HOW I TEACH INEQUALITIES ASSOCIATED WITH FUNCTIONS USING GEOGEBRA.
A CASE OF GRADE 10.
Dr Collen Manganyana
Lowveld High School
TARGETS AUDIENCE:
FET Phase
DURATION: 30 Minutes
INTRODUCTION
My presentation today is about how I teach inequalities associated with functions using GeoGebra in
Grade 10. According to (DBE, 2021, p. 192), “Basic interpretation of graphs should start in Grade
10. Learners should then be able to approach questions in Grades 11 and 12 with a little more
confidence”.
In this presentation I will demonstrate how GeoGebra as a powerful visualisation tool allows learners
to explore mathematical concepts, properties and ideas involving inequalities associated with
functions. GeoGebra`s visual, interactive and dynamic impact can be used to help learners with
developing learners` skills required to be able to write inequality solutions for questions on functions.
Questions asked are Level 3 (L3) type classified as Complex Procedures problems according to
Bloom`s Taxonomy levels and normally carries marks ranging from 2 to 5 depending on the
complexity of problem (DBE, 2011).
The latest South African curriculum encourages the use of technology in mathematics classrooms
(DBE, 2011). Research has shown that effective teaching in schools through the use of technology
has the potential to improve learner achievement and learning (Manganyana, van Putten, & Rauscher,
2020; Mushipe & Ogbonnaya, 2019). However, reports have revealed that most questions requiring
learners to provide inequality solutions for a number of questions in National Senior Certificate
(NSC) examination papers have been performed poorly (DBE, 2020, 2021).
CONTENT
As an in-service teacher, I integrated the use of GeoGebra in my teaching of Euclidean Geometry in
Grade 10 mathematics classes. Learners I taught previously had struggled with understanding the
content in Euclidean Geometry, and this forced me to be innovative as a teacher. Since I introduced
GeoGebra in this section of the curriculum, in 2013, my class average has moved from 32% to 46%
in 2014 and 56% in 2015. I did not teach Grade 10’s in 2016 and 2017(Manganyana, 2020). I am
proud that I have extended the use of GeoGebra to other mathematics topics where it is applicable.
Illustrating the four important inequality regions in the Cartesian Plane using GeoGebra
1. 𝑦 ≥ 0 {The plane below the horizontal axis including the value of 𝑦 = 0}.
Table 1: Summary of the condition to be satisfied
Variable(s)
𝑦
Possible combination(s)
Condition to be satisfied
+𝑣𝑒 (𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠)
𝑦≥0
2. 𝑦 < 0 {The plane below the horizontal axis excluding the value of 𝑦 = 0}.
Table 2: Summary of the condition to be satisfied
Variable(s)
𝑦
Possible combination(s)
−𝑣𝑒 (𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠)
Condition to be satisfied
𝑦<0
Figure 1: GeoGebra illustrations of conditions shown in in Tables 1 and 2.
3. 𝑥 > 0 {The plane to the right of the vertical axis excluding the value of 𝑥 = 0}.
Table 3: Summary of the Condition to be satisfied
Variable(s)
Possible combination(s)
Condition to be satisfied
+𝑣𝑒 (𝑥 − 𝑣𝑎𝑙𝑢𝑒𝑠)
𝑥
𝑥>0
4. 𝑥 ≤ 0 {The plane to the left of the vertical axis including the value of 𝑥 = 0}.
Table 4: Summary of the Condition to be satisfied
Variable(s)
𝑥
Possible combination(s)
−𝑣𝑒 (𝑥 − 𝑣𝑎𝑙𝑢𝑒𝑠)
Condition to be satisfied
𝑥≤0
Figure 2: GeoGebra illustrations of the conditions shown in Tables 3 and 4.
