Tuesday, August 31, 2010

Yearly Plan – Additional Mathematics Form 4 (2009)


Yearly Plan – Additional Mathematics Form 4 (2009)


 


 


 

Involve algebraic functions only.


 


 


 

Images of composite functions include a range of values. (Limit to linear composite functions).

Define composite functions


 


 


 


 


 


 


 


 


 

!
 


 


 


 

Week

No 

Learning Objectives

Pupils will be taught to..... 

Learning Outcomes

Pupils will be able to…

No of Periods< /span> 

Suggested Teaching & Learning activities/Learning Skills/Values 

Points to Note 

Topic/Learning Area Al : FUNction --- 3 weeks

First Term 

   


 


 

2


 

5-9/1/09


 


 

1. Understand the concept of relations.

1.1 Represent relations using

  1. arrow diagrams
    1. ordered pairs
    2. graphs
  1. Identify domain, co domain, object, image and range of a relation.

1.3 Classify a relation shown on a mapped diagram as: one to one, many to one, one to many! or many to many relation.


 


 

1


 


 


 


 


 


 


 

1


 


 

Use pictures, role-play and computer software to introduce the concept of relations.


 


 


Skill : Interpretation, observe connection between domain, co domain, object, image and range of a relation.


 


 


 

Discuss the idea of set and introduce set notation.

 


 


 

2.    Understand the concept of functions.


 

2.1 Recognise functions as a special relation..


 

2.2 Express functions using function notation.


 

2.3 Determine domain, obje! ct, image and range of a function.

2.4 Determine the image of a function given the object and vice versa.


 


 

1


 


 


 


 

1 


 


 

  • Give examples of finding images given the object and vice versa.
  1. Given f : x ® 4x – x2. Find image of 5.
  2. Given function h : x ® 3x – 12. Find object with image = 0.


 

Use graphing calculators and computer software to explore the image of functions.
  • Represent functions using arrow diagrams, ordered pairs or graphs, e.g.


     

    "" is read as "function f maps x to 2x".

  • ""is read as "2x is the image of x under the function f".

Include examples of functions that are not mathematically based.

Examples of functions include algebraic (linear and quadratic), trigonometric and absolute value.

Define and sketch absolute value functions.


 

3


 

12-17/1/09 


 

3. Understand the concept of composite functions.


 

3.1 Determine composition of two functions.


 


 

3.2 Determine the image of composite f! unctions given the object and vice versa

 60;

3.3 Determine one of the functions in a given composite function given the other related function.


 

1


 


 


 


 

1


 


 


 


 

2


 


 


 


 


 


 

  • Use arrow diagrams or algebraic method to determine composite functions.
  • Give examples of finding images given the object and vice versa for composite functions


 

For example :

Given f : x ® 3x – 4. Find

  1. ff(2),
  2. range of value of x if ff(x) > 8.


 

  • Give examples for finding a function when the composite function is given and one other function is also given.


 

Example :

Given f : x® 2x – 1. find function g if

  1. The composite function fg is given as fg : x ® 7 – 6x
  2. composite function gf is given as

gf : x ®
5/2x.


 


 


 

4


 

19-23/1/09 


 

4. Understand the
concept of inverse
functions.


 

4.1 Find the object by inverse mapping given its image and function.


 

4.2 Determine inverse functions using algebra.


 

4.3 Determine and state the condition for existence of an inverse function

Additional Exercises


 

1


 

1


 


 


 

1

1 


 


 


 


 


 

Use sketches of graphs to show the relationship between a function and its inverse.


 

Exa mples :

Given f: x, find


 


 

  • Limit to algebraic functions.
  • Exclude inverse of composite functions.


 


 


 


 

  • ! Emphasise that the inverse of a function is not necessarily a function.


 

5


 

26-30/1/09

 


 

PUBLIC HOLIDAY

(CHINESE NEW YEAR)!

   

Topic A2 : Quadratic Equations ---3 weeks

   


 


 

6


 

2-6/2/09


 

1. Understand the concept of quadratic equations and their roots.


 

  1. Recognise a quadratic equation and express it in general form.


 


 

1. 2 Determine whether a given value is the root of a quadratic equation by

  1. substitution;
  1. inspection.


 

1.3 Determine roots of quadratic equations by trial and i! mprovement method.


 

1


 


 


 


 


 


 

1


 


 


 


 


 


 

!
 


 


 

Use graphing calculators or computer software such as the Geometer's Sketchpad and spreadsheet to explore the concept of quadratic equations


 


 


 


 


 

Values : Logical thinking

Skills : seeing connection, using

trial and error methods 


 

Questions for 1..2(b) are given in the form of ; a and b are numerical values.


 


 


 


 


 


 


 


 


 

7


 

9-14/2/09


 

2. Understand the concept of quadratic equations.


 

2.1 Determine the roots of a quadratic equation by

  1. factorisation;
  2. completing the square

c) using the formula.


 


 

2.2 Form a quadratic equation from given roots.
!


 


 


 

1


 

1


 


 


 

2


 


 

If x = p and x = q are the roots, then the quadratic equation is , that is .

Involve the use of:

and

where α and β are roots of the quadratic equation


 

Skills : Mental process, trial and error


 

Discuss when

, hence or .

Include cases when p = q.


 

Derivation of formula for 2.1c is not required.


