TUTORIAL – PROBLEM SOLVING

Work in groups of two or three

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Tutorial 1.1

Question 1: The King’s Elephant Problem.

Once upon a time the King of Siam asked his warriors to measure the weight of an elephant. However the weight of the elephant was too heavy to be measured by the biggest weighing machine available. Your task is to help the warriors to solve the problem.

Question 2: Abu Nawas’s Problem

In one of the stories in ‘One Thousand and One Night’, Abu Nawas would only allow any traveler to fetch water from the only well in the desert after solving the following problem.

The travelers were given a 3-litre container and a 5-litre container. They have to measure exactly four and seven litres of water using the above containers. You are required to help the travelers solve the problem.

Tutorial 1.2: Creative Problem Solving.

Identify the strategy that you used to solve the problem in Tutorial 1. Create another problem that can be solved using different strategy / strategies. Group Presentation.

Tutorial 2.1:

In groups of 2 – 3, discuss:o What is Numeracy?o Relate Numeracy to National Mathematics Curriculum.o As a teacher, how can you help your students to acquire this proficiency?

Present results of your group’s discussion to the whole class.

Tutorial 2.2:

Physical manipulative materials such as counters are commonly used to promote the learning of computation among primary school children. In recent years, the advancement in computer technology has resulted in the creation of virtual manipulative material that can be seen and manipulated on a computer screen.

Compare and contrast the effectiveness of using virtual manipulative materials and physical manipulative materials in the teaching and learning computation

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Tutorial 2.2:

Physical manipulative materials such as counters are commonly used to promote the learning of computation among primary school children. In recent years, the advancement in computer technology has resulted in the creation of virtual manipulative material that can be seen and manipulated on a computer screen.

Compare and contrast the effectiveness of using virtual manipulative materials and physical manipulative materials in the teaching and learning computation

Tutorial 2.3:

In groups of 2 – 3, discuss:

o Why and when do we need rough estimation?o What are the advantages and disadvantages of applying estimation in real

life?

Tutorial 3.1:

1. Discuss and identify all measurable properties of the following objects:

(a) A bulletin board,(b) A table,(c) An extension cord,(d) A bowl.

2. Suggest a suitable standard unit for each of the following measurements:

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Tutorial 3.1:

1. Discuss and identify all measurable properties of the following objects:

(a) A bulletin board,(b) A table,(c) An extension cord,(d) A bowl.

2. Suggest a suitable standard unit for each of the following measurements:

Tutorial 3.2:

1. What is the relationship between gram and liter?

2. What are the advantages of the metric units of measurement?

3. Given that 1 mile = 1.6 km. Use a calculator to verify that a hectare is about 2.5 acres. Show all the steps in your calculation.

4. Cullianan diamond, the largest diamond found in 1906 at the Premier Mine in S. Africa, weighed 3106 carats. Estimate the weight of Cullianan diamond in pounds. (Given that: 1 carat = 200 mg, 2.2kg = 1 lb)

Tutorial 4.1:

In small groups, discuss on the following:

Words, vocabulary, and language used to describe 2-D shapes and 3-D solids.

Similarities and differences between:

(a) Cube, cuboid, cylinder and prism,(b) Prism and pyramid,(c) Pyramid and cone.

The relationships between:(a) Cube and cuboid,

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Tutorial 4.1:

In small groups, discuss on the following:

Words, vocabulary, and language used to describe 2-D shapes and 3-D solids.

Similarities and differences between:

(a) Cube, cuboid, cylinder and prism,(b) Prism and pyramid,(c) Pyramid and cone.

The relationships between:(a) Cube and cuboid,

Tutorial 4.2:

Paper-folding activities

Construct the following 2-D shapes by folding an A4 paper.

A scalene triangle, An isosceles triangle, An equilateral triangle, A rectangle, A square, A kite, A rhombus, A parallelogram, A trapezium, A regular pentagon, A regular hexagon.

Tutorial 4.3: Area & Perimeter of 2-D Shapes

1 unit

1 unit

This is a unit square tile. The length of each side of this tile is 1 unit. The area of this tile is 1 unit2.

Task A: Below are ALL the possible shapes constructed by joining 4 units of square tiles.

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Tutorial 4.3: Area & Perimeter of 2-D Shapes

1 unit

1 unit

This is a unit square tile. The length of each side of this tile is 1 unit. The area of this tile is 1 unit2.

Task A: Below are ALL the possible shapes constructed by joining 4 units of square tiles.

