2007 Math Prelim Papers

When I received the recent prelim exam papers, I was taken aback by some of the questions set. Not only were some of these ‘cheem’ questions extremely lengthy (some can have as many as 8 sentences in a question), the person reading them must be strong in their math concepts just to be able to piece the question together to ‘see’ the clues. Surprisingly, these “cheem” questions were not only from our elite schools but a growing numbers were originated from neighbourhood schools!

Judging from the growing trend of lengthy questions with many conditions attached, all students must not only know the Fundamental Concepts of each topic well, but must also be strong in their Cognitive Skills (the skills needed to process information mentally) in order to score well. These skills include Visual Processing (ability to rotate or flip a diagram in mind), Working Memory (ability to hold and process a group of information mentally) and Logic & Reasoning (ability to make see logics in questions).

I have selected some recent prelim questions for discussion as I personally felt that despite the fact that some students might be strong in problem solving, their learning styles and cognitive abilities might cause them to under perform for these questions.

Nanyang Primary School 2007 Math Prelim
Question:
A rectangle piece of paper is folded along AC at Corner B as shown in Figure 1 so that the line BC lies on the centre line.

Next, it is folded along DC as shown in Figure 2 so that the line AC lies on the centre line.

Finally, it is folded along EF as shown in Figure 3 so that the line GD lies on the centre line.

Find angle b.


Can you imagine this question is actually an Angle question?

Frankly, I read this question 3 times but for every attempt, I was not able to complete the entire question because whenever I reached figure 3, I cannot proceed on as the diagram shown in the question is different from the one I had in my mental mind!

(What I was suffering from is commonly known as ‘mental block’. This will occur to most learners who are strong in their Logic & Reasoning skill. Whenever their perceived mental picture or logic differs from those shown in the question, the mind will become extremely ‘noisy’ and will be reluctant to move on until the differences are resolved.)

Finally, to help me move on, I took an A4 paper and started folding and only then, I was able to move on and complete the question as my folded paper revealed the same image as the one I had in my mind!

From an educator’s point of view, this question is a very good question as it tests a student’s true understanding of symmetry as well as the geometrical properties of triangles, four-sided figures and angles.

In order to solve this question, besides having a good grasp of the fundamentals of Symmetry, geometrical properties of Triangles, four sided figures and angles, a student must also be strong in their Visual Processing and Logic Reasoning skills.

1. To help you see and understand better, first fold an A4 paper into half as shown in figure 1
2. Next, fold a triangle ABC (as shown in figure 2). Noticed angle BAC and angle BCA are the same (line BC = line BA and triangle ABC is an isosceles triangle)

3. Then, fold such that line AC lies on the centre line of the A4 paper. You will notice that your folded paper is different from figure 3 because line AC and line AD should be of the same length! (this is the ‘cause’ of my mental block as instead of drawing line AC to be the same length as line AD, the picture showed line AC to be shorter than line AD)

To prove my point, referring to figure 2, you should spot a rhombus before folding DC. Therefore, in figure 3, angle ACD equals angle ADC (22.5 degree) as line AC equals line AD. Thus, angle CAD is 135 degree (180 – 22.5 – 22.5 = 135).

4. Before you fold your paper to form figure 4, you will notice that line GD is parallel to line AC. Therefore, angle CAD equals angle ADG (135 degree) due to alternate angles.

5. Now, I believe you can confidently solve for angle b as it is the sum of angle ADG (135 degree) and angle ADC (22.5 degree).

Answer: angle b = 157.5 degree

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Singapore Chinese Girls School 2007 Math Prelim
Question
When Mrs Lee was 40 years old, her son was twice her daughter’s age. Mrs Lee will be twice her son’s age when her daughter is 28 years old. How old will Mrs Lee be when her daughter is 20 years old?

Unlike the Nanyang question, I was ‘shocked’ when I first read this question, for what first comes to my mind is …….. ‘Is this a primary school question?’

I have a reason for feeling this way. My reaction was caused by my immediate choice of approach to solve this question. As there were too many variables or unknowns in this question, I used what most adults will use to solve this question, Algebra.

