maths.pdf

PST201F/102/3/2015

Tutorial letter 102/3/2015 Mathematics and Mathematics Teaching

PST201F

Semesters 1 & 2

Department of Mathematics Education

IMPORTANT INFORMATION

You will need this tutorial letter to do assignment 02.

BAR CODE

Learn without limits UNISA

Author: Ronl Paulsen

IMPORTANT NOTICE Dear student

You will have to work through this tutorial letter to do assignment 02.

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Contents

3.1 Numbers, numerals and digits .................................................................................................. 3

3.1.2 Development of Hindu-Arabic digits ............................................................................................. 4

3.2 The Hindu-Arabic numeration system ....................................................................................... 4

3.3 Understanding place value ....................................................................................................... 5

3.3.1 Place value, face value and total value ........................................................................................ 5

3.4 Models to illustrate place value ................................................................................................. 7

3.4.1 Base 10 blocks (Dienes blocks) ................................................................................................... 7

3.4.2 Unifix cubes ................................................................................................................................. 9

3.4.3 Sticks or matches ......................................................................................................................... 9

3.4.4 Beans ......................................................................................................................................... 10

3.4.5 The hundred chart ...................................................................................................................... 11

3.4.6 Number cards ............................................................................................................................ 11

3.5 Operations on whole numbers ................................................................................................ 12

3.5.1 Addition and subtraction ............................................................................................................ 12

3.5.2 Dienes blocks ............................................................................................................................. 14

3.5.3 Number cards ............................................................................................................................ 15

3.5.4 Vertical and horizontal algorithms .............................................................................................. 15

3.5.5 Multiplication and division .......................................................................................................... 18

3.5.6 Dienes blocks ............................................................................................................................. 19

3.6 Large numbers ........................................................................................................................ 23

3.7 Illustrating numbers on the number line .................................................................................. 25

3.8 Rounding off ........................................................................................................................... 26

3.9 Prime numbers ....................................................................................................................... 27

3.10 Rules of divisibility .................................................................................................................. 27

3.11 Multiples ................................................................................................................................. 29

3.12 Factors .................................................................................................................................... 29

3.12.1 The factor tree ............................................................................................................................ 29

4.1 Basic fraction concepts ........................................................................................................... 31

4.2 Fraction models ...................................................................................................................... 32

4.2.1 Area models ............................................................................................................................... 33

4.2.2 Set models ................................................................................................................................. 33

4.2.3 Length models ........................................................................................................................... 33

4.3 Fraction notation ..................................................................................................................... 34

4.3.1 Understanding fraction notation ................................................................................................. 34

4.4 Nonunit fractions ................................................................................................................... 35

4.5 Number line presentations ...................................................................................................... 36

4.6 Equivalent fractions ................................................................................................................ 37

4.6.1 Continuous wholes (area model) ............................................................................................... 37

4.6.2 Discontinuous wholes (set model) ............................................................................................. 38

4.6.3 Number line ................................................................................................................................ 38

4.7 Comparing fractions ................................................................................................................ 39

4.7.1 Comparing non-unit fractions ..................................................................................................... 39

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4.7.2 Which is bigger? ........................................................................................................................ 40

4.8 Addition of fractions ................................................................................................................ 41

4.8.1 The three stages of teaching the addition of fractions ............................................................... 41

4.9 Subtraction of fractions ........................................................................................................... 42

4.9.1 The three stages of teaching the subtraction of fractions........................................................... 42

4.10 The meaning of of ............................................................................................................... 44

4.11 Multiplication of fractions......................................................................................................... 45

4.11.1 The three stages of teaching the multiplication of fractions ....................................................... 45

4.11.2 The area model. ......................................................................................................................... 47

4.11.3 An algorithm for multiplication of fractions: ................................................................................ 49

5.1 An introduction to shapes ....................................................................................................... 50

5.2 The Van Hiele levels of Geometric thought ............................................................................. 51

5.1 Comments about the thought levels ........................................................................................... 52

5.2 Consequences of the Van Hiele theory for learning ................................................................... 53

5.3 Flat shapes ............................................................................................................................. 53

5.4 Polygons ................................................................................................................................. 54

5.4.1 Naming polygons ....................................................................................................................... 55

5.5 Triangles ................................................................................................................................. 56

5.5.1 Types of triangles ....................................................................................................................... 56

5.6 Quadrilaterals ......................................................................................................................... 57

5.6.1 Concepts dealing with quadrilaterals ......................................................................................... 57

5.6.2 Classification of quadrilaterals ................................................................................................... 58

5.7 Space shapes ......................................................................................................................... 59

5.7.1 Special space shapes ................................................................................................................ 60

Polyhedrons .................................................................................................................................. 60

Prisms ........................................................................................................................................... 61

Pyramids ....................................................................................................................................... 62

5.7.2 Regular polyhedra ...................................................................................................................... 63

5.8 Practice how to draw 3 D objects ......................................................................................... 63

5.9 Nets of polyhedra .................................................................................................................... 64

5.10 Drawings from different views ................................................................................................. 65

Appendix Unit 3: Activity for Dienes blocks ....................................................................................... 67

Appendix Unit 4: Fraction resources .................................................................................................. 73

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UNIT 3: NUMBERS AND OPERATIONS

This unit covers chapters 11 13 in the text book (pages 204 273)

In this unit, we will discuss:

The Hindu-Arabic numeration system

Understanding the concept of a number

Models to represent numbers

Understanding operations on numbers

Factors and multiples

Prime numbers

Rules of divisibility

3.1 Numbers, numerals and digits

A number is a count or measurement - that is really an idea in our minds

A numeral is a symbol or name that stands for a number

A digit is a single symbol use to make up numerals

So the number is an idea, the numeral is how

we write it

number

The name "digit" comes from the fact that

the 10 digits (ancient Latin digiti meaning

fingers) of the hands correspond to the 10

symbols of the common base 10 number system,

i.e. the decimal (ancient Latin adjective dec.

meaning ten) digits.

We have 10 digits (or symbols) in our numeration

system: 0; 1; 2; 3; 4; 5; 6; 7; 8 ; 9

4

3.1.2 Development of Hindu-Arabic digits

The digits we are using today did not always look like the ones you are used to. It took centuries to develop

the digits to what we are using today. And who knows, in a couple of hundred years they might look different

again.

3.2 The Hindu-Arabic numeration system

The elegant numeration system which we use today is thought to have been invented by the Hindus from

approximately 1 000 BC onwards, to have spread via trade with the Arabs over centuries to their world, and

hence by trade and conquest via the Moors in Spain, to Europe.

Using only ten symbols, including a zero-symbol, and the concept of place value, we can

represent any number we please.

The main features are summarised below: (for example the number 4 213)

Employs place value

Is a denary system

The base is ten and the place values powers of 10

Has digits to count how many times a particular grouping occurs

Thousands Hundreds Tens Ones

10 10 10

10 10 10

10 10 10

10 10 10

10 10

10 10

10 1 1 1

4 2 1 3

Indian, 1st century AD

Indian, 9th century

East Arabic, about 11th century Indian, about 11th century West Arabic, about 11th century

15th century

16th century

These are very similar to the symbols we are using

today

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As the grouping number is ten, there will never be more than nine groups in each place.

Our numeration system:

uses a zero, to count the number of an empty set

is multiplicative: 4 (10 10 10) + 2 (10 10) + 1 10 3 1

and additive: (4 103) + (2 102) + (1 101) + 3 100

Activity 3.1

1 Explain the difference between the concepts: digit, numeral and number by giving

examples.

2 What is a numeration system?

3 What is the role of zero in our numeration system?

3.3 Understanding place value

Make two numbers using the digits 1 and 3

What numbers can you make?

o 13 and 31

How are they different?

o Thirteen is 1 ten and 3 ones

o Thirty- one is 3 tens and 1 one

How many numbers can you make from the digits 3, 5 and 8?

