This study developed a mathematical evaluation model, attempting to evaluate the capability of the curriculum (N1 to N2 mathematics) to equip students with higher-order thinking skills (HOTS) in the Technical and Vocational Education and Training (TVET) Colleges. The data set was collected at eMnambithi TVET College during March 2022 and October 2022. The two most crucial elements contributing to students attaining HOTS during teaching and learning are the lecturer's content delivery ability and the curriculum. In that regard, this study began by partially evaluating the lecturer's content delivery ability using students' perspectives through a questionnaire. It was found that N1 and N2 mathematics lecturers' content delivery abilities from the perspective of the students might be adequate. That left the curriculum as the only major contributing factor in a case where students were found to have poor HOTS at eMnambithi TVET College. Then, the SVHIR model (susceptible S(t), vaccinated V(t), healthy H(t), infected I(t), and recovered R(t)) was successfully developed to attempt the evaluation of students' HOTS. The model indicated poor HOTS in students at eMnambithi TVET College. That ultimately meant that the curriculum might be incapable of equipping students with HOTS, since the lecturers' content delivery abilities were deemed to be adequate by the participants of this study.
Keywords: HOTS; TVET; SIR model; curriculum
In the 1980s to early 1990s in South Africa, the Technical and Vocational Education and Training (TVET) national accredited technical education diploma (NATED) programs (in Report 191) were combined with apprenticeships through the Manpower Training Act (No. 56 of 1981) [[
Currently in South Africa, a student who acquires N2 TVET training is permitted to undergo trade-test preparation and testing to become an entry-level artisan. This presumes that the N1 and N2 curriculum is sufficient to equip students with the necessary thinking skills required by the industry or real-world problems space. Among those required thinking skills is the higher-order thinking skills (HOTS). HOTS are defined to be the three highest thinking levels of cognitive skills that includes analyzing, evaluating, and creating [[
Therefore, the current study aimed to find an optimal mathematical model which could be used to evaluate the capability of the curriculum (N1 to N2 mathematics) to equip students with higher-order thinking skills (HOTS) in the TVET Colleges. In that regard, we will discover how efficient the SIR model is (susceptible (S), infected (I), and recovered (R)) in evaluating the curriculum. This will be achieved by applying the SIR model and assuming that low-ability students can affect negatively the HOTS students directly or indirectly. However, we are not nullifying that the opposite case is also true—where HOTS students can impact low-ability students positively. According to Budsankom et al. [[
As the topic of HOTS has become more dominant in teaching and learning, an effective valid and efficient learning model to enhance students' HOTS has become a necessity. Hence, there are several scholars who have adopted a modeling approach into the HOTS evaluation concept. In Indonesia, two scholars conducted research to develop a HOTS learning model [[
The approach of involving models in higher-order thinking skills has been around for more than two decades [[
The data collection instrument was developed by the applying the work of Brookhart [[
- ➢ Working systematically through cases in an exhaustive way;
- ➢ Interpret and extend solutions of problems;
- ➢ Identifying possible applications of mathematics in the surroundings;
- ➢ Translate a worded or graphically represented situation to relevant mathematical formalisms;
- ➢ Use with reasonable skill the available tools for mathematical exploration.
The HOTS questions in the instruments were selected from the curriculum under investigation. However, one may be concerned about using existing knowledge to evaluate HOTS. According to Maharaj and Wagh [[
According to some scholars, there are many variables that can contribute to students attaining HOTS but the most critical are lecturers' content delivery ability and the curriculum [[
This sub-section focuses on the development of the new mathematical model that can be used to evaluate the capability of the TVET curriculum to equip students with HOTS. Hence, it is structured as a review of the SIR (susceptible, infected, and recovered) model; SVHIR model development for the current study, determination of the new model's parameters; validation of the SVHIR model; derivation of the basic reproduction ratio, association of the actual data with the SVHIR model; and the SVHIR model application instructions.
A major assumption of many mathematical models of pandemics is that the population can be divided into a set of distinct compartments. These compartments are defined with respect to disease status. The simplest model is the SIR model described by Kermack and McKendrick [[
Susceptible: Individuals that were never infected, and they can catch the disease. Once they have it, they move into the infected compartment.
