Sequences

TERM:-                                                                                                                                   Grade:-

Week:-                                                                                                                                     Periods:-

LESSON TITLE: Sequence

Period:-1 – 2

                 Lesson Objectives:-

                                                Students will be able to continue the given number sequence

Key Vocabulary:-Sequence, Terms, Pattern, Next term.

Key questions:- 

(i)     What do you mean by sequence?

(ii)   What you mean by pattern?

(iii) How can we find the next terms of a given sequence?

Teacher Activity	Student’s activity
Introduce sequence Give suitable example such as Even numbers sequence, multiples of three numbers sequence, multiples of five numbers sequence. Show the students how to find the next terms of the given sequence.                 Give more example questions for practice.	Listen to the teacher, understand  the meaning of the word sequence and Understand Pattern of numbers sequence given        Finds the next term of the given sequence

Period:-3

                 Lesson Objectives:-

                                                Students will be able to find the Arithmetic Progression and continue the sequence calculate the n^th term of the sequence

Key Vocabulary:- Arithmetic Progression, Common difference, n^th term.

Key questions:- 

(i)     How can you find that a given sequence is an Arithmetic Progression?

(ii)   Give the significance of the word common difference?

(iii)  How can you find the n^th term of tan A.P?

Teacher Activity	Student’s activity
Introduce Arithmetic Progression Show how to find the common difference and find the next term of the sequence and Give suitable example to find the  nth term of A.P and extend this idea to find a particular term of a given sequence.               Give more example questions for practice	Listen to the teacher, understand  the sequence A.P             Find the next three terms, a particular term and the nth term of an A.P

Period:-4 – 5 

                 Lesson Objectives:-

                                                Students will be able to find the next terms of a sequence by applying the First successive difference& Second successive difference method

Key Vocabulary:-First Successive difference, Second successive difference

Key questions:- 

(i)     What is the first successive difference of the given sequence?

(ii)   What is the second successive difference of the given sequence?

Teacher Activity	Student’s activity
Introduce First successive and second successive difference type sequence and Show how to find the successive difference of the sequence and find the next terms          Give suitable example to find the next terms	Listen to the teacher, understand  the significance of the First successive and second successive difference        Find the next three terms  of  the sequence by using the First successive and second successive difference type sequence.

Period:- 6 – 7

                 Lesson Objectives:-

                                                Students will be able to find the sequence idea involved in the diagram pattern and solve problems involving diagrams

Key Vocabulary:- Squares, rectangles, Triangles, Dots

Key questions:- 

(i)     How many triangles are there in the first figure?

(ii)   How many dots would come in the 10 the diagram?

Teacher Activity	Student’s activity
Give Suitable questions to find the pattern in which the given diagrams are arranged Explain how to find the pattern and using it how to find the required solutions           Give more example questions related to real life situations for practice.	Listen to teacher, understand  the significance of the given diagram.       Find the number of dots ,number of squares ext. Draw conclusions.

Materials:-IGCSE Mathematics, Ric Pimentel  and terry Wall Page numbers From 25 to 29

                 Extended Mathematics for IGCSE, David Rayner, Page numbers from 6 to 7

                 Core Mathematics for IGCSE, David Rayner, Page numbers from 42 to 44

Teaching Aids:-  Work sheet, black board.

Evaluation:-   Oral Questions, class Work, Completing class work

Reflection:-

………………………………………..

                                                                         ……………………………………….

                                                                                                                             (Supervisor)

PROPERTIES OF QUADRILATRALS;

LESSON PLAN                          

MATHEMATICS                                                                                 

Term                                                                                                                Grade

Week                                                                                                                          Periods

From :

LESSON TITLE:  PROPERTIES OF QUADRILATRALS;

                               ANGLESUM PROPERTY OF TRIANGLE AND QUADRILATERALS

PRE-REQUISITE KNOWLEDGE:, angle at a point, angle in a straight line and  angles in parallel lines                                              

