As students transit from lower secondary to upper secondary, most of the students are offered Additional Mathematics (A Maths). This is in addition to the compulsory Elementary Mathematics (E Maths) they have to study. Many of the skills gained from lower secondary are still applicable, and teachers expect students to know them before starting on new chapters.
Which means, if your child has gaps in concepts for lower secondary school maths, he/she may struggle in the upper secondary school maths.
Identification of gaps is simple: As long as your child is not scoring above 80 for lower secondary school math, there is a high likelihood there are some gaps missing. If you need some help to identify the gaps, feel free to contact us for a free diagnostic consult to identify concept gaps or if you are looking for secondary school tuition with experienced tutors.
How to approach upper secondary maths questions systematically
Maths is a systematic subject – the learning, the presentation, are all linear. It requires students to think in a logical flow. This means that conquering mathematics questions requires a systematic and logical thinking process.
At Eton tuition, we guide students through five steps thinking process when teaching students how to approach questions. Maths, be it A maths or E maths is never just about the questions and solutions, but the thinking process to reach the solution. That is the mindset we want our students to have, and it does not just apply to studies, but to the outside of school life as well. It is a life skill.
In a nutshell, this is the 5 step thinking process we use for mathematics questions:
- Identification – Identify the question’s requirements through reasonings.
- Recognition – Recognize contents related to the question.
- Analysis – Analytical approach through figures and equations.
- Regulation – Regulate thinking process through a reflection of contents with applications learned.
- Connection – Identify the relationship between answer and question’s objective.
Here is an example of a question involving calculus.
The diagram shows cardboard rolled into the shape of a cylinder of radius r cm and height h cm. The shape is held together by three pieces of adhesive tape whose width and thickness may be ignored.
One piece of tape forms the shape of a rectangle. The other two pieces form the shape of circles. The total length of tape is 300 cm.
(i) Show that the volume, V cm³, of the cylinder, is given by 𝜋𝑟²(150−2𝑟−2𝜋𝑟).
(ii) Given that r can vary, show that V has a stationary value when 𝑟=𝑘/(1+𝜋), where k is a constant to be found and find the corresponding value of h.
By applying the 5-steps thinking process,
- We would identify that the question requires us to find the volume of the cylinder in terms of r and to identify the corresponding value of h given it has a stationary value.
- The objective of the question is related to the understanding of the minimisation and maximisation problems under calculus.
- Analyse that the expression quoted in part (i) requires us to understand that h must be presented in terms of r for us to be able to show the final expression. The second part requires the student to be able to understand how it is related to the first part and to be able to use the expression to find the stationary value of V.
- Systemic thinking required with the approach of this question through prior practice and understanding of the topic on minima and maxima problems and its applications.
- Understanding the objective of the question and its answer will assist to define a more contoured thinking process for handling of future questions.
Where necessary, the thinking process often boils down to how we can relate variables in such questions to having only 2 variables (O level maths) in an equation before differentiation is carried out.
For students who are taught here, the way to assist them in their thinking process was to zoom in on the expression wanted in part (i) where we rely on the fundamental of finding the volume of the cylinder [V = 𝜋𝑟²ℎ] and we can analyse that ℎ=150−2𝑟−2𝜋𝑟 naturally. Students will also find such skills useful in deriving the expression and gain confidence and assurance as they go through the question looking for specific figures such as “three pieces of adhesive tape” and “total length of tape is 300 cm”.
The same thinking process can also be easily applied to every other chapter in maths. It then becomes a matter of practice of the thinking process.
If your child requires assistance beyond math, contact us for secondary school tuition.