Infected: Individuals that can spread the disease to susceptible individuals. The time they spend in the infected compartment is the infectious period, after which they enter the recovered compartment.
Recovered: Individuals in the recovered compartment are assumed to be immune for life.
Note that this study only considers the SIR model without demography, which means the model excludes the births and deaths of the population's individuals. Hence, the three compartments form a closed system. That simply means the total size (N) of the population does not change and is given as follows [[
(
The SIR model is easily written using ordinary differential equations (ODEs), which implies a deterministic model (no randomness is involved, the same starting conditions give the same output). Analogous to the principles of reaction kinetics, the model assumes that encounters between infected and susceptible individuals occur at a rate proportional to their respective numbers in the population. The SIR model is given as follows [[
(
The SIR model possesses the following quantities, parameters, and rates of change [[
- S(t): number of susceptible individuals;
- S′(t): rate of change in S;
- I(t): number of infected individuals;
- I′(t): rate of change in I;
- R(t): number of recovered individuals;
- R′(t): rate of change in R;
- β: disease transmission rate;
- γ: recovery rate.
The SIR model also has the following assumptions about the nature of the disease [[
- The duration of infection is the same for everyone;
- Once recovered, you are immune, and can no longer infect anyone;
- Only a fraction of contacts with the disease cause infection;
- The units of S, I, and R are persons;
- The units of time are days;
- The units of S′, I′, and R′ are persons per day, written person/day.
One of the most vital parameters in epidemiology is the basic reproductive ratio (
(
The concept of the SIR model has been a center of attention for many scholars to model pandemic diseases. In 2021, a study using the SIR model determined the dynamics of tuberculosis (TB) in Turkey as to how much it will affect the future and the impact of vaccine therapy on the disease. The obtained results revealed that the basic reproduction ratio for the model is less than 1. Also, recently some went as far as to improve the SIR model to best suit the situation of the COVID-19 outbreak. For instance, Diagne et al. [[
The current study adopted the notion of model development from the SIR model [[
Some studies revealed that study peer groups have an influence on students, and such an influence could be positive or negative on their academic achievement [[
To investigate the ability of the N1 to N2 mathematics curriculum to equip students with HOTS, we will use the above assumption to develop a mathematical model. The curriculum is learned in six months (180 days), in which it is also expected to equip students with HOTS during that period. During this period, we will assume there is a pandemic called DHOTS within students enrolled for the N1 to N2 curriculum, and the curriculum acts as a vaccine against the pandemic. In that context, we use the help of Kermack and McKendrick [[
With the help of Kermack and McKendrick [[
(
where,
(
and
(
All the parameters in the SVHIR model which have not been described before are described in Table 1.
Since we know the time interval (1 to 180 days), we can determine the finite parameters as follows:
From (
(
Dividing (
Substituting (
(
Dividing (
Substitute (
(
From (
(
Substituting (
(
Add (
(
Substituting (
(
where
It is part of the model development procedure to validate the model before its first application. The general notion to validate any model is to compare it with the actual data. Unfortunately, in this study we have very limited data in terms of data points of different times. The current study's data will not allow us to apply that notion into our model in (
Since we intend to find general prediction functions, indefinite integrals will be applied; hence, the four parameters will also be general in this subsection.
From (
(
Then solving (
(
where
Dividing (
(
where
Substituting (
(
where
In the above,
Dividing (
(
Substituting (
(
where
Then,
From (
(
When adding (
(
Substitute (
(
(
Substitute (
(
In the above,
(
where
In the previous sub-section, we arrived at four different constants of integration but three of them remain partially unknown, given we only know their lower limits. Determining these constants is crucial for this study since they play a very important role towards achieving the goal of the study on HOTS, as it will be seen towards the end of this chapter.
From Equation (
(
If we refer to the above sub-sections, S′, H′, I′, and R′ are rate of change according to their respective compartments, where their units are persons per day. It is true that each compartment rate of change cannot be greater than the total number of the population (N given by (
(
In (
(
Substituting (
(
Substituting (
(
Therefore, combining (
(
Again, in (
(
Substituting (
(
Substituting (
(
Therefore, combining what (
(
where we differentiate (
(
Substituting (
(
Equating (
(
Substituting (
(
Solving (
(
Substituting (
(
Hence,
(
where
Therefore from (
(
Note that from (
It has no influence at all in the final conclusions of this study. If all other prediction functions are valid except the vaccinated prediction function, the SVHIR model will still be effective for this current study. However, it should be clear that only the prediction function is insignificant not the compartment itself, given that the vaccinated compartment plays a vital role to other compartments more especially on the healthy and infected.
