Algebra Lesson Plans - Steps for Skill and Concept Development

The algebra  lesson plans below and in a following page More Algebra Plans provide, we hope,  a systematic path for developing algebraic skills and concepts, a path to ease or avoid many common difficulties and to enrich and clarify comprehension for beginners and experts alike. This path provides smaller steps for skill and concept development to ease and enable mastery. These smaller steps will make algebra more accessible and so help more students. That is to be appreciated.   But as always, smaller steps still or alternative routes may be required. 

Steps to introduce and develop algebraic reasoning skills.

1. Formula Evaluation Format - A Standard to improve performance and to imply format is important
2. Solving Linear Equations with Stick Diagrams - consolidate fraction skills and sense, set an example of exact arithmetic in algebra, further show the importance and advantage of good notation and performance.
3. Solving Systems of Linear Equations (triangular and essentially one).  Make Word Problems Easier. Consolidate and Extend Exact Algebra Skills, Set the stage for solving general systems.
4. Words Besides Symbols.  Word may describe numbers, measures and further quantities as known or not, constant or not, variable or not, varying in one direction (temporal or spatial) but not in another.  Then letters or symbols that denote (stand-for) numbers, measures and quantities will be described as constants, variable, known, unknown in the same sense as the corresponding number, measure, quantity, amount or value.  That should get rid of the formal nonsense of saying a variable in mathematics is a letter, and vice-versa. The foregoing use of word variable should come before the modern mathematics or logic description or characterization of a variable as a function.  
5. Forward and Backward Use of Formulas and Equations - Introduce a universal and unifying theme in mathematics and science.  All formulas will be used forward and backwards, numerically first and algebraically (literally) second. Many formulas could appear here to illustrate that forward and backward use.  Stay with the ones in your current courses.
6. Arithmetic with Units.  The algebraic deployment of units in calculations appears in chemistry, physics and engineering, if not in pure mathematics.  Arithmetic with units prepares for that deployment and provides a foretaste of operations on polynomials. And in proportional reasoning,  rates and further proportionality constants can represent using monomials in units of measure, alone or as the numerator and denominators of fraction like expressions. Hint: Cover this topic fully or partially as permitted by course design.  Cover it alone or embedd it in your representation and manipulation of rates (ratios of unlike quantities) and of proportionality constants. 
7. Proportional Reasoning.  The forward and backward use and manipulation of proportionality equations Y = KX and/or Y = KXZ provides the algebraic key and perspective for solving proportionality problems.
8.  Ordinary and Multiple Ratios, - A different  perspective. Coverage of ordinary (double) ratios a:b is required, but how is not yet certain. 

Co-Readings: The earlier written advice on Secondary II Mathematics, a year of algebra and proportional reason, offers more insights and in particular more details to illustrate skill and concept development.  See too these algebra lesson plans.  Both references are must reads for this page, companions which add ideas to include in the sequence below and/or in the further, more algebra, skill and concept development. 

Food for thought - Not all certain:  Step 6 on arithmetic with units is optional - I think it valuable, and I would include if time permit.  Step 8 on ordinary and multiple ratios is also optional.  This step, not essential,  explores an optional alternative to treatments I have seen in high school, treatments that have not been to my liking.  The alternative is technical correct, but I am not sure how to reconcile with what may be found it high school and college textbooks.  

Step 1. Formula Evaluation Format

Setting standards for notation and written work in mathematics will speed learning and teaching,  and help in the evaluation and documentation of student progress.  

Second skill for Algebra: we can describe calculations which we want to do or avoid or have someone else do, without doing any arithmetic. The description gives a recipe or a formula for doing a calculation. The description can be done with words alone or with shorthand notation. This shorthand notation is worth a thousand words. The first service of mathematics to other subjects lies in the description of calculations that can be done or repeated as needed. There is more to mathematics than just doing arithmetic well.

Formula evaluation requires letters to be replaced by numbers or quantities to obtain an arithmetic expression to calculate.   

This format for formula evaluation (discussion aimed at teachers, readable by students too) provides a standard.   Good format and good  notational habit, easily understood and repeated, speed comprehension and reduce errors.  Require your students to adopt the format for better marks, for clear communication of comprehension and  reason, and for a solid base for thinking and problem solving. Have them adopt a similar format for evaluation of arithmetic expressions.  Note the format employs the equal properly and so avoids its abuse.

