Secondary Mathematics - Core Elements

Original: 10-03- 2008. Revised 18-03-2008.  Eight core elements are indicated below. More details of  them, and more core elements should follow.

Introduction

In primary school, there should be some mastery of decimal column methods for addition, subtraction, multiplication and long division of whole numbers and all unsigned numbers with terminating decimal expansions.  Students should also meet counting methods, and be given examples to imply that the number of objects is or should be independent of how counted. 

In primary school, there should also be some mastery of fractions - what is a proper or improper fraction; how improper fractions may be written as (is equivalent to) a whole number plus a proper fraction; how to add and subtract fractions using like or common denominators; and how to multiply fractions by forming products of numerators and denominators.  See the format evaluation standard below. However, many students leave primary with a poor command of arithmetic with fractions and the significance or meaning of fractions. That is inconsistent with the long term objective of mathematics education.

1 Formula Evaluation Format
(explanation and reasons for a standard, all level)

In teaching mathematics after arithmetic, and the mastery of the latter via drill and practice, quality is more important that quantity.  We need to decide what skills and concepts are important, or are key for further learning in a repeatable, reproducible and efficient manner.  Learning enthusiasm should not be wasted by needless or aimless work. Good habits need to be developed and maintained.

The formatting rules, convention or standard below tells students how to present their work. 

Follow a variant of this format in evaluating arithmetic expressions.

Formatting Rules for Formula Usage and Evaluation

When the computation of a number or quantity requires the evaluation of a formula, the evaluation itself and the the reasoning that supports the evaluation can be written and well-formatted using the following step by step.

1. Write or state the formula with words or algebra.
2. Give or indicate the meaning and values of numbers or quantities that appear in the formula. That indication may be in form of equations written on a diagram or written in the text of a solution.
3. After the values have been been indicated, substitute the values of the numbers or quantities in the formulas
4. Substitution in the formula (the previous step) gives an algebraic or numerical expression to evaluate.  That evaluation may be done with a calculator. But it is good practice for further learning in mathematics to simply the expression as much as possible with a minimal use or none of a calculator.  It is good practice for further learning in mathematics to encourage students to arrive at an exact answer in simplified form, if such an answer is available. That being said, there are situation where immediate use of a calculators is warranted.

An Example

This example involves the rectangle area formula.  But we could illustrate the above format and standard for formula evaluation.

The area of a rectangle

 

is given by its length times its width. We can write this as

A = L×W

where latter diagram shows, defines and determines the meaning or significance of W and L - the denote the lengths of sides - as a rhetorical practice in class and in writing, we may write a sequence of words with similar meaning to convey the underlying message and to extend or refine the vocabulary of our students or readers.

The net result of the above formatting should be a solution that looks like the following.

Observe how the equal signs are present, and how they are vertically aligned.  The above solution communicate the logic (use a formula) and communicates the reasoning process or steps in that use or evaluation. 

Formatting Advantages: The above format for formula usage or evaluation provides a model for students to follow not for rectangle area evaluation, for also for the evaluation of formulas triangle, trapezoidal, parallelogram and circle area and perimeter.  There-in lies a model for showing work and for showing and recording comprehension in mathematics, science and further quantitative arts and disciplines, where formula evaluation questions.

Units may be carried through calculations. Seeing how would be useful for later courses in science. Seeing how would be useful too in provide a concrete setting for working with monomials.  If the letters x, y and z are meaningless to student in monomials, one may use units of measurement as well without loss and without gain in rigour.

Quantity versus Quality: Students will find security and confidence in following the model.  Where the model appear in student written examples, there is proof of comprehension and an addition to the notes met or given by books and teachers.  Yes, the model takes a little more time to present but it completeness most likely obviates the need for a large number of practice problems.

Benefits for further Learning: Meeting formatting standards for evaluation of formulas may get students to appreciate the role of format and position in mathematical notation and solutions as a tool for recording and developing mathematical reason and steps on paper (or on further media that may appear).

Calculation and Simplification of Arithmetic Expressions

The calculation of arithmetic and algebraic expressions could also follow a similar format:

in which equal signs = are aligned vertically.

Notation is Everything

While students in their further studies should see and mastered acceptable variations in the format advocated above,  in the first instance, we as teachers should insist on a format easily followed and repeated. Free-form gives too much freedom to students. Offering a format and enforcing it provides good habits and a lower bound for note taking and student work.

