Finished

1. Solving Linear Equations

The site section Solving Linear Equations with and then without Stick Diagrams covers solution of (i) linear equations ax+b = cx + d in one unknown, (ii) triangular systems of equations and (iii) systems of equations in essentially one unknown before covering the solution of systems of equations in two unknowns by the (a) substitution, (b) comparison and (c) row-addition-multiplication method. The site section emphasize exact arithmetic with whole numbers and fractions, and the checking or verification of solutions. With the latter students can check and/or correct their work before submission for grading.  Warn the student that if a check fails, the error is somewhere between the start of their solution and the end of their check.

Links: www.purplemath.com offers an introduction to Solving Linear Equations. It discussion of solutions of equations in one unknown (solution method, no solution case, infinite solution case) may go beyond course requirements. The notes  below  call for a format in solving equations in one unknown that will help students in solving linear equations in two unknowns by the equation addition and multiplication below. The foregoing purplemath lesson (4 pages) gives examples of the recommended format.

Notes

The remarks or their suggestions, hindsights,  will be incorporated into site lessons on Solving Linear Equations with and then without Stick Diagrams.

 Requiring the exact solution of linear equations in one and then two unknowns and requiring students to check their answers forces students to review and master operations with whole numbers, fractions and signs; and forces students to master the distributive law

 a(b+c) = ab+ac

and its generalizations

a(b+c+d+ ... + z) = ab + ac + .... + az 

n expanding and factoring expressions. Factoring is present when we write

8x+ 4x + -9x = (8+4+-9)x = 3x

You should Insist that students do or be able exact arithmetic with whole numbers and fractions, and signs too,  without the use of calculator. 

Remember to include a lesson on Proper Usage of the Equal Sign.  Say the equal sign (=) means "has the same value as" instead of saying "is the same as".  The former usage lends itself to a greater ease of use.

1 Fraction Sense versus Fraction Skills

Reference: Solving Linear Equations with and without Stick Diagrams,

Students at this level should not need the stick diagrams. However, if you see that students have a weak command of fractions,  examples if not exercises with the stick diagrams may develop the missing fraction sense. Students may need to review the fraction summary and more in the site section Fractions,  Ratios, Rates, Proportions   & Units. Explain to students that efficient arithmetic skills with fractions is a must. Test and test students on their comprehension of fractions and their mastery of efficient ways to add, multiply and divide fractions in proper, improper and mixed form.

Teachers: Coefficients in solving one equation in one or more unknowns should be chosen to imply integer coefficients in the first instance and fraction coefficients and even fraction coefficients in the second instance. Algebra requires an efficient command of exact arithmetic with whole numbers and fractions. Some students may need to review the fraction summary and more in the site section Fractions,  Ratios, Rates, Proportions   & Units 

2  Notation

 For solving ax + b = c, in place of or besides stick diagrams, I would use the column format

a x + b =   c
      b =   b    _
ax      = c - b

as a hint of and preparation the format seen in the row multiplication and addition method for solving linear equations in two unknowns. The site section Solving Linear Equations with and then without Stick Diagrams does not yet use this format in its algebraic solution of equations. A correction or improvement may follow later.

Mention the following in class:  if expression A has the same value as expression B, that is, if A = B and C is a third expression then A+C should have the same value as B+C, that is A+C = B + C. We take this equality and the equality AC = BC as as assumption in manipulating equations.

I would also use a similar format for reducing ax+ b = cx + d to the form
Ax + B = C just considered.  That is

  ax +  b =   cx+d
  cx      =   cx    _
(a-c)x +b =     d

3.  Triangular Systems.

In  Solving Linear Equations with and then without Stick Diagrams,  introduction of triangular systems (optional in 436)  provides a quick and easy way to meet the concept of simultaneous equations.   Students in meeting triangular systems of equations may be surprised that one unknown, say x or y, has the same value in two or more simultaneous equations. The objective here is to minimize that surprise. Solving triangle and scramble triangle systems of equations is not yet part of the course, so it inclusion is optional. But inclusion may develop students algebraic and arithmetic skills.

4. Systems of Equations in Essentially One Unknown
 
The solution of systems of equations in essentially one unknown introduces the elimination of one to several variables by the substitution method. Substitution is simplest in systems of equations like the following

A = 4x
B = 5x
C = 2x
3A+ 2B + 4C = 44

where there no need for the expansion use of the distributive law a(b+c) = ab + ac. (The contraction use  ax+bx = (a+b)x is present, and might be used without mention.)  Students will have to be reminded to calculate A, B and C after obtaining the value of the unknown x, otherwise their solution is incomplete.  Substitution is more complicated in systems of equation like the following

A = 4x + 6
B = 5x
C = 2x - 1
2A+ 3B -5 C = 30

where there is a need for the expansion use of the distributive law a(b+c) = ab + ac and the collection of  terms involving the essentially unknown x. C choosing coefficients so solutions are whole numbers or integers will ease or avoid difficulties for students in the first instance. Students may need some practice here to ensure or check that their use of the distributive law is correct. The ability to check their solutions will lead students to question their own calculations and so seek help in repairing faulty arithmetic skills.