Formulas for geometric volumes, areas and perimeters describe calculations that might be done, that could be done or be postponed.  A single formula represent many, many, possible calculations - emphasize that to your students.  

Examples of Formulas and Shorthand Notation:  Formulas for area of triangles, squares, rectangles, circles, trapezoids, parallelograms and polygons; for volumes of spheres, cylinders, cones, pyramids, and boxes (parallelepipeds); and for perimeters of triangles, rectangles, circles and so on, provide opportunities to illustrate and reinforce the format, and to illustrate exact and approximate arithmetic with the format.  Formulas for simple interest, compound growth and decay (compound saving accounts included), and the geometric formula provide further examples.  All or some of these formulas can be employed in a plug and play manner without or with indication of why they might hold.  Mention that the proof of formulas for volumes, for circle area and perimeter may be justified in calculus. 

Modern pedagogy talks emphasizes communication, reason and problem solving skills. Here written communication and reason (showing work) are two sides of the same coin.  In decimal, place value, column methods for addition, subtraction, multiplication and long division,  the format helps student obtain results in a repeatable, reproducible and observable manner.  Each step of the solution process (aka student work or reason) is recorded as done to provide a base for the next step.  That record in full communicates the reasoning involved in the arithmetic in an observable manner that can be checked or corrected by the doer or another - a peer or an instructor. Learning the format and practicing it automates a skill and build confidence, while setting the stage for further mastery of mathematics as a discipline in which  there are rules and methods to master in observable manner. Observability means that the steps are developed and recorded on paper for the sake of verification or correction.  Here and below, wherever a format is given, it or an alternative can be prescribed to help student master the routines and methods of mathematics.  

Mechanical and efficient mastery of given rules and patterns in mathematics in arithmetic and beyond, in a repeatable, reproducible and observable manner is  a sign of intelligence for  the common person in the street as well as for college instructors in mathematics, science and law think otherwise. If you want to prepare your students for college mathematics, mechanical mastery of the rules and methods of arithmetic, algebra, geometry and eventually logic is a must.  Moreover, if we provide a standard format for the neat development and record of steps in mathematical methods and patterns,  the record itself documents student progress for themselves, for their teachers and their parents. Problem solving in mathematics should be first developed and tested with problems with routine solution methods to provide a base for non-routine problem solving while being part of the preparation for college mathematics (calculus).  Mathematics is an art and discipline with a core part or channel focused on preparation of for calculus. That channel mastery of key skills and concepts or key methods and routines with as much understanding as possible. That being said, the first part of those channels may focus on the mastery of rules and patterns, steps included, in a way that leads to repeatable and reproducible results on paper for checking or correction, while the next part may include a greater emphasis on the thought-based development of skills and concepts.  

Students need to learn how to evaluate formulas in a manner that records and show mastery of the evaluation process. The pre-requisite for formula evaluation is the ability to evaluate arithmetic expressions exactly or with a calculator, involving numbers and quantities with or without units being present. I recommend that units be carried through some  calculations to minimize the need for unit conversions and to take advantage of the latter. If you want to prepare your students for college mathematics, calculus, then emphasize exact arithmetic in formula evaluation.  Exercises in formula evaluation are also exercise in arithmetic and in the art or discipline of following format.  Emphasizing a proper format for the evaluation of arithmetic and algebraic expressions will provide a standard, easily understood and appreciated by students and parents. Neatness count. Many students and students do not realized that good notational habit, and work of recording all steps in a calculation carefully in a way that others can follow, will ease or avoid difficulties, and enable learning and teaching to go further. Focus on quality of the work. 

  The question of why a formula holds is separate from the question of whether or not a student will follow a clear or proper format in its evaluation, a format that help record and develop the steps in that evaluation. 

More on Formula Evaluation - elements of algebra in covering or reviewing fractions)

A format for this second kind of formula use is not specified. 

 Besides formulas describing areas, volumes and perimeters,  formula may also describe calculations that may be done: For example, algebraic description of fraction addition is often described by the sub optimal addition formula. 

