5]. The calculation of the greatest common divisor (g.c.d) of two whole numbers finds an application and thus its motivation in this reduction: the simplification of fractions. The calculation of the g.c.d requires mastery of division with decimals and the two further notions of (i) prime numbers and (ii) prime number decomposition.

The addition of fractions can also be introduced by taking pieces of a pie. The question of how to take the fraction [1/3]and also the fraction [2/5] of a pie can be resolved by cutting a pie (or pies) into 15ths. Here students may see that 5 times 3 is fifteen. The question of how to take the fraction [3/4] along with the fraction [1/6] of a pie can also be resolved by cutting the pie into 6 times 4 = 24 equal portions or into 12 portions. The latter number is the least common multiple of the denominators. The addition of fractions together provides motivation for the discussion and computation of the lowest least common multiple (l.c.m) and the computation of the latter via prime number decomposition.

Products of Fractions

Taking [2/5] of a whole pie, an amount or a group of objects, involves a division of the pie, amount or group into 5 equal portions and then taking two portions. Now if the original amount is given by [2/3] of a pie, we need to divide the [2/3] into five equal pieces. Here

Examples like this may suggest the rule.

Digression. When the factors [(a)/(b)] and [(c)/(d)] are both in reduced form, a shortcut for the simplification of the product comes from observing the product also equals [(a)/(d)] ·[(c)/(b)], where the fractions may be further reduced or simplified using the l.c.m of a and d, and the l.c.m of c and b.

Chapter 11: Notational Conflict.

The improper fraction [5 ½] equals 5 + [½]. But with the plus sign omitted, the latter is often written 5[½]. Now in algebra, the product a×b of two numbers is also written as ab. There is a notational hazard here. Thus the arithmetic notation for 5 plus [½] coincides with the notation for 5 times [½], and only attention to the context or to convention indicates the intended meaning. Students have to be alerted to this conflict. And it is not an error if one becomes initially confused by the conflict. The common work-around is to write 5([½]) or 5·[½] for the product and reserve the notation 5[1/2] for the sum. This conflict should be mentioned in the curriculum, but not eliminated. It points to the human development of arithmetic and algebraic notation.

Chapter 11: Reciprocals Etc

Reciprocals of Whole Numbers. The division of the fraction [2/3] into 5 equal portions results in the same result as the product [2/3][1/5] = [2/15]. It can be easily visualized starting with 10 fifteenths of a pie or dividing each third into 5. The division of the fraction [2/3] into 5 parts gives the same result physically as 2-thirds divided by 5 if each third is divided into five first. Thus the division of a fraction by a whole number is easily represented.

Reciprocal of Fractions. The reciprocal of a fraction R = [(p)/(q)] can be arbitrarily introduced to students as S = [(q)/(p)]. Then the identity

Multiplication by Reciprocals. The number of times 4 pies goes into 11 pies can be obtained by dividing the 11 into groups of four and then counting them. The answer is two and a three quarter times, or the improper fraction [11/1]×[1/4] = [11/4] simplified.

The number of times 3 goes into 7 pies can be obtained by dividing the 7 into groups of three and then counting them. The answer is two and a third times, or the improper fraction [7/3] simplified.

The number of times 1.5 or [3/2] pies goes into 5 pies can be visualized by dividing the 5 each into halves, and then dividing the resulting halves into groups of three, and then counting the groups. The answer is then three groups with one half leftover or the answer is three and a third times. The third leftover is one third of a group of three halves. The latter gives the same result as [5 ×2/3] = [5/1] ×[2/3] = [10/3] = 3[1/3] as before. So division of 5 by [3/2] gives the same result as multiplication by the reciprocal [2/3].

The foregoing [4] inductively suggests that division by a fraction gives the same result as multiplication with the fraction’s reciprocal.

Now an answer to the question, what fraction when multiplied by [5/2] yields the result [2/3] is obtained as follows. Multiply the desired result [2/3] by the reciprocal [2/5] of [5/2], and simplify. The answer to the question is [2/3]×[2/5] = [4/15]. This may provide some motivation for the further or later notion that division and multiplication (thanks to the associative law) are inverse operations.

Chapter 11: Decimal Notation and Decimal Fractions

The positional decimal representation of whole numbers is easily extended to decimal fractions via the introduction of a decimal point and the introduction of positions for tenths, one hundredths and one thousandths, and so on. A decimal fraction by definition is a multiple of one tenth, one hundredth, and so on. Here we can even speak of improper and proper decimal fractions. The fraction [(p)/(10^k)] is improper if p > 10^k.