2. Operational Command of Fractions and Factorization of Whole Numbers

The Standard: An efficient command of exact arithmetic operations with whole numbers and with fractions is needed for algebra and higher mathematics. Have students master the rules and methods for this efficiently, with brief explanations of why the rules and methods work, if possible.  But for the sake of algebra, aim for efficient, exact command of arithmetic with fractions with results being repeatable and reproducible, and clearly and carefully written, so the the work of each student is legible and so that all necessary steps are written fully, completely and precisely. Learning to get results in a repeatable and reproducible way will give students confidence.  Efficiency and exactness in computations (avoid decimal approximations) has a much greater priority than number sense or why the operations work. 

Recognition of Composite Number

Efficiency in exact arithmetic with fractions requires students to recognize multiples of 2, 3, 5 and 10 using divisibility rules. Students may also need to recognize when a whole number less than 100 is prime or not.

Teach this Rule to All: A whole number < 121 = 11² is prime if it is not a whole number multiple of the first four primes 2, 3, 5 and 7, and it can be factored if it is divisible by one first four primes.

Now if a number < 121 is composite (not prime), it will equal one of the first four primes (or 11) times second factor < 121. The above rule provides a simple basis for factorization and prime factorization of all whole numbers < 121 or if you prefer < 100. To test divisibility by 7 encourage students to learn all multiples of 7 < 121 or let them use a calculator.

Prime number decomposition are useful for finding least common multiples or denominators, and in finding greatest common divisors. The expression of whole numbers in factored formed, prime or not, may be employed in calculation of fraction products to identify and cancel common factors.  More over, prime factorization may be employed to find exact expressions for square and n-th roots of whole numbers, alone or in fractions.

Teach this Rule to Advanced Students: In general, for each prime number p, if N is whole number < p² then N is prime if is not divisible by all primes less than p.

Efficient, Exact Arithmetic with some Fractions:

Students should be able to efficiently add, subtract, multiply and divide fractions efficiently when the numerators and denominators are say whole numbers from say 1 to 100 or 121.

Students should master the following multiplication and division skills efficiently and exactly.

1. How to simplify fractions by cancellation of common factors in numerators and denominators. Here recognition of composite and prime numbers < 121 helps with efficiency.
2. How to multiply fractions, form products, via the rule multiply the numerators (tops) and multiply the denominators (bottoms).
3. How to simplify products of fractions (two factors) by using the factors of their numerators and denominators. Then for the sake of efficiency, how to (cross) cancel common factors in products of fractions before the product. How or why the latter aids the simplification of the product - the expression in a form where the numerator and denominator have no common divisors.
4. How to compute the reciprocal of any whole number or fraction, and how to use the latter in division. Include here the reciprocal of a reciprocal of a whole number.

Students should see and master addition and subtraction skills:

1. How to add and subtract using a common denominator.
2. Then for the sake of efficiency how and why addition and subtraction with the aid of least common denominators results in fractions with small numerator and denominators, fractions that are thus easier to reduce. There-in lies compensation for the greater work required to find the least common denominator.

For details, see these fraction starter lessons. I recommend putting efficiency with multiplication and division first since that only involves recognition factorization and their cancellation, operations that are simpler to master than recognition of least common denominators.

Remark: In the foregoing, students should be shown how to present and record addition, subtraction, division multiplication and simplification of fractions clearly and properly.  Vertical alignment of equal signs may be required here to avoid proper use of the equal sign.

1. When fractions are present, the center of the equal sign should be aligned horizontal with the division bar.  The division bar should be horizontal and not slanted so that in students writing, the slanted division bar / as in 5/4 does not become confused with the digit 1 as in 514.
2. When addition, subtraction, multiplication and/or division signs are present in the top level of a numerical expression, these the center of these signs should also be aligned horizontally with the center of the equal sign.

3. Solving Linear Equations

The site introduction of solving linear equations uses a letter a or x to denote the the unknown length of a line segment.  The introduction then employs fractional operations on line segments (stick diagrams) to find the unknown length.  The same operations are described algebraically. That being said, the two fold aim of introducing stick diagrams and fractional operations on them is to provide concrete framework for solving linear equations and also for reinforcing and developing fraction skills.