A B

+

C D

=

AD +BC BD

where BD is not always the least common denominator. In contrast, in the LCD oriented, addition formula  

A BE

+

C ED

=

AD +BC BED

the common denominator BED will be least if  1 (one) is the greatest common whole number divisor of B and D.  The latter formula is optimal for efficient, exact arithmetic with fractions.  The presence of formulas to describe fraction subtraction, multiplication, division, fraction reduction (lower terms) and equivalent fraction generation (raising terms)  all provide a hint of algebra, or the role of algebraic expressions (formulas) in describing calculations that are done, or could be done. That hint can be made stronger by giving examples in which  the letters in the formulas are identified with numerators and denominators by giving instances of the formulas and saying give the correspondence between letters and numbers.  Here the role of the numerators  A and C is played by the number  ...  and ... respectively. That attention to detail and illustration of the formulas may slow the development of fraction skills while preparing for algebra.  I leave it to you to decide whether or not that is worthwhile trade-off.   Good luck. 

In sum, Algebraic Shorthand Description of Rules for Fraction Arithmetic, Calculation Formulas for Sums, Differences, Products and Division of Fractions, provide hints of algebra in the pre- or co-algebra development or re-enforcement of fraction skills and sense as in the site area on  Fractions Ratios Rates Proportions, Units 

Step 2. Solving Linear Equations With Stick Diagrams 
Consolidate Fraction Sense and Introduce or reinforce algebra skills

Following the evaluation of formulas for areas, perimeters and volumes, students are accustomed to letter denoting lengths that will be given.  The letters that appear in the stick diagrams represent a length that is to be found.   The use of letters that denote lengths or geometric measures appears more concrete and easier to digest than the more abstract, statement Let x etc denote numbers explicit in senior or college mathematics, and implicit in the statement of linear equations, 

Background Information: Role of letters as shorthand for lengths in formulas for areas of triangles, rectangles and circles.

Stick Diagrams is a site invention for visually providing a context for the solution of equations in one unknown. Worked examples follow with and then without stick diagrams 

The objective of the stick diagram method and its format is to lead students to solving linear equations properly without the stick diagrams.  Fractional operations on the sticks (line segments) may improve fraction sense and skills. 

The stick diagram method here employs only subtraction, division and replication of segment lengths.  Magnification and reduction of diagrams is also useful to fit them in the width of a column. Example equations are chosen so that all coefficient and terms in the stick diagram method remain non-negative.  

Show students how check their solutions so that they correct they work (time permitting) before submission. Include marks (20%) for showing the check in their work. 

Students should be required to check that the solution they obtained satisfies the original equation, and be told explicitly if the right hand side does not equal the left hand side for your solution that they have to look for the error (or if time is short, acknowledge their solution is wrong). Finding that the the right hand side does not equal the left hand side and saying nothing, or worse claiming to have done the problem points to a lack of comprehension.

If a solution check fails, tell students the error in their work will be somewhere between the start of their solution and the end of their check. 

Step 3. Solving Linear Equations - Triangular or Essentially One Unknown
Make Word Problems Easy,  Set the stage for solving general systems of equation

After students can solve linear equations in one unknown, and are in the habit of verifying solutions,  they may quickly learn to solve and check the solutions of the following types of equations. 

• systems of equations that are triangular as is or after a permutation of the equations. Here the unknowns can be found in sequence.
• systems of equations in essentially one unknown with the solution following from a substitution and the proper use (whether student know it or not) of the associative property of real numbers.
• systems of equations in essentially one unknown with the solution following from a substitution and the proper use (whether student know it or not) of the distributive and/or associative property of real numbers 

Remind students that 

If a solution check fails, then the error in their work will be somewhere between the start of their solution and the end of their check. 

A word problem that can be solved through a mental gymnastic recognition of a key unknown in the rewriting of the word problem in terms of a linear equation in that key unknown can also through the introduction of more unknowns, be written as a system of equations in essentially one unknown - the key unknown that would otherwise be obtained via mental gymnastics.  The latter approach of writing a system of equations and recognizing that is a system in essentially one unknown should be less challenging. It may also provide motivation for the immediate or later solution of general systems in two or more unknowns that are not triangular, and are systems in essentially one unknown.  

Remark: The substitution method employed in the solution of systems in essentially one unknown gives a foretaste of the substitution method for solving general linear systems.  

Possible Continuation:  For the solution of  general systems in two unknowns, see three elimination  methods for solving systems (sets) of  linear equations 

• Substitution
• Comparison
• Equation (or Row) Addition-Multiplication

Here is an example of equation addition-multiplication method for 3 equations in 3 unknowns.  Note: Students on becoming aware that there are three different methods for solving systems of equation may decide to learn only one. Students have to be warned against that option. An effective method to warn to give student a system of equation and specify the solution method to be used. 

