INTERVENTION LESSON 1 (GROUP A)

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 first intervention lesson was performed with Group A and named Intervention Lesson 1.

The introduction of the lesson comprises of the general form of the quadratic function in equation form and the learners were required to identify the key

features of the function emanating from this algebraic representation. At first, the learners did not understand what was meant by the key features until they were given an example of the importance of the parameter ‘ ’ in this general form of the quadratic function.

In order to focus attention on the importance of the parameter ‘ ’, separation was used as the pattern of variation where the learners were required to give the shape of the parabola where examples of graphs of different signs of the value of the parameter ‘ ’ were shown on the chalkboard and concluded in the slide. This enabled learners to discern the effect of the sign of the parameter ‘ ’ on the shape of the parabola. The learners were then able to conclude that if ‘ ’, the parabola was concave upwards and that if ‘ ’, the parabola was concave downwards. The parameters ‘ ’ and ‘ ’ were kept constant when this variance of the parameter ‘ ’ was done.

After these examples, where only the parameter ‘ ’ was discussed, the learners were asked what the parameter ‘ ’ represented and they answered in unison by giving the other critical feature as represented by the parameter ‘ ’ which is the -intercept of the parabola. After the general form of the quadratic function was discussed with the learners in this form, they were given a different form of the general quadratic function. They were presented with ( )( ) from which they were required to extract the key features of the parabola. The key features of a parabola that the learners extracted were the x-intercepts of the

parabola. These features from this form of the parabolic function were then revealed from the slide on the computer.

The next form of the general equation of the quadratic function was then presented to the learners on the next slide. The general form of the quadratic function was revealed from the slide as ( ) The learners were required to extract the key features from this form of the equation of the parabola which they identified as the turning point and the axis of symmetry. The learners’ response was then confirmed by revealing the correct answers from this form of the parabolic function on the slide on the computer.

The pattern of variation that was applied was that of separation by keeping the equations of the parabola in their general forms and varying the forms of the general equations.

From the general forms of the equations of the parabola, the learners were then given a specific equation of a parabolic function in the form: which was in line with the form of the general equation . Learners were asked to extract the key features of the specific equation of a parabola as they did with the general form. The specific key features were revealed in the slide on the computer.

The learners were then asked to factorise the specific equation of the parabolic function and they found the answer of: ( )( ) which was the form of the second general equation of the parabola that the learners were exposed to at the beginning of the lesson. The learners could extract the specific key features of the parabola from be the specific equation of the parabola. The key features that were revealed from the slide were the -intercepts of and .

The learners were then required to complete the square on the standard form of the same specific equation of the quadratic function which resulted in the following form of the equation: ( ) which was the form of the third general equation of the parabola that the learners were exposed to at the beginning of the lesson. They were required to interpret the key features emanating from this form of the quadratic function which are the turning point and the axis of symmetry of the parabola.

The learners were asked to plot the points they had discovered from this specific equation of a parabola after completing the algebraic manipulations that they were required to perform to arrive at the different forms of the same parabolic function. They were asked to join the points they had plotted in their books and compare their sketches with the sketch that was revealed from the slide. Two more examples were revealed from the slides and the same process that was discussed above was repeated.

The learners were then given the graphical representation of the same specific function and they were required to find the equation of the parabolic functions. The first graphical

representation of the function had the points of the x-intercepts and another point labelled on the quadratic function. The second graphical representation of the same parabolic graph was presented to the learners with the turning point and another point on the parabola and the third graphical representation of the same function had three points labelled on it. The learners were required to find the equations of these graphical representations.

The learners requested the teacher to go back to the slides that contained the three forms of the quadratic function before they could find the equations of each one of the given graphical representations. After the learners found the equations of the first three graphical

representations of the first graph, one learner asked why the equations of the three different graphical representations were the same and yet they used three different general forms of the equation of the parabola. The presenter then asked the learners to look at the three graphical representations and tell her whether the graphs were the same or not.

The learners discovered that the three graphs were the same and only the labelled points were different.

Another two different examples of graphical representation of the graphs were given to the learners. In each example the same parabolic graph was presented with the different points labelled on the graphs as discussed above. In each case the learners requested the presenter to go back to the slides where the general forms were given. From each graphical representation with the three different points labelled on the graphs, the learners found the same equation. The learners’ results were confirmed by the appearance of the same equations from the PowerPoint slides.

The notion of variance using the dimension of variation prominent in Lesson 1 was separation. The specific equation of the quadratic function was kept the same and the

different forms from the algebraic manipulations of the specific equation of the parabola were the result. The algebraic manipulations of factorisation and completing a square were

performed on the same specific equation of the parabola to produce the different forms of the same specific equation, which forms the pattern of variation called separation as the equation was kept the same and the forms of the equation were kept the same.

With regards to the graphical representations of the function, the pattern of variation that was applied in Lesson 1 was that of separation. The same function was presented to the learners, but the points labelled on the graphical representation were varied. This variation of the points given in the graphical representation of the same parabola required learners to be able to use the appropriate form of the equation of the quadratic function in order to find the equation of the same parabola.

A detailed table of the intervention Lesson 1 for Group A is presented in the following section.