Note: The use of stick diagrams is a crutch, a vehicle introduced to develop the stick-free algebra skill sufficient (a) to solve linear equations in one unknown with integral and fractional coefficients, with integral and fractional solutions; and (b) to test, verify or check whether or not a given or derived number is indeed a solution.  Students should be taught to check the solutions they have found. Then they may catch and correct their own mistakes before submitting an answer, or on an exam, indicate their method is good, but implementation bad. 

Following stick method introduction and development of algebraic methods for solving linear equations, the site area on solving linear equations develops the abilities to

1. solve systems of equations in essentially one unknown. 
2. solve systems of equations that are triangular or essentially triangular, that is permutation of a triangular system)
3. solve systems of equations in two or more unknowns.

The solution here of systems of equations in essentially one unknown requires students to meet and operationally mastered the associative law for multiplication and the distributive law for multiplication over addition. But, the associative laws and distributive laws are not explicitly stated - you may mention them if you like.   Words problems met while students are studying the solution of one equation in one unknown either lead to the latter directly or through the recognition of an essential variable in the problem.  I think the latter kind of problems are best treated, made more accessible, by showing students how to recognize and solve systems of equations in essentially one and unknown.  That will provide students the algebraic notation and tools for solving word problems in essentially one unknown, and take advantage of the power of algebra in a simple manner, more accessible that identifying the essential unknown before and without the use of algebra.

The solution of triangular systems of equation introduces students to the notion that a letter or variable may have the same value in the other (simultaneous) equations.  Following that, student will quickly grasp how to solve triangular systems, by calculating one unknown at a time, one after another.

We can mention here that for any given figure, the the measurements and further quantities are fixed, constant, unchanging.  We may also mention that the measurement and further quantities vary between figures. So the latter sense, the letters that appear in the geometric formulas are variables.   See the lesson What is a Variable to learn more.  Include that lesson or aspects of it in your present or future classes.

4. Forward and Backward Use of Equations with (i) arithmetic or numerical; and with (ii) literal or algebraic solution of equations

Chapter 14 in Volume 2, Three Skills for Algebra, introduces the concept of forward and backward use of formulas and equations in arithmetic or numerical style and in algebraic or literal style.  All formulas in high school mathematics are employed forwards or directly, and backwards or indirectly.  By naming this phenomena and identifying it in example after example, we provide ourselves and students with a unifying theme in the development of algebraic writing and reasoning.  The forward and backward use of formulas can be seen and emphasized in the determination and employment of proportionality constants, in the backward use of formulas for areas, volumes and perimeters to finding missing dimensions of geometric figures, and in the determination and employment of scale factors (proportionality constants) for similar figures in two and three dimensions.  The Pythagorean theorem also provides a relation if not a formula which can be used forwards and backwards to find missing lengths in a right triangle.

Three Skills for Algebra

Chapters 8 to 12 in Volume 2, Three Skills for Algebra, describe the three skills at length and in that include advice on the use of notation (lower and upper case letters, with or without subscripts).  Talking about the three skills for algebra (and the fourth in chapter 13) adds an oral dimension to mathematics. The site essay on what is a variable goes further into the use of words in mathematics, and into some nuances that may clarify the use and overuse of the term variable in mathematics.

Origins of the Shorthand Role of Letters: The use of geometric formulas for perimeters,  areas and even volumes introduces letters as shorthand symbols for geometric measurements and quantities. Those geometric roles for letters - denoting the value of a measurement, known or not, provides a first step in algebra, or the use of letters to denote numbers with or without a geometric significance. The expression

let x be a real number

in which x has no geometric significance is enough for some students to say, give me the number, while they fail to grasp the meaning of the phrase. That being said, there need be no rush. Mastery of the phrase or its meaning can come latter.

5. The Pythagorean Theorem, Forwards and Backwards 

The Chinese dissection proof of the Pythagorean theorem points to the use of assumptions about area calculations in deriving and mixing algebraic and geometric results. The algebraic forward use of the Pythagorean theorem algebraic identity employs the lengths of the legs to calculate the length of the hypotenuse. The algebraic backward use of the algebraic identify employs the length of one leg and the hypotenuse to find the length of the other leg. In the associated calculations, equal signs should not be abused. A vertical alignment of equal signs would be consistent with previous formatting habits for the evaluation of formulas. While students may use calculators to evaluate square roots approximately, students may be shown how to obtain exact expressions for lengths of legs or hypotenuses when the given sides (legs or hypotenuse) have integral lengths.