TEACHING AND LEARNING PROCESS                                               

PERIOD   1 Learning Objectives: Students will be able to                                         Appreciate the properties of quadrilaterals and use them to solve problems
Key Vocabulary:  parallel; diagonal; bisect; quadrilateral; parallelogram; square; rhombus; kite; trapezium
Key Questions: 1.      What are the properties of various quadrilaterals? 2.      How will you use them to solve problems?
TEACHER ACTIVITY	STUDENT ACTIVITY
Explains & Describes properties of quadrilaterals	Understands the properties of quadrilaterals
Explains the questions and helps the students in finding out the solution.	Tries to solve the question and asks if they have any doubt.
Gives more questions for practice and explains the questions so that the student can easily find out the solution using all the concepts	Understands and tries to find out the solution from the method taught by the teacher
Gives some more questions as home-work and some in the weekend assignment	Complete their work and ask the teacher if they have any doubt

                                                                                                                                                                                   

PERIOD   1 Learning Objectives: Students will be able to                                         Solve problems involving angles at a point, angles on a straight line
Key Vocabulary:  point, straight line, sum
Key Questions: 1.      What is the sum angles at a point? 2.      What is the sum of angles on a straight line?
TEACHER ACTIVITY	STUDENT ACTIVITY
Explains & Describes the sum angles at a point and sum of angles on a straight line	Understands the properties to solve problems.
Explains the questions and helps the students in finding out the solution.	Tries to solve the question and asks if they have any doubt.
Gives more questions for practice and explains the questions so that the student can easily find out the solution using all the concepts	Understands and tries to find out the solution from the method taught by the teacher
Gives some more questions as home-work and some in the weekend assignment	Complete their work and ask the teacher if they have any doubt

                                   

PERIOD   1 Learning Objectives: Students will be able to                                         Solve problems involving angles in parallel lines
Key Vocabulary:  alternate, allied, corresponding
Key Questions: 1.      How many angles can be created on parallel lines? 2.      How will you use parallel line properties to solve problems?
TEACHER ACTIVITY	STUDENT ACTIVITY
Explains & Describes alternate, allied, corresponding angles	Understands the properties to solve problems.
Explains the questions and helps the students in finding out the solution.	Tries to solve the question and asks if they have any doubt.
Gives more questions for practice and explains the questions so that the student can easily find out the solution using all the concepts	Understands and tries to find out the solution from the method taught by the teacher
Gives some more questions as home-work and some in the weekend assignment	Complete their work and ask the teacher if they have any doubt

                                                                                                           

PERIOD  5 Learning Objectives: Students should be able to know ·         Sum of interior angles of a triangle is 180˚ ·         Sum of interior angles of a Quadrilateral  is 360˚
Key Vocabulary:   sum of interior angles,
Key Questions: How many triangles can be formed from a quadrilateral simultaneously? What is the sum of exterior angles of a triangle/quadrilateral?
TEACHER ACTIVITY	STUDENT ACTIVITY
Asks the students about triangle and quadrilateral.	Tries to recall a triangle and quadrilateral
Asks about angle sum property of triangle and quadrilateral.	Recall  the angle sum property of triangle and a quadrilateral.
Shows that any quadrilateral can be drawn as combination of two triangles.	Recognize that one diagonal separates a quadrilateral into two triangles.

………………………………………….

                  (Subject Teacher)

percentage and money exchange - Japan

1. Japan's population is estimated at around 127 million.

  (a) Write  Japan's population in standard form

  There are about 136,000 Western expatriates.

  (b) What percentage of the Western expatriates are in Japan?

   About 19.5 percent of the population was over 65 years of age.

  (c) Find the number of people who are below 65.

The Japanese population is rapidly aging as a result of a post–World War II baby boom followed by a decrease in birth rates. A growing number of younger Japanese prefer not to marry or have families. Hence Japan's population is expected to drop to 100 million by 2050 and to 64 million by 2100.

(d) Calculate the expected percentage decrease in population in the year 2050 and 2100.

Japan's legislative organ is the National Diet, a bicameral parliament. The Diet consists of a House of Representatives with 480 seats, and a House of Councilors of 242 seats.

(e) Express the number of House of councilors seats as a percentage of the total number of seats.
........

2. The Asakusa Samba Carnival is held annually, on a Saturday towards the end of August, in Asakusa, Tokyo.  The cost of tickets for Adults     -  350 JPY and Children -  150 JPY

The carnival starts at 7.15 am everyday and closes at 11.30 pm.