Therefore, the four resulted prediction functions from (
(
Let us consider the model (
(
Note that there are two types of the four parameters derived in this study. Firstly, there is the indefinite type found from (
(
If we can accurately predict the respective compartments in the actual data by using (
The basic reproductive ratio of the SVHIR model is given as follows:
(
The basic reproductive ratio at
(
According to our model, at
(
where
In this study, we have two HOTS tests (Appendix A and Appendix B), and we use their test scores or marks (
- A score of less than or equal to 5% cannot be used to define the status of a student: it is a nil, given that this score is highly possible to be obtained by a person who guessed the answers without being exposed to the curriculum. Therefore, we equivalate this person as someone who never took the test; hence, this score is associated with the susceptible compartment;
- A student with a score between 5% and 50% counts as a failed; hence, this score is associated with the infection compartment;
- A student with a score at 50% and above counts as a pass; hence, this score is associated with the healthy or recovery compartment.
There are 15 possible combination outcomes if a student takes the two HOTS tests (Appendix A and Appendix B), and each outcome defines the SVHIR model compartment as shown in Table 3. Those outcomes are explained as follows:
- A student who received nil in the first test and nil in the second test is considered susceptible. The first test shows symptoms of susceptibility (neither infected nor healthy but at risk of infection), and towards the end of the curriculum the second test confirms the symptoms remained the same, which means the student did not move to the vaccine compartment. Hence, the student stays in the susceptible compartment. Nonetheless, this does not mean the curriculum was not presented to the student but rather means it was presented and did not make any significant impact or sink in for the student. Therefore, the student is the same as the time of arrival, which happens at the susceptible stage.
- A student who received nil in the first test and failed in the second test is considered infected. The first test show symptoms of susceptibility, and towards the end of the curriculum the second test confirms that the student is infected. Hence the student will move from susceptible S(t), vaccinated V(t) and to infected I(t) compartment. In this case, the curriculum was presented and did make an impact to the student but not enough.
- A student who received nil in the first test and passed in the second test is considered recovered. The first test show symptoms of susceptibility, and towards the end of the curriculum the second test confirms the symptoms have improved. Hence, the student will move from the susceptible S(t) or vaccinated V(t) and to the healthy I(t) compartment. When the curriculum is presented to this student, it is very impactful.
- A student who failed in the first test and received nil in the second test is considered infected. The first test shows symptoms of infection, and towards the end of the curriculum the second test confirms the symptoms of being at risk of infection. This student is considered infected. In the model, this student will move from the susceptible S(t) or vaccinated V(t) and to the infected I(t) compartment. In this case, the curriculum was presented and did make an impact on the student but not enough.
- A student who failed in the first test and failed in the second test is considered infected. The first test shows symptoms of infection, and towards the end of the curriculum the second test confirms the symptoms remained the same. Hence, the student will move from the susceptible S(t) or vaccinated V(t) and to the infected I(t) compartment. In this case, the curriculum was presented and did make an impact on the student but not enough.
- A student who failed in the first test and passed in the second test is considered in recovery. The first test shows symptoms of infection, towards the end of the curriculum the second test confirms the symptoms have gotten better. Hence, the student will move from the susceptible S(t), vaccinated V(t) or infected I(t) and to the recovered R(t) compartment. In this case, the curriculum was presented and did make an impact on the student.
- A student who passed in the first test and received nil in the second test is considered infected. The first test shows symptoms of being healthy, and towards the end of the curriculum the second test confirms the symptoms of being susceptible. For a student to be moved from healthy to susceptible is the indication of degradation of the skill; and that can only happen when someone is infected. Hence, the student will move from the susceptible S(t) or vaccinated V(t) and to the infected I(t) compartment. In this case, the curriculum was presented and did make an impact on the student but not enough.
