notes mte3111

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 between objects 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 of objects 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.



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 solids

o 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.



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



Key Issues in Teaching Shapes and Spaces Young students can define shapes, 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 same perimeter.

Students have a difficult time communicating visual information, especially if the task is to communicate a 3-D environment (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 a shape that help students generalize their conceptual image of that shape in any size or orientation (Shelton, 1985). E.g. : Geometer’s 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 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 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 direct comparison

Measure and compare length using uniform non-standard units

Measure and compare length using standard units

Measure, writing and estimate length

Conversion of units of length

Operation of units of length

Daily life problem



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 to measure the length of pencil

- Measure using other non-specific object (without scale) E.g.: using eraser to measure 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 (cm³) or cubic metres (m³).

o The Imperial system - cubic feet (ft³). 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 units

Standard 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

- Measure using other 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)



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.

- Measure using other 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 week 30 / 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 the day

to name the days of the week

to name the months of the year

to read and write the time



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 calculations o 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 values represented by the coins and notes.

Using different denomionations to represent the values of money

Converting between ringgit and sen

Performing basic arithmetic operations involving money

Applying their knowledge to solve daily problems involving money.



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 initial measurement

concepts, 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 to understand 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 a length 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 determining volumes 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 vocabulary associated with measurement activities 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. It’s 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.



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 are quantifying 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 are commonly used in everyday life 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 printed materials

classify, sort and analyse data

organising data in a table, chart or graph

carrying out simple surveys to collect data



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 Pictograph

1. 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.



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 of

basketballs 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 Sold o 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 three quantities.

Determine the formula for average.

Calculate the average using formula.

Calculate the average of up to five numbers.

Solve problem in real life situation involving average.



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 average of 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 is very 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 Mohd Ridzuan al-Kindy™

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

Central Tedency

Mean (Average)

ModeMedian

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푵풖풎풃풆풓풐풇풅풂풕풂

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