Iam a student pursuing my Master of Teaching (Secondary Level) - majoring in Maths and Science. For my current semester, the unit/subject that I'll be taking is " ECM 562- Teaching the Curriculum:...

Calculus – Differentiation
Lessons
i. Derivatives – finding derivatives, table of derivatives
ii. Increasing decreasing functions, Maxima and minima
iii. Real life applications, relative changes of rates.
Mathematics is always a puzzle which creates interst in one group of students while onlyfear in another group of students. As far as Mathematics is concerned we can find two extreme sets of students mutually exclusive for this subject. One set will like this very much and will study this full time without seeing time and other things, while others will postpone knowing these terms also as a last resort.
Psychologists say that we are afraid of what we do not understand. Once we understand any thing the fear automatically goes and familiarity comes. Thus it becomes the duty of any Maths tutor to ride away the anxiety/fear of the second set of students from their minds. Only after the mind becomes free from fear, it would receive the matter taught and keep in memory. For this the duty of the tutor is to make the subject as crystal clear as possible with simple terms and give examples in a very interesting and practical manner.
In Australia, we find four main branches of Mathematics namely Essential mathematics, General mathematics, Mathematical Methods, and Specialist mathematics. For senior secondary level, we can start with Calculus as this is an important subject.
Calculus is the door to open the higher level Mathematics such as engineering, Fourier series, Z transformations, etc. Calculus is a term newly heard by the students for the first time.
Description of concept
Calculus starts with differentiation. Differntiation is nothing but the rate of change of one variable with respect to another variable. Normally we have learnt previously the concept of relations and functions, A function is one relation from one set to another in which each element in the first set has a unique image in the second.
Here we learn about functions. When we say function, we find that there is one independent variable and there may be one or more dependent variables. Here in our lesson, we learn about y the dependent variable and x the independent variable
We write y = f(x). Normally we have x to vary freely while y changes according to change in x.
Pre requisites for this lesson are to understand the rules of exponents, sum and product of algebraic terms, etc. Also three formulae on limits would be continuously applied in derivative calculation which are given below for ready reference and use.
Also we will be applying the three formulae form Pre calculus for limits as
What is differentiation? How it can be found out?
Consider a function f(x) = y . Let x be a small increment given to x, i.e. x is increased by h. Corresponding y becomes f(x+h). We find rate of change of y for change h and check what happens if h is small. This is explained below:
Differentiation of y with respect to x which is also called derivative of y denoted by y’ or y1 or f’(x) or is nothing but
We select h as very small and find the increase of f due to increase in x.
The term is called difference quotient and it finds out the rate of change of f when x is increased by a small number h.
Concept Map
A concept map refers to a picture that portrays an association between the concepts. Further, it is a graphical technique that is used to establish and visualize the content of the lesson. Concept map has a hierarchical structure that is used to demonstrate the students about numerous concepts that are associated with a particular topic (Watson et al., 2016). The concept map for Calculus is demonstrated as follows.
(You can zoom and see these clearly)
For this we separate differentiation or derivative and draw concept map separately.
Concept map:
Real life applications:
Target group
The target group is the students in the secondary school level. These students are expected to have completed pre calculus such as concepts of limits, pre algebra such as rules of exponents, operations on exponents, and also functions such as exponential, trigonometric and algebraic functions, The total strength of the students in the class would be approximately 15-20 students. Furthermore, all these 20 students belong to diverse ethnicity, culture, and possess different economic background. Additionally, there will be students international also. Each student has different competencies and abilities to learn the mathematical concepts, fractions, in this case. The major objectives of conducting the activities for this topic are to enhance the knowledge of students regarding numbers in year 15-16.. Further, it aims to develop the efficiency and fluency of students in the concept of functions, rate of changes, applications of derivatives in lives etc.
Lesson plans:
I. Derivatives (40 minutes)
i) Explaining the meaning of derivatives (1 min)
ii) Showing different types of functions and methods to plug in values for different values of x. (6 min)
iii) Explain the arbitrary increment to x, as a quantity h very small.(3 min)
iv) Write the difference quotient (7 min)
v) Explain how to find f(x+h) and then find difference quotient (5 min)
vi) Explain how limit is applied for h, as h tends to 0. ( 5 min)
vii) Using the three formulae in limits wherever necessary. (5 min)
viii) Example problems (5 min)
ix) Doubt clarification (5 min)
II. Increasing and decreasing functions: (40 minutes)
i) Explain the meaning of increasing and decreasing functions for variables x and y ( 1 minutes)
ii) Showing different graphs such log, exponential, etc ( 2 min)
iii) Explaining the concept of intervals, types of intervals (5 min)
iv) Graphs, wherever necessary to get more clear idea (3 min)
v) Using derivatives as measures to check whether increases or decreases (4 min)
vi) Explain about critical points (5 min)
vii) Finding intervals from f’(x) by giving mid values for x and determining the signs of f’(x) ( 2 minutes)
viii) Activities (10 minutes) (1 and 2)
ix) Find maxima, minima by using II derivative test (conclusion part) (3 min)
x) Doubt clarification and discussion (5 min)
Activity I:
A firm manufacturing iron bolts have the sale price as 25 dollars. Its cost average for one unit from the past history has been found as 10 dollars. Fixed costs such as rent, insurance, loan interest etc amount to 100000 dollars. The firm now produces 100 bolts and is worried about the loss.
The firm wants to know whether it can stop production. Kindly guide the firm.
You are to give different values to production units and find out the profits in table as follows:
    No of units x
    Var cost
    Fixed cost
    Total...