Let us take a look:

Hundreds Tens Units (ones)

3 5 8 = 358

3 8 5 = 385

5 3 8 = 538

5 8 3 = 583

8 5 3 = 853

8 3 5 = 835

3.3.1 Place value, face value and total value

Place value

The place values are the value of the PLACE where the digit is in the numeral. In a three digit number, there

are three places, the hundreds, the tens and the units.

In a basic digital system, a numeral is a

sequence of digits, which may be of

arbitrary length. Each position in the

sequence has a place value, and each

digit has a value.

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The value of the numeral is computed by multiplying each digit in the sequence by its place value, and

summing the results.


3 5 8

The numeral 358 has the value of 3 hundreds plus 5 tens plus 8 ones.

OR: 3 100 + 5 10 + 8 1 = 358

"Listen" how we read it: three hundred and fifty eight or just three hundred fifty eight.

Face value

The face value of a digit in a numeral is simply the number that you see.

3 456

The face value of the numeral in the hundred place is 4.

Total value

The total value (some text books refer to "the value" only) of a digit in a numeral is the

face value the place value

So the total value of the 4 in 3 456 is 4 100 = 400

In the tens place of the numeral 234, we have a digit with a face value of 3, and a place value of 10, giving us a total value of 30.

Activity 3.2

1 Our numeration system employs place value. What is your understanding of place

value?

2 Write down the place value of the underlined digits:

54 982

459 234

3 Write down the total value of the underlined digits:

54 982

459 234

Fifty is the English for 5 tens

These are the place values

The place value of the 3 is

HUNDREDS

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3.4 Models to illustrate place value

Firstly, go back to the section on relational understanding in your text book (page 24).

Read the section "Base-ten models for place value" in your textbook (page 207)

Young children need models to develop an understanding of place value. We will discuss a few here.

3.4.1 Base 10 blocks (Dienes blocks)

Zoltan Dienes (1916 - 2014) developed these base 10 blocks to

teach place value.

Dienes' place is unique in the field of mathematics education

because of his theories on how mathematical structures can be

taught from the early grades onwards using multiple

embodiments through manipulatives, games, stories and dance.

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Pictures

Manipulative

models

Real-world

situations

Written

symbols

Oral language

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This diagram shows five

different representations

of mathematical ideas.

Translation between and

within these can help to

develop new concepts.

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Examples

Illustrate the following numbers using Dienes blocks (also called base 10 blocks)

Number Place value chart Representation

326

H T U

3 2 6


2476

TH H T U

2 4 7 6

We call this a BIG or a CUBE We call this a FLAT We call this a LONG We call this a TINY or a ONE

When we draw them, we try to be as accurate as we can. You can make copies of these, laminate them and have them ready for the blackboard.

three flats two longs six tinies

300

twenty six three hundred

20 6

two thousand

four hundred

thousand

seventy six

AnikkiComment on Text

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3.4.2 Unifix cubes

Unifix cubes are plastic connecting cubes which learners can stack

up to bars of 10. They are useful for working with numbers under

100, otherwise the piles become quite big. The benefit is that they

can be placed together or taken apart. Number bonds can be

illustrated using Unifix cubes.

Example

Illustrate the following numbers using Unifix blocks


135

H T U

1 3 5

See also the website: http://www.themeasuredmom.com/math-activities-unifix-cubes/

3.4.3 Sticks or matches

Bundles of sticks, grouped in tens, and then tied with a rubber band

These cubes are

stacked up to

show number

bonds of 10

10

Activity 3.3

What numbers are represented in the grouping of sticks above?

3.4.4 Beans

Beans in bottle tops and empty match boxes

Example

Illustrate the following numbers using beans


148

H T U

1 4 8

1 100 + 4 10 + 8 1 = 100 + 40 + 8 = 148

A bottle top holds 10 small beans A matchbox holds 100 small beans

100 beans

4 10

beans

8 beans

The number represented here:

2 100 + 6 10 + 3 = 263

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3.4.5 The hundred chart

The hundred chart has many advantages in teaching number concepts.

Activity 3.4

1 Find a row or column where all the unit digits have a face value of 3

2 Find a row or column where 9 of the tens digits have a face value of 3

3 Find the numbers where the face values of the tens digits are the same as the units

digits. What do you notice? Draw a line through them.

4 Find the numbers where the sum of the tens digit and the unit digit is 9. What do you

notice?

5 What do the numbers in the last column have in common?

3.4.6 Number cards

Below are number illustrations of number cards, and how a number can be built up

We may think of 385 as written on three cards: 3 0 0 8 0 5 fitted

one behind the other to look like this: 3 8 5

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

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3.5 Operations on whole numbers

There are four basic operations in mathematics, namely addition, subtraction, multiplication and division. Once

you understand how numbers are made up (hundreds, tens units, etc.) you will have a better understanding of

operations on numbers.

3.5.1 Addition and subtraction

Work through pages 230 246 in your text book.

Read the section "student invented strategies" on page 232 233, then do the following activity.

Activity 3.5

You want to buy a book priced R105, but you find you only have R89 in your purse.

Think of ways in which you can find how much money you still need. You do not have pen and paper or

a calculator with you. You do not want to rely on the salesperson!

There are various ways in which you can reason to find the answer.

One way could be:

Say you had R100, how much would you have been short? R5

Say you had R90, how much would you have been short? R15

But now you have R89, so you have R15 + R1 short R16

Now write down another way in which you could do the calculation

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Activity 3.6

Read through pages 232 233 in your text book, and then do the activity

1 What is an algorithm?

2 Investigate and report on the differences between standard algorithm and students' (leaners')

own invented strategies.

3 Write down the benefits of student-invented strategies in your own words

Using models to explain concepts of addition and subtraction

Example:

Use Dienes blocks to explain how to add: 8 + 6. This is a very simple example, but it will show you how to

exchange the blocks.

Pack out 8 tinies and 6 tinies.

Take 2 tinies from the second group and place it with the 8 tines.

You now have 10 tinies in the first group and 4 tinies in the second group. Exchange the 10 tinies for

one long.

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T U

8

6

1 4

T U

8

6

14

3.5.2 Dienes blocks

Use Dienes blocks to show 367 + 134

Pack out 3 flats, 6 longs and 7

tinies. Then pack out 1 flat, 3

longs and 4 tinies

H T U

3 6 7

1 3 4

Group 10 tinies together

H T U

3 6 1 7

1 3 4

1

The ten tinies from

one long (one ten)

H T U

3 6 1 7

1 3 4

1

This is the one long that you exchanged for 10 tinies

Put 10 longs together to make 1 flat (10 tens is 1 hundred) H T U

3 1 6 1 7

1 3 4

5 0 1

This is the one flat

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3.5.3 Number cards

Use number cards to illustrate the addition:

370 + 425 + 86

300 + 70 + 400 + 20 + 5 + 80 + 6

= 300 + 400 + 70 + 20 + 80 + 5 + 6

= 700 + 170 + 11

= 700 + 100 + 70 + 10 + 1

= 881

3.5.4 Vertical and horizontal algorithms

The two examples above demonstrated vertical and horizontal algorithms for addition

In a horizontal algorithm, you will break up the numbers into 100's, 10's etc., and place them in a row.

In a vertical algorithm, you will place the numbers underneath each other.

Horizontal algorithm for addition

For the horizontal algorithm, you need separate the hundreds, tens and units, from the number, and then add:

Hundreds to hundreds

Tens to tens

Units to units

Vertical algorithm for addition

Traditionally, you might know this algorithm for addition as "carrying". To "carry" is another way to talk about

exchange:

10 units to one ten

10 tens to one hundred

10 hundreds to one thousand, and so on.

Here is another example which will illustrate the "carrying" from one place value to the next.