Illustrating important region(s) in the Cartesian Plane using GeoGebra
Examples:
Given a straight line 𝑓(𝑥) = 4𝑥 + 8 and a parabola 𝑔(𝑥) = −2𝑥 2 + 8. For which
value(s) of 𝑥 will:
1. 𝑥 ∗ 𝑔(𝑥) ≤ 0 {One part of the graph above the horizontal axis and 𝑥 −values negative
OR vice-versa}
Table 5: Summary of the condition(s) to be satisfied
Variable(s)
𝑥 − 𝑣𝑎𝑙𝑢𝑒𝑠
Possible combinations
−𝑣𝑒 (𝑥 − 𝑣𝑎𝑙𝑢𝑒) +𝑣𝑒 (𝑥 − 𝑣𝑎𝑙𝑢𝑒)
𝑔(𝑥) − 𝑣𝑎𝑙𝑢𝑒𝑠
+𝑣𝑒 (𝑦 − 𝑣𝑎𝑙𝑢𝑒)
Condition to be satisfied
𝑥 ∗ 𝑔(𝑥) ≤ 0
2. 𝑔(𝑥) > 0
−𝑣𝑒 (𝑦 − 𝑣𝑎𝑙𝑢𝑒)
{One graph above the horizontal axis}.
Table 6: Summary of the condition(s) to be satisfied.
Variable(s)
𝑔(𝑥)
Possible combinations
−𝑣𝑒 (𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠) +𝑣𝑒 (𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠)
Condition to be satisfied
𝑥 ∗ 𝑔(𝑥) ≤ 0
𝑓(𝑥)
+𝑣𝑒 (𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠)
−𝑣𝑒 (𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠)
Figure 3: GeoGebra results from Tables 5 and 6.
3. 𝑔(𝑥) ∗ 𝑓(𝑥) ≤ 0
{One graph above and other one below the horizontal axis}.
Table 7: Summary of the condition(s) to be satisfied.
Variable(s)
𝑔(𝑥)
Possible combinations
−𝑣𝑒 (𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠) +𝑣𝑒 (𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠)
Condition to be satisfied
𝑔(𝑥) ∗ 𝑓(𝑥) ≤ 0
𝑓(𝑥)
+𝑣𝑒 (𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠)
−𝑣𝑒 (𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠)
4. 𝑔(𝑥) > 𝑓(𝑥) {One graph is strictly above the other graph}.
Table 8: Summary of the condition(s) to be satisfied.
Variable(s)
𝑔(𝑥)
Possible combinations
−𝑣𝑒 (𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠) +𝑣𝑒 (𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠)
𝑓(𝑥)
+𝑣𝑒 (𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠)
Condition to be satisfied
𝑔(𝑥) > 𝑓(𝑥)
−𝑣𝑒 (𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠)
Figure 4: GeoGebra results from Tables 7 and 8.
5. Inequalities in functions using GeoGebra sliders.
Figure 5: GeoGebra results from sliders.
CONCLUSION
I strongly believe that conceptual growth in the learners involving inequalities in functions is
observed to improve after exposing them to GeoGebra lessons. As such, as an agent of change in the
classroom, the most effective teacher enhances effective learning by allowing learners to be actively
involved in the knowledge construction while his/role is to provide support where it needed
(Manganyana, 2020).
REFERENCES:
DBE. (2011). Curriculum and Assessment Policy Statement Grade 10-12. Pretoria: Government
Printing Works.
DBE. (2020). National Senior Certficate 2020 Diagnostic Report. Pretoria: Government Printing
Works.
DBE. (2021). Diagnostic Report part 1: Content Subjects National Senior Certificate 2021. Pretoria:
Government Printing Works.
Manganyana, C. (2020). The influence of GeoGebra training on teachers and learners in rural
geometry classrooms. PhD Thesis. University of Pretoria. Retrieved from
https://repository.up.ac.za/bitstream/handle/2263/78470/Manganyana_Influence_2020.pdf
Manganyana, C., van Putten, S., & Rauscher, W. (2020). The use of geogebra in disadvantaged
rural geometry classrooms. International Journal of Emerging Technologies in Learning,
15(14), 97–108. https://doi.org/10.3991/ijet.v15i14.13739
Mushipe, M., & Ogbonnaya, U. I. (2019). Geogebra and Grade 9 Learners’ Achievement in Linear
Functions. International Journal of Emerging Technologies in Learning (IJET), 14(8), 206–
219.