Step 4. Words Besides Symbols
(How to clarify and expand Use of Words in mathematics)

Algebra employs words, formulas and equations (i) to describe numbers, amounts and quantities; (ii) to describe how to calculate them; and (iii) to describe relations between them. The word variable in mathematics may refer to a letter, but outside of mathematics, the word variable refers to variation and change. The mathematical use of terms and words would be clearer if the commonality is found between that use and the usage exterior to mathematics.  While the shorthand roles of letters and symbols provide a key part of algebra,  words have a role in algebra in describing and talking about numbers and equations.

First Skill For Algebra: We can talk about & describe numbers and quantities without doing any arithmetic and without using any letters or symbols.  For instance, numbers and quantities may be big, small, known, measured, never known, changing or unchanging, private, top-secret, confidential, embarrassing, or simply forgotten. A number, measurement or quantity may be known to you but not to me. We can speak about numbers and quantities in many ways. Talking about numbers and quantities is an ability we all have.  It is a part of mathematics that does not require us to do arithmetic. There is more to mathematics than just doing arithmetic carefully.

The Greek letter p usually denotes a constant  3.1416 approximately. Letters a, b, c, ... at the start of the Roman alphabet often denote numbers or quantities that will not change in the problem at hand. In contrast, letters z, y, x at the end of the alphabet often denote numbers or quantities that are unknown or may vary. So some letters in mathematics or algebra denote constants - numbers that will not change, while others denote numbers or quantities that may vary. 

This first skill for algebra need not first in an algebra lesson or course, but it should be included.  Reference: Chapter 9
Talking about Numbers or Quantities in Volume 2, Three Skills For Algebra. The viewpoint that a letter in mathematics denotes a variable or is variable, without connection to the dictionary meaning of the word variable is not to my liking. If you object, see Words Before Symbols: What is a Variable? [ A number or quantity which may change in the circumstances of interest to us is called a variable. The common idea that all variables have to be given by letters has mislead many. As just suggested, talking about variables, that is numbers or quantities which may change or vary, can be done without from any reference to letters and symbols. That is the notion of a variable can be clarified or explained before any linkage to algebraic shorthand or symbols used to write and record calculations and further parts of algebraic thought.]

Words are absent in mathematics.  Formulas and equations are better seen and read in a glance than read aloud. Naming Formulas or describing them with descriptive phrases is way to end the silence:  Include phrases like the following:  

Rectangular Area Formula,  Geometric Sum Formula,  Trapezoid Area Formula, Compound Interest or Growth Formula,  Quadratic Formula, Complete the Square,  Difference of two squares, sphere volume formula,  cone surface area formula,  circle perimeter formula, Prime Number Decomposition, 

in your courses and test your students on their comprehension of these phrases and names. End the silence. Teach student to talk.  

Remark:  The compact, shorthand and even cryptic description of calculations with algebraic formulas leads to a silence in mathematics communication that be offset by the deliberate use of names, descriptive phrases and temporary labels, formula (A) and rule (B)  for example,  But we can go further in developing a mathematical rhetoric for reading aloud and describing formulas that could be useful for speaking over the phone, or the online description of arithmetic and algebraic expressions with words or texts linearly where or where the latter cannot be written (drawn) and seen.   A few examples of this rhetoric follow.  The examples point to the shorthand advantages of algebraic and arithmetic shorthand notation while providing an alternative.

• The area of a rectangle is the product of its dimensions, height and width, in any order.
• The area of a triangle is one half the product of its base length and height - more precisely, it one half the product of the distance between two of its vertices, and of the distance of the third vertex to the line through the latter two.
• The volume of a cylinder with a horizontal base and all cross-sections congruent to the base (or  with area equal to the base area) is given by the product of the cylinder height and base area. 
• A ball has surface area given by  4 p times the square of the radius or the radius squared times 4 p where p is a real number with an infinite decimal expansion, 
• A ball has volume   fourth thirds of the product of the number p times the square of the radius.
• The quadratic formula for calculating the solution of a quadratic equation with real coefficients is given by a fraction in which the denominator is twice the coefficient of the square term in the quadratic, and in which the denominator is given the additive inverse of the coefficient of the linear term  plus or minus the square root of an expression called the discriminant.  When the discriminant is positive, use of the plus sign gives a solution, the most positive one, and while the negative sign gives the least positive one.  When the discriminant is zero, both signs give the a single solution.  When the discriminant is negative the quadratic equation has not real solution.  The discriminant itself is given by the square of the linear term coefficient in the quadratic minus 4 times the product of constant and square terms coefficients. 
• The expression on the left hand side of an equation is the sum of five terms. The first term is a fraction with top given by ... and bottom given by ....  The second term is the product of three factors.  The first factor is the sum of two terms, namely ... and .... The second factor is given by ... .   The third term is a rational   etc, etc.    
  