This was how I solve the question for my first attempt.

Let daughter’s age = x
When Mrs Lee is 40, daughter = x and son = 2x

‘y’ years later, when daughter becomes 28 years old
Mrs Lee will be 40 + y, daughter = x + y and son = 2x + y

From above, x + y = 28 --> equation 1
and since Mrs Lee will be twice son’s age, that is 2(2x+y)
therefore, 2(2x+y) = 40 + y --> equation 2

Using the above 2 equations and the substitution technique, I found daughter’s initial
age was 4 years old (x=4) and thus when she is 28 years old (16 years later),
Mrs Lee will be 40 + 16 = 56 years old

If you noticed my above solutions (does not matter if you don’t understand because you will be taught them when you are in secondary school), my default mode was to use Algebra to solve.

After solving, the next question that followed my mind was “How many primary 6 students, who were not taught algebraic expansion and simultaneous equations, can solve this question?”

Staring at the question for a while, I resumed the resource of a primary 6 student and re-approached the question using techniques like listing of tables, drafting of models and finally, after many minutes, I obtained the correct answer through Guess and Check (yeah)!

Not satisfied with the Guess and Check approach as the time taken was way too long, I began to explore the model drawing approach again. After overcoming the challenge of not being able to draw the models for this sentence, Mrs Lee was 40 years old, the son is twice the daughter’s age, I finally arrived the answer.

The following Step-by-Step model solution might not be easily accepted by some students, especially students who belong to the sequential learner group.

1. Construct the first set of models of ‘when Mrs Lee was 40 years old, son is twice the daughter’s age’ by JUST listing and not drawing Mrs Lee’s model first, drawing 1 unit for the daughter and 2 units for son. (for this step, sequential learners, students who need to do things in sequence, will feel uneasy when told not to draw Mrs Lee’s model first)

2. To draw the next step correctly, one needs to be strong in their language. ‘When the daughter is 28 years’, for this sentence, you will need to agree that you have to add some more years instead of subtracting a number of years from the current age. (you will really need to read the ‘English’ of the sentence, to pick up the present and past tense of the sentence for you to agree that it should be some years later from the initial age for her daughter to become 28 years old.)

My reasoning is highlighted in red:

‘When Mrs Lee was 40 years old, her son was twice her daughter’s age. Mrs Lee will be twice her son’s age when her daughter is 28 years old. How old will Mrs Lee be when her daughter is 20 years old?

Therefore, I shall construct a second set of models by adding an additional block to daughter and son to represent some years later. I shall name this additional block as ‘y’.

3. Following the condition of the sentence ‘When daughter is 28, Mrs Lee will be twice her son’s age’, I am now able to draw Mrs Lee’s model by copying twice the son’s model.

4. Now, you need to spot for patterns from the 2 sets of models. If you noticed the first and second set of the models, the model of the son and daughter in the first set is ‘y’ block shorter than the model of the son and daughter in the second set. Therefore, logically, the model of Mrs Lee in the first set must be ‘y’ block shorter than the model of Mrs Lee in the second set. Therefore, we can now construct the model of Mrs Lee in the 1st set.

5. When we begin to label the models, we will arrive the following equation:

4 units + y = 40 years
1 unit + y = 28 years

3 units = 40 – 28 = 12 years
Therefore, 1 unit = 4 years

The daughter was 4 years old when Mrs Lee was 40 years old.
For her daughter to become 20 years old, we need to add 16 more years to her
daughter. Therefore, Mrs Lee will be 40 + 16 = 56 years old when her daughter is
20 years old.

I hope my sharing has benefited you and your child. Before I end this posting, I would like
you print out an elite school’s 2007 math prelim paper for your child to work on. An answer
key is included for marking purposes. From the results, you can have a good gauge as to
how prepared your child is for the coming PSLE. I hope your child will benefit from this
2007 prelim paper.

To get a copy of a FREE PSLE 2007 Math Prelim Exam Paper, click here

Norman Tien
http://www.pslemath.com/