2000 Adults and 6000 Children attended on the first day

a) Calculate the amount of money they got by selling the tickets

               (i) To adults                                                                                                                                   

              (ii) To children  

              (iii) The total amount           

  b) Calculate the percentage of the amount of money collected from children.

 c) Convert the above total amount to dollars, if $1 = 80.86 JPY.

 d) Convert the timing to 24 hr clock time.                          

 e) Calculate the duration of the carnival in a day                

 f) Mrs. Carolene has got some dollars with her. How many dollars she has to pay to enter the           carnival along with her two kids. If  $1 = 80.86 JPY

FUNCTIONS

FUNCTIONS


1. The functions f,g and h are defined by
f : x--->4x-5
g : x---> 2x2
h : x--->x6+4x

a) find the value of (i) f(4) (ii)g(-3) , (iii) h(1 /2)

b) solve the equations (i) f(x) = 3, (ii) g(x) = 32, (iii) h(x) = 7

c) express the composite functions (i) gf (ii)fh in the form of gf  : x---> ……., fh : x--->
you need not simplify your answers
d) express the inverse f -1and h -1 in the form of f -1: x---> , h -1 : x--->

e) show that the equation g(x) = 5[f(x)] + 7 can be written in the form ax2 + bx + c = 0 and state the values of a,b and c.

2. f and g are defined by
f : x--->4x –1 and g : x---> 2x2 + 3

a) find the value of f(2)

b) express the inverse of the function f in the form f -1: x--->
c) express fg in the form fg : x--->
d) solve the equation fg(x) = f(x) + 24.

3. The functions f and g are defined by
f : x---> 2x2 + 1, g : x---> 7x – 5
a) find the value of x for which f(2) = g(x)
b) copy and complete the following
i) fg : x---> ii) inverse of g : x--->
c) find the values x for which f(x) = g (x)


4. Given that f : x---> x2 +3
a) find f(4), b) complete and simplify the statement ff : x--->

5. f : x ---->3x2 + 1 , g ; x 2x –1
copy , complete and simplify the following
a) gg : x---> b) gf : x---> c) g -1 : x--->

6. f : x---> 5x + 4, g : x---> 4x + 3
a) find g(2)
b) solve, for x, the equation f(x) = g(2)
c) copy and complete the following, simplifying as appropriate
i) f -1: x---> ii) ff : x--->
d) show that f (g(x)) = g (f(x))
e) solve the equation g(x) = 2 giving your answers to 2 decimal places.

7. The functions f, g and h are defined by
f : x---> 3x + 2, g : x---> x2 – 4, h : x---> x + 1
a) find the value of i) f(2) ii) g(-3) iii) h(1/4)
b) solve the equation i) f(x) = 12 ii) f(x) = h(x)
c) express the composite function gf in the form gf : x ….,simplify your answer.
8. f : x x2 – 5x≠
a) write down the value of f (-2)
b) write down the value of ff(2)
c) find the range of f when the domain is {-1, 0, 1}

9. The functions f and g are defined by
f : x---> 2x – 1 , g : x---> x2 + 8,
a) find the values of i) f(3) ii) g(-2)
b) solve the equations i) g(x) = 12 ii) g(x) = 4 f(x)
c) Express the composite functions i) fg, ii) gf in the form fg : x--->, gf : x--->
d) Show that the equation fg(x) = gf(x) can be expressed in the form x2 + bx + c = 0. Where b and c are integers, and state the values of b and c.
e) Express the inverse function, f –1, in the form, f –1 : x--->


10. The functions f,g and h are defined by f : x 2x + 1, g : x 3x2 , h : x , x ≠ 1
a) find the value of i) g (-4) ii) h(1 /2)
b) solve the equations i) g(x) = 75 ii) h(x) = 4, iii) f(x) = g(x)
c) express hf(x) in terms of x, simplifying your answer.


11. f : x---> , x ≠ -2

Given that f(x) = 3, a) find x b) find f (f(4))
Given that the domain of f is the set s = {-1,0,1}
c)find the range of s under f.