Group the hundreds, tens and units together

Add hundreds to hundreds, tens to tens and units to units

This illustrates the HORIZONTAL algorithm for addition

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Understanding "carrying": using Dienes blocks to add

347 + 176

Set out:

3 flats, 4 longs and 7 tinies and 1 flat, 7 longs and 6 tinies

Add the tinies

Exchange 10 tinies for 1 long

You get 1 long and 2 tinies

Carry the long to the other longs

Add the longs

There are 12 longs

Exchange 10 longs for one flat

Carry the flat over to the other flats

You are left with 5 flats, 2 longs and 3 tinies

3 4 7 1 7 6 3

3 4 7 1 7 6 2 3

3 4 7 1 7 6 5 2 3

Vertical algorithm for subtraction

Traditionally, you might know this algorithm for subtraction as "borrowing". To "borrow" is another way to talk

about exchange:

one ten to 10 units

one hundred to 10 tens

one thousand to 10 hundreds, and so on.

3 tinies

1

1 long

1 1

2 longs

1 flat

1 1

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Understanding "borrowing": using Dienes blocks to subtract

342 176

Set out:

3 flats, 4 longs and 2 tinies and 1 flat, 7 longs and 6 tinies

Subtract the tinies (2 6 cannot do)

Exchange 1 long with 10 tinies

Take away the 6 tinies

You are left with 6 tinies

Subtract the longs (3 7 cannot do)

Exchange 1 flat with 10 longs

Take away the 7 longs

You are left with 6 longs

Subtract the flats (2 1 = 1)

You are left with 1 flat, 6 longs and 6 tinies

342 176 __6

3 4 2 1 7 6 _6 6

3 4 2 1 7 5 1 6 6

13 1

1

2 13

Exchange 1 long for 10 tinies

3

1

2

6 longs left

1 flat left

13 longs minus 7 longs = 6 longs

6 tinies left

18

Activity 3.7

Use Dienes blocks to illustrate the following operations:

1 24 + 57 2 196 + 105

3 44 17 4 416 109

Use number cards to illustrate the following operations:

5 458 + 263 6 458 263

3.5.5 Multiplication and division

Work through pages 252 272 in your text book.

Multiplication

Read the section "student invented strategies" on page 253 257, then do the following activity.

Activity 3.8

The excerpt below was taken from page 255 in your text book. Explain in your own words how each of

the three children solved the problem:

There were 35 dog sleds. Each sled was pulled by 12 dogs. How many dogs were there in all?

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3.5.6 Dienes blocks

Use Dienes blocks to show 23 7

Multiplication by 10

Often teachers would say to their learners: When you multiply by 10, "you must simply put a zero at the

end". The problem is that this "rule" only works for multiplication of whole number by 10. When dealing with

decimals, that rule does not work. Now teachers often say: "move the comma for every zero in the

multiplier".

What really happens when we multiply whole numbers by 10?

We know that 10 fives will be 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5+ 5 + 5 = 50

Let us look at the place value chart.

Activity 3.9 1 Show 329 100 by drawing a place value chart. Explain why there are now 2 zeroes at the end.

2 Mrs Tesfaya has 6 boxes of markers. Each box has 19 markers in it. If she sold each marker for

R2.70, how much money would Mrs Tesfaya earn?

T U

5

5 0

10

The unit digit moves to the tens place.

10

Set out 2 longs and 3 tinies

2 longs 7

= 14 longs

3 tinies 7 = 21 tinies

Exchange 10 longs for a flat: 100

Exchange 20 tinies for 2

longs: 20

The answer is: 100 + 60 + 1 = 161 = 60 + 1

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Division

Division is an operation that splits a quantity into smaller, equal sized quantities. It is important that you

understand the two different concepts of division, namely sharing and grouping.

Consider the two examples.

1 Patsy wants to share 30 sweets between 5 children. How many will each child get?

2 We have to transport 70 children to a function. Each mini-bus can take 10 children. How many mini-

buses do we need?

Let us look at the action involved in each of the above:

SHARING

Sharing is usually the first concept of division that learners encounter. It is used to share items out equally

amongst a number of people, such as sweets.

In sharing, the number of groups is known. The quantity of items in each group is unknown. The

answer is found by sharing the items equally between the groups. Here one would ask the question:

How many items will each person gets?

GROUPING

In grouping, the quantity in each group is known. The number of groups is unknown.

We know how many children there are. We ask: How many sweets will each child get?

We know how many learners can go into one bus. We ask: how many buses will be needed?

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Solutions:

1 Sharing can be a one- by- one action. Patsy can share her sweets by handing them out one at a time.

Each child will get 5 sweets.

2 When grouping is involved, we have to make groups of 7 and see how many groups we need to make 70.

Ten buses will be needed.

Long division algorithm

Let us illustrate division of 234 3 using Dienes blocks to show how the algorithm can be understood.

First set out 2 flats, 3 longs and 4 tinies.

234 = 200 + 30 + 4

First set out 2 flats, 3 longs

and 4 tinies.

We start with the hundreds. We cannot divide the 2 hundreds by 3, so we exchange them for 10's. We now have 23 tens.

We can make 7 groups of 3 tens each and there will be 2

tens remaining.

Put the longs (tens) in

groups of 3

22

Read the section on Standard algorithms for addition in your textbook on page 261.

:

Activity 3.10

1 Use the above method for division to find:

228 12

642 6

Now split up the 2 tens in twenty tinies. There are 24

tinies left.

Put the tinies in groups of three. There are 8 groups of 3 each.

Solution: 234 3 = 78

Another strategy: Trial and improvement:

continuous subtraction to get the quotient.

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2 Mrs. Tesfaya learned that R1 340 worth of tickets were sold at the carnival. If tickets cost 4 for

R10, how many tickets were sold?

3 A company donates 935 pencils to a school. The pencils are divided evenly among 9

classrooms. The rest of the pencils are given to the library. How many pencils were donated to

the school and to the library?

4 You have R15 in 5c and 10c pieces. If you have the same number of each kind of coin, how

many 5c pieces do you have?

5 In the summertime, you can earn R4 a day by cutting the grass. How many days will it take you

to earn R184?

6 The goat in the village weighs 145 kg. It is five times heavier than the baby goat. How much

does the baby goat weigh?

7 Three hundred children are divided into two groups. There are 50 more children in the first group

than in the second group. How many children are there in the second group?

8 Three thousand exercise books are arranged into 3 piles. The first pile has 10 more books than

the second pile. The number of books in the second pile is twice the number of books in the third

pile. How many books are there in the third pile?

3.6 Large numbers

In recording and reading large numbers, we adopt certain powers of ten as provisional units. The face values

of the various digits in the resulting numeral are the results of counting these provisional units. The total of the

place values times the face values of the digits is the number of units we want to count (the total value).

We can use number cards to help leaners understand the place values of larger numbers

24

You should be able to read large numbers. To help reading large numbers, we choose certain collective nouns

to name provisional units.

1018 1015 1012 109 106 103 100

Quintillions Quadrillions Trillions Billions Millions Thousands Ones

H T U H T U H T U H T U H T U H T U H T U

103 is 1 with 3 zero's 1 000 One thousand

105 is 1 with 5 zero's 100 000 One hundred thousand

108 is1 with 8 zero's 100 000 000 One hundred million

Can you complete the rest?

1013 is 1 with ___ zero's ______________ _________________

1019 is 1 with ___ zero's ______________ _________________

Activity 3.11

Write the following numbers in the table below:

1 234 567 890 Read the number.

2 1 011 110 111 Read the number

3 70 010 001 002 Read the number

4 Four million, five hundred and one thousand and one

5 Twenty five quadrillion, three hundred and ten billion six hundred and twelve

Quintillion Quadrillion Trillions Billions Millions Thousands Ones

H T U H T U H T U H T U H T U H T U H T U

Complete

6 What is one more than a million? _______________________

7 What is one million more than 999 million? _____________________

8 What is one more than 999 million? _______________________

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9 A thousand thousands is _______________________

10 A thousand millions is _______________________

11 A million millions is _______________________

12 A thousand billions is _______________________

13 Read the following numbers: (write in words)

386 030 800 000 __________________________________________

1 020 300 040 500 006 __________________________________________

Activity 3.12

Write the following numbers in symbols:

1 Four hundred and fifty-two thousand and twenty _______________________

2 One hundred and seven million five hundred and nine ____________________

3 Fifty billion two million and one hundred thousand ______________________

4 Two million four hundred and eight thousand __________________________

5 Three trillion four hundred and eight million and eight thousand

3.7 Illustrating numbers on the number line

To show a number on the number line is an important way to teach learners to scale the number line in an

appropriate way.

