Maths Olympiad 61191x
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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
Singapore Mathematics Curriculum (1992, 2001)
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
Item Types in PSLE What is the value of 84 7 – 4 2? (1) 56 (2) 16 (3) 14 (4) 4
PSLE Items Computation must not be tedious
Item Types in PSLE A piece of wire is bent to form the rightangled triangle shown below. Find the area of the triangle. 16 cm
Answer: _________ cm2
20 cm
12 cm
PSLE Items Selecting data is now required
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
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
PSLE Items Concepts are tested alongside procedures
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?
Item Types in PSLE
(1) 24 cm2 (2) 40 cm2 (3) 48 cm2 (4) 56 cm2
PSLE Items Expects hands-on learning in the classroom
Singapore Mathematics Curriculum (1992, 2001)
More Than Computational Fluency What is the reading indicated on the weighing scale shown?
PSLE Items Practical skills are tested too
More Than Computational Fluency What is the reading indicated on the weighing scale shown?
More Than Computational Fluency The figure shows a line XY and three points R, S and T.
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 ing through point T.
Computation? What is the missing number in the box?
4 2 2 11 11 11
1 11
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
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
Singapore Mathematics Curriculum (1992, 2001)
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.
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 problemsolving proficiency.
The PSLE Format There are 20 short-answer tasks for 20%. They are all 1mark 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.
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.
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.
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.
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
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
Structured Questions How far was Town P from Town Q? How many minutes later than Lee did Chan start this journey?
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
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.
Sam
$72 Bob $24
$48 Number of weeks = 24 3 = 8
Sam
$72 Bob $24
$48 How much pocket money does Bob get each week?
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?
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?
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?
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
PSLE Items Expect pupils to realise that a situation differs from familiar ones and requires different strategies to solve
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?
Strategies to Help Pupils (1) Read the text (2) Retell the story (3) Pose questions based on the story (4) Answer comprehension questions
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?
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?
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?
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?
Thinking Completely
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?
Using Knowledge (1) 25 (2) 38 (3) 48
(4) 63
Draw a Model 1 3 In a class, of the pupils are girls and of the girls wear 6 5 spectacles. 3 4
If of the boys wear spectacles, what fraction of the pupils wear spectacles?
Guess and Check B A
D
C
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. 3 Which of the following two parts will add up to form 8 of
the figure?
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?
Visualization A carpenter uses identical blocks to make low stools. Each block is 44 cm long, 15 cm wide and 9 cm thick.
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.
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?
Educating the Next Generation How many cubes are there in this stack?
Educating the Next Generation Cubes of the same size are stacked in a corner of a box as shown.
How many cubes are there?
Discussion [email protected]
Thank You Association of Mathematics Educators Singapore Polytechnic
Singapore Mathematics Curriculum (1992, 2001)
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
Item Types in PSLE What is the value of 84 7 – 4 2? (1) 56 (2) 16 (3) 14 (4) 4
PSLE Items Computation must not be tedious
Item Types in PSLE A piece of wire is bent to form the rightangled triangle shown below. Find the area of the triangle. 16 cm
Answer: _________ cm2
20 cm
12 cm
PSLE Items Selecting data is now required
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
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
PSLE Items Concepts are tested alongside procedures
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?
Item Types in PSLE
(1) 24 cm2 (2) 40 cm2 (3) 48 cm2 (4) 56 cm2
PSLE Items Expects hands-on learning in the classroom
Singapore Mathematics Curriculum (1992, 2001)
More Than Computational Fluency What is the reading indicated on the weighing scale shown?
PSLE Items Practical skills are tested too
More Than Computational Fluency What is the reading indicated on the weighing scale shown?
More Than Computational Fluency The figure shows a line XY and three points R, S and T.
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 ing through point T.
Computation? What is the missing number in the box?
4 2 2 11 11 11
1 11
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
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
Singapore Mathematics Curriculum (1992, 2001)
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.
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 problemsolving proficiency.
The PSLE Format There are 20 short-answer tasks for 20%. They are all 1mark 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.
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.
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.
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.
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
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
Structured Questions How far was Town P from Town Q? How many minutes later than Lee did Chan start this journey?
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
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.
Sam
$72 Bob $24
$48 Number of weeks = 24 3 = 8
Sam
$72 Bob $24
$48 How much pocket money does Bob get each week?
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?
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?
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?
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
PSLE Items Expect pupils to realise that a situation differs from familiar ones and requires different strategies to solve
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?
Strategies to Help Pupils (1) Read the text (2) Retell the story (3) Pose questions based on the story (4) Answer comprehension questions
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?
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?
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?
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?
Thinking Completely
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?
Using Knowledge (1) 25 (2) 38 (3) 48
(4) 63
Draw a Model 1 3 In a class, of the pupils are girls and of the girls wear 6 5 spectacles. 3 4
If of the boys wear spectacles, what fraction of the pupils wear spectacles?
Guess and Check B A
D
C
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. 3 Which of the following two parts will add up to form 8 of
the figure?
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?
Visualization A carpenter uses identical blocks to make low stools. Each block is 44 cm long, 15 cm wide and 9 cm thick.
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.
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?
Educating the Next Generation How many cubes are there in this stack?
Educating the Next Generation Cubes of the same size are stacked in a corner of a box as shown.
How many cubes are there?
Discussion [email protected]
Thank You Association of Mathematics Educators Singapore Polytechnic
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