12. f : x---> x2 + 3x – 4 , g : x---> x +1
a) find i) f(-2) ii) g -1(1/2 )
b) solve the equation fg(x) = 0
c) solve the equation fg(x) = gf(x)


13. f : x---> 1 – 3x , g : x x ≠ 0

find the values of a) f(3) b) gf(1)


14. f : x---> 2x2 – 3 , g : x---> 3x + 5, copy , complete and simplify
a) gg : x---> b) fg : : x---> c) g -1 : x--->

15. f : x---> x2 – 3x – 5, g : x---> 2x + 1
a) find the value of I) f(2) ii) fg(-1 /2 )
b) express the inverse function g -1 in the form g -1 : x--->
c) express the composite function fg in the form fg : x--->
simplifying your answer
d) find the values of x for which f(x) = g(x)

16. The functions f and g are defined by
f : x---> 2x – 3, g : x , x ≠ 0

a) find the value of I) f(3), ii) gf(2)
b) find and simplify I) f -1 : x---> ii) fg : x--->
c) solve the equation fg(x) = gf(x)


FUNCTIONS ( continued)

17. The functions f and g are defined by f: x---> 3x – 2 , g : x--->2x+8 , x ≠ 2
a) find the value of I) f(2) ii) gf(10)
b) express
i) the inverse function g -1, in the form g -1 : x--->
ii) the composite function fg, in the form fg : x--->
c) solve the equation f(x) = g(x)


18. The function f is defined by f : x , x ≠3
Given that f(x) = 2
a) find the value of x
b) find the value of ff(0).

19. f : x 3x – 4 , g : x x2 + 1
copy, complete and simplify
a) fg : x ……. b) gf : x …….
c) find the two values of x for which fg(x) = gf(x)

20. Given that f : x 3x –1 ,
a) evaluate f(-1/3), b) find f -1 (x)

21. Write down the range of each of the following functions
a) f : x x2 , -3< x < 3,
b) g : x sinxo 0 < x < 90


22. f : x a 5x + 3, g : x a + 15

a ) calculate i) f (3) , ii) fg(4)
b) solve i) f(x) = 1, ii) f(x) = g(x)

23. (i) Given that f : x 6 – x2
a) state the maximum value of f(x).
b) write down the range of f.
(ii) Given that g : x , state the value of x which must be excluded from the domain of g

24. The functions f and g are defined by f : x 1 – x, g : x 2x2 + 3,
a) find the values of I) f(-2) ii) g( 2)
b) Express the inverse function f -1 in the form , f -1 : x ……
c) Express the composite function gf in the form gf : x ………simplifying your answer
d) solve the equation
i) g(x) = 53 ii) ff(x) = f(x)

25. f : x x 2 – 1, g : x 3x – 4 ,
a ) complete the following statements, simplifying your answers where appropriate
i) g-1 : x …….. ii) fg : x ……..

b) find the values of x which satisfy the equation fg(x) = 9 – 3x

FUNCTIONS

FUNCTIONS


The idea of functions is used in almost every branch of Mathematics.
The two common notations are
a) f(x) = x2 + 4
b) f : x2 + 4

Some more notations used in functions are as follows

Simple Functions : f(x)
Inverse Functions : f-1(x)
Composite Functions : fg(x) or f(g(x))

Grade 9
Example:-
1) Given f(x) = 3x - 1 and g(x) = 1 – x2
Find the following:-
a) i) f(2) = 5 b) i) g(2) = - 3
ii) f(0) = - 1 ii) g(-2) = -3
iii) f(-3) = -10 iii) g(1/2) = ¾
iv) f(-x) = -3x – 1 iv) g(m) = 1 – m2
v) f(k) = 3k + 1 v) g(1/t) = 1 – 1/t2

Exercise

1) Given the functions h : x x2 + 1 and g: x 10x + 1
Find the following:-

a) h(2) b) h(-1) c) h(0)
d) h(+5) e) h(1/2) f) g(k)
g) g(-m) h) g(k) i) g(1/10)
j) g(-5)



2) Given the functions f(x) = 2x – 4 , g(x) = , h(x) = (7 – 3x)2
Use the above functions to find the following

a) f -1(x)
b) f -1(8)
c) g -1(x)
d) g -1(16)
e) h-1(x)


3) Given the functions f(x) = 2x2/3 , g(x) =10 – x2 , h(x) = 2x2 + 1
Use the above functions to solve the following
a) f(x) = 5
b) f(x) = x
c) h(x) = 0
d) g(x) = 6
e) g(x) = h(x)
f) h(k) = k + 1

Grade 10

Take all the above questions also for Gr 10
These questions are also included

4) Given the functions f: x x/4 , h : x x2 + 1 and g: x 10x + 1
Find the following

a) fg(2) b) gh(3) c) f-1g(4)

d) f(x) = g(x) e) h-1f(-2) f) hg(5)