For example:

To show a number between 0 and 10, we will scale the number line from 0 to 10 (using 1cm for a unit)

To show a number between 0 and 100, we will scale the number line from 0 to 100 (using 1cm for ten)

Showing 4 on the number line 1 2 3 4 5 6 7 8 9 10 0

Showing 47 on the number line 10 20 30 40 50 60 70 80 90 100 0

26

To show a number between 0 and 1000, we will scale the number line from 0 to 1000 (using 1cm for

hundred)

3.8 Rounding off

Rounding off of numbers is mostly used in measurement, but it is also used in estimation when we do mental

calculations. If we have to round off a number, we are required to work to a certain degree of accuracy.

Suppose 23 533 tickets were sold for a cricket match, what will the most appropriate way to say how many

people attended the match (provided they all attended of course).

There were about 23 530 people at the match (rounded off to the nearest ____)



Activity 3.13

Suppose 34 467 tickets were sold for a cricket match, round this figure off to the nearest:

10 __________ Look at the unit digit (if it is 5 or more, the tens digit becomes

one more) 34 467

100 __________ Look at the tens digit (if it is 5 or more, the hundreds digit

becomes one more) 34 467

1 000 __________ Look at the hundreds digit (if it is 5 or more, the thousands

digit becomes one more) 34 467

10 000 __________ Look at the thousand digit (if it is 5 or more, the ten

thousands digit becomes one more) 34 467

Explain your answers.

Can you come up with a rule for rounding off?

Showing 747 on the number line

100 200 300 400 500 600 700 800 900 1000 0

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3.9 Prime numbers

Before looking any further, can you write down a definition of a prime number?

________________________________________________________________

A prime number is a number which has only two different factors, of which 1 is one

of them. 1 is not a prime number.

Write down all the prime numbers between 1 and 10

_________________________

The Sieve of Eratosthenes is a well-known way to find prime

numbers. In this example, we will find all the prime numbers

between 1 and 100.

Sieve of Eratosthenes

3.10 Rules of divisibility

If learners can count in twos, threes, fives and tens, hundreds and thousands, with understanding, they should

easily be able to recognise the multiples of these numbers.

1 What are we actually doing when we count in fives? ____________________________

What can you notice about all these numbers?

2 What are we actually doing when we count in twos? ____________________________

What can you notice about all these numbers?

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

METHOD

Cross out 1 (1 is not prime) X

Cross out all the multiples of 2, 3, 5, 7, except these numbers themselves.

The numbers that are not crossed out will be prime.

Circle all the prime numbers

Answer

2 ; 3 ; 5 ; 7

28

A number is divisible by 2 if the last digit is even:

Examples: 234 ; 456 028

A number is divisible by 3, when you add all the digits and the sum is a multiple of 3.

Example: 3 567 3 + 5 + 6 + 7 = 21 and 2 + 1 = 3 .

3 567 is divisible by 3

A number is divisible by 4, when the last two digits are divisible by 4.

Examples: 11 124 24 is divisible by 4

A number is divisible by 5, when the last digit is a zero or a five.

Examples: 123 455 ; 340

A number is divisible by 6, when both 2 and 3 can divide into it.

Example: 45 612 last digit even divisible by 2

4 + 5 + 6 + 1 + 2 = 18 and 1 + 8 = 9 (a multiple of 3)

45 612 is divisible by 6

A number is divisible by 9, when you add all the digits and the sum is a multiple of 9.

Example: 45 612 4 + 5 + 6 + 1 + 2 = 18 and 18 is a multiple of 9

A number is divisible by 11, when you add every second digit, then add the others, and then subtract

the two sums. If the answer is 0 or a multiple of 11, then the number is divisible by 11.

Example:

1 2 3 4 2 Add 1 + 3 + 2 = 6

1 2 3 4 2 Then add 2 + 4 = 6

A number is divisible by 10, when the last digit is a zero.

Example: 230 ; 988 500

6 6 = 0

29 PST201F/102

Activity 3.14

Test the following numbers for divisibility by the given number. You may not do the actual division, and

no calculators are allowed.

1 345 890 for divisibility by 2 ; 3; 4 ; 5 ; 6 and 10

2 246 789 by 9 and 11

3 108 108 by 9 ; 11 and 12.

3.11 Multiples

You all know what a multiple of a number is.

M3 = 0 ; 3 ; 6 ; 9 ; 12 ; ..

M4 = 0 ; 4 ; 8 ; 12 ; 16 ;

The Lowest Common Multiple (LCM) is the lowest number in which two or more numbers can divide.

What is the Lowest Common multiple of 3 and 4? (ignoring zero)

M3 = 0 ; 3 ; 6 ; 9 ; 12 ; ..

M4 = 0 ; 4 ; 8 ; 12 ; 16 ;

Can you see that 12 is the LCM of 3 and 4?

Activity 3.15

What is the LCM of 2 ; 3 ; and 5?

M2 = ________________________

M3 = ________________________

M5 = ________________________

3.12 Factors

A factor of a number is a number that can be divided into the number without leaving a remainder.

F12 = 1 ; 2 ; 3 ; 4 ; 6 ; 12 the prime factors are 2 and 3

F30 = 1 ; 2 ; 3 ; 5 ; 6 ; 10 ; 15 ; 30 the prime factors are 2 , 3 and 5

3.12.1 The factor tree

An easy and fun way to show learners how to find the prime factors of a number, is by making use of the factor

tree

0 is a multiple of any number

The LCM of 2 ; 3 ; and 5 is

__________

30

Factor trees for 60

The prime factors of 60 are 2 ; 2 ; 3 and 5

60 can be written as 2 2 3 5

Activity 3.16

1 Complete the factor trees

32 can be written as _____________ 135 can be written as _____________

2 A remainder of 1 is left when you divide 61 by 2, 3, 4 and 5. What is the lowest number which

leaves a remainder of 1 when divided by each of the numbers from 2 to 10?

Websites that can be useful:

Subtraction using base 10 blocks

https://www.youtube.com/watch?v=NXCsEkMLWtY

Addition using base 10 blocks

https://www.youtube.com/watch?v=I0dAjSj6q64

Division of 3 digit number by 1 digit number

https://www.youtube.com/watch?v=qJxt-kSzfbo

Long division

https://www.youtube.com/watch?v=AXQNeP6NN44

60

3 20

4 5

2 2

60

2 30

6 5

2 3

60

4 15

3 5 2 2

32

2

8

4

135

3

5

31 PST201F/102

UNIT 4: FRACTIONS

This unit covers chapters 15 17 in the text book (pages 310 378)


Fraction concept formation

Models to represent fraction concepts

Fraction notation

Number line presentations

Equivalent fractions

Operations on fractions

Activity 4.1

Read the section "Why Fractions are so Difficult" in your textbook (pp 311 312)

Now reflect on what you have read. Explain in your own words what the difficulties with fractions might be.

4.1 Basic fraction concepts

Start with simple examples, involving halves. Learners first have to be able to distinguish between "objects"

and "non-objects" representing halves.

Let learners trace these shapes and fold along the dotted line.

W vd

Questions to ask learners:

Into how many parts is each

shape divided?

What do you notice when you put

one part on top of the other?

Which shapes are divided into

two equal parts?

What name do we give to each

equal part?

32

NB NB NB Do not use the fractional notation at this stage

2

1.

Let learners just use the verbal expression: half.

Learners must say: My whole is a (circle, rectangle, etc.). It is divided into two equal parts. Each part

is a half of the whole.

Do the same exercise for thirds, fourths, etc.

Do not use the fractional notation at this stage ( 3

1 ).

