5.4.1 INTRODUCTION.
With the object of learning being ’discerning the features of a quadratic function and
identification and use of the appropriate form of the equation to generate the equation of the quadratic function from the graph(s) or any information about the quadratic function’ as the focus of the lessons, the third intervention lesson was performed with Group C and named Intervention Lesson 3.
Intervention Lesson 3 followed the same format as the other two intervention lessons except for the introduction. For Group C in Intervention Lesson 3, the lesson began with the specific quadratic function which appeared in the three forms of the equation. The three forms of the
specific quadratic function were given to the learners on the same slide. This same specific quadratic function used in its different forms was
, which represents the standard form of a quadratic function. Followed by ( )( ), which represents the factorised form of the equation and
( ) which represents the form of the equation of a quadratic function resulting from completing the square. This constitutes the pattern of variation called simultaneity.
From the standard form of the equation of a parabola, the learners were required to discuss the key features of this specific standard form of the parabola. They were reminded that in order to find the -intercept of the function, they had to put in the parabola’s equation. This specific equation was then associated with its general form. The parameters ‘ ’, ‘ ’ and ‘ ’ of the equation of the general form of the parabola were equated to the numerical values of the coefficients of the different variables ‘ ’, ‘ ’ and the constant value ‘ ’of the specific equation.
The generalisation pattern of variation theory was introduced from the specific equations. The standard form of the specific quadratic function was analysed and its features highlighted. Thereafter the general form was given for the standard form of this specific quadratic function as .
The learners were then required to factorise the specific equation of the parabola. They were then reminded that in order for them to find the -intercepts of the specific equation, they had to put in the specific equation of the parabola and solve for . The resulting form of the specific equation was then compared with the general form of the equation of the parabola. The parameters ‘ ’ and ‘ ’ of the general form of the quadratic equation were compared to the values of the specific equation, and , which represented the - intercepts of the specific equation.
The use of the same specific equation of the parabola in its different forms represented the pattern of separation as discussed in Lesson 1.. The specific quadratic equation was factorised to get the factors as follows:
( )( ). The key features of the specific equation were highlighted as the - intercepts where and . This was followed by the presentation of the general form of ( )( ).
The completion of the square was then performed on the specific equation and the resulting form of the specific equation compared with the general form of the equation of the parabola to show the turning point ( ) represented by the values ( ) in the specific equation. The learners were reminded that in order for them to find the equation of the axis of symmetry for the parabola, they had to solve the equation as represented in the specific equation as . to yield the constant equation from the general equation of the parabola and from the specific equation of the parabola.
A step-by-step approach was used with these learners in order for them to see how the
equation of the general form of the parabola was derived from the use of the specific equation of a parabola.
Finally, the completion of the square was performed on the same specific quadratic function which resulted in the form of the quadratic function ( ) . This form of the equation highlighted the axis of symmetry as and the turning point ( ) followed by the appropriate general form of ( ) . The details of Lesson 2 are shown in Appendix I. From this point onwards, the lesson was similar to Lessons 1 and 2. As this was not the first time that the learners saw the general forms of the quadratic function, they could associate the different specific forms of the quadratic function with their
appropriate general forms as follows:
as corresponding to the general form ;
( )( ) as corresponding to the general form ( )( )
( ) as corresponding to the general form ( ) This constituted the pattern of variation called generalisation. By using the processes of factorisation and completing the square on the specific quadratic function as discussed above, the learners were afforded the opportunity to connect the general forms of the different quadratic functions on their own before being given the general form of the quadratic function. After the learners had undergone the processes of factorisation and completing the square, the learners were then shown how the general form of the quadratic function was derived and written as this was the weakest group of the three groups.
The lesson was planned in such a way that the specific quadratic functions in their different forms should first be given to the learners representing simultaneity as the pattern of
variation. From this point onwards, the learners were given an opportunity to derive the different forms themselves by applying the processes of factorisation and completing the
square. From these results, the learners were afforded the opportunity to generate a form of the quadratic functions that they could associate with the generalised forms of the different forms of the quadratic functions by themselves. This process represented generalisation as a pattern of variation.