Many word problems students meet in secondary I and II, if not secondary IV, can be written as systems of equations in essentially one unknown. Word problems become much easier if students learn to formulate them as one systems of equations in essentially one unknown instead of trying directly formulate them as a single equation in one unknown.  This simple extra would greatly benefit earlier mathematics at the junior high school level.

 5 Solving Systems of Equations in two Unknowns. 

All three methods below result in or employ a single equation in one unknown to find a first unknown. The single equation result from the elimination of one unknown by one method or another. So all three methods below are elimination methods. For a textbook to identify only one as an elimination method or the elimination method is a bit absurd.

Teaching Students to check their solutions and providing a format for this may extend their algebraic thinking skills and lead students to correct their work before handing if a check fails. That being said, tell student that when a check fails, there is an error between the start of the solution and the end of the check. The error may lie in the solution or the check.

(5a) The substitution method introduced for systems of equations in essentially one unknown applies to equations of systems of the form

 y  = 2 x + 5
 3x  + 2y = 27       

I might give the above system as one question and thene the equivalent system

 y  - 2 x  =  5
 3x  + 2y = 27       

as another system in the same lesson or same exercise set.  So the substitution method for reducing the number of unknowns (here eliminating the y) to obtain an equation in one unknown appears as special case of the method for solving systems of equations in one unknown. After find the "essential" unknown, students need to calculate the other from one equation, and then check the find values for x and y satisfy both equations.  In the case of equations of the form

 y  = 2 x + 5
 3x  + 2y = 27       

the first equation is satisfied automatically by the computation of y. So only the second equation 3x + 2y = 27 needs to be checked. That is a shortcut. If students do not understand it, have them check for both equations. Use the following format:

Check Format:

Check: when (x,y) = (a,b), 

LHS₁ = y = b and RHS₁ = 2x+5 = 2a+5 =  ...
LHS₂ = 3x  + 2y = 3a  + 2b = ...  and RHS₂ = 27

(5b) The comparison method gives another method for eliminating one unknown to obtain a single equation in one unknown. Here two simultaneous equations

y = ax+b
y = cx+d

imply the two expressions for y should have the same value. So

ax + b = cx + d

Here the ability of students to solve one equation in one unknow (step 1) becomes useful. After finding x, student need to check that the two right-hand side expression ax + b and cx + d give the same value for y.

Check Format:

Check:  RHS₁ = ax+b = and RHS₂= cx+d = ...

(5c) The equation-multiplication-addition method is yet another method for eliminating one unknown and obtaining one equation in one unknown.  It can be applied to systems of equations of the form

y = ax+b
y = cx+d

to obtain  0 = (a-c)x + (b-d).   It can be applied to systems of equations of form

ax + by = e
cx + dy = f

In the case where a, b, c and d are integers, the choice of multipliers can be based on comparison of the least common multiples of the x coefficients a and c, and the least common multiples of the y coefficient b and d.  If the former is smaller than the latter lcm, eliminate x and obtain an single equation in y, while if the former is greater than the latter, eliminate y to to obtain an equation in x. The foregoing elimination decisions result in smaller coefficients. Have the students to use the follow check format:

Check Format:

Check: when (x,y) = (p,q), 

LHS₁ = ax + by = ap+bq = ...
RHS₁ = e
LHS₂ = cx + dy = cp+dq = ...
RHS₂ = f 

www.whyslopes.com
Lesson & Lesson Plans for
Sec IV (Maths 436)

a reference for learning and teaching functions, polynomials, solving linear systems, 
powers + exponents + bases + radicals (roots) , quadratic formulas, equations of straight lines

1A. Master Logic
1B. Problems Solving Method
2A Solve Linear Equations i
2B.Solve Linear Equation II
2C Use Equal Sign Properly
2D. Perfect Arithmetic Skills
3 Words & Symbols
3 Goals to Set for Students
4 Use Equations Backwardly
5. Master Functions & Relations
6. Exponents & Radicals I
7. Straight Lines
8. Polynomials (x,/,+/-)
9. Quadratics
10 Prove it
13 Similarity Scale Factors
12 Trig & Triangles
14 Statistics
MEQ Intermediate Objectives
Remarks for Teachers

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Sit down and study - no one else can do that for you.

Advice and Directions
What to do in School   & Why
How to Study Maths & Why

Preparing for Science 

Good News: If you can learn to follow a multi-step methods in any subject precisely, you should be able to do so in other subjects, as well. Hint: Start with arithmetic

Words Before Symbols: 
What is a Variable?
Level:  Secondary II to VI, or Grades 7 to 12)
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent or Independent
Variable, a Matter of Choice

Complex number starter lesson  

Arithmetic Videos
Fractions
Primes
Greatest Common Divisors
Least Common Multiples
Square Root Simplification

Arithmetic Videos

Decimal Addition Methods
Decimal Subtraction Methods
Decimal Multiplication Methods
Decimal Division Methods

Fraction Starter Lesson
(simplify, multiply, divide & 
then add or subtract)

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