112275963-NOTA-MTE3111-ENGLISH-VERSION.pdf

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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 dimensionalworlds. (Kennedy and Tipps, 2006)

Geometric Thinking(a) Visual spatial thinking

Happened on the right hemisphere of the brainthat 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. Analysi s classify or group according dependingon the characteristics of shapes or figures but theycannot visualize the interrelationship between them.

3. Informal Deduction established or seesinterrelationships between figures.

4. Deduction mental thinking and geometric thinkingdeveloped significantly. They can understand thesignificant of deduction, the role of postulates,theorem and proofs. They are able to write proofwith understanding.

5. Rigor make abstract deduction and understandhow to work in axiomatic system even non-Euclidiangeometry 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 theproperties of objects in plane (2D) or in space (3D).

(b) Coordinate Geometry about location shapes oncoordinate or grid systems. Describe location ofobject on planed coordinate of vertical and

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

(c) Transformation Geometry about geometry inmotion. It describes the movement of shapes orobject in a plane or in space.

(d) Topological Geometry describes the location ofobjects and their relation in space or recognition ofobjects in the environment.


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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 relevantsolids (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 includingclassify and grouping shapes according to

similarities and differences.

Able to assemble and explaining types of shapesused to build models and relate models to solidshapes in real life.


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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 aroundand observing the environment and describe inwords what they have seen.

o Exploring and experimenting shapes (visualimages) in order to gain insight into propertiesand its uses

o Analysing shape informally, observing size andposition in order to make inferences then torefine and extended out knowledge that developfrom 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 equalsides and three equal angle

Isosceles triangle 2 equal sidesand 2 equal angle

Scalene triangle no equal sidesand no equal angle

Right-Angle Triangle One angleis 90

Acute angled triangle All three

angles are acute (< 90

)

Obtuse angled triangle Oneangles is obtuse (> 90 )

Quadrilaterals

Curved Shapes


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Key Issues i n Teaching Shapes and Spaces Young students can define shapes , but then not use

their definitions when asked to point out examples ofthose shapes.

Young students discriminate some characteristicsof different shapes, often viewing these shapesconceptually 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 visualinformation, especially if the task is to communicatea 3-D environment (e.g., a building made from smallblocks) via 2-D tools (e.g., paper and pencil) or thereverse.

Appl ications of Geometry in Technoogy A computer environment can generate multiple

representations of a shape 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 specificattribute with suitable vocabulary (short, shorter,tall, taller, etc.)

Transitivity principle o 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 orientationof the object is changed.

Measuring principl e 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 linealong 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


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Standard and non-standard units

Standard Non-Standard

- any fixed length that hasbeen accepted as astandard internationally(SI)

- any arbitrary lengthused as a unit

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

inches metres and

kilometres, etc.

- E.g.: body parts such

as span, foot,pace and armlength

objects such aspen, paper clip,etc.

- Measure using specificapparatus (with scale)such ruler, tape, etc.

E.g.: using ruler tomeasure the length ofpencil

- Measure using othernon-specific object(without scale)

E.g.: using eraser tomeasure the lengthof 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 spaceoccupied 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 acube 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 thathas been accepted asa standardinternationally (SI)

- any arbitrary volumeused as a unit

- E.g.: Millilitre, litre

- E.g.: A cup, jug, bottle Other containers

- Measure usingspecific apparatus(with scale) such ruler,tape, etc.E.g.: using beaker tomeasure water

- Measure using othernon-specific object(without scale)E.g.: using a jug tomeasure 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)


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Can be used to measure the volume of a solidobject, 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 onthat mass.

It is normal to refer weighing of an object as aprocess to find its mass.

Standard and non-standard unitsStandard Non-Standard

- any fixed mass /weight that has beenaccepted as astandardinternationally (SI)

- any arbitrary mass /weight used as a unit

- E.g.: Kilogram, gram Ounce,

- E.g.: Marbles, battery

- Measure usingspecific apparatus(with scale) suchweighing scale.E.g.: using weighingscale to measure themass of watermelon

The mass of

watermelon is 3 kg.

- Measure using othernon-specific object(without scale)E.g.: using a marblesto measure the massof bottle

The mass of bottle is 7marbles 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. Ituses 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 Weeko Months of the Yearo Relationship between Units of Time

60 seconds = 1 minutes60 minutes = 1 hour24 hours = 1 day

7 days = 1 week30 / 31 days = 1 month12 months = 1 year10 years = 1 decade10 decades = 1 century

o Operation involving Units of Timeo 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


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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

saleo 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 thevalues of money

Converting between ringgit and sen

Performing basic arithmetic operations involvingmoney

Applying their knowledge to solve daily

problems involving money.


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Using Coins to Model Decimal (Sen) Recording amounts in Ringgit and sen does involve

decimal fractions, but care must be taken on howthe children see the connection between the senand 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, childrenneed to think of 10 sen and 1 sen as fractional partsof a Ringgit.

RM 1.00 = 100 senRM 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 initial measurementconcepts, students need extensive experiences withseveral fundamental ideas prior to introduction to theuse of rulers and measurement formulas.

Number assignment: Students need to understandthat the measurement process is the assignment of anumber to an attribute of an object (e.g., the length ofan object is a number of inches).

Comparison: Students need to compare objects onthe basis of a designated attribute without usingnumbers (e.g., given two pencils, which is longer?).

Use of a unit and it eration: Students need tounderstand and use the designation of a special unitwhich is assigned the number one, then used in aniterative process to assign numbers to other objects(e.g., if length of a pencil is five paper clips, then theunit is a paper clip and five paper clips can be laidend-to end to cover the pencil).

Add it ivit y property: Students need to understandthat the measurement of the join of two objects ismirrored by the sum of the two numbers assignedto 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 t ools used to help teach number

concepts and operations are inexorably intertwinedwith the ideas of measurement.

The improved understanding of measurementconcepts is positively correlated with improvementin computational skills

Students are fluent with some of the simplemeasurement concepts and skills they willencounter outside of the class, but have greatdifficulty with other measurement concepts and skills(e.g., perimeter, area, and volume)

Students initially develop and then depend onphysical techniques for determining volumes ofobjects that can lead to errors in other situations.

o E.g.: students often calculate the volume of a box bycounting the number of cubes involved. When thisapproach is used on a picture of a box, students tendto count only the cubes that are visible.

The vocabulary associated with measurementactivities is difficult because the terms are eitherentirely new (e.g., perimeter, area, inch) or may havetotally 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 concepto 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 ofthe sun, hands on a clock face, digits changing ina display, changing seasons, etc.

o Clocks come in all sorts of styles and designs -some with all 12 numerals, (some Romannumerals), others with only 12, 3, 6 and 9numerals, and still others with no numerals at all.


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TOPIC 3: DATA HANDLING

Data handling deals with the processes involved inselection, collecting, organising, recording,summarising, describing and representing data forease 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, chartsand diagrams that are commonly used in everydaylife to display information.

Two main process in collecting:o combinatorial counting (to determine all thepossible outcomes)

o tallying (to organise the data under thecategories)

Data collected can be organise using:1. Table

o Simple table

o Regular table the matrix style table wherethere are more than two columns (more thancolumn of data).

2. Charts less regular in terms of rows andcolumns. They attempt to display informationmore visually, to relate the display to whatactually occurs.o The strip map

o Branch map - combination of strip maps,involving branching as in a tree.

3. Diagrams visual ways to represent membershipin 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 barlines.

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


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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 ofdata.

The scatter graph is made for data which isnot in sequence (in terms of the horizontalaxis) 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 onthe vertical axis.

3. Write the name of the items below thehorizontal axis.

4. Draw the bars vertically according to thequantity given for each item. Then colour thebars.

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 thebars.

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


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Interpreting data Data analysis and interpretation is the process of

assigning meaning to the collected information anddetermining the conclusions, significance, andimplications of the findings.

Interpretation of Pictograph

The questions above will lead your students tounderstand 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 Sallyhas 3 balls, Ken has 2 balls, Kamal has 1 ball andlastly, Ben has 4 balls.

This means that one picture can represent one ormore 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 scalesin 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 onMonday, 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 developedthrough using concrete materials and visualmanipulation (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.


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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 sameheight, using only the cubes they have used toconstruct the towers.

3. Guide pupil to find the total number ofinterlocking cubes used in building both towers.

7 + 5 = 124. Next, the pupils will have to divide the total

number of cubes by two.12 2 = 6

5. By doing the calculation, the pupils willunderstand the concept of average and alsothe method of calculating averages.

6. Use same strategy in determining the averageheights 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 reinforcetheir understanding of averages.

Measures of Central Tendency

Mean (Average)o The average can be useful for comparing things.

Modeo The most common item in a set of data.o It's the number or thing that appears most often.

Mediano 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 average of a data set

correctly, either by hand or with a calculator, and stillnot understand when the average (or other statisticaltools) is a reasonable way to summarize the data.

Introducing students prematurely to the algorithmfor averaging data can have a negative impact on

their understanding of averaging as a concept. It isvery difficult to pull students back from thesimplistic add-then-divide algorithm to view anaverage as a representative measure for describingand comparing data sets. Key developmental stepstoward understanding an average conceptually areseeing an average as reasonable, an average as amidpoint, and an average as a balance point.

Prepared by:Cg M o hd R i d z u a n a l -K i n d y

Mohd Ridzuan bin Mohd Taib(Facebook - Cg Mohd Ridzuan al-Kindy)

http://jilmuallim.blogspot.com PISMP Mathematics Semester 6IPG Kampus Dato Razali Ismail.

Copyright 2010

CentralTedency

Mean(Average)

ModeMedian

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