Learners must say: My whole is a (circle, rectangle, etc.). It is divided into three equal parts. Each

part is a third of the whole.

Here we are establishing the concept of a WHOLE being cut or divided into three EQUAL parts.

Each part is a third of the whole

Activity 4.2

Draw three diagrams which can be used to show wholes that are divided into four parts, but

which do not all represent fourths.

4.2 Fraction models

Read the section "Models for fractions" on pages 312 315 in your text book. W vd

Which of these are thirds of the whole?

33 PST201F/102

4.2.1 Area models

The wholes for area models are continuous. That means they are not single pieces. We usually use diagrams,

clay, pattern blocks that can fit together or paper folding for area models. The whole is "cut up" or partitioned

into several equal sized pieces. You will find examples of these models on page 313 of your text book.

Language pattern: My whole is a circle. To find one eighth of the whole, I divide it into eight parts of

equal size, and shade one part. The shaded part is one eight of the whole

4.2.2 Set models

The wholes for set models are discontinuous. That means that the whole consists of several separate equal

sized pieces. Each piece makes up a part of the whole. Examples of set models would be bottle tops, hard

sweets or counters. See more examples on page 317.

Language pattern: My whole consists of 12 bottle tops. I divided them into three parts of equal size.

Each part is one third of the whole. Each part has four bottle tops. So one third of

12 is four.

4.2.3 Length models

Length models differ from the above models, as it relates to the number line. Do not use the number line as a

model too soon. We will use length models such as paper strips and Cuisenaire rods.

One-sixth

One-fifth One-eighth

34

The diagram below is called a fraction wall, to show how the whole can be divided into equal parts.

4.3 Fraction notation

Once learners have a solid understanding of the concept of a fraction, we can move towards the fraction

notation. Please study the section "Fraction notation" on page 322 of your text book.

4.3.1 Understanding fraction notation

What does the fraction 3

1 mean?

The bottom part of the fraction tells us into how many

parts the whole is divided. The bottom part is called the

denominator.

The top part of the fraction tells us how many of the

parts we shade. The top part is called the numerator.

It is important that learners will still see the relation between a concrete example and the notation. Thus it is a

good idea to still involve drawings, or concrete apparatus, like bottle tops or clay, or paper.

Unit fractions are fractions with the numerator 1. It shows ONE part of the whole, such as

.

The whole

Two halves

Three thirds

Four fourths

Twelve twelfths

Six sixths

The numerator counts The denominator tells what is being counted

The whole divided into two equal parts.

1 2

The 1 shows how many parts of the whole is shaded

The 2 shows into how many parts the whole is divided

The shaded part is written as

. We read it: one half

35 PST201F/102

4.4 Nonunit fractions

Once learners have a thorough grasp of unit fractions, we can move on to non-unit fractions, for example,

etc8

4,

5

3,

3

2

We go through the same processes as before.

Example: Shade

of the triangle

Language: My whole is an equilateral triangle. To shade 3

2

of the triangle, I divide the triangle into 3 equal

parts, and shade 2 of these parts.

Examples

Shade

of the pentagon

The whole is divided into five equal parts. Each part is one fifth of the whole. Two

parts are shaded. So two-fifths of the whole is shaded.

In fraction notation, we write, 5

2

In the fraction5

2, what does the 5 mean? _________________ [the number of parts into which the whole is divided]

What does the 2 mean? _______________________________ [the number of parts shaded]

Activity 4.3 Shade the required parts of the given wholes.

1 Shade 5

4 of the whole

In the fraction5

4, what does the 5 mean? _________________

What does the 4 mean? _______________________________ 2

Shade 8

5 of the whole

In the fraction8

5, what does the 8 mean? _________________

What does the 5 mean? _______________________________

Write down in words what you will do to shade 8

5of the whole

Learners will need to count the bottle

tops (beans or other objects) to find out

how many there are. So he/she will

need to know their tables to divide into

8 equal parts. Make 8 groups.

36

4.5 Number line presentations

Let us look at the number line from 0 to 1. (This is one unit). The way in which we demarcate (iterate) the number line, will tell us into what fraction parts the unit is divided. We will now practice to place fractions on the number line.

Examples: Remember if we talk about

, we actually mean

of 1. So where is

on the number line?

Where is

and

Where is

,

on the number line?

Activity 4.4 1 Into how many parts is this unit divided? Label each of the parts. 2 Count in thirds (place your pencil on the numbers as you are counting) One-third, two-thirds, ______________________________________

3 Show the following on the number line:

3

153

1037

33

31

4 Show the following on the number line below:

4

1048

47

43

41

0 1

0 1 1

3

2

3

0 1 1

4

2

4

3

4

0 1 2 3

0 1

1 0 2 3 4 5 6 7

0 1 2

37 PST201F/102

5 Then show the following on a number line: 5

1058

57

53

51

Activity 4. 5 Help the boys:

Sipho has a piece of string that is exactly 2m long. He wants to divide it equally among three friends. What part of the string will each one get?

Each friend gets

of the string

Each friend gets

of 2 metres

Each friend gets ___________ of a metre

4.6 Equivalent fractions

Please study the section "Equivalent fractions" on page 325 330 in your text book.

4.6.1 Continuous wholes (area model)

The activity below will guide you to understand the meaning of equivalence.

Activity 4.6 1

The whole Shade

of the whole Shade

of the whole

What do you notice? 2

Shade

of the whole Shade

of the whole

What do you notice?

0 1 2

1

3 2

2

3 1

Important

38

4.6.2 Discontinuous wholes (set model)

Who sold the most goats?

What can you say about

and

of the same whole?

4.6.3 Number line

This number line shows that the unit is divided into 6 parts.

What does A on the number line represent? A represents

as well as

. So we can say that

=

Activity 4.7

1 Use a number line to illustrate the equivalence of 62

31 and

2 Use the same number line to illustrate the equivalence of 64

32 and

3 Use a number line to illustrate the equivalence of 86

43 and

Famer Bobo has 24 goats. He wants to sell

of

his goats. So he puts them in 3 camps and chooses the goats in two camps Complete the drawing.

He sold ___________ goats

Famer Xomo has 24 goats. He wants to sell

of

his goats. So he puts them in 6 camps and chooses the goats in four camps Complete the drawing

He sold ___________ goats

0 1

A 1

2

2

2

2

6

1

6

3

6

4

6 5

6

6

6

0 1

The number line is used for the action of counting. Here we count in halves and in sixths.

39 PST201F/102

Look carefully at the fractions 62

31 and and the fractions

64

32 and

62

31 and

64

32 and

Activity 4.8

Fill in the missing numbers to make the fractions equivalent:

105

3and

102

1and

4

3

1and

123

2and

4.7 Comparing fractions

By comparing fractions, we will decide which part of the same whole is bigger or smaller than another part.

Remember that when you compare fractions, the whole must be kept the same size.

4.7.1 Comparing non-unit fractions

The following activity will guide you towards an understanding of ordering of fractions. We will work with two

types of fractions: ones with the numerators the same, and ones where the denominators are the same.

Activity 4.9

1 Use the wholes given below, shade the given fraction parts. Then arrange the fractions from big to small.

Same numerators:

62

52

42

32

Same denominators:

54

53

52

multiply by 2

multiply by 2

multiply by 2

multiply by 2

= 1

We multiply each fraction by ONE!

The whole must be of the same size

Complete the drawings and

shade

40

4.7.2 Which is bigger?

Examples

1 Compare the following two fractions, using blocked paper. Which is bigger?

?or85

43

Firstly you have to remember that you can only compare fractions if they are parts of the same whole.

Choose a whole that can be divided in 4 as well as 8 equal parts. So we will choose the whole to be 8 blocks.

8

5

4

3

Making use of equivalence: 8

6

4

3

8

5

8

6

2 Compare the following two fractions, using blocked paper. 5

3

3

2and

Choose a whole that can be divided into 3 as well as 5 equal parts. So the whole has to consist of 15 blocks.

5

3

3

2

Making use of equivalence: 15

10

3

2 and

15

9

5

3

15

9

15

10

Activity 4.10 Draw up a worksheet to compare fractions. A set model has to be used.

The whole: 8 blocks

Three-quarters of the whole is shaded

Five eighths of the whole is shaded

The whole: 15 blocks

Two thirds of the whole is shaded

Three-fifths of the whole is shaded

41 PST201F/102

4.8 Addition of fractions

SIMPLE ADDITION

Can you add the following?

?

?

3

1

3

1

?

?

4

1

4

1

4

1

4.8.1 The three stages of teaching the addition of fractions

Stage 1 Denominators the same: (like fractions)

Illustrate 7

4

7

2

7

6

7

4

7

2

Stage 2 One denominator a factor of the other (unlike fractions)

Illustrate 61

31

Into how many parts must the whole be divided?

The algorithm for addition

2

1

6

3

6

1

6

2

6

1

3

1

or

You can only add "like fractions". They have to be the same parts of the whole. The DENOMINATORS have to be the

SAME

Show the whole: 7 blocks

Shade 7

4

7

2 of the whole

Choose a whole that both 3 and 6 can divide into


Shade 6

1

3

1

of the whole

Make denominators the same,

using equivalence

Add numerators. Denominator stays the same.

42

Stage 3 One denominator NOT a factor of the other (unlike fractions)

Illustrate: 41

31


The algorithm for addition

12

7

12

3

12

4

4

1

3

1

Activity 4.11 Do the following examples on quad paper:

1 54

51 2

103

51

3 21

31 4

52

31 5

65

41

4.9 Subtraction of fractions

4.9.1 The three stages of teaching the subtraction of fractions

Stage 1 Denominators the same: (like fractions)

Illustrate 103

107

The algorithm for subtraction 10

4

10

3

10

7


Choose a whole that both 3 and 4 can divide into


using equivalence

Add numerators. Denominator

stays the same.

Shade

of the whole


Shade

of the whole

Shade

of the whole

Subtract

from

=

Subtract numerators. Denominator stays the same.

43 PST201F/102

Stage 2 One denominator a factor of the other (unlike fractions)

Illustrate 52

107 (One fraction must be altered)


The algorithm for subtraction

10

3

10

4

10

7

5

2

10

7

Stage 3 One denominator NOT a factor of the other (unlike fractions)

Illustrate: 21

53 (Both fractions must be altered)


The algorithm for subtraction

10

1

10

5

10

6

2

1

5

3


Shade

of the whole

Shade

of the whole

Subtract

=

Subtract numerators. Denominator

stays the same.


using equivalence


Shade

=

of the whole

Shade

of the whole

Subtract

=

Subtract numerators. Denominator stays the same.

Make denominators the same, using equivalence

44

4.10 The meaning of of

A mathematical explanation

What is

of 8? ______________________

What is 8

? ______________________

The commutative property for multiplication: 3 4 = 4 3

of 8 is the same as 8

which is the same as

8

Therefore

of 8 =

8

A little pizza problem

Princess invited her two friends to come and eat a pizza at her house. When they got there, they found that her brother had already eaten one quarter of the pizza. Shade the part of the pizza that is left What part of the pizza is left to share among the three of them? Each one will get one-third of _________

In fraction notation we write 43

31 of

Illustrate the following, using drawings (Remember that the whole must be kept the same if you want to compare the answers!)

1 21

31 of of the rectangle

2 31

21 of of the rectangle

We usually tell learners that of means times. Do you know why? How will you explain it to your

learners?

Shade

Shade

of the total rectangle

is shaded

Shade

of the total rectangle

is shaded

Shade

45 PST201F/102

4.11 Multiplication of fractions

Can you multiply the following?

3

12

4

13

Complete:

32

3332

311143

Solutions

+

2

3

Pre-knowledge

multiplication of whole numbers

multiplication as repeated addition

commutative property for multiplication

the meaning of "of"

the notion of area

4.11.1 The three stages of teaching the multiplication of fractions

Stage 1 Multiplier a natural number and the multiplicand a fraction

Illustrate 324

Repeated addition

324 = ---- + ---- + ---- + ----

= ----- Solution:

Multiplier multiplicand = product

SHADE!

SHADE!

324 =

3

8

3

2

3

2

3

2

3

2

46

Stage 2 Multiplier a fraction and the multiplicand a natural number Illustrate

432

What does 432 mean? It is the same as 4

3

2of

Here they must understand the concept of "of"

Show 432 of

Shade 432 of wholes

What part is shaded? Solution:

Stage 3

Fraction fraction Pre-knowledge:

the area of a rectangle

3 4 = 12 (units)2

Illustrate: 21

21 of Illustrate

4

1

2

1of

2

1

3

2

3

1

2

1

4 units

3 units

One square unit

Shade a half of the circle.

Divide the half into 3 parts

Double shade 2 of the 3 parts

You have now shaded one third of the

circle

47 PST201F/102

Activity 4.12 Use diagrams to show the following

1 A R5 coin is 2

cm wide. If you put seven R5 coins end to end, how long would they be from beginning

to end?

2 You have

of a pumpkin pie left over from Sunday lunch. You want to give

of it to your sister. How

much of the whole pumpkin pie will this be?

3 Erick Hazan gave

of his money to his wife and spent

of the remainder. If he had R300 left, how

much money did he have at first?

4 David spent

of his money on a storybook. The storybook cost R20. How much money did he have at

first?

5 Penny had a bag of marbles. She gave one-third of them to Rebecca, and then one-fourth of the

remaining marbles to John. Penny then had 24 marbles left in the bag. How many marbles were in the

bag to start with?

4.11.2 The area model.

We know that the area of a rectangle is length times breadth

So if you have to multiply: 3

1

2

1 , we can illustrate this

as follows: Divide the one side into halves, and the other side into thirds.

The shaded part is one sixth of the one-by-one square.

This is the area of a rectangle measuring 3

1

2

1by

Solution: 6

1

3

1

2

1

If you have to multiply: 3

2

2

1 , we can illustrate this as

follows: Divide the one side into halves, and the other side into

thirds. The shaded part is two sixths of the one-by-one

square.

This is the area of a rectangle measuring 3

2

2

1by

1

1

1

1

This is a 1

by 1 square

This block is one-sixth

48

Solution: 6

2

3

2

2

1

Blocked paper makes it easier to use the area model Use the blocked paper to illustrate the following:

1 3

4

8

3 2

5

6

3

5

Solutions

1 3

4

8

3

2

1

24

12

3

4

8

3

2 5

6

3

5

1

1 1

1

1

1

Demarcate the horizontal axis in eights Demarcate the vertical axis in thirds Each little block will now be 1/24 Mark off 3/8 on the horizontal axis Mark off 4/3 on the vertical axis Shade the rectangle that is formed

Demarcate the horizontal axis in thirds Demarcate the vertical axis in fifths Each little block will now be 1/15 Mark off 5/3 on the horizontal axis Mark off 6/5 on the vertical axis Shade the rectangle that is formed

1

1

1

1

49 PST201F/102

215

30

5

6

3

5

4.11.3 An algorithm for multiplication of fractions:

Teachers should work with concrete apparatus (manipulatives) and drawings extensively before leaners get

involved with the algorithm.

From the two examples above, we can see the following:

2

1

24

12

3

4

8

3 and 2

15

30

5

6

3

5

Can you see what happened?

24

12

3

4

8

3

15

30

5

6

3

5

Activity 4.13

Use the multiplication algorithm to find the product of

1

2 15

3 1

8

4

1

Multiply numerators

Multiply denominators

Multiply numerators

Multiply denominators

50

UNIT 5: SHAPE AND SPACE

This unit covers chapter 20 in the text book (pages 426 - 456)


the van Hiele levels of Geometrical thought

Flats shapes

Polygons: properties of triangles and quadrilaterals

Space shapes

Polyhedrons: properties of prisms and pyramids

Nets of polyhedra

Views

5.1 An introduction to shapes

Geometry is possibly one of the most neglected topics in schools. Yet, it is one of the most interesting topics,

which could be made easily understandable to learners, provided they have the proper material to work with.

In this unit, we will make use of different types of materials. We also give this unit a theoretical underpinning,

because teachers should understand how learners learn geometry, and why it is important to follow a definite

line of development and sequence when teaching the concepts of shapes to young learners.

Let us look at the different geometric shapes we see in our daily lives.

All shapes can be classified into two major parts:

Space shapes

o Space shapes are objects that protrude in space (they stand up or stick out; they take up

space) Most of the shapes that we see in our daily lives are space shapes. You yourself is a

space shape.

Flat shapes (we also call them plane shapes)

o Flat shapes are shapes that lie flat. You can put it flat on a desk, and it will not stick up in

space.

What is a plane?

51 PST201F/102

Activity 5.1

Which of the following objects are space shapes, and which are flat shapes?

A telephone

A page in your book

A soccer ball

A stop sign

5.2 The Van Hiele levels of Geometric thought

Study the section on page 427 onwards in your text book.

For us to understand the way learners think about shapes, we will look at research that had been done years

ago by van Hiele. In 1957, Dutch educators Dina van Hiele-Geldof and Pierre van Hiele proposed that the

development of a student's understanding of geometrical concepts, developed through five distinct levels.

These levels are important for us to understand, because it will influence the way in which we teach shape to

learners.

In short, the levels are labelled:

Level 0 Visualization

Level 1 Descriptive

Level 2 Abstract/Relational

Level 3 Formal deduction and proof

Level 4 Rigor

In primary school, we hope learners to achieve level 1, but they will seldom move to level 2. Note that a

learner cannot be taught level 1 information before he or she has not yet achieved level 0. Learners will simply

not be able to make the connections if they have not proceeded with proper teaching on level 0.

Let us look at the first three levels, zero, one and two. Although we might be teaching Intermediate Phase, it

will be very useful to test your learners to decide on which level they find themselves.

LEVEL 0 VISUALIZATION

Level zero deals with what shapes look like.

Learners recognises and name figures on their visual characteristics. They will say: this is a square because it

looks like a square.

Example:

Point out the square(s):

W vd

52

If you place a square in a different position, they might see the square as a diamond, and no longer as a

square.

They might identify a rectangle as a "door shape".

Learners identify and reason about shapes and other geometric configurations based on shapes as visual

wholes rather than on geometry properties. Some properties of the shapes are included in this level, such as

right angles, parallel sides, but only in an informal manner.

LEVEL 1 DESCRIPTIVE

Learners recognise and characterise shapes by their properties

For example, they can identify a rectangle as a shape with opposite sides parallel and four right angles.

When learners investigate a certain shape they come to know the specific properties of that figure. For

example, they will realise that the sides of a square are equal and that the diagonals are equal. Students

discover the properties of a figure but see them in isolation and as having no connection with each other.

Learners at this level still do not see relationships between classes of shapes (e.g., all rectangles are

parallelograms), and they tend to name all properties they know to describe a class, instead of a sufficient set.

LEVEL 2 ABSTRACT/RELATIONAL

Learners are able to form abstract definitions and distinguish between necessary and sufficient sets of

conditions for a class of shapes, recognizing that some properties imply others. When learners reason about

and compare the properties of a figure they realise that there are relationships between them.

The relationships being perceived:

exist between the properties of a specific figure, and

exist between the properties of different figures.

5.1 Comments about the thought levels

Each thought level has its own language, grammar and symbols.

The subject matter that is implicit at one level becomes explicit at the next level.

Memorising is at no level.

Learners pass through the levels in order without skipping any of them.

Not all learners progress through the levels at the same rate.

Learners reasoning at one level will not be able to understand the explanations or answer the

questions given at a higher level.

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5.2 Consequences of the Van Hiele theory for learning

Movement from the first level (visual) to the second level (descriptive) implies the movement from a

level without any relation network (visual) to one with a relation network (descriptive).

At the visual level the child uses language. The function of this is mainly to give the object or situation

a name. This can be called the social knowledge that the child acquires. The child is not in a position

to elaborate on any functions of the object that is called by the specific name. For example, the child

will call a rhombus by its name at this level merely on the overall visual appearance of the shape. The

child will not be able to defend this decision of calling it a rhombus through any logical reasoning

where the properties of the rhombus are required.

One of the main differences between the reasoning at the visual level and that at the descriptive level

lies in the difference in judgement that the child makes. Learning at the visual level relies mainly on

an intuitive understanding of the object or situation. That is why the child does not see the need to

reason about what is experienced. The child will not see the need to reason about the relationships

between a rhombus and a square. The child is so strongly bound by the intuitive knowledge that (s)he

will argue that a square is also a rhombus.

The reasoning that takes place at the third level (abstract relational) relies quite heavily on the

structure of the descriptive level. The judgement that the child is making does not rely on the fact that

there are links between the relation networks but on the relationship between these links.

The different thought levels have a hierarchical development. This implies that thinking on the

descriptive level is not possible unless the visual thought level had been well established.

You as a teacher in the intermediate and/or senior phase should particularly take note of the descriptors of

Level 0, Level 1 and Level 2. That will give you an idea of the types of learning activities that your learners

should be involved. Levels 4 and 5 descriptors are not applicable to learners in the above mentioned two

phases. Consult you text book on page 431.

5.3 Flat shapes

The first activity that young learners should engage in will involve Level 0 of the van Hiele levels of geometric

thought.

Each group of learners will receive a box with a variety of shapes, carefully chosen so that each group has the

correct variety of shapes to classify them according to the teachers request.

In this activity, learners will sort the shapes according to what they SEE. There are not really correct or wrong

answers, because learners might visualize the shapes differently. what is important is that they should be able

to explain to the group or the teacher, why they classify a particular shape in the way they do.

An assortment of flat shapes given to a group of learners. It is important that each learner in your class should

have at least one opportunity to bring a shape to the front of the class, where the teacher can stick the shape

on the blackboard using Prestik.

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Examples of the variety of shapes that could be used. (see van de Walle page 428)

Activity 5.2 This is an exercise for level 0 learners

Classify the shapes above according to the following criteria:

1 Shapes with curved edges

2 Three sides

3 Four sides

4 Opposite sides "go the same way" (parallelograms)

5 Shapes with "dents" (concave)

5.4 Polygons

A polygon is a closed plane (flat) shape made up of line segments. These line

segments must touch only once at their endpoints.

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Activity 5.3

Which of the following are polygons?

Learners should be able to identify, classify and sort. This should be done before the names of the

shapes are taught.

5.4.1 Naming polygons

3 sides: triangle

4 sides quadrilateral

5 sides: pentagon

6 sides hexagon

7 sides: heptagon

8 sides octagon

9 sides nonagon

10 sides: decagon

12 sides dodecagon

20 sides icosagon

many polygon

A polygon is a two dimensional shape with sides made up from line segments. They are simple, closed curves. (page 436, vd Walle)

The affix "gon" means "sides". So a

decagon is a polygon with 10 sides.

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Activity 5.4

Classify the shapes as polygons or non-polygons

5.5 Triangles

Learners should be able to recognise, classify and sort

Which of these are triangles?

5.5.1 Types of triangles

Read the section "Categories of two dimensional shapes" on page 435 436 in vd Walle.

Activity 5.5

Draw the following triangles

1 As right isosceles triangle

2 An acute scale triangle

3 An obtuse scale triangle

4 An obtuse isosceles triangle

5 An equilateral triangle

6 A right angled scalene triangle

A

E

B

F

C

G

D

H

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5.6 Quadrilaterals

Learners should be able to recognise, classify and sort

Which of these are quadrilaterals?