- A student who passed in the first test and failed in the second test is considered infected. The first test shows symptoms of being healthy, and towards the end of the curriculum the second test confirms the symptoms of infection. For a student to be moved from healthy to susceptible is an indication of degradation of the skill; and that can only happen when someone is infected. Hence, the student will move from the susceptible S(t) or vaccinated V(t) and to the infected I(t) compartment. In this case, the curriculum was presented and did make an impact on the student but not enough.
- A student who passed in the first test and passed in the second test is considered healthy. The first test shows symptoms of being healthy, and towards the end of the curriculum the second test confirms the symptoms have remained the same. Hence, the student will move from the susceptible S(t) or vaccinated V(t) and to the healthy H(t) compartment. This student is presumed to have arrived already equipped with HOTS; hence, when the curriculum is presented to them, it is very impactful.
- A student who received nil in the first test and did not get a chance to participate in the second test is excluded in the current study. The reason is that with the three possible scores (nil, fail, and pass) the student could have obtained in the second test, the student could either be Outcome 1 or 2 or 3 in Table 3, which are three different compartments (susceptible, infected, and recovered) the student could possibly belong to, and the study is unable to conclude about the student's compartment between the three in the absence of the second test score. Hence, the student is excluded.
- A student who failed in the first test and did not get a chance to participate in the second test is excluded in the current study. The reason is that with the three possible scores (nil, fail, and pass) the student could have obtained in the second test, the student could either be Outcome 4 or 5 or 6 in Table 3, which are two different compartments (infected and recovered) the student could possibly belong to, and the study is unable to conclude about the student's compartment between the two in the absence of the second test score in that case. Hence, the student is excluded.
- A student who passed in the first test and did not get a chance to participate in the second test is excluded in the current study. The reason is that with the three possible scores (nil, fail, and pass) the student could have obtained in the second test, the student could either be Outcome 7 or 8 or 9 in Table 3, which are two different compartments (infected and healthy) the student could possibly belong to, and the study is unable to conclude about the student's compartment between the two in the absence of the second test score in that case. Hence, the student is excluded.
- A student who did not participate in the first test and received nil in the second test is excluded in the current study. The reason is that with the three possible scores (nil, fail, and pass) the student could have obtained in the first test, the student could either be Outcome 1 or 4 or 7 in Table 3, which are two different compartments (susceptible and infected) the student could possibly belong to, and the study is unable to conclude about the student's compartment between the two in the absence of the first test score in that case. Hence, the student is excluded.
- A student who did not participate in the first test and failed in the second test is considered infected. The reason is that with the three possible scores (nil, fail, and pass) the student could have obtained in the first test, the student could either be Outcome 2 or 5 or 8 in Table 3, which are all the infected compartments.
- Lastly, this is a student who only participated in the second test and passed. This student will also be excluded in the current study. The reason is that with the three possible scores (nil, fail, and pass) the student could have obtained in the first test, the student could either be Outcome 3 or 6 or 9 in Table 3, which are two different compartments (recovered and healthy) the student could possibly belong to, and the study is unable to conclude about the student's compartment between the two in the absence of the first test score in that case. Hence, the student is excluded.
From the SVHIR model, we produced the basic reproduction in (
(
where
For this study, the data were collected at a TVET college and categorized according to Table 3. For the SVHIR model application, (
Case 1 (
This means the curriculum has failed to equip students with HOTS.
Case 2
This means the curriculum has equipped students with HOTS.
We collected our data at eMnambithi TVET College, Ezakheni E-section campus. It was collected from the 47 students using Appendix A as the pre-assessment and Appendix B as the post-assessment. Then, Appendix C was used to score the students' responses. The final processed data of the current study are presented in Appendix D. The first part of data collection took place in March 2022 and the second part in October 2022. Normally, N1 classes start in January but, due to COVID-19, classes were disturbed. N1 ended up starting in March 2022. Nonetheless, students were given an equal opportunity of 180 days to complete the N1 to N2 curriculum and all of them received the same teaching approaches.
This section presents the results of the partial evaluation of lecturers' content delivery ability and the SVHR model application results from the data under investigation.