Tutorial 5.1:

1. When flipping a coin, there are only two possible outcomes: a ‘heads’ or a ‘tails’. So, we say that the probability of getting a ‘heads’ is one out of two, that is ½. When you sit for the examination at the end of this semester, there say that the probability of you passing the examination is also ½ ? If yes, why? If no, why not?

2. A person in a small foreign town applies for a marriage permit at age 18. To obtain the permit, the person is handed six strings, as shown in Figure (a). On one side (top or bottom) the ends are picked randomly, two at a time, and tied, forming three separate knots. The same procedure is then repeated for the other set of string ends, forming three more knots, as in Figure (b) . If the strings form one closed ring, as in Figure (c), the person obtains the permit.

Source: Billstein, R., Libeskind, S. & Lott, J. W. (1990). A Problem-solving approach to mathematics for elementary school teachers. 4th . Redwood City, CA: Benjamin/ Cummings. P. 402.

JAWAPAN :

Tutorial 1.1

1.1 Weighing an elephant

1. Load the elephant onto a boat large enough to carry it.

2. The boat will sink slightly, and you mark the level of the water on the

side of the boat.

3. Then you offload the elephant and fill the boat with bags until the boat

sinks to the level marked.

4. The bags can be individually weighed using beam scales and the weight

of the elephant is the sum of the weight of the bags.

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Tutorial 5.1:

1. When flipping a coin, there are only two possible outcomes: a ‘heads’ or a ‘tails’. So, we say that the probability of getting a ‘heads’ is one out of two, that is ½. When you sit for the examination at the end of this semester, there say that the probability of you passing the examination is also ½ ? If yes, why? If no, why not?

2. A person in a small foreign town applies for a marriage permit at age 18. To obtain the permit, the person is handed six strings, as shown in Figure (a). On one side (top or bottom) the ends are picked randomly, two at a time, and tied, forming three separate knots. The same procedure is then repeated for the other set of string ends, forming three more knots, as in Figure (b) . If the strings form one closed ring, as in Figure (c), the person obtains the permit.

1.2 Abu Nawas’s Problem

1.2.1 measure exactly four litres of water:

1. Fill 3ltr can, pour into 5ltr can.

2. Fill 3ltr can and pour into 5ltr can again, this would leave 1ltr in 3ltr can.

3. Empty 5ltr can and pour the 1ltr into it.

4. Fill 3ltr can again and pour into 5ltr can.... making 4ltrs!

1.2.2 measure exactly 7 liters of water:

1. Fill the 5 liter container, then use that 5 liters to fill the 3 liter container.

2. Dump out the 3 liter container.

3. What was left in the 5 liters container is 2 liters. Pour this into the 3 liter

container.

4. Fill the 5 liter container again.

5. Now you have 7 liters : 5 in the large container and 2 in the smaller one.

Tutorial 2.1:

2.1.1 What is Numeracy?

1. Numeracy is the effective use of mathematics to meet the general

demands of life at school and at home, in paid work, and for

participation in community and civic life. (MCEETYA Benchmarking

Task Force 1997:4)

2. From Wikipedia, the free encyclopedia: Numeracy is the ability to reason

with numbers and other mathematical concepts. A

numerically literate person can manage and respond to the

mathematical demands of life. Aspects of numeracy include number

sense, operation sense, computation, measurement, geometry,

probability and statistics.

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2.1.2 Relation Numeracy to National Mathematics Curriculum.

The national mathematics curriculum will be the basis of planning, teaching,

and assessment of school mathematics, and be useful for, and useable by,

experienced and less experienced teachers of K–12 mathematics.

2.1.3 As a teacher, how can you help your students to acquire this

proficiency?

In planning the teaching of a topic, a mathematics teacher should consider

how and when the following factors could be effectively used

a) Activities that give meaningful learning experiences

b) The use of mathematics in real life situations

c) The effective use of problem solving skills

d) Instilling of Malaysian societal values

e) Imparting and appreciating the elements of history of mathematics

To achieve the aims of the curriculum, several factors are given priority.

a. Students active involvement in the learning process is emphasised.

b. The learning activities, the types of questions asked and the guides

given to students should be geared towards upgrading the ability to

think and assisting students learning through real life experiences.

c. The simulated experiences should involve activities that encourage

inquiry and provide opportunities for students to reach certain

conclusions or solve problem independently.

d. These experiences could also include the use of mathematics in

situations that are meaningful to students.

