topic1 whole number

INTRODUCTION

Number is a basic concept in mathematics. Why do we teach pupils the concept of number? A good understanding of numbers is important to build the foundation for computational skills. To develop number sense of your pupils, you will teach them to recognise, read and write whole numbers, compare and arrange numbers. Pupils in Year Three learn about whole numbers up to 10 000 and basic number concepts. Understandings of number concepts include knowing place value of numbers to the thousands, and round off numbers to tens, hundreds, and thousands. You have to make sure that the pupils are very comfortable with place value and estimation. These are the two most important topics in this chapter. In addition, your pupils need to acquire computational skills such as addition

TTooppiicc 11 Whole

Numbers

LEARNING OUTCOMES

By the end of this topic, you should be able to:

1. Define the vocabulary related to whole numbers correctly;

2. List the major mathematical skills and basic pedagogical content knowledge, related to whole numbers;

3. List the major mathematical skills and basic pedagogical content knowledge, related to addition and subtraction; and

4. Apply teaching and learning activities for the topic of whole numbers for Year Three.

TOPIC 1 WHOLE NUMBERS 2

and subtraction operations within the range of 10 000 as well. Another thing, we have to tell the pupils why we need to do additions and subtractions of big numbers. Addition and subtraction have been taught in Year One and Two. These two operations are subsequently taught every year by reviewing operations learnt previously and extending algorithms for work with bigger numbers. