nota mte3111 (english version)

7/31/2019 NOTA MTE3111 (ENGLISH VERSION)

1/12

Compilation of Notes of MTE3111

By Cg Mohd Ridzuan al-Kindy (IPG KDRI)

MTE3111 TEACHING OF GEOMETRY,

MEASUREMENT AND DATA HANDLING

TOPIC 1: GEOMETRY

Spatial Sense

Spatial is spatial perception or spatial visualization,

helps students understand the relationship betweenobjects and their location in three dimensional

worlds. (Kennedy and Tipps, 2006)

Geometric Thinking

(a) Visual spatial thinking

Happened on the right hemisphere of the brain

that associate with literature

Occur unconsciously without being aware of it

Simultaneously processing.

(b) Verbal logical thinking

Lies on the left hemisphere of the brain that is of Continuous processing and always aware of it

Operate sequentially and logically and to

language or symbol and numbers.

Van Hiele, five levels of geometric thought:

1. Visualization recognized figures by looking at

their appearance.2. Analysis classify or group according depending

on the characteristics of shapes or figures but they

cannot visualize the interrelationship between them.

3. Informal Deduction established or sees

interrelationships between figures.

4. Deduction mental thinking and geometric thinking

developed significantly. They can understand the

significant of deduction, the role of postulates,

theorem and proofs. They are able to write proof

with understanding.

5. Rigor make abstract deduction and understand

how to work in axiomatic system even non-Euclidian

geometry can be understood at this level.

Geometric System

(a) Euclidean Geometry the geometry of shape and

objects in plane (2D) or in space (3D). Describe the

properties of objects in plane (2D) or in space (3D).

(b) Coordinate Geometry about location shapes on

coordinate or grid systems. Describe location of

object on planed coordinate of vertical and

horizontal axis for 2D shapes or positioning ofobjects on grid systems for three dimensional

spaces.

(c) Transformation Geometry about geometry in

motion. It describes the movement of shapes or

object in a plane or in space.

(d) Topological Geometry describes the location of

objects and their relation in space or recognition of

objects in the environment.


2/12



Geometry in Mathematics KBSR

Teaching Shapes and Space

Teaching 3D Shapes

Teaching in Pre School (Level 1 & 2):

Early geometric sense:

o Identify shapes (surface area) and the relevant

solids (explore)

o Match and label each shape and solids

(discover)

o Identify similarities and differences between

shape and solidso Use correct vocabulary and language

Teaching in Year 1 Primary (Level 1, 2 & 3):

Name, labelling and use correct vocabulary for each

solid 3D shape

Describe features or parts of solid shapes including

classify and grouping shapes according to

similarities and differences.

Able to assemble and explaining types of shapes

used to build models and relate models to solid

shapes in real life.


3/12



Teaching in Year 2

Understanding and using vocabulary to name and

label two dimensional shapes.

Describing and classifying two dimensional shapes

Building models using three dimensional and two

dimensional shapes

Understanding and using vocabulary to name and

label three dimensional shapes

Describing and classifying three dimensional shapes

Teaching in Year 3

Understanding and using vocabulary related to two

and three dimensional shapes

Describing and classifying two and three

dimensional shapes

Building two and three dimensional shapes

Understand and recognising lines of symmetry

Sketching lines of symmetry.

Teaching in Year 4

Identify two dimensional shapes

Drawing geometrical drawing of two dimensional

shapes.

Identify perimeter

Calculation on perimeter of various two dimensional

shapes and combined two dimensional shapes.

Teaching 2D Shapes

Suggested teaching and learning activities:

o Contextual learning children looking around

and observing the environment and describe in

words what they have seen.

o Exploring and experimenting shapes (visual

images) in order to gain insight into properties

and its uses

o Analysing shape informally, observing size and

position in order to make inferences then to

refine and extended out knowledge that develop

from various learning activities

Introduction of three-dimensional shape must be

earlier or before the teaching of shapes.