Finite decimal expansions (decimal fractions, proper or not) have a role in the approximate recording of measurements. The uncertainty of the last digit should be less than or equal half a unit if no error bound is specified.

Decimal Methods for Division

The Euclidean algorithm for division is represented in decimal computations. The algorithm is discussed more generally in higher mathematics. Given a whole number m, one can ask what is the largest multiple kn of another whole number n, the divisor, which equals or does not exceed m. The Euclidean division algorithm provides an answer. The computed number k is called the quotient and the difference r=n-kn is called the remainder. The first examples can be constructed so that the remainder is zero. And when the remainder is nonzero, the division is not exact. The computation can be checked by computing the product kn and the difference r=n-km is less than the divisor m

A modification of the Euclidean division algorithm yields the long division process of decimal arithmetic for ratios of whole numbers and then for ratios of other real numbers.

(1) To compute the decimal expansion of m/n where both m and n are whole numbers, the decimal division algorithm can be continued until a zero remainder is obtained or sufficient accuracy is obtained or until patience is exhausted.

(2) To compute the decimal expansion of m/n where n is represented by a decimal expansion with p places after the decimal point, an equivalent fraction M/N with N a whole number can be obtained by setting M = 10^pm and N = 10^pn This corresponds to shifting the decimal point in the numerator and denominator.

(3) In general, the decimal expansion of m/n can be computed to any desired precision by approximating m and n by decimal expansions of sufficient precision.

Note the foregoing represents an exercise in rule and pattern-based reason. Calculators can be used in place of teaching the division algorithm. But teaching the division algorithm has one advantage. It can be recalled in the discussion of polynomial division algorithm that the decimal expansion of a whole number can be viewed as a special polynomial in powers of 10. Decimal notation just provides a shorthand way of writing the polynomial expansion.

Chapter 11: Ordering by Size or Magnitude

The concept of greater than or more than is easily understood by students when dealing with counts or whole numbers. The decimal place representation of numbers is accompanied by the notion of magnitude. We first note that 10⁰ < 10¹ < 10² < 10³ < ... Comprehension of the decimal place representation of whole numbers > 0 implies each whole number

Before the introduction of signs, that is negative and positive numbers, finite decimal expansions extend this idea of greater than or more than. A finite decimal expansion in particular counts the number of units, tenths, thousandths and so on that the number it represents can be divided into. Beyond this, students may be shown or pointed to the comparison of (unsigned) numbers with infinite decimal expansions, albeit such comparisons may be rare due to the prevalence in everyday computations of finite decimal representations and expansions. When dealing with unsigned numbers, the ideas of greater than and more than imply that the larger number can be obtained from the lessor number through the addition of a (unsigned) number.

Chapter 11: Positive and Negative Numbers

the coordinate perspective

Thermometers with temperatures above or below a reference point labelled zero provide an example of a numbered line where the numbers have positive and negative signs in front and a physical significance. A positive temperature indicates so many steps or units above the zero mark while a negative sign indicates some many steps below the zero mark. The addition of a positive number now corresponds to be and may be defined as an upward movement of so many steps. The addition of a negative number and the subtraction of a positive number corresponds to a downward movement. These additions and subtractions can be done at any point on the scale.

The subtraction of a negative number in the first instance is undefined. But one can define negation for a number as the reversal of direction, and regard subtraction of a number as the addition of it negation. Students can be shown that this applies to the subtraction of a positive number before the subtraction of negative numbers is considered.

Two negations or reversals further result in the original number. Here the negative -a of a number a will be the number or point obtained by subtracting the number a from 0. The foregoing provides a physical concept of addition and subtraction.

Note that multiplication of a number a by a whole number n, can be viewed as the result of the addition of a to itself, a whole number n times. Multiplication by nonnegative proper and improper fractions, and then positive decimals can also be physically interpreted. Next every negative number b is the negation of a positive number a. In consequence, multiplication by a negative number b = -a can be defined as the multiplication by a followed by a negation (reversal of direction).

Coordinates along a horizontal line (the real numbers) can be represented by signed decimal numbers

More on Subtraction.

Subtraction of n from m yields a number k with the property that n+k = m. When m and n are given, subtraction of n from m answers the question, what number k when added to n yields m? Examples for subtraction when m > n can be revisited in this context. Such examples imply that k equal m - n when n + k = m.

The calculation k = -(n-m) can be given when n > m as a means for computing k = m-n. Then again n + k = m. Here n-m is computed using previously taught methods for the subtraction of decimals or fractions. (When m and n are fractions, subtraction answers the question: what fraction k when added to n gives m? The fraction can be signed.)

