Class with Jurek ppt about different topics

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High School Mathematics
Teaching Methodology

High School Advanced Math
Algebra 1
Geometry
Algebra 2
Trigonometry
Pre-Calculus
Calculus

Algebra 1
• Expressions
• systems of equations
• functions,
• real numbers
• linear questions
• Polynomials
• Quadratic equations
• Functions.

Changing the Subject of a
Formula (Simple Formula)
Substitute Numbers for Variables in Algebraic Expressions

Simple Formulae Involving the Four Basic
Operations
The objectives of this lesson serves to accomplish the following:
1. Provide an explanation of the concept ‘changing the subject of a formula’
2. Demonstrate the procedure related to ‘changing the subject’ in formulae
involving
Addition and Subtraction
Multiplication and Division
A combination of the aforementioned operations
3. Introduce a shorter method for ‘changing the subject of a formula’ or
‘transposing.’

The Concept
Given the formula Area of a rectangle = length x width (written A = lw),
we say that A is the subject of the formula and this is so because:
 It is on the left hand side (the subject usually is)
 The coefficient is 1
 The power is 1
 It is in the numerator
If, however, we are interested in finding the length of the rectangle (l),
then we would get . We say that the subject of the formula has been
changed to l.

The Concept
The topic ‘changing the subject of a formula’ therefore implies that:
• A formula will be given
• It will have a subject
• The subject must be changed to something else

Procedure Related to “changing the subject”
Think of the following process:
 I walk two steps forward, jump three steps left, make a 180° turn
clockwise, then throw a ball to Ralph.
Question: How do I undo that process to get back
to the starting point?
Consider this:
Ralph would throw the ball back to me, I would make a 180° turn
anti-clockwise, jump three steps right, step two steps backward.
(All things being equal, I should be back at the starting point)

Procedure Related to “changing the subject”
In order to get back to the starting point, we had to ‘reverse’ which
involved ‘undoing’ that is ‘doing the opposite of what was done’.
In the context of mathematics, we will be doing the inverse of what was
done. Importantly, we must identify what is done to the subject in
order to determine what must be ‘undone’.
THUS: THE SAME RULES USED TO SOLVE AN EQUATION WILL BE
APPLIED

Operation Inverse Operation
Addition/Positive Number Subtraction/Negative Number
Subtraction Addition
Multiplication Division
Division Multiplication
Operations and Their Inverses

Subject appears once
t
d
s 
s is the Subject
To make t the Subject
Multiply by t d
ts 
Divide by s
s
d
t 

c
mx
y 
 y is the Subject
To make x the Subject
Subtract c mx
c
y 

Divide by m x
m
c
y


m
c
y
x


Subject to LEFT

h
r
v 2
3
1

 v is the Subject
To make h the Subject
Multiply by 3 h
r
v 2
3 

h
r
v

2
3

Subject to LEFT
Divide by 2
r

2
3
r
v
h



h
r
v 2
3
1

 v is the Subject
To make r the Subject
Multiply by 3 h
r
v 2
3 

2
3
r
h
v


Divide by h

r
h
v


3
Square root
h
v
r

3


Subject appears twice
x
t
tyx 7

 To make x the Subject
Add 7x t
x
tyx 
7
Factorise LEFT t
ty
x 
 )
7
(
)
7
( 

ty
t
x
Divide by (ty+7)

Subject appears twice
r
t
p
7
3

 To make t the Subject
Multiply by t
r
t
tp
7
3

Multiply by r t
r
tpr 7
3 

Subtract 7t r
t
tpr 3
7 

Factorise LEFT r
pr
t 3
)
7
( 

)
7
(
3


pr
r
t
Divide by (pr-7)
Subject ONLY ONCE
Subject NOW TWICE

New Subject Raised to a Power
x
r
p
y 
 y is the Subject
To make x the Subject
Subtract p
x
r
p
y 

x
r
p
y 
 2
)
(
SQUARE
r
p
y
x 
 2
)
(
Multiply by x
2
)
( p
y
r
x


Divide by (y-p)²

Examples
Change the subject of the following formulae to the subject indicated in
brackets:
1. V = u + t (u)
2. -
3. (F)

Solutions
1. v = u + t (u)
Identify what was done to the required subject, u, and ‘undo it’.
t was added to u so:
Subtract t from both sides
, on the left
NB: The rules used for solving equations were applied

Solutions
2. -
Rewrite to get:

A Shorter Method
Remember the principle:
In order to ‘undo’ what was done to the subject
we go in reverse order and use the inverse
operation.

Example
Make c the subject in the formula:
Solution:
Inverse
Remember that dividing by a fraction is the same as multiplying by its reciprocal
(turn the fraction upside down )

Consider this alternative solution
Make c the subject in the formula:
)
REMEMBER:

Operation Inverse Operation
Square Root, e.g Square, e.g ( = 3
Cube Root, e.g. Cube, e.g = 5
nth Root, e.g Raise to nth power, e.g.(= a
Squaring, e.g Square Root, e.g = 3
Cubing, e.g Cube Root, e.g = 5
nth power, e.g nth root, e.g = b
Operations and Their Inverses
NB: The inverse operation eliminates the operation leaving the number/symbol alone

Examples
Transpose the following formulae for the subject indicated in brackets:
1. T = (e)

Solutions
OR
positive 2as becomes negative 2as on the other side
(remember 2as will be eliminated from the RHS)
In order to eliminate the square, we square root, but show it on the other side:
(remember the square root eliminated the square leaving u alone on the RHS)
Rewrite to get:

Solution (cont’d)
NB :
REASON:
.
.

Class with Jurek ppt about different topics

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