Vocabulary and Classification of 2D Shapes

Triangle

Equilateral triangle three equal

sides and three equal angle

Isosceles triangle 2 equal sides

and 2 equal angle

Scalene triangle no equal sides

and no equal angle

Right-Angle Triangle One angle

is 90

Acute angled triangle All three

angles are acute (< 90)

Obtuse angled triangle One

angles is obtuse (> 90)

Quadrilaterals

Curved Shapes


4/12



Key Issues in Teaching Shapes and Spaces

Young students can defineshapes, but then not use

their definitions when asked to point out examples of

those shapes.

Young students discriminate some characteristics

of different shapes, often viewing these shapes

conceptually in terms of the paths and the motions

used to construct the shapes. Student misconceptions in geometry lead to a

depressing picture of their geometric understanding

(Clements and Battista, 1992). Some examples are:

o A square is not a square if the base is not

horizontal.

o Every shape with four sides is a square.

o A figure can be a triangle only if it is equilateral.

o The angle sum of a quadrilateral is the same as

its area.

o The area of a quadrilateral can be obtained by

transforming it into a rectangle with the sameperimeter.

Students have a difficult time communicating visual

information, especially if the task is to communicate

a 3-Denvironment (e.g., a building made from small

blocks) via 2-D tools (e.g., paper and pencil) or the

reverse.

Applications of Geometry in Technoogy

A computer environment can generate multiple

representations of ashape that help students

generalize their conceptual image of that shape inany size or orientation (Shelton, 1985). E.g. :

Geometers Sketchpad

TOPIC 2: MEASUREMENT

Basic Principle of Measurement

Comparison principle

o Comparing and ordering of objects by a specific

attribute with suitable vocabulary (short, shorter,

tall, taller, etc.)

Transitivity principleo Comparing and ordering of three or more objects

using appropriate language (tallest, shortest,

lightest etc.)

Conservation principle

o States that the length of an object does not

change even when the position or the orientation

of the object is changed.

Measuring principle

o Measurement involves stating how many of a

given unit match the attribute (e.g. length,

volume, mass) of an object.

Teaching of Length

The length of an object refers to the number of

standard unit which can be laid in a straight line

along or beside the object.

Teaching Length in Primary School:

Use vocabulary related to length

Compare length of object by directcomparison

Measure and compare length using uniformnon-standard units

Measure and compare length using standardunits

Measure, writing and estimate length

Conversion of units of length

Operation of units of length

Daily life problem


5/12



Standard and non-standard units

Standard Non-Standard

- any fixed length that has

been accepted as a

standard internationally

(SI)

- any arbitrary length

used as a unit

- E.g.: yards, miles, feet,

inches

metres and

kilometres, etc.

- E.g.: body parts such

as span, foot,

pace and arm

length

objects such as

pen, paper clip,

etc.

- Measure using specific

apparatus (with scale)

such ruler, tape, etc.

E.g.: using ruler tomeasure the length of

pencil

- Measure using other

non-specific object

(without scale)

E.g.: using eraser tomeasure the length

of pencil

Conversion of units

Involve metric unit of length:

Conversion of unit:

Area and Perimeter

Area

o Amount of surface enclosed in a plane.

Perimeter

o Distance all the way round its edges.

Teaching of Volume

Volume is a measure of the amount of space inside

a three-dimensional region, or the amount of space

occupied by a three-dimensional object.

Measured in:

o SI unit - cubic centimetres (cm) or cubic metres

(m).

o The Imperial system - cubic feet (ft).

One cubic centimetre (cm3) is the measure of a

cube having an edge with a length of 1 cm.

Liquid capacity / Volume of Liquid

Quantity of liquid that fills up a container.

Standard and non-standard unitsStandard Non-Standard

- any fixed volume that

has been accepted as

a standard

internationally (SI)

- any arbitrary volume

used as a unit

- E.g.:

Millilitre, litre

- E.g.:

A cup, jug, bottle

Other containers

- Measure using

specific apparatus

(with scale) such ruler,tape, etc.

E.g.: using beaker to

measure water


non-specific object

(without scale)E.g.: using a jug to

measure water

Half of jug

Volume Displacement

Displacement occurs when an object is immersed in

a fluid, pushing it out of the way and taking its place.