Chapter 11: Ordering by More Positive Than

Renaming the Greater Than Symbol

The symbol > traditional has been called the greater than sign. Technically, given two real numbers a and b we write a > b if and only if there is positive number c such that a = b+c. The tradition is read a > b as the statement that a is greater than b. To avoid a conflict and to align mathematical terminology with the common usage, the symbol > should be renamed the more positive than symbol. This new name corresponds precisely to the technical meaning. With this new convention, the phrase a greater than b can revert to the common usage and mean |a| > |b|. Similarly, a < b can be read not as a is less than b but as a is more negative than b. This new terminology means there is a positive number c such that a = b-c or equivalently such that a+c = b. The signs £ and ³ now may be read respectively as more negative or equal to and as more positive or equal to.

A number b is said to between two other numbers a and c if and only if there is a positive number l < 1 such that b = la + (1-l) c.

Chapter 11: Complex Numbers

By means of measurement coordinates locate points on line and in the plane with both rectangular and polar coordinates. Navigational examples can be employed to introduce, illustrate and motivate vector and displacement addition. The earlier described operation on vectors of multiplying lengths and adding angles can be used to define multiplication in the plane and extend the concept of multiplication on real line from pairs of positive numbers, to any pair of real numbers – points on a horizontal axis. The foregoing defines the complex numbers (minus a representation of products in terms of real and imaginary parts). The polar coordinate definition of multiplication multiply lengths and add angles applied to real numbers provides or agrees with the law of multiplication and law of signs.

This development is pre-algebraic, but rule-based. It is in accordance with the expositional principle of putting the material easiest to understand first. Before this development only a familiarity with addition, subtraction and multiplication of positive numbers (fractions or decimal representation) is assumed. Negative numbers need only be employed as coordinates. Operations with them need not be defined before the exposition of complex numbers. An elaboration of these ideas follow. This development represents an innovation for elementary mathematics instruction, and it has some consequence for intermediate mathematics instruction.

This new perspective introduces operations on real numbers (signed decimals, signed fractions or points on a horizontal axis of the complex plane) without relying on algebra or algebraically described properties of real numbers. It is computational and pre-algebraic.

Coordinates On the Line

Before the introduction of negative numbers, the notion of positive numbers is not emphasized. Students may have a knowledge of unsigned (positive) numbers. These numbers can be employed as coordinates on an infinite half-line to locate points. After this, signed numbers, positive and negative can be employed to locate points, that is to serve as coordinates on the bi-infinite real line. This allows students to graphically comprehend the role of positive and negative numbers, and zero too, as coordinates or marks on a coordinate line. No arithmetic is required. Examples of coordinate lines are provided perhaps by temperature scales, by water levels (the signed height of tides, reservoirs or river waters above or below a zero mark) and by bank account balances. Accountants today employ parentheses to avoid writing negative signs. Prior to the 15th century, negative numbers were thought to be imaginary – figments of the imagination.

Coordinates in the Plane

Ordered pairs of positive and then arbitrary real numbers can be introduced as rectangular coordinate for the plane after the selection of an orthogonal pair of axes. Following Descarte, ordered pairs of positive or unsigned number locate points in the first quadrant. Following Newton (or others before Newton), signed coordinates can be employed to locate points in all four quadrants. This role of signed coordinates offers another motivation for having and employing positive and negative numbers.

Displacement and Vector Addition

Points in the plane can be identified with vectors (issuing from the origin). The transport of these vectors and the head-to-tail addition of vectors can be described graphically, and then in terms of rectangular coordinates. The rules for this can be drawn from examples in an inductive fashion. Motivation can be provided by the problem of planar navigation, and moving from point to point on a map. Here the addition of vectors representing displacement on a map can be introduced. The resultant of two successive displacement can be declared to be the linear displacement between the initial point of the first displacement and the terminal point, following the second displacement. Students will find from examples and exercises that the addition methods appear to be repeatable and reproducible, and thus verifiable in a pre-algebraic and pre-deductive fashion.

Restriction to An Axis

Addition of vectors or points on a coordinate axis or coordinate line can then be viewed as a special or restricted case of the more easily visualized situation in the plane, an application of the head to tail vector addition method to pairs of points, alias vectors, on the coordinate line. This will lead via examples to easily visualized rules for addition of positive and negative numbers, that is points on the horizontal axis with positive and negative coordinates. Rules for the addition and subtractions of numbers, vectors or displacements in the horizontal coordinate line, can now be extracted from the planar case: regarded as the special or limiting case of motion restricted to a single line in the plane. This provides another means to visualize mathematics.