All the foregoing may imply to student that the use of arithmetic and algebraic expressions is far simpler than the use of mathematical rhetoric in the description of arithmetic or possible arithmetic.  That being said, it remains a challenge to optimize mathematical rhetoric for the description of arithmetic and algebraic calculation being done now or be left for later.  That optimization might help mathematics education of students, blind in full or part. 

Step 5. Forward and Backward Use of Formulas and Equations
Introduce a universal & unifying theme in the mathematics and science 

In Volume 2, Three Skills for Algebra, Chapter 14 employs the Compound Interest formula directly and indirectly (forwards and backwards), and compares arithmetic (numerical) and algebraic (literal) ways for this.  Every formula you met in high school and college mathematics and science is likely to be used backwards and forwards. The arithmetic approach to this may be easiest for students in the first instance, but the algebraic approach and it ability to solve many problems at once should be emphasized. Chapter 14 provides a model for introducing this unifying & essential theme in high school mathematics, and the arithmetic properties used in it, numerically or algebraically.  See too chapter 10 for an example of the forward and backward use of the rectangular area calculation formula A = WL.

Formulas for area of triangles, squares, rectangles, circles, trapezoids, parallelograms and polygons; for volumes of spheres, cylinders, cones, pyramids, and boxes (parallelepipeds); and for perimeters of triangles, rectangles, circles and so on, provide opportunities to illustrate and reinforce the forward and backward us or equations. Readers are left to identify and provide their own examples. 

The two equivalent phrases Forward and Backward Use (or Direct and indirect use) voice, identifies and emphasized what has hitherto been a silent theme in the teen and adult mathematics education. The phrases spoken repeatedly in the classroom will alert students to this common thread and the need to understand and master it. 

Consumer Mathematics: Formulas for simple interest, compound growth and decay (compound saving accounts included), and the geometric formula as well may be mentioned. In case of money matters, the formulas for present value and future value are consequences of the direct and indirect use of the geometric sum formula. The finer discussion of compound interest and geometric sums in connection with credit cards, loans, mortgages and annuities,  could provide students with an application of mathematics of interest, useful in its own right,  and of service in their preparation for a possible study of calculus.  See if the online chapters 

21 What's Next  22. Geometric and Arithmetic Sums 23 Summation Notation 24 Investments, Loans, Pensions - Personal Money Calculations 25 Mathematical Induction and Recursion - Proofs, Product Notation, & Factorial Notation

are useful in all or part for your consumer math lesson planning. 

For example, direct or forward use of the rectangle area formula A = WL where W denotes the width and L denotes the length of a rectangle  calls for the value of A to be found from given value of W and L. One backward use of this formulas will find the value of the width W from the values of area A and length L. See chapter 10 and 14 in Three Skills for Algebra to learn more and to see how numerical (arithmetic) and literal (algebraic) analysis and backward use may be presented in class to build skills and confidence. Again, the forwards and Backward use of formulas is a unifying theme for teen and adult education in the mathematical deployment  of formulas.  

Pythagorean Example From Geometry:  For Right triangles, the Pythagorean identity c² = a²+b²   between leg lengths a and b, and hypotenuse length c.  The forward use would obtain c from the principal square root  of a²+b² before or after substitution of values for a and b. The arithmetic solution would involve substitution first, while  algebraic solution would involve substitution after.  A backward use find a, given b and c values,  would obtain a from the principal square root  of c²- b² before or after substitution of values for a and b in the identity.  The backward use, find b, given a and c is similar.  

Step 6. Arithmetic with Units

 In pure and applied mathematics, saying how to do a calculation defines it.  Formal mastery of arithmetic  operation with expression involving units is needed to represent rates and proportionality constants and to use proportionality relations forwards and backwards.  The complete theory is developed in site pages. 

