Problem-Solving Items in PSLE Mathematics

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Problem-Solving Items in PSLE Mathematics

• Yeap Ban Har
• National Institute of Education Nanyang
  Technological University

Organised by Association of Mathematics Educators
Department of Science and Mathematics Singapore
Polytechnic
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Singapore Mathematics Curriculum (1992, 2001)
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New Directions in Assessment
? Examinations are here to stay
? Changes in emphasis

• Changes in format
• Not necessarily paper-and-pencil
• Not necessarily individual
• Not necessarily independent

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Item Types in PSLE
What is the value of 84 ? 7 4 ? 2?
(1) 56
(2) 16
(3) 14
(4) 4
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PSLE Items

• Computation must not be tedious

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S
M
B
O
D
A
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Item Types in PSLE
A piece of wire is bent to form the right-angled
triangle shown below. Find the area of the
triangle.
20 cm
16 cm
Answer _________ cm2
12 cm
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PSLE Items

• Selecting data is now required

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Item Types in PSLE
The figure is made up of four identical squares
each of side 2 cm. What is the perimeter of the
figure?
(1) 16 cm
(2) 20 cm
(3) 24 cm
(4) 32 cm
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Item Types in PSLE
The figure is made up of four identical squares
each of side 2 cm. What is the perimeter of the
figure?
(1) 16 cm
(2) 20 cm
(3) 24 cm
(4) 32 cm
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PSLE Items

• Concepts are tested alongside procedures

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Item Types in PSLE
A rectangular piece of paper, coloured on one
side, is folded to form the shape shown
below. What is the area of the rectangular
piece of paper before it was folded?
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Item Types in PSLE
(1) 24 cm2
(2) 40 cm2
(3) 48 cm2
(4) 56 cm2
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PSLE Items

• Expects hands-on learning in the classroom

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Singapore Mathematics Curriculum (1992, 2001)
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More Than Computational Fluency
What is the reading indicated on the weighing
scale shown?
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PSLE Items

• Practical skills are tested too

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More Than Computational Fluency
What is the reading indicated on the weighing
scale shown?
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More Than Computational Fluency
The figure shows a line XY and three points R, S
and T.
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More Than Computational Fluency
Draw a straight line from point X to one of the
points R, S or T so as to form an angle between
50o and 70o at X.
Draw a perpendicular to XY passing through point
T.
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Computation?
What is the missing number in the box?
?
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Computation?
1 2 3 4 .. 94 95 96 97 When the
first 97 whole numbers are added up, what is the
digit in the ones place of this total?
(1) 1
(2) 2
(3) 3
(4) 8
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Computation?
1 2 3 4 .. 94 95 96 97
1 2 3 4 .. 94 95 96 97
1 2 3 4 .. 94 95 96 97
1 2 3 4 .. 94 95 96 97
1 2 3 4 .. 94 95 96 97 98
99
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Singapore Mathematics Curriculum (1992, 2001)
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Life becomes harder for those who perceive these
to be computational items. There are pupils who
perceive mathematics to be computations. Those
who are a little critical would ask themselves if
there are more elegant methods to get the answer.
Those who are a little creative would be able to
figure different ways to do the same tasks.
Perception, critical thinking and creative
thinking are part of habits of mind.
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The PSLE Format

• It lasts for 2 hours 15 minutes.
• There are 15 multiple-choice questions for 25 of
  the total marks.
• 5 1-mark questions (5)
• 10 2-mark questions (20)
• The first format tests really basic knowledge.
  The second format tests a range of competencies
  including problem-solving proficiency.

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The PSLE Format
There are 20 short-answer tasks for 20. They are
all 1-mark item. If units are required, they are
indicated. If working is necessary, they can be
done but will not be considered for credit. They
are meant to test basic skills. Many can be done
mentally.
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The PSLE Format

• There are 15 structured and long-answer tasks for
  55.
• Three 2-mark tasks (6)
• Three 3-mark tasks (9)
• These two types tend to test basic skills and
  simple problem solving.

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The PSLE Format

• There are 15 structured and long-answer tasks for
  55.
• Five 4-mark tasks (20)
• Four 5-mark tasks (20)
• These two types tend to test problem-solving
  proficiency. Some tasks are demanding.