 


 


 

8


 

16-20/2/09


 

3. Understand and use the conditions for quadratic equations to have

a) two different roots;

b) two equal roots;

c) no roots.    a)dua punca berbeza;


 


 


 

3.1 Determine types of roots of quadratic equations from the value of .


 

3.2 Solve problems involving in quadratic equations to:

a) find an unknown value;

b) derive a relation.


 

Additional Exercises


 

2


 


 


 

2


 


 


 


 


 

2 


 

Giving quadratic equations with the following conditions :

,

and ask pupils to find out the type of ro! ots the equation has in each case.


 

Values: Making conclusion, connection and comparison


 

Explain that "no roots" means "no real roots".

Topic A3 : Quadratics Functions---3 weeks 


 

!


 

 


 


 

9


 

23-27/2/09


 

1. Understand the concept of quadratic functions and their graphs.


 

1.1 Recognise quadratic functions


 

1


 

1) Use graphing calculators or computer software such as Geometer's Sketchpad to explore the graphs of quadratic functions.

  1. f(x) = ax2 + bx + c
  2. f(x) = ax2 + bx
  3. f(x) = ax2 + c

* pedagogy : Constructivism

Skills : making comparison

& making conclusion

 
  


 

1.2 Plot quadratic function
graphs:

     a)based on given tabulated
values;

  1. b) by tabulating values
    1. based on given functions.


 


 


 

2  


 

1) Use examples of everyday situations to introduce graphs of quadratic functions.


 

  • Contextual learning


 

  


 

1.3 Recognise shapes of graphs of quadratic functions.


 


 

1  


 


 


 

Discuss the form of graph if

a > 0 and a < 0 for


 

Explain the term parabola. 


 

10


 

2-6/3/09

 

1.4 Relate the position of quadratic function graphs with types of roots for .


 

2  

Recall the type of roots if :

  1. b2 – 4ac > 0
  2. b2 – 4ac < 0
  3. b2 – 4ac = 0


 

Relate the type of roots with the position of the graphs.

 


 

2. Find the maximum and minimum values of quadratic functions.


 

2.1 Determine the maximum or minimum value of a quadratic function by completing the square.


 

2  


 

Use graphin! g calculators or dynamic geometry software such as the Geometer's Sketchpad to explore the graphs of quadratic functions


 

Skills : mental process , interpretation

Students be reminded of the steps involved in completing square and how to deduce maximum or minimum value from the function and also the corresponding values of x.


 

11


 

9-13/3! /09


 

3. Sketch graphs of quadratic functions.


 

3.1 Sketch quadratic function graphs by determining the maximum or minimum point and two other points.


 

2  


 

    Use graphing calculators or dynamic geometry software such as the Geometer's Sketchpad to reinforce the understanding of graphs of quadratic functions.

Steps to sketch quadratic graphs:

a) Determining the form"È" or
"Ç"

b) finding maximum or minimum
point and axis of symmetry.

c) finding the intercept with x-axis
and y-axis.

d) plot all points

e) write the equation of the axis of symmetry 


 

Emphasise the marking of maximum or minimum point and two other points on the graphs drawn or by finding the axis of symmetry and the intersection with the y-axis.

Determine other points by finding the intersection with the x-axis (if it exists).

 


 

4. Understand and use
the concept of quadratic inequalities.


 

4.1 Determine the ranges of values of x that satisfies quadratic inequalities.


 


 


 


 

2 


 

Use graphing calculators or dynamic geometry software such as the Geometer's Sketchpad to explore the concept of quadratic inequalities.


 

Emphasise on sketching graphs and use of number lines when necessary.

12


 

16-20/3/09


 


 


 


 

SCHOOL HOLIDAY 

   

Topic A4: Simultaneous equations---2 weeks

  


 


 

13


 

23-27/3/09


 

1. Solve simultaneous equations in two unknowns: one linear equation and one non-linear equation.


 

1.1 Solve simultaneous equat! ions using the substitution method.


 

4 


 

Use graphing calculators or dynamic geometry software such as the Geometer's Sketchpad to explore the concept of simultaneous equations.

Value: systematic

Skills: interpretation of mathematical problem


 

Limit non-linear equations up to second degree only.


 


 

14


 

30-3/4/09

 


 

1.2S olve simultaneous equations involving real-life situations.


 


 


 

Additional Exercises


 

2


 


 


 


 


 

2


 

Use examples in real-life situations such as area, perimeter and others.


 

Pedagogy: Contextual Learning

Values : Connection between mathematics and other subjects


 

 

Topic G1. Coordinate Geometry---5 weeks

   


 


 

!

15


 

6-10/4/09


 

1. Find distance between two points.


 

  1. Find the distance between two points, using formula


 

1 


 

Skill : Use of formula 

Use the Pythagoras' Theorem to find the formula for distance between two points.

 


 

2.    Understand the concept of division of line segments    


 

2.1Find the midpoint of two given points.


 

2.2Find the coordinates of a point that divides a line according to a given ratio m : n.

1


 

2 


 

Skill : Use of formula

Value : Accurate & neat work 

Limit to cases where m and n are positive.

Derivation of the formula

is not required.


 

16


 

13-17/4/09


 


 

3    Find areas of polygons.


 

3.1 Find the area of a triangle based on the area of specific geometrical shapes.

 

3.2 Find the ar

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