Contributed by Foo Chee Juan, Head Of Secondary Mathematics, United Square.
Additional Mathematics is one of the most challenging subjects at the ‘O’ Levels.
One of the common difficulties experienced by students is that they are able to do well in class tests in schools but when faced with high stakes examinations like the preliminary exams or the ‘O’ Level exam, they may experience mental blocks or are unable to complete the paper on time.
Find out how to strategise for revision, manage the time crunch in the exam and tackle challenging questions from the 2017 ‘O’ Level Additional Mathematics Paper 1.
This guide is split into two key sections:
• Focus on Exam Strategies
• Focus On Essential Heuristics
Section 1: Focus On Exam Strategies
Understanding key exam strategies helps students to rise above the challenges of Paper 1.
Being well-prepared for the exam begins with understanding how much revision needs to be done and how best to go about revising for each examination.
1. Master Relational Understanding
Students understand content through two main ways.
Students know of different techniques that can solve a particular problem.
Students actually understand why certain techniques work and how concepts relate to one another.
As a result of the increase in emphasis in deep learning of concepts in Mathematics, the 'O' Level syllabus is moving away from questions which solely test students’ instrumental understanding.
The syllabus is now moving towards questions which require students to have relational understanding across topics.
What does this mean for students? It is now no longer sufficient to only apply formulae to questions in order to solve them.
Instead, students must demonstrate a clear understanding of the concepts behind the formulae they use and be flexible in how they apply it in different contexts whilst keeping in mind the connections between concepts.
Refer to the example below.
The 2017 'O' level Additional Mathematics Paper 1 Question 1
A curve is such that d2 y/ dx2 = 8 - 6x and the point P(2, 8) lie on the curve.
The gradient of the curve at P is 3.
Find the equation of the curve.
In order to solve this question, students have to:1. Make the connection that the second derivative is actually the differentiating the first derivative
2. Understand the meaning of a point lying on a curve and its implications
3. Make sense of why the question gives the value of the gradient
4. Know that the equation of a curve is not a linear function
From the above example, we can see how students not only had to know the techniques involved in differentiation but also had to understand how the concepts of gradient and equation connects to differentiation.
2. Manage Time Wisely
Time management is a constant struggle for students in the Additional Mathematics examination.
Students can start training to complete the paper within the given time limit by completing timed trials and practice tests.
Students may want to time themselves as they solve a problem that involves one topic, followed by a problem that involves two topics and so on.
This helps them to gauge how long they need to complete each type of question without committing careless errors in calculations and workings.
TLL Bonus Tip: Remember, when you start timing yourself to complete a whole paper, simulate examination conditions by skipping a question when you are stuck on it for more than 10 seconds.
Section 2: Focus On Essential Heuristics
Have you ever wondered why most students think that Algebra is the only method to use when solving secondary Math questions?
Let’s clear up one myth: Algebra is definitely not the only way to solve mathematical problems.
In fact, using Algebra is often known as a heuristic called working backwards.
In order to beat the time constraint of exam conditions, it is useful for students to equip themselves with as many strategies to solve mathematical problems as possible.
These strategies form a safety net, allowing students who may be experiencing mental blocks to find a solution to the problem.
Ample knowledge of key strategies allows students to develop a more flexible approach to thinking during the exam.
3. Master The Heuristic: Working Backwards or Intelligent Guessing
The heuristic, “working backwards”, requires students to begin solving the question by considering the end of a problem.
Based on the topics of Quadratic Equation and Prime Numbers, refer to the question below which requires the use of “working backwards” and “intelligent guessing”.
In the flower dome situated within Gardens by the Bay, a certain species of roses is being planted in a square array such that the number of rows is equal to the number of columns.
The caretaker decides to increase the size of the bed of roses equally in the number of rows and columns.
His new bed of roses contains 223 additional roses.
How many roses did he have in one row of the original square array?
Students should use the heuristic “working backwards” to solve the question.
In order to do so, students should set up an equation to represent the information in the question.
Consider the method below:
We let x represent the original number of rows and the number of columns and b represent the additional roses for each row and column. Therefore, the new number of roses is (x + b)2.
In the question, we know that there are 223 additional roses after the caretaker increases the size of the bed of roses. This will give us the equation x2 + 223 = (x + b)2
This equation can be simplified:
x2 + 223 = (x + b)2
x2 + 223 = x2 + 2bx + b2
223 = b2 + 2bx
However, we only have one quadratic equation with two unknowns.
Students can substitute values for the valuables and solve the equation, but this is not a very efficient method. Students should use the strategy “intelligent guessing and checking”.
- Let's continue solving the question below:
- Since we know that 223 is a prime number, we can factorise the equation 223 = b2 + 2bx
- to obtain the equation 223 = b(b + 2x).
- As 223 is a prime number, it only has two factors: 223 and 1.
- The equation should can be seen as 223= 1(223)
- Therefore, b must be equal to 1 and b + 2x must be equal to 223.
- Hence, 2x = 222 and x = 111.
Answer: There are 111 plants in one row of the original bed of roses.
Hence, it can be seen that even though algebra is involved in the question, students should can use key heuristics such as working backwards as well as intelligent guessing or checking to solve such questions.
Would You And Your Child Like To Learn More?
Download our full guide on “5 Ways To Manage 'O' Level Additional Mathematics” where Chee Juan shares more about how to fully prepare for Paper 1 with question analysis skills and key heuristics.
Click here or on the image below to download the full guide.
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