As mentioned, there are many variables that can contribute to students attaining HOTS but the most critical are lecturers' content delivery ability and the curriculum [[
The current study used Question 6 in Appendix B to partially investigate the N1 and N2 mathematics lecturers' content delivery ability at eMnambithi TVET College and the results are presented in Table 4. Note that we used the word "partially", given that it takes more than what we conducted to evaluate the lecturers' content delivery ability. However, what we performed is the beginning of that evaluation, but it has the potential to indicate the possible outcome of the overall evaluation. Hence, we will build our assumption about the lecturers' content delivery ability upon it.
This sub-section presents the validation and application of the SVHIR model by using the actual data in Appendix D. Note that the categorization of students according to the SVHIR model compartments in Appendix D was accomplished by applying Table 3. From those categorizations, Table 5 was produced.
In the validation section, our validation notion deduces that the SVHIR model is valid if and only if (
(
The compartment rates calculated using (
As mentioned in Section 2.2.7, the HOTS is investigated by applying the extension of the SVHIR model called the basic reproductive ratio. Substituting all necessary variables taken from Table 5 into (4.50), we obtain
(
The basic reproductive ratio relates to Case 1
As a partial evaluation of the lecturer's content delivery ability at eMnambithi TVET College, students were asked if they recognize the questions in Appendix A and Appendix B from the class lessons. The results indicate that the first portion of 55.6% responded yes, the second portion of 4.3% responded no, and the last portion of 40.4% did not answer the question. The last portion (40.4%) will be excluded since their stand is unclear. Therefore, when we only consider the first and second portion, we find that the majority of the students considered the questions in Appendix A and Appendix B to be relevant to what they have learnt. That indicates a possible adequate N1 and N2 mathematics lecturers' content delivery ability. In that regard, the current work establishes an assumption that the lecturers' content delivery ability might be adequate at eMnambithi TVET College. That leaves the curriculum as the only major contributing factor in the case where students are found to have a poor HOTS. Hence, in that case, the impact of the curriculum remains the only variable to be evaluated in Appendix A and Appendix B regarding students' responses.
We found the SIR model to be inefficient to evaluate the curriculum. However, after modifying the SIR model to an SVHIR model, it was found to be efficient. Therefore, to investigate the capability of the curriculum to equip students with HOTS, an extension of the SVHIR model called the basic reproductive ratio was applied. The resulting basic reproductive ratio in Section 3.2.2 indicated that students were not helped to improve their HOTS during the period of 6 months that they were exposed to the N1 to N2 mathematics curriculum. In other words, the SVHIR model indicates that the curriculum might be incapable of equipping students with HOTS at eMnambithi TVET College.
The study reported on was an exploratory study, where the authors explored a new potential approach of evaluating HOTS in the TVET college mathematics curriculum. Mostly, the study was focused on the development of the evaluation model and less on the application. This was because the application of the model is a separate study on its own. Also, the data were very limited because of the reason mentioned in the next section: the limitations. Hence, all the findings of the study indicate a possible situation rather than a definite one.
Looking at Appendix D or Table 5, at the end of the curriculum about 77% of students were indicated to possibly remain with DHOTS. That is a huge portion of the population, which will make the basic reproductive ratio to be expected far beyond 1, as a confirmation of a large portion of population remain with DHOTS. Indeed, the basic reproductive ratio from the SVHIR model was as anticipated. Hence, the N1 to N2 mathematics curriculum was found as being potentially incapable of equipping students with HOTS.
A further study should be conducted in the future, where the model could be applied to larger-scale data, given that in the current study the model development was a priority rather than its application. Hence, the application was performed on smaller-scale data so as to test the model. However, the model produced a sound conclusion within the context of the available data. Also, in a further study we will have an opportunity to eliminate some of the assumptions. Currently, the study has significant assumptions which has the possibility of affecting the study to some extent.
During the period of the current study's execution, we identified a common challenge in most KZN TVET colleges at a distance. There was a lack of proper leadership, which ultimately affects many areas including teaching and learning within each TVET college. Some of the gatekeepers were aware of the challenge, but instead of rectifying it they shielded that challenge. Hence, they prevented any research focusing on teaching and learning from taking place in their institutions. That resulted in the current study having limited data from TVET colleges, given that we were only able to obtain eMnambithi TVET College data. In that regard, the study changed to an exploratory study. Hence, the developed model was only tested to be valid on limited data.