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Tutorial 2.2

2.2.1 Compare and contrast the effectiveness of using virtual manipulative

materials and physical manipulative materials in the teaching and learning

computation among primary school children.

In math classrooms today, teachers are using manipulatives to help students

learn mathematics. Manipulative materials are any concrete objects that allow

students to explore an idea in an active, hands-on approach. Manipulatives

can be almost anything – base ten blocks, tangrams, spinners, rulers, fraction

barsshapes, algebra tiles, geometric plane, solids figures, geoboardsor even

paper that is cut or folded.

The power of using manipulatives is that they let the student connect

mathematical ideas and symbols to physical objects, thus promoting better 

understanding. With many experiences building and representing

using manipulatives, students can deepen their understanding of abstract

math concepts.

Manipulatives can also be tools to help students solve problems. By using

physical models to represent their thinking, they can move and adapt the

materials as they explore possible solutions to problems. In real life, many

people use models to help solve problems, such as an architect who might

construct a model of a building or an engineer who might build a

prototype of a piece of equipment.

2.2.2 Identify appropriate and inappropriate uses of calculator and computer in

the teaching of computation among primary school children

Appropriate

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Inappropriate

No ideas

2.3.1 Why and when do we need rough estimation

Estimation theory is a branch of statistics and signal processing that deals with

estimating the values of parameters based on measured/empirical data that has a

random component. The parameters describe an underlying physical setting in such

a way that their value affects the distribution of the measured data.

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An estimator attempts to approximate the unknown parameters using the

measurements.

For example, it is desired to estimate the proportion of a population of voters who will

vote for a particular candidate. That proportion is the unobservable parameter; the

estimate is based on a small random sample of voters.

In estimation theory, it is assumed the measured data is random with probability

distribution dependent on the parameters of interest. For example, in electrical

communication theory, the measurements which contain information regarding the

parameters of interest are often associated with a noisy signal. Without randomness,

or noise, the problem would be deterministic and estimation would not be needed.

2.3.2 What are the advantages and disadvantages of applying estimation in

real life?

Advantages

1. Estimates (i.e. budget, schedule, etc.) become more realistic as work

progresses, because important issues are discovered earlier. 

2. It is more able to cope with the (nearly inevitable) changes that software

development generally entails. 

3. Software engineers (who can get restless with protracted design processes)

can get their hands in and start working on a project earlier. 

Disadvantages

1. Highly customized limiting re-usability 

2. Applied differently for each application 3. Risk of not meeting budget or

schedule 4. Risk of not meeting budget or schedule

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Tutorial 3.1:

1. Discuss and identify all measurable properties of the following objects:

(a) A bulletin board,(b) A table,(c) An extension cord,(d) A bowl.

Answers

( a.) A bulletin board (perimeter ,area ,height ,width)

(b). A table (width ,metre ,cm ,height ,area)

( c ) An extension cord ( metre )

( d ) A bowl ( perimeter, weight, area)

1. ( a) a bulletin board

by using ruler

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(b) A table ( c ) An extension cord

By measurable tape By measurable tape

 

3.2

3.2.1What is the relationship between gram and liter?

One liter is the volume of a cube with 10cm sides (refers to illustration below).

The spelling of the word used by the International Bureau of Weights and Measures is "litre" and this is also the usual one in most English-speaking countries, but in American English the spelling is "liter", being endorsed by the United States.

Litre

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A gram and a liter are units of two separate measurements. Grams are a measure of mass which most widely used unit of measurement for non-liquid ingredients, whereas liters are a measure of volume, so a liter of aluminum will weigh fewer grams than a liter of gold.

A liter of water has a mass almost exactly equal to 1000 gram of water. An early definition of the 1000 gram was set as the mass of one liter of water. Because volume changes with temperature and pressure, and pressure uses units of mass, the definition of 1000 gram was changed. At standard pressure, one liter of water has a mass of 999.975 gram at 4°C, and 997 gram at 25°C

There is an indirect relationship between them, and that is that any material's net mass will increase with it's volume. The units themselves though, are not related.

3.2.2What are the advantages of the metric units of measurement?

Metric is one group of units used to measure items such as length, temperature, time and weight is known as the metric system. Some units that come from the metric system you may have heard of such as the meter, the kilogram and etc.

Several advantages/benefits come from the metric Unit of Measurements (UoM) adoption, 3 of them are listed as below:

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1) The metric system has been adopted by most major countries around the world. By the mid-1970s, most countries had converted the metric system or had plans to do so. Hence metric system adoption is able to standardize the measurement around the globe actually. When it comes to measurement, the United States is the only major country who has not adopted the metric system.