For this group, more than one pattern of variation was applied to the lesson as is discussed above. The table of the first part of Intervention Lesson 3 for Group C is represented in Table 5.3 in the following section.
5.4.2 A TABLE OF THE INTERVENTION LESSON 3 FOR GROUP C.
The detailed table of the intervention lesson for Group C adapted from the PowerPoint presentation in Appendix I is presented in Table 5.3 below.
LESSON 3 (GROUP C) DURATION 60 MINUTES. Object of learning.
Discerning the features of a quadratic function, specifically identification and use of the appropriate form of the equation to generate the equation of the quadratic function from the graph(s) or other information about the quadratic function.
Input (Object) Process (Action) Output
The three general forms of the quadratic equation were presented to the learners:
( )( ) ( ) The specific equation was thereafter presented to the learners.
Specific equation is given as:
Discerning the general form of the equation to be: .
For the y-intercept, put in the specific equation to get:
The key features are:
The shape of the parabola: the graph is
concave upwards , and the intercept of the parabola is at
Factorising the specific equation we get: ( )( ).
which can be written in the general form of : ( )( ).
Putting and solving the equation we get: ( )( )
, which represent the -intercepts of the parabola.
The key features are:
The intercepts of the parabola are and which are and for the specific equation.
Completing the square we get: ( )
Discerning the general form of the equation to be: ( ) .
For the equation of the axis of symmetry, put
We get as the equation of the axis of symmetry.
The key features are:
The turning point of the parabola = ( ) and the axis of symmetry is .
In the general form, the turning point is ( ) and the equation of the axis of symmetry is .
Sketching the parabolic graph with the help of the key features.
From this point onwards, the lesson is similar to lessons 1 and lesson 2.
Table 5.3: A detailed table of Intervention Lessons 3 adapted from the Microsoft PowerPoint presentation.
After the lesson, the learners were immediately given the same post-test as was given to the previous groups. The learners’ answer sheets together with the question papers were collected from the learners. I marked the scripts as per memorandum in Appendix G and then the panel members moderated the answer sheets. The results of the test were analysed by the entire team and deliberation on these results, the lesson, the video and the audio recording of the lesson ensued. The results of the post-test are discussed in the next chapter.
5.4.3 REFLECTIONS AND DISCUSSION OF LESSON 3.
As for lesson 2, the three colleagues from the Department of Mathematics gave their input on the lessons, the results of the test, the video and audio recordings of the third lesson were reviewed. The results of the pre-test for all three groups were compared with the results of the post-test in all three groups and the of the tests of all the groups in order to discuss and reflect on all three lessons. From the feedback of my colleagues and my reflection on the lessons, the
importance of starting the lesson with the specific quadratic functions followed by the general forms of the quadratic functions became evident.
The incorporation of more than one pattern of variation, as discussed above, was seen as the only way in which all the problems that the learners were experiencing across the lessons, could be dealt with all at once.
Planning for separation was important in that with the specific parabolic function, the learners are able to clearly see the varying forms of the parabolic equation while the parabolic
equation itself remains unchanged. The importance of planning for simultaneity is that the forms of the parabolic equation can be seen at the same time and the learners are able to choose the appropriate form of the quadratic equation to be used in the relevant context. Planning for generalisation was important so as to enable the learners to choose the appropriate form of the general quadratic function when required to find the equation of a parabola given certain features of the parabola.
This approach made the learners aware of how the general formulae were derived and thus made the application of these general forms of the equations easy for them to use in appropriate situations. Lesson 3 for Group C incorporated the pattern of variation theory called fusion. Many patterns of variation were used, which constitutes a fusion of the patterns of variation.
The results of all the groups are discussed in detail in the next chapter.