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Proper Use of Equal Sign
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Solving Linear
Equations
(Feb 14, 2005)
with & then without stick diagrams plus   solving word problems; and solving systems: -  essentially one unknown, essentially triangular & general

 


Read them in order

(i) x + 20 = 29 WS
(ii) 2x + 6 = 24 WS
(iii) 3x + 10 = 32 WS
(iv) 5a + 16 = 3a+ 24 WS
(v)  (½)x + 8 = 24½ WS
(vi)  (¾)a + 16 = (¼)a+ 24 WS
(vii) (¾)q + 17 = 32 WS
(viii) 13 =[2/3]x +7 twice WS
(5/6)q + 8+(5/6) = 14 + (2/3) WS
Proper Use of Equal Sign



 

2.1  Use of the Equal Sign

Here are a few words about the equal sign. The equal sign = is used to say or suggest the following.
  1. two different symbols (or expressions) are shorthand for the same number and quantity.
  2. two different calculations or expressions give the same result when done, or
  3. the value of a number or quantity can be computed using another expression.
The suggestion in question can be true or false depending on circumstances. Examples follow:
4+5
=
7+2
r2
=
r·r
3x+1
=
x+7
x+4
=
x+6

Here the first equation or equality holds (meaning is true) since both 4+5 and 7+2 are expressions giving the value 9. The second equation r2 = r·r always holds, no matter what value you give to r. It tells us how to compute the number or quantity described by the expression r2. The third equation 3x+1 = x+7 holds (is true) when and only when x = 2. When x has a value other than 2, the statement (suggestion or assertion) that 3x+1 gives the same result as x+7 is false. The fourth statement x+4 = x+6 is always false. No value given to (or substituted for) x will make this statement true. Adding 4 and adding 6 to the same number give different results, no matter what the number is.

Abuse of Equal Sign

The solution of the equation

3
4
x
4

is given x =3. But  it is an error, a mistake, a major misuse of the equal sign,  to insert an = 3 besides the x in the above equation  and thus write

3
4
x =3
4     

in place of writing x = 3. While a person who writes 

x  = 3
3         

may mean x = 3, for mathematicians

x  = 3
3         

actually means a third of x  is 3.  

Proper Format:  For the sake of observable skill development, the statement of the equation  to solve

3
4
x
4

should have been followed by a steps like the following

4 × 3
4
4 × x
4

and then the conclusion   

3 = x, 

 and then the conclusion

 x = 3.   

The difference between the last two equations is cosmetic:  We prefer to say   x is 3 or equals 3 in place of saying 3 is or equals x.  In general, when two different expressions are opposite sides of an equal sign,

first expression = second expression

then should be read as follows.  The first expression in the circumstances at hand has or will have when evaluated, the same value as the second expression, when or if it too is evaluated. 

Format for Evaluation of  algebraic and arithmetic expressions, a standard.

Note vertical alignment of equal signs

Steps:

A Simple Example

  1. Write the geometric formula properly.
  2. Draw or sketch the diagram, and on it indicate the values of the letters or quantities in the formula.
  3. Substitute the latter values in the formula,
  4. After substitution, simplify as much as possible without the aid of a calculator.
  5. Lastly, if wanted, evaluate the simplified expression with or without a calculator.

In all the foregoing, keep the = signs vertically aligned,  At this stage, students may not know how to use the equal sign.=, or see the necessity for it, but the format above introduces a proper use of the equal sign.  Mastering this format ensures proper use of the equal sign is ensured. Teaching this format show students how to present their work or reasoning on paper in a systematic instead of ad hoc manner..

Note: Students may do work in their head. But  teachers cannot read students mind. The use of this format provide an observable skill that can be seen and verified, or corrected.  Teaching may provide students with food for thought, but in the end in practice, skill and knowledge development has to be observable to be believed and guided. 

Problem: Find the area of a 12 cm by 5 cm rectangle

Solution:

area A 

L × W  (the formula written above)
=  (12 cm) ×(5 cm)  (substitute)
=  12 × 5 cm2  (simplify)
=  60 cm2  use a calculator if need-be.
Note: The above solution communicate the logic (use a formula) and communicates the reasoning process or steps in that use or evaluation. Observe how the equal signs are present, and how they are vertically aligned. That provide a good habit to follow in the evaluation of algebraic and arithmetic expressions - one that avoids abuse or improper use of the equal sign - a parallel topic to cover in class. The vertical alignment of equals signs should be required except at the bottom of a page or blackboard where a horizontal alignment of equal signs may permit the evaluation or simplification underway to be written without jumping to a new page or to a distant location of the black board. 
 

Another Formatting Example

 Observe how curve arrow shows the next step in the calculation. 
Calculations should be shown in sequence and not in place.


Formatting Advantages:
The above format for formula usage or evaluation provides a model for students to follow not for rectangle area evaluation, and 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.

 

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