2) Metric system is a logical and exact system as during its invention, scientist has designed the metric system to fit their needs.

3) The metric system was designed to be simple. When making measurements of all kinds, it is only necessary to know a few metric units. There are only 7 base units in metric UoM if compared to 20 base units found in inch-pound system of measurement. The metric system also follows the decimal number system, each metric unit increases or decreases in size by 10 (example: 1 meter = 10 decimeters; 1 decimeter = 10 centimeters and etc).

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3.2.3Given that 1 mile = 1.6km. Use a calculator to verify that a hectare is about 2.5 acres. Show all the steps in your calculation.

The hectare is a metric unit of area defined as 10,000 square metres (100 m by 100 m), and primarily used in the measurement of land.

The acre is a unit of area in a number of different systems, including the imperial and U.S. customary systems. The most commonly used acres today are the international acre and, in the United States, the survey acre. The most common use of the acre is to measure tracts of land. One international acre is equal to 4046.8564224 square metres.

Basic of UoM:

1mile = 1.6km

1mile² = 1km²

1km² = 100hectare

2.56km²=632.56 acres

Relation between km² and acre

2.56km²=632.56 acres

1km²=632.56÷2.56

=247.09km²

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Unit conversion

1km²=100hectare

=247.09acres

100hectare=247.09acres

1heactare = 247.09÷100

=2.47acres (proven)

Cullianan diamond the largest diamond found in 1906 at the Premier Mine in S.Africa, weighed 3106 carats. Estimate the weight of Cullianan diamond in pounds.(Given that: 1 carat = 200mg, 2.2kg = 1lb)

Carat to mg conversion:

1carat = 200mg

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3106carat = 200 x 3102

= 621200 mg

Mg to Kg conversion:

621200 mg ÷ 1000 = 621.20g

= 0.62120kg

Kg to Pound conversion:

0.6212kg ÷ 2.2kg = 0.282lb.

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SUGGESTED ANSWER.

1) Words, vocabulary, and language used to describe 2-D shapes and 3-D

solids.

2-D shapes.

These shapes are flat and can only be drawn on paper.

They have two dimensions – length and width.

They are sometimes called plane shapes.

3-D solids.

These shapes are solid or hollow.

They have three dimensions – length, width and height.

2) Similarities and differences between:

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Tutorial 4.1:

In small groups, discuss on the following:

Words, vocabulary, and language used to describe 2-D shapes and 3-D solids.

Similarities and differences between:

(d) Cube, cuboid, cylinder and prism,(e) Prism and pyramid,(f) Pyramid and cone.

The relationships between:(c) Cube and cuboid,(d) Cylinder and prism.

Definition of polygon and classifications of polygons according to number of sides.

Properties of n-sided polygons for 3 ≤ n ≤ 10.

Cube, cuboid, cylinder and prism,

Shapes Similarities Differences

Cube A three-dimensional shape which has

6 square faces all the same size.

6 square equal size flat

faces.

Cuboid A three-dimensional shape which has

6 rectangular faces.

6 rectangular flat faces.

Cylinder A three-dimensional shape with

circular ends of equal size.

2 circular ends of equal

size.

Prism A three dimensional shape that has

the same cross-section all along its

length.

The same cross-section

all along its length.

3) The relationships between:

(a) Cube and cuboid,

CUBE CUBOID

(a) The dimensions of a cube

are length, breadth and

height.

(b) All the edges of a cube are

equal.

(c) Volume of a cube

= length x breadth x height

(a) The dimensions of a cuboid

are length, breadth and

height.

(b) Volume of a cube

= length x breadth x height

(b) Cylinder and prism.

CYLINDER PRISM

(a) A cylinder has 3 edges and

no corners.

(a) A shape that has the same

cross-section all along its

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(b) It has 2 circular ends of

equal size.

length.

(b) No curved sides.

4) Definition of polygon and classifications of polygons according to

number of sides.

Definition of polygon.

A polygon is a two-dimensional figure consisting of at least 3 vertices

(points) and at least 3 sides (straight segments) such that:

1) each vertex is the endpoint of exactly 2 sides

2) each pair of vertices are the endpoints of exactly one side

3) if 2 sides intersect then their intersection is a vertex.

Is it a Polygon?