South Africa has nine provinces, where KZN is only one of them. Therefore, the next study should mostly focus on the application of the developed model to a wider scale of data, to be based on the other eight provinces.
Graph: Figure 1 Susceptible, infected, and recovered model (SIR model).
Graph: Figure 2 Susceptible S(t), vaccinated V(t), healthy H(t), infected I(t), and recovered R(t) model (SVHIR Model).
Table 1 SVHIR model parameters and their descriptions.
Parameters Description Vaccination rate Healthy individuals' discovery rate Disease transmission rate Recovery rate Initial or 1st day Final or 180th day . Susceptible individuals on the 1st day/ Initial susceptible individuals Susceptible individuals on the 180th day/ Final susceptible individuals Infected individuals on the 1st day/ Initial infected individuals Infected individuals on the 180th day/ Final infected individuals Healthy individuals on the 1st day/ Initial healthy individuals Healthy individuals on the 180th day/ Final healthy individuals Recovered individuals on the 1st day/ Initial recovered individuals Recovered individuals on the 180th day/ Final recovered individuals Vaccinated individuals on the 180th day/ Final vaccinated individuals Total number of individuals on the 1st day/ Initial total number of individuals
Table 2 Description and compartmental categorization of students based on HOTS tests scores range.
Order Scores Description Compartment 1. Nil susceptible 2. fail infection 3. pass healthy or recovery
Table 3 Compartmental categorization of students based on HOTS tests scores.
Outcome Test 1 Marks Test 2 Marks Resultant Compartment 1 Susceptible 2 Infected 3 Recovered 4 Infected 5 Infected 6 Recovered 7 Infected 8 Infected 9 Healthy 10 None Excluded 11 None Excluded 12 None Excluded 13 None Excluded 14 None Infected 15 None Excluded
Table 4 Validation of the evaluation instrument with Appendix B Question 6.
Students' Response Number of Students Percentage Yes 26 55.6% No 2 4.3% No comment 19 40.4%
Table 5 SVHIR compartment values from the actual data in Appendix D.
Compartment Parameters Actual Data Values Initial time (in days) Final time (in days) Initial susceptible individuals Final susceptible individuals Initial infected individuals Final infected individuals Initial healthy individuals Final healthy individuals Initial recovered individuals Final recovered individuals Final vaccinated individuals Initial total number of individuals Final total number of individuals
Table 6 SVHIR compartment rates calculated from the actual data.
Compartment Rate Names Calculated Values Vaccination rate Healthy individuals' discovery rate Disease transmission rate Recovery rate
Table 7 Predicted versus actual SVHIR compartments.
Integration Constant Predicted Compartment at Actual Compartment at -
Conceptualization, G.N.M.; methodology, G.N.M. and A.M.; validation, G.N.M. and A.M.; formal analysis, G.N.M. and A.M.; investigation, G.N.M.; resources, G.N.M.; data curation, G.N.M.; writing—original draft preparation, G.N.M.; writing—review and editing, G.N.M. and A.M.; visualization, G.N.M.; supervision, A.M.; project administration, G.N.M.; funding acquisition, A.M. All authors have read and agreed to the published version of the manuscript.
The approval is waived due to the University of KwaZulu-Natal has indicated that the protocol has been granted EXEMPTION FROM ETHICS REVIEW.
Informed consent was obtained from all subjects involved in the study.
The relevant data is included in the manuscript.
The authors declare no conflict of interest.
We would like to convey our appreciation to eMnambithi TVET College for granting us the permission to collect data. A special thanks to Beatrice Mpangase (in the principal's office), Phelelani Buthezi (campus manager), and all the Ezakheni E-section campus stuff for their support throughout the period of data collection. TESP for making available financial [
- Simplify the following:
Choose the correct option from the following:
- (a)
- (b)
- (c)
- (d)
- (e) None of these.
- 2. The following equation has not more than two roots/solutions:
- 2.1. Why the above equation has not more than two roots/solutions?
- (a) It is a cubic equation.
- (b) It has no solution.
- (c) It is a quadratic equation.