The converse to the Pythagorean theorem indicates if the sides of a triangle satisfy the Pythagorean identity then the triangle is a right triangle..

6. The Distributive Law and its consequences

The distributive law in the first instance can be regarded as consequence of assumptions about the calculation of areas or, in the case of whole numbers and fractions, of assumptions about counting. The site coverage of polynomials shows how methods for multiplication and addition of polynomials can be introduced (albeit not fully justified) with the aid of geometric assumptions about area, or how the area of a rectangle may be obtained from a sum of areas of covering subrectangles. There-in lies justification for column methods for multiplication, addition and subtraction of decimals and polynomials, full and partial. 

The foregoing provides a first use and perspective of the distributive law.

7. Maps, Plans and Coordinates

Maps and Plans can be used to describe figures and displacements in the plane or in space. Drawing figures and circles to scale changes lengths but preserves angles. That being said displacements in the plane can be represented by arrows or vectors. The head to tail addition of arrows or vectors then corresponds to successive displacements in the plane. This addition is clearly associative. The head to tail addition of collinear vectors is also commutative.  The assumption that the selection of unit directions, a perpendicular pair, and the selection of a unit length determines a coordinate system in which signed numbers are employed as coordinates, and in which head to tail addition can be described leads the option of adding vectors with the aid of horizontal and vertical components.

 For rectangular coordinate systems, the assumption that the addition of horizontal and vertical components commute together with the commutativity of addition for collinear vectors yields the commutativity of vector addition.  The commutativity and  associativity of vector addition in the plane implies the coordinate description of vector addition is also commutative and associative. Whence the properties of signed numbers and arithmetic operations on them is implied by the applied mathematics assumptions made to permit the use of coordinates implicitly define methods for addition and subtraction of signed numbers, and imply associative and commutative laws for addition and subtraction of coordinates.  Finally, the assumption that addition of vectors is independent of the coordinate system in which it may be described implies the distributive law for addition of signed numbers, that is real numbers. 

Details are indicated in the site areas on number theory. But there is a complex number extension of the foregoing in which the field properties of both real and complex numbers stems from the rectangular coordinate representations of the head to tail addition of vectors and from the definition of products of vectors in the plane using polar coordinates. See the site starter lesson for complex numbers, and the easy consequences. in the site area on complex numbers.

8. Why Slopes, Geometric and Algebraic Calculus Previews

Calculus employs high school mathematics (fraction and algebra skills, trigonometry, etc) at full strength. Algebra in particular is employed at full strength too suddenly for many. This geometric why slopes lesson and algebraic   chapters 2 to 6, from in Volume 3, Why Slopes and More Math, provide two previews of calculus. Those geometric and algebraic why slope previews of calculus could provide the end of secondary mathematics and the start of college mathematics.   The algebraic previews provide a context for factorization of polynomials and associated factor dependent sign analysis to determine where the polynomials are positive, negative or zero. Both previews together will help ease or avoid algebraic shock in differential calculus.

Calculus in the first instance is the subject of slope-related computations and their reversal for linear and nonlinear functions y = f(x).  The reversal leads to the formulas for areas and volumes for circles, spheres, pyramids and prism given in earlier instruction. Calculus is the basis or doorway (a filter) for advanced studies in accounting and business; in nursing and medicine; in science and technology; and for  teaching earlier mathematics courses. Site pages include methods for easing or avoiding difficulties in mathematics before and during calculus.

The foregoing path or identification of core elements of high school mathematics does not depend on Euclidean Geometric to arrive at results. It depends on the applied mathematics assumptions needed to deploy rectangular coordinates in the plane.  In the particular, the development of complex numbers does not depend on (i) Euclidean Geometry nor (ii) the Pythagorean theorem. However the Pythagorean theorem is a consequence of the equality of two different ways to form products of complex numbers, namely (i) through the use of real and imaginary parts; and (ii) through the use of polar coordinates (modulus and argument). The equality of two different ways to form products of complex numbers has many further easy consequences for students to meet and master.

Remark: The lean site coverage of Euclidean geometry is not necessary for the core high school mathematics program indicated above.  The site coverage is sufficient for an operational command of Euclidean geometry.  The lean coverage does not include any proof of the Pythagorean theorem. But the coverage does include various characterization of parallel lines alone and of parallelograms.  There are not hard proofs in the site treatment.

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