Polygons are 2-dimensional shapes. They are made of straight lines,

and the shape is "closed" (all the lines connect up).

Polygon

(straight sides)

Not a Polygon

(has a curve)

Not a Polygon

(open, not closed)

Polygon comes from Greek. Poly- means "many" and -gon means

"angle".

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Types of Polygons

Simple or Complex

A simple polygon has only one boundary, and it doesn't cross over

itself. A complex polygon intersects itself!

Simple Polygon

(this one's a Pentagon)

Complex Polygon

(also a Pentagon)

Concave or Convex

A convex polygon has no angles pointing inwards. More precisely, no

internal angles can be more than 180°.

If there are any internal angles greater than 180° then it is concave.

(Think: concave has a "cave" in it)

Convex Concave

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Regular or Irregular

If all angles are equal and all sides are equal, then it is regular,

otherwise it is irregular

Regular Irregular

More Examples

Complex Polygon

(a "star polygon", in

this case, a pentagram)

Concave Octagon Irregular Hexagon

Classifications of Polygons

These terms classify a polygon by the number of sides. In fact, the common names of polygons with 3 to 10 sides are:

#sides name examples

3 triangle

4quadrilatera

l

5 pentagon

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6 hexagon

7 heptagon

8 octagon

9 nonagon

10 decagon

You may be surprised that an object

like this is called a pentagon:

We are accustomed to seeing

pentagons that look like this:

Both these figures are polygons with 5 sides according to our definition. What

is peculiar about the first figure is it "caves in" at the top. Such a polygon is

called concave. In fact, a definition of "concave" is this:

DEFINITION: A polygon is concave if there are two points somewhere inside it

for which a segment with these as its endpoints cuts at least 2 of the sides of

the polygon.

For

example,

is concave, because the segment with points

A and B as its endpoints cuts two sides:

A polygon that is not concave is called convex. Some important theorems that

apply only to convex polygons will be stated in the next lesson.

Regular Polygons

A special kind of convex polygon is called a regular polygon:

DEFINITION: A regular polygon is a convex polygon in which all sides are congruent.

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Another way to define a regular polygon is to state that its sides are congruent and its angles are congruent. If a polygon is not regular, then it is called irregular.

Examples:

Concave Polygons:

Convex, Irregular Polygons:

Regular Polygons:

Properties of n-sided polygons for 3 ≤ n ≤ 10.

NAME SIDES(n) SHAPE

Triangle(or Trigon)

3

Quadrilateral(or Tetragon)

4

Pentagon 5

Hexagon 6

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Heptagon(or Septagon)

7

Octagon 8

Polygon Names

Generally accepted names

Sides Name

n N-gon

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

Names of Polygons

Name Sides Angles

Triangle 3 3

Quadrilateral 4 4

Pentagon 5 5

Hexagon 6 6

Heptagon 7 7

Octagon 8 8

Nonagon 9 9

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5.1 probability / kebarangkalian

1. No, passing the examination is depends on our own preparation before exam. I must work smart to pass this coming exams

2a. Probability of the person getting the marriage permit is lower than the probability of not getting the permit.

2b. A B C D E F

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Tutorial 5.1:

3. When flipping a coin, there are only two possible outcomes: a ‘heads’ or a ‘tails’. So, we say that the probability of getting a ‘heads’ is one out of two, that is ½. When you sit for the examination at the end of this semester, there say that the probability of you passing the examination is also ½ ? If yes, why? If no, why not?

4. A person in a small foreign town applies for a marriage permit at age 18. To obtain the permit, the person is handed six strings, as shown in Figure (a). On one side (top or bottom) the ends are picked randomly, two at a time, and tied, forming three separate knots. The same procedure is then repeated for the other set of string ends, forming three more knots, as in Figure (b) . If the strings form one closed ring, as in Figure (c), the person obtains the permit.

(d) Do you think the probability of the person getting the marriage permit is higher or lower that the probability of not getting the permit?

(e) Carry out an experiment to check your guess in (a).

(f) Determine the theoretical probability that the marriage permit will be obtained on the first try.

A B C D E F

G LH I J K

A √ √ √ √ √B √ √ √ √ √C √ √ √ √ √D √ √ √ √ √E √ √ √ √ √F √ √ √ √ √

G H I J K LG √ √ √ √ √H √ √ √ √ √I √ √ √ √ √J √ √ √ √ √K √ √ √ √ √L √ √ √ √ √

2c. formula getting marriage permit isP( A )=

n( A )n (S )

= 160

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