5.6.1 Concepts dealing with quadrilaterals

Activity 5.6 Explain each of the following concepts and draw an example

Explanation Drawing Line segment

Parallel lines

Equal sides

Diagonals

Perpendicular diagonals

Right angles

Opposite sides

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Opposite angles

Bisecting diagonals

Bisecting angles

Adjacent sides

5.6.2 Classification of quadrilaterals

The minimum set of properties that will identify a quadrilateral.

Trapezium One pair of opposite sides parallel

Parallelogram Two pairs of opposite sides parallel

Rhombus All sides equal

Rectangle All angles equal

Kite Two pairs of adjacent sides equal

Square All sides and angles equal

You must understand this classification

Rectangle

Quadrilateral

Trapezium

Parallelogram

Kite

Rhombus

Square The properties below will help you with the classification

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5.7 Space shapes

A space shape protrudes in space. Also called 3D objects or solids

Examples of space shapes

Learners should be able to identify, classify and sort. This should be done before the names of the shapes are

taught.

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Activity 5.7 Classify (draw one example of each)

1 Shapes that will roll

2 Shapes that have triangles

3 All the faces are rectangles

4 Shapes that have a "point"

5 Shapes with parallel faces

6 Make up four more categories

5.7.1 Special space shapes

Polyhedrons

A figure that is not a plane figure, is a space figure. Space figures "take up space".

Some space figures are made up of plane surfaces.

They have:

Faces the flat surfaces (they are all polygonal regions)

Edges where the faces meet (they are all straight lines)

Vertices where the edges meet (they are all points)

A polyhedron is a three dimensional object, whose faces are polygons

We name polyhedra according to the number of faces they have. We use the same prefixes as for polygons,

but the names end in the word "hedron". (penta , hexa , hepta , octa , nona , deca , dodeca , icosa ,

poly ) E.g. a polyhedron with six faces is called a hexahedron.

The smallest number of faces a polyhedron can have is ______. This is called a tetrahedron.

A polyhedron on is a three dimensional shape with specific characteristics.

Activity 5.8

Write down the definition of a polyhedron.

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Activity 5.9

Classify the following as polyhedra or non-polyhedra

Polyhedra are made up of the following:

Faces (they must be polygons)

Edges (where the faces meet)

Vertices (where the edges meet)

Prisms

A prism is a polyhedron with two parallel, identical bases. The Lateral faces are parallelograms. In a RIGHT

prism, the lateral faces are rectangles.

Activity 5.10 When is a polyhedron a prism?

It must have:_______________________

The side faces must be: _________________

Activity 5.11 What is the BASE of each object?

A B C D E

A B

C

D E F

G

Base

Lateral face

Base

Lateral face

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A is called a __________ prism

B is called a __________ prism

C is called a __________ prism

D is called a __________ prism

E is called a __________ prism

Pyramids

A pyramid has BASE and all the other faces are triangles. The vertices of all the

triangles meet in one point, called the apex.

Activity 5.12

A B C D E

What shape is the BASE of each object?

A is called a __________ pyramid (also called a tetrahedron)

B is called a __________ pyramid

C is called a __________ pyramid

D is called a __________ pyramid

E is called a __________ pyramid

In summary:

A triangular prism

A triangular

pyramid

A pentagonal

pyramid

A pentagonal

prism

A square

pyramid

A square

prism

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5.7.2 Regular polyhedra

There are only five regular polyhedra. This was discovered by Plato, and so they are called Platonic solids.

On the following page you will see the five regular polyhedra.

Tetrahedron

Hexahedron

Octahedron

___ faces _____ faces _____ faces

Dodecahedron

Icosahedron

_____ faces _____ faces

5.8 Practice how to draw 3 D objects

Always us a ruler when you draw shapes with edges or face which are polygons.

Some tips on sketching 3D solids:

Cylinder

A rectangular prism

A square pyramid

Cone

Draw base

copy base

Join

Join Draw base

plot a point Draw

base plot a

point Join

Draw base

copy base

move sideways

Join corners

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A tetrahedron (triangular pyramid)

Cube

5.9 Nets of polyhedra

A net is a fold-out (flat) shape that can be folded up into a space shape. We can also make nets for other

space shapes that are not polyhedra.

Here are the nets of the five platonic solids.

Draw base

plot a point

Join Draw

base copy

base Join

Triangular prism

Net of triangular prism

Triangular pyramid

Net of triangular pyramid

Octahedron Tetrahedron Icosahedron

These are the flaps that you need to paste the pieces after

folding

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5.10 Drawings from different views

Learners often find it difficult to draw three dimensional objects. They would draw a square for a cube, or a

triangle for a cone (see section 5.6 in this guide).They also need to develop the ability to draw or recognise

views from different directions.

Start with simple blocks. The best would be to use real cubes and allow

learners to draw the cubes from different perspectives. Allow learners

to look at a cube from top, bottom, left, right, etc. Show them that when

you look at the cube from the corner, you actually see 3 faces, but they

do not all look like squares.

Example

Draw the front, side and top views of the stack of cubes.

Front view: Side view Top view

Dodecahedron

Cube

These two face look like

parallelograms

Top view

Front view

Side view

Top view

Front view

Side view

T

T

F

F

F

S

S

It helps to mark the front, side and top views, or colour them

with different colours

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Activity 5.13

1 Draw the front, side and top view of the following structures

2 Two of the three views of a solid is shown

What is the greatest number of cube units in the solid?

What is the least number of cube units in the solid?

Draw the front views of the solid parts mentioned above.

3 Draw a solid with the following front, side and top views

Top view Side view

Top view Side view Front view

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Appendix Unit 3: Activity for Dienes blocks

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http://www.curriculumsupport.education.nsw.gov.au/secondary/mathematics/assets/pdf/literacyy7/s4placevalu

e2.pdf

72

Dienes Blocks 10

0 B

lock

F

LAT

10 B

locks (cut out) LO

NG

(6)

1 B

lock

s T

INY

(60

) C

ut o

ut Laminate and cut out

10 Blocks (cut out)

LON

G (6)

100

Blo

ck

FLA

T

Make enough copies for all learners

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Appendix Unit 4: Fraction resources

LAMINATE AND CUT OUT

FOR USE IN CLASSROOM

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Teaching of fractions concepts.

How to use the shapes in the classroom

Hand out shapes to learners. Each learner must have at least one shape.

Each learner must show his/her shape and use the correct language pattern. Allow learners to swop shapes

and to repeat the language pattern with the new shape.

Learners have to know the correct vocabulary when dealing with fraction concepts. Learners must say the

following over and over, until they have mastered the correct vocabulary.

The following language patterns must be taught:

Examples

My whole is a triangle.

My whole is divided into two equal parts.

(Now the learner has to show how the two equal parts fit into the triangle).

Each part is one half of my whole.

So two halves make one whole

My whole is a square.

My whole is divided into four equal parts.

(Now the learner has to show how the four equal parts fit into the square).

Each part is one quarter (or one fourth) of my whole.

So four quarters make one whole

My whole is a hexagon.

My whole is divided into three equal parts.

(Now the learner has to show how the three equal parts fit into the hexagon).

Each part is one third of my whole.

So three thirds make one whole My whole is a circle.

My whole is divided into eight equal parts.

(Now the learner has to show how the eight equal parts fit into the circle).

Each part is one eighth of my whole.

So eight eights make one whole My whole is a rectangle.

My whole is divided into six equal parts.

(Now the learner has to show how the six equal parts fit into the rectangle).

Each part is one sixth of my whole.

So six sixths make one whole

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My whole is a pentagon.

My whole is divided into five equal parts.

(Now the learner has to show how the five equal parts fit into the pentagon).

Each part is one fifth of my whole.

So five fifths make one whole

The activity can be extended to ask learners to show:

Two thirds

Three fifths, etc

When teaching fraction concepts, the teacher should refrain from using the symbolic form of a fraction, such

as

,

, etc. Children should learn the correct pronunciation of the fraction in WORDS, and not by saying one

over three or one over five.

maths.pdf

Documents

blocks dienes blocks

face value

total value

understanding place

arabic digits

large numbers

number cards

hinduarabic numeration