- (d) It has a constant number 1.
- 2.2. Elaborate further what is meant by something being a root/solution of a particular equation?
- (a) It is any integer number.
- (b) It is a number that when substituted in a given equation satisfies it.
- (c) It is a number that when substituted in a given equation leaves the result undefined.
- (d) It is any constant number found in the equation.
- 3. One chocolate and one apple cost a total amount of ZAR 50 while four chocolates and three apples cost a total amount of ZAR 190. How much is each chocolate and each apple?
- 4. Write down the following sentences/statements in a form of mathematical equations.
One red car together with a black bicycle costs ZAR 150 000. Also, the price of three red cars and a black bicycle is ZAR 430 000.
- 5. Given the following diagram
- 5.1. Mention the method/s that can be used to find the distance AB.
- 5.2. Use the above-mentioned method/s to calculate the distance AB (e.g., if you mentioned two methods, find the distance of AB by the first method and after that use the second method).
- Simplify the following:
Choose the correct answer in the following:
- (a)
- (b)
- (c)
- (d)
- 2. The following equation has no more than two roots/solutions:
- 2.1. Why does the above equation have no more than two roots/solutions?
- (a) It is a cubic equation.
- (b) It has no solution.
- (c) It is a quadratic equation.
- (d) It has a constant number 10.
- 2.2. Elaborate further what is meant by something being a root/solution of a particular equation?
- (a) It is any integer number.
- (b) It is a number that when substituted into a given equation satisfies it.
- (c) It is a number that when substituted into a given equation leaves the result undefined.
- (d) It is any constant number found in the equation.
- 3. One chocolate and one apple cost a total amount of ZAR 40, while four chocolates and three apples cost a total amount of ZAR 150. How much is each chocolate and each apple?
- 4. Write down the following sentences/statements in a form of mathematical equations.
One red car together with a black bicycle costs ZAR 200 000. Also, the cost of three red cars and a black bicycle is ZAR 580 000.
- 5. Given the following diagram
- 5.1. Mention the method/s that can be used to find the distance BC.
- 5.2. Use the above-mentioned method/s to calculate the distance AC (e.g., if you mentioned two methods, find the distance of AC by the first method and after that use the second method).
- 6. Do you think all the above questions from 1–5 are familiar or relevant to what you have learnt from the N1 to N2 curriculum and class lessons?
• (a) YES
• (b) NO
Correct answer [1 mark] Correct answer [1 mark] Correct answer [1 mark] Formulation of the first equation [1 mark] Formulation of the second equation [1 mark] Solving for Variable 1 (chocolate/ apple) [1 mark] Solving for Variable 2 (apple/ chocolate) [1 mark] Labelling the variables [1 mark] Formulation of the first equation [1 mark] Formulation of the second equation [1 mark] Identifying the required method [1 mark] Stating Pythagoras theorem formula [1 mark] Solving for AB [1 mark]
1 0 23 Infected 2 69 0 Infected 3 54 0 Infected 4 53 77 Healthy 5 53 15 Infected 6 46 0 Infected 7 38 46 Infected 8 54 0 Infected 9 38 0 Infected 10 38 0 Infected 11 38 0 Infected 12 46 0 Infected 13 38 0 Infected 14 0 0 Susceptible 15 15 0 Infected 16 69 51 Healthy 17 23 0 Susceptible 18 23 8 Infected 19 15 0 Infected 20 15 31 Infected 21 31 15 Infected 22 38 0 Infected 23 38 31 Infected 24 46 31 Infected 25 54 31 Infected 26 23 31 Infected 27 69 62 Healthy 28 15 23 Infected 29 84 100 Healthy 30 69 92 Healthy 31 15 31 Infected 32 54 77 Healthy 33 38 0 Infected 34 53 0 Infected 35 46 0 Infected 36 54 23 Infected 37 85 0 Infected 38 31 0 Infected 39 0 85 Recovered 40 0 100 Recovered 41 0 46 Infected 42 0 23 Infected 43 0 85 Recovered 44 0 38 Infected 45 0 38 Infected 46 0 46 Infected 47 0 46 Infected
By Godfrey Nkululeko Mazibuko and Aneshkumar Maharaj
Reported by Author; Author