|
Solving Linear
Equations
(Feb 14, 2005)
with & then without stick diagrams plus
solving word problems; and solving systems: - essentially one
unknown, essentially triangular & general
|
A. Letters and Lengths B. Solving Linear Eq'ns - WS C. Solving Linear Eq'ns - No Sticks D. Almost One E Substitution Methods F. Two More Methods G. Larger Systems H. Exercises to Try
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
Up A. Letters and Lengths B. Solving Linear Eq'ns - WS C. Solving Linear Eq'ns - No Sticks D. Almost One E Substitution Methods F. Two More Methods G. Larger Systems H. Exercises to Try
| |
Role of Letters in formulas for Areas of Rectangles, Triangles and
Circles
Welcome. In some formulas for areas and perimeters, letters denote and
serve as shorthand for lengths of sides or heights. We can describe calculations
with or without knowing the measures of the lengths until the last possible
moment.
If we said, let x or W be a number, you might panic.
But if we say or imply that x or W denotes a length, there may be no
panic. The roles of letters in denoting
length and areas is easier to grasp that of a letter denoting a number . Some examples follow.
1.1 Rectangles
Recall the area of a rectangle is given by its length times its width. We can
write this as
| Area of a rectangle = its length ×
its width |
|
This is a longhand description of the computation of the area of a rectangle. If
you give me the values of the length and the width, I can compute the area.
The computation of the area of the rectangle can be rewritten with shorthand
notation as follows. To introduce shorthand notation, we say the area A
of a rectangle is given by its width W times its length L. Here,
we use A as shorthand for the area of a rectangle, L as shorthand
for its length and W as shorthand for its width. The formula (recipe) for
calculating the area A of a rectangle can be written more briefly as
or
The shorthand notation takes less space to write than the word-only description.
Read the symbols × and · as times or multiply. The symbols ×
and · are both shorthand codes for times or multiplication. The dot symbol ·
is preferably to the times symbol × when the latter could be confused with the
letter x. Confusion can occur because the letter x which many
people write is too similar to the multiplication symbol ×.
Shorthand notation provides a code for the description of calculations.
Formula decoding is required. The shorthand formula A = L ·W
is more compact (takes less room) than the word-only description. This formula
is meaningless for us when the role of the letters in this shorthand description
is not explained. To understand and to use the shorthand description or formula,
you need information. You need to know or find what numbers or quantities the
symbols mean or represent. In the above formula, L stood for the length
of a rectangle. This has to be said to you or you have to ask. To anyone without
this information, the formula remains mysterious.
Talking about and describing computations almost gives us the power to do
them. In the area calculation, the area A is obtained from the recipe A
= L×W provided the length L and W are given or can
be found. Without this information, we can describe or understand a calculation
but not use it. The above rectangle example reminds us of the following:
- We can talk about quantities or numbers without doing any arithmetic. We
can speak about numbers and quantities even if we have not measured them or
do not know their values exactly.
- We can describe calculations without performing them. This description can
be done with words alone (throw out the letters) or with mathematical
shorthand notation, as convenient.
We will describe a few more calculations before starting to change them.
1.2 Triangles
In words, the area of a triangle is given by one half the length of a base of
the triangle multiplied by the height of the triangle. This formula can be
justified but at this moment we will not worry about why it holds. We may also
write more briefly
| (Area of triangle) = |
1
2 |
[ (base length) ·(height of the
triangle)] |
|
We may write still more briefly that the area of a triangle is
given by
This involves some shorthand notation: the letters A, B and H.
When you read or decode this shorthand notation, remember B stands for
the length of a base of the triangle. Also remember H stands for the
height of the triangle above this base. Lastly, remember A stands for the
area of the triangle.
We have used single letters in this shorthand
description of the calculation. Any mark or squiggle or symbol you can draw and
name can serve as shorthand for some number or quantity.
Perhaps, we should use Atriangle or another symbol, since
we have already used the letter A in the previous rectangle example.
Alternatively, we adopt the following rule: while you are reading this triangle
example, we use the letter A here as shorthand for the area of the
triangle only. More will be said on using and reusing (recycling) shorthand
symbols (for example, letters) and the roles they take. Think of them as actors
which can perform many parts. They may take only one role in any scene, except
for stories and scenes involving identical twins or cases of mistaken
identities.
1.3 Circles
The symbol for the Greek letter called Pi is p. In
words, the area of a circle is given by the number p
times the square of the circle's radius. The square of a number or quantity
refers to the number or quantity times itself. The square10 of 5 for instance is 52 = 5 ×5 =
25. We can also more briefly write
| Area of a circle = p·radius ·radius |
|
Here we are using a letter, the Greek letter p to
stand for and be shorthand for a constant, invariable, unchanging number. The
number p is approximated by 3.14159
The value of to two decimal places is 3.14 - the latter
value is not exact, it is approximation. Many primary school teachers and
some high school math teachers do not know that.
To rewrite or encode this formula in shorthand form, we will first describe
the code. Let A be shorthand for the area of a circle.11 Let r be our
shorthand for the radius of the same circle. Then the previous word-only formula
for the area of a circle is written A = p·r
·r or as
In the latter expression, the multiplication signs have been left out (omitted)
and r2 is shorthand for r·r = the radius r
multiplied by itself. The shorthand form of the formula, namely A = pr2, takes up less space than the
word-only form: the area of a circle is given by the number p times the square of the radius of the circle. Here
one must ask which is the easiest to understand, the above shorthand or the
just-given word-only form?
The value

| |
|
|
|
www.whyslopes.com
site
search
Parents: Help
your Child/Teen Learn covers Speaking
Skills, Reading
& Writing,
Preparing for Science &
Having Patience, etc
Math How-TOs
1. Arithmetic
2. Algebra
3. More
Algebra 4. Geometry
5 More
Geometry 6. Calculus
>> densely written
>> use as skill checklists
Online
Volumes (orders)
1, Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3 .Why.Slopes.&.More.Math.1995
Skill
& Concept
Review or Development
1. Decimal
Arith - Video Based ]
2 Fractions
3. Fractions
with Units
3. Solving
Linear Equations -
making alg easier
4. Formulas
forwards & Backwards - unifying theme for Algebra
5. Proportionality,
Back- & For-wards - theme at work.
6. Logic
- Math Free, good for precision in work & studies
7. Euclidean-Geometry
(leanly)
8. Slopes
and Lines
9. Why
Study Slopes - a context
10. Quadratics
11 Polynomials
12 Factored
Polys - a context
13 Functions
- For-& Back -wards
14 Number
Theory, Richly
15. Exponents,
Radicals & logs.
16 Calculus
- Examples & Advice
17. Real
Analysis
18
Electric
Circuits Etc (So So)
19 Maps,
Similarity & Trig, (alt view)
20 Complex
numbers
21
Logic with Symbols+truth tables
22 Consistent
Story Telling
23. Even
More Logic
|
|