16  Units in Arithmetic   16 Longer Explanation  16 Change Units  16 Products of Quantities 16. Fractions with Units 16. Division+Reciprocals. 

Arithmetic  with monomials involving units, their products and  quotients takes on utility if not  meaning  in the subsequent appearance as rates and proportionality constants.  

Remark 1.  Operations with monomials involving units and their quotients resemble and provide a foretaste of  operations on monomials in variables x, y, z etc  and their quotients. The latter too (ouch) may represent formal operations on expressions that have no meaning for students other being marks on paper, albeit operations on monomials in  variables x, y, z etc could represent operations on potential calculations - the calculations that would result by replacing the variables by numbers or quantities. Inclusion of this topic will help later in  examples of exponent addition and subtraction with monomials in one to several variables x, y, z, ... and in their products or quotients.

Remark 2.    An operational command of calculations with units could be sufficient for further use in the representation of rates and proportionality constants and for further use in calculation in chemistry, physics and money matters. 

Remark 3.. In this arithmetic with units of measurement,  products and quotients of monomials may be formed and simplified with monomials that contain units to unlike powers. In contrast, sums and difference of monomials may be formed and simplified only with monomials that contain units to like powers.

Step 7. Proportional Reasoning - Or, the forward and backwards
use of proportionality formulas

Definition (1). A single quantity Y is proportional to a second quantity X  when and only when  there is a non-zero constant K such that Y = K X.

Here the direct use of Y = KX is to calculate the value of Y from those of K and X. But the typically two step problem gives the values (X₁, Y₁) first, from which the value of the proportionality K can be computed via a backward use of the formula. And after K is known, the formula Y = K X can be used directly or indirectly to compute Y or X respectively. The foregoing represents a two step recipe for finding and then using the proportionality constant K.  

The discussion of rates of changes can be included in this subject along with development of algebraic computation skills with units.  See the site section Fractions,  Ratios, Rates, Proportions   & Units. 

Constant speed and rate examples give equations of the form Y = K X with K = rate or speed.

Definition (2). A single quantity Y is jointly proportional to a second quantity X and a third quantity Z  when and only when  there is a non-zero constant K such that Y = K X Z.

The backward use of the equation Y = K X Z may give the value of the proportionality constant K in terms of the quantities X, Y and Z; or may give expressions for X or Z in terms of the other quantity, Y and K. The latter expression imply inverse proportionality relations.  Thus direct and inverse proportionality relations can be obtained and generated from each other via forward and backward manipulation of proportionality equations.

Senior High School, Proportionality Example From Geometry:  For similar plane figures, the ratio of corresponding lengths and areas (absolute measures) equals a scale factor K or its square K². For similar 3D figures, the ratio of corresponding lengths, areas and volumes equals a scale factor K,  its square K² or its cube K³.  Student may be asked to find and/or use the length, area and/or volume scale directly or indirectly.  From the algebraic viewpoint, the corresponding proportionality equations, relations or formulas (whatever you would like to call them) are being used forwards and backwards.  

A dozen proportionality situations or examples appear in the site area  Fractions,  Ratios, Rates, Proportions   & Units.  See too the site description of secondary II mathematics as the year of algebra and proportionality. The site area says or shows how to carry units in fraction like calculations. The latter is useful in the algebraic analysis of physical situations when quantities are expressed as numerical multiples of a units of measurements and their powers, alone or in fraction format. 

Remark: 

Step 8: Ordinary and Multiple Ratios  

Fraction and ratios are overlapping concept and have overlapping roles in arithmetic, but they are not identical even though fractions a/b where a and b are whole numbers may be called ratios. In mathematics ordered pairs of whole numbers a and b may appear in coordinate form (a,b) or [a,b]; in ratio form a:b and in fraction form. The following treatment emphasizes the difference. 

The following treatment (mathematically correct) below and in the  pages  Two Term  (ordinary) Ratios  Implied Ratios & Multiple Ratios  provides food for thought rather than a lesson plan.    The information here may clarify comprehension of ordinary and multiple ratios for tutors and teachers, or gifted students, but what elements of it should be present in class and how will not be prescribed here.  Here is a necessary topic whose depth and extent of coverage is puzzle for me, the site author. Pieces of this topic and puzzle follow. What can be done without doing any harm? That is the question.