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Problem Solving
Lee and Chan both drove from Town P to Town Q.
They started their journeys at different times.
Lee drove at an average of 45 km/h and took 40
minutes. Chan drove at an average speed of 72
km/h and reached Town Q at the same time as Lee.
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Problem Solving
Lee and Chan both drove from Town P to Town Q.
They started their journeys at different times.
Lee drove at an average of 45 km/h and took 40
minutes. Chan drove at an average speed of 72
km/h and reached Town Q at the same time as Lee.
Speed Distance ? Time 45 Distance ? 2/3
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Problem Solving
Lee and Chan both drove from Town P to Town Q.
They started their journeys at different times.
Lee drove at an average of 45 km/h and took 40
minutes. Chan drove at an average speed of 72
km/h and reached Town Q at the same time as Lee.
60 minutes --- 45 km 20 minutes --- 15 km 40
minutes --- ?? km
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Structured Questions
How far was Town P from Town Q?
How many minutes later than Lee did Chan start
this journey?
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Problem Solving
Lee and Chan both drove from Town P to Town Q.
They started their journeys at different times.
Lee drove at an average of 45 km/h and took 40
minutes. Chan drove at an average speed of 72
km/h and reached Town Q at the same time as Lee.
72 km --- 60 minutes 12 km --- 10 minutes
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Problem Solving
Sam gets 3 more pocket money than Bob each
week. They each spend 15 per week on food and
save the rest. When Sam saves 72, Bob only saves
48.
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Sam
72
Bob
24
48
Number of weeks 24 ? 3 8
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Sam
72
Bob
24
48
How much pocket money does Bob get each week?
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Word Problems
Vani was given 4 to spend during recess. She
spent 90 cents on a chicken wing and 65 cents on
a bottle of mineral water. How much did she have
left?
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Typical Word Problem
Mrs Wong has 24 tarts. She packs all of them into
boxes. Each box holds 4 tarts. What is the total
number of boxes she used?
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Typical Word Problem
Mrs Wong has 26 tarts. She packs all of them into
boxes. Each box holds 4 tarts. What is the total
number of boxes she used?
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PSLE Item
Mrs Wong has 26 tarts. She packs all of them into
boxes. Each box can hold up to 4 tarts. Which of
the following cannot be the total number of boxes
she used?
(1) 5 (2) 7 (3) 8 (4) 10
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PSLE Items

• Expect pupils to realise that a situation differs
  from familiar ones and requires different
  strategies to solve

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PSLE Item
A box of greeting cards was shared equally among
a group of 35 pupils. 7 of them gave all their
cards to the rest of the pupils. As a result, the
rest of the pupils received 2 more cards each.
How many cards were there in the box at first?
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Strategies to Help Pupils
(1) Read the text
(2) Retell the story
(3) Pose questions based on the story
(4) Answer comprehension questions
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PSLE Item
A box of greeting cards was shared equally among
a group of 35 pupils. 7 of them gave all their
cards to the rest of the pupils. As a result, the
rest of the pupils received 2 more cards each.
How many cards were there in the box at first?
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PSLE Item
A box of greeting cards was shared equally among
a group of 35 pupils. 7 of them gave all their
cards to the rest of the pupils. As a result, the
rest of the pupils received 2 more cards each.
How many cards were there in the box at first?
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PSLE Item
Miss Tang went to a supermarket to buy exactly 44
apples for her class camp. The apples were priced
at 45 cents each or in bags of 5 at 2.00 per
bag. What was the smallest amount of money that
Miss Tang could have spent on the apples?  
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Modified Item
Miss Tang went to a supermarket to buy exactly 44
apples for her class camp. The apples were priced
at 45 cents each or in bags of 5 at 2.00 per
bag. What was the smallest amount of money that
Miss Tang could have spent on the apples?  
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Thinking Completely
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Using Knowledge
Each of the three cards shown is printed with a
different whole number. The smallest number is
23. When these numbers are added two at a time,
the sums are 61, 71 and 86. What is the largest
number on the cards?
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Using Knowledge
(1) 25
(2) 38
(3) 48
(4) 63
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Draw a Model
In a class, of the pupils are girls and of
the girls wear spectacles. If of the boys
wear spectacles, what fraction of the pupils wear
spectacles?
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Guess and Check
The figure is a square made up of four parts, A,
B, C and D. C and D are squares and each is ¼ of
the figure.
Which of the following two parts will add up to
form of the figure?
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Draw a Diagram Solve Part of the Problem
A toy-maker has a rectangular block of wood 30 cm
by 14 cm by 10 cm. He wants to cut as many 3-cm
cubes as possible. How many such cubes can he
cut?
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Visualization
A carpenter uses identical blocks to make low
stools. Each block is 44 cm long, 15 cm wide and
9 cm thick.
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He cuts the length of the block into three parts
A, B and C in the ratio 5 3 3.
He then nails B and C to A to make a stool such
that there is a gap between B and C. The stool is
shown on the right.
Find the width of the gap.
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He cuts the length of the block into three parts
A, B and C in the ratio 5 3 3.
He then nails B and C to A to make a stool such
that there is a gap between B and C. The stool is
shown on the right.
There are a number of ways the carpenter can
stack up to 10 completed stools one on top of
another. What is the lowest possible height of
the stack of 10 stools?
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Educating the Next Generation
How many cubes are there in this stack?
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Educating the Next Generation
Cubes of the same size are stacked in a corner of
a box as shown.
How many cubes are there?
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Discussion

• bhyeap_at_nie.edu.sg

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Thank You

• Association of Mathematics Educators
• Singapore Polytechnic