Past Paper Questions

1. (P2May/June- 2001)
Given f(x) = for x>0 and g(x) = 3 – 3x for any value of x.
a) find f( ) , giving your answer as a fraction
b) if f(x) = g(x), find the value of x
c) find f-1(x) and g-1(x)
d) g-1(18)



2.(P2, May/June 2001)


f(x) = x1/3 and g(x) = 2x2 – 5 for all values of x.
a) Find
i) g(4)
ii) f(27)
b) Find an expression for g-1(x) in terms of x.
c) Find f1(x)


3.



a) calculate i) ii)



b) find and simplify as a single fraction
of the composite function
find the inverse h-1, of the function.
c) solve for x if
Give your answer , correct to, 2 decimal places.


4. The functions f(x) and g(x) are defined as follows.

a) find the value of g (- 4)
b) find and simplify as a single fraction f(x) – g (x).
c) find an expression for f -1.
d) find an expression for g -1.

Algebraic Functions

  LESSON PLAN                              

MATHEMATICS                                                                                         

Semester                                                                                                           Grade

Week                                                                                                                 Period

From ............... to ..........

LESSON TITLE:                       Functions

TEACHING AND LEARNING PROCESS

LESSON OBJECTIVES:            

Students will be able to

            use and interpret simple functions and function notations

such as                f(x) = 5x + 4;           f:x   5x +4

PERIOD 1 Learning Objectives : Students will be able to use and interpret simple functions and function                                     notations such as          f(x) = 5x + 4; f:x   5x +4
Key Vocabulary:         function, variable, value
Key Questions:           How will you find the value of a  given function?
TEACHER ACTIVITY	STUDENT ACTIVITY
Show how to use and interpret simple functions and function notations such as               f(x) = 5x + 4;  f:x          5x Show how the value of a function varies when the value of its variable is varied. Explain how to find the value of a function.	use and interpret simple functions and function notations such as                f(x) = 5x + 4; f:x          5x +4 find the value of a  given function.

LESSON OBJECTIVES:            

Students will be able to

            describe the inverse of a function f (x) ; use and interpret it.

PERIOD 2 – 3 Learning Objectives : Students will be able to find the inverse of a function
Key Vocabulary:         inverse, change the subject.
Key Questions:           How will you find inverse of a function?
TEACHER ACTIVITY	STUDENT ACTIVITY
Show how to find the inverse of a function	find the inverse of a function

LESSON OBJECTIVES:            

Students will be able to

use and interpret composite functions and function notations such as  g f(x) or  g [f(x)]                           

PERIOD 4 – 6   Learning Objectives : Students will be able to use and interpret composite functions and function notations such as   g f(x) = g [f(x)]
Key Vocabulary:         composite function
Key Questions:           How will you find composite of a function?
TEACHER ACTIVITY	STUDENT ACTIVITY
Show how to use and interpret composite functions and function notations such as   g f(x) = g [f(x)]	use and interpret composite functions and function notations such as   g f(x) = g [f(x)]

LESSON OBJECTIVES:            

Students will be able to

            solve simple and quadratic functions where f(x) = g (x)

            Do combined problems.

PERIOD 7 Learning Objectives: Students will be able solve simple and quadratic functions where f(x) = g (x)                                   Do combined problems.
Key Vocabulary:         quadratic funtions
Key Questions:           How will you solve simple and quadratic functions where f(x) = g (x)                                   Do combined problems.
TEACHER ACTIVITY	STUDENT ACTIVITY
Show how to solve simple and quadratic functions where f(x) = g (x) and do combined problems.	solve simple and quadratic functions where f(x) = g (x) Do combined problems.

MATERIALS:                IGCSE Mathematics/ Ric Pimentel and Terry Wall

                                    Ex 1.2 – 1.5 Student assessments 1 – 2

                                    Ex 2.2 – 2.3 Student assessments 1 – 4

TEACHING AIDS:         Black board, Work Sheet

ASSESMENT AREAS:                Understanding of the information

                                                Extraction of values from the information

                                                Accuracy of the answer.

EVALUATION:                         Oral questions

                                                Class work

                                                Home work

REFLECTION:

………………………………………….

       (Subject Teacher)

………………...………… …. ……                                      …………………….…………….

 ( Head of the Department)                                                                (Supervisor)