An object that sinks displaces an amount of fluid

equal to the object's volume (Archimedes principle)


6/12



Can be used to measure the volume of a solid

object, even if its form is not regular.

Teaching of Mass and Weight

The measure of the amount of matter in an object

whereas weight is the gravitational force acting on

that mass.

It is normal to refer weighing of an object as a

process to find its mass.

Standard and non-standard units

Standard Non-Standard

- any fixed mass /

weight that has been

accepted as a

standard

internationally (SI)

- any arbitrary mass /

weight used as a unit

- E.g.:

Kilogram, gram

Ounce,

- E.g.:

Marbles, battery

- Measure using

specific apparatus

(with scale) such

weighing scale.

E.g.: using weighing

scale to measure the

mass of watermelon

The mass of

watermelon is 3 kg.


non-specific object

(without scale)

E.g.: using a marbles

to measure the mass

of bottle

The mass of bottle is 7

marbles mass.

Teaching of Time

Major skills in measurement of time:

Development of measurement of time:

o Time of the Day start learning about time by

telling time of the day, i.e. day time and night. It

uses phrase that common into their everyday life.

o Telling Time Introduce to clock face clockwise direction

Introduce the concept of minute hand and

hour hand.

Relate to time of the day

o Time duration difficult to teach

Elapsed time for:

eating (fried rice, pizza, donut)

running around the field (and other

distance)

sleep

Longer times:

a baby to be born

o Days of the Week

o Months of the Year

o Relationship between Units of Time

60 seconds = 1 minutes

60 minutes = 1 hour

24 hours = 1 day

7 days = 1 week30 / 31 days = 1 month

12 months = 1 year

10 years = 1 decade

10 decades = 1 century

o Operation involving Units of Time

o Problem solving

to tell the time and events of theday

to name the days of the week

to name the months of the year

to read and write the time


7/12



Hour system

Teaching of Money

Skill development:

Mental Computation of Money

Estimation and mental computations on money can

help pupils:

o Save time doing long calculationso Judge the reasonableness of prices of items on

sale

o Solve problems when exact answers are not

required

Integrated Learning in Teaching Money

Responsibility

Family values and attitudes

Decision-making

Comparison-shopping

Setting goals and priorities Managing money outside the home.

Identiying and recognizing the valuesrepresented by the coins and notes.

Using different denomionations to represent the

values of money

Converting between ringgit and sen

Performing basic arithmetic operations involvingmoney

Applying their knowledge to solve daily

problems involving money.


8/12



Using Coins to Model Decimal (Sen)

Recording amounts in Ringgit and sen does involve

decimal fractions, but care must be taken on how

the children see the connection between the sen

and the fractional part of a decimal number.

E.g.: children do not readily relate RM 75.25 to RM

75 and 25 hundredths of a Ringgit or 10 sen to one-tenth of a Ringgit.

If money is used as a model for decimals, children

need to think of 10 sen and 1 sen as fractional parts

of a Ringgit.

RM 1.00 = 100 sen

RM 0.75 = 75 sen

Key issues in teaching measurement

Young children lack a basic understanding of the

unit of measure concept.

When trying to understand initialmeasurementconcepts, students need extensive experiences with

several fundamental ideas prior to introduction to the

use of rulers and measurement formulas.

Number assignment: Students need to understand

that the measurement process is the assignment of a

number to an attribute of an object (e.g., the length of

an object is a number of inches).

Comparison: Students need to compare objects on

the basis of a designated attribute without using

numbers (e.g., given two pencils, which is longer?).

Use of a unit and iteration: Students need tounderstand and use the designation of a special unit

which is assigned the number one, then used in an

iterative process to assign numbers to other objects

(e.g., if length of a pencil is five paper clips, then the

unit is a paper clip and five paper clips can be laid

end-to end to cover the pencil).

Additivity property: Students need to understand

that the measurement of the join of two objects is

mirrored by the sum of the two numbers assigned

to each object (e.g., two pencils of length 3 inches

and 4 inches, respectively, laid end to end will have alength of 3+4=7 inches)

The manipulative tools used to help teach number

concepts and operations are inexorably intertwined

with the ideas of measurement.

The improved understanding of measurement

concepts is positively correlated with improvement

in computational skills

Students are fluent with some of the simple

measurement concepts and skills they will

encounter outside of the class, but have great

difficulty with other measurement concepts and skills

(e.g., perimeter, area, and volume)

Students initially develop and then depend on

physical techniques for determiningvolumes of

objects that can lead to errors in other situations.

o E.g.: students often calculate the volume of a box by

counting the number of cubes involved. When this

approach is used on a picture of a box, students tend

to count only the cubes that are visible.

The vocabularyassociated with measurementactivities is difficult because the terms are either

entirely new (e.g., perimeter, area, inch) or may have

totally different meanings in an everyday context

(e.g., volume, yard).

Measurement of Time

Some aspects of time measurement which make it

difficult to learn among your children. Its because:

o Time is an abstract concept

o Time is measured using a mixture of base 12 and

base 60 systems, and when extended to days,months and years, it uses base 4, 7, 365 and 28,

29, 30 and 31 systems

o Time is measured indirectly - the movement of

the sun, hands on a clock face, digits changing in

a display, changing seasons, etc.

o Clocks come in all sorts of styles and designs -

some with all 12 numerals, (some Roman

numerals), others with only 12, 3, 6 and 9

numerals, and still others with no numerals at all.


9/12



TOPIC 3: DATA HANDLING

Data handling deals with the processes involved in

selection, collecting, organising, recording,

summarising, describing and representing data for

ease of interpretation and communication.

Data that we get and use may be discrete or

continuous depending on whether we arequantifying by counting or measurement.

Teaching of Data Handling

Collecting and organizing data

Appropriate methods for primary pupils is

interpreting and constructing simple tables, charts

and diagrams that arecommonly used in everydaylife to display information.

Two main process in collecting:o combinatorial counting (to determine all the

possible outcomes)

o tallying (to organise the data under the

categories)

Data collected can be organise using:

1. Table

o Simple table

o Regular table the matrix style table where

there are more than two columns (more than

column of data).

2. Charts less regular in terms of rows and

columns. They attempt to display information

more visually, to relate the display to what

actually occurs.

o The strip map

o Branch map - combination of strip maps,

involving branching as in a tree.

3. Diagrams visual ways to represent membership

in different sets and subsets.

o Venn diagram

o Carroll diagram

Displaying Data

Types of Graph:

o Bar Graph facilitate comparisons of quantities.

Bar graphs can be vertical as well as horizontal.

They can also be the forms of blocks, or bar

lines.

understanding what data is

collecting data from printedmaterials

classify, sort and analyse data

organising data in a table, chart orgraph

carrying out simple surveys tocollect data


10/12



o Picture Graph

Can also facilitate comparisons of quantities

just like bar graphs.

Can easily be updated.

Also called pictographs and isotypes.

o Line Graph

Can be used for comparisons and for

expressing allocations of resources.

It seems particularly useful for communicating

trends.

o Circle Graph

Also known as pie charts.

Can be used to picture the totality of a

quantity.

To indicate how portions of the totality are

allocated.

o Scatter Graph

It similar to line graphs which show the

relationship between two different sets of

data.

The scatter graph is made for data which is

not in sequence (in terms of the horizontal

axis) and is unsuitable for a line graph.

Constructing Graph

Pictograph1. Draw a horizontal or a vertical line as a baseline.

2. Write the names of the items that you have.

3. Put a symbol to represent the number of items

you have in each category.

4. Put in the key to represent the quantity of items.

(Means: 1 symbol = ? items).

5. Then finally, give a title to the graph.

Vertical Bar Graph:

1. Draw vertical and horizontal axes. Give them

names.2. Determine the correct interval to be marked on

the vertical axis.

3. Write the name of the items below the

horizontal axis.

4. Draw the bars vertically according to the

quantity given for each item. Then colour the

bars.