Multiplication of Planar Points

Both polar and rectangular coordinates with respect to a pair of axes can be determined (measured) for points. Given or measured values of polar or rectangular coordinates can also be used to locate points. The foregoing geometrically suggests that polar and rectangular coordinates are interchangeable. It provides a method, geometric measurement, for obtaining polar coordinates from rectangular, and vice-versa. This approach is hands-on, physically dependent and while not deductive, it is repeatable, reproducible, and thus secure. Angles in polar coordinates can be computed, modulo 360 degrees.

Given a pair of nonzero vectors issuing from the origin, that is two points in the plane, their angles can be measured, and their lengths measured and represented by an unsigned decimal number – a unit-free length. Adding the angles together, modulo 360 degrees, and multiplying the unit-free lengths together yield the angle and unit-free length of third vector, their product. This defines via polar coordinates, the multiplication of points or vectors in the plane.

Multiplication of Real Numbers

Following the identification of the horizontal axis with the real number line, a polar-coordinate representation or visualization of the product of real numbers follows. This may define for students such products. This definition implies the law of signs for the product of real numbers. Moreover, the identification also provides a context and location for the definition of square roots of negative numbers. These square roots can be found on the vertical, alias imaginary, axis.

Consequences for Intermediate Instruction

The foregoing offers in elementary instruction a computational and visual comprehension of arithmetic with real and complex numbers. In intermediate level instruction, there is choice of how expressions for the real and imaginary parts of the product of two complex numbers are to be obtained. Assumption of the distributive law of multiplication over addition in the complex plane immediately implies expressions for the real and imaginary parts of the product in terms of those of the factors. Ease of exposition may justify the assumption: Intermediate instruction need only offer strands of reasoning. Threading them together in a purely deductive fashion may be left to advanced courses. But the distributive law can be seen or justified via geometric arguments:

The distributive law itself can be geometrically implied or suggested by viewing multiplication by a nonzero complex number as the consequence of multiplying by a positive length and following up by a rotation. The operations commute. Both are distributive over addition. Reasons for the latter follow.

For just a rotation, one can physically show the distribution law by considering the rotation of a parallelogram.

For just a positive stretch factor, one can show this in the special case of small whole numbers, and then proceed inductively to the case of rational numbers. The case of irrationals now follows by an assumption of and intuitive appeal to continuity.

The geometric argument has the appeal that it applies to real multiplication as well.

As a third alternative, the distributive law can be assumed for real numbers only and then later on in a trigonometry course, the distributive law for complex numbers can be obtained from the angle sum formulas. The rotate-a-triangle proof of these formulas may be less of a surprise and more accessible to students who have seen the add the angles, multiple the lengths polar coordinate method for complex multiplication. Against this third approach, I suspect that many students on learning the distributive law for real numbers will apply it to complex numbers without a second thought, and with little patience for the notion that it should be derived. They may be correct.

For ease of exposition, and to provide a greater command of mathematics, the distributive law for complex numbers can be assumed, and from it the distributive law for real numbers obtained as a special. In the derivation of mathematics from set theoretic foundations for arithmetic (axiomatic set theory), both distributive laws, the one for reals and the one for complex numbers, are almost equidistant from the axioms in terms of the work required for their respective derivation. The first exposition of complex numbers like that of trigonometry and calculus may mixed algebraic and geometric arguments which illustrate the deductive aspect of mathematics.

The geometric argument, the first alternative outline above, avoids the semantic problem of which distributive law to assume first. A second chapter on complex numbers in the companion book Why Slopes and More Math explores these possibilities in more detail. A fourth alternative is to present the distribution law as a theorem and leave its proof as an intellectual IOU.

Footnotes:

1. This educational writer has no first hand experience of the elementary school classroom as teacher. The image here is based on home-based observations of the children of others, how they learn, and this author’s memory as a student in the classroom. This author as a child was an adult in waiting – mentally alert and observant, if not informed, attending the future and a reason for being.
2. Numbers may be just adjectives that become objects when discussed separately from the description of other objects.
3. Another context for subtraction is provided by the coordinate line. Addition of a number n corresponds to n steps or a movement in a forward or positive direction. Subtraction corresponds to step in another direction.
4. flimsy evidence perhaps
5. Technical detail: if a measurement is known or seen to lie between a lower limit L and an upper limit U, the measurement can be recorded as equal
6. Technical Note: Arithmetic operations with the decimal expansions of whole numbers is a modification of the polynomial multiplication process which takes into account the carrying and borrowings which avoids coefficients with values >9 the base -1.
7. A Caution: Isolation in its use should be avoided.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science