5. Lastly, give a proper title to for the graph.

Horizontal bar graph:

1. Draw vertical and horizontal axes. Give them

names.

2. Determine the correct interval to be marked on

the horizontal axis.

3. Write the names of items on the left of the

vertical axis.

4. Draw the bars horizontally according to the

quantity given for each item. Then colour the

bars.

5. Lastly, give a proper title to for the graph.


11/12



Interpreting data

Data analysis and interpretation is the process of

assigning meaning to the collected information and

determining the conclusions, significance, and

implications of the findings.

Interpretation of Pictograph

The questions above will lead your students to

understand that pictograph :

o What is the title of the pictograph?

o What picture is being used here?

o What does the key mean?

o How many people are involved in the data?

o Who has the most basketballs?

o Who has the least basketballs?

o If one basketball represents 2 balls, how many

balls are there altogether?

The data in that pictograph shows the number ofbasketballs each person has. It tells us that Sally

has 3 balls, Ken has 2 balls, Kamal has 1 ball and

lastly, Ben has 4 balls.

This means that one picture can represent one or

more quantities.

Interpretation of Bar Graph

Let us check in detail the information on it.

o Title of bar graph: Curry Puffs Soldo Vertical axis on the left: Shows the number of

curry puffs sold.

o Markings on the vertical axis: Shows the scales

in a specific range. The interval is 5 in this case.

o Horizontal axis: Shows the days Monday,

Tuesday, Wednesday

o The bars: Show the number of curry puffs sold on

Monday, Tuesday and Wednesday.

Teaching Average

As the middle point of a set of numbers.

Finding the average helps do calculations and also

makes it possible to compare sets of numbers.

Averages supply a framework with which to describe

what happens.

Understand the Concept and Deriving Formulae of

Average

An understanding of average can be developed

through using concrete materials and visual

manipulation (Rubenstein, 1989).

E.g.: Interlocking cubes,

Describe the meaning of average.

State the average of two or threequantities.

Determine the formula for average.

Calculate the average using formula.

Calculate the average of up to fivenumbers.

Solve problem in real life situationinvolving average.


12/12



Steps on building pupils understanding:

1. Build a tower with seven cubes and another

with five cubes.

2. Discuss on how to make both towers the same

height, using only the cubes they have used to

construct the towers.

3. Guide pupil to find the total number of

interlocking cubes used in building both towers.

7 + 5 = 12

4. Next, the pupils will have to divide the total

number of cubes by two.

12 2 = 6

5. By doing the calculation, the pupils will

understand the concept of average and also

the method of calculating averages.

6. Use same strategy in determining the average

heights of three and four towers.

7. The formulae of average than derived as:

8. Once the pupils understand the concept,

provide them with more activities that reinforce

their understanding of averages.

Measures of Central Tendency

Mean (Average)

o The average can be useful for comparing things.

Mode

o The most common item in a set of data.

o It's the number or thing that appears most often.

Median

o The middle number in a set of numbers.

o It is the mid-point when the numbers are written

out in order.

Key issues in teaching graphs and average

Students can calculate the averageof a data set

correctly, either by hand or with a calculator, and still

not understand when the average (or other statistical

tools) is a reasonable way to summarize the data.

Introducing students prematurely to the algorithm

for averaging data can have a negative impact on

their understanding of averaging as a concept. It isvery difficult to pull students back from the

simplistic add-then-divide algorithm to view an

average as a representative measure for describing

and comparing data sets. Key developmental steps

toward understanding an average conceptually are

seeing an average as reasonable, an average as a

midpoint, and an average as a balance point.

Prepared by:

Cg M ohd Ridzuan a l -K indy

Mohd Ridzuan bin Mohd Taib

(Facebook - Cg Mohd Ridzuan al-Kindy)

http://jilmuallim.blogspot.com

PISMP Mathematics Semester 6

IPG Kampus Dato Razali Ismail.

Copyright 2010

CentralTedency

Mean(Average)

ModeMedian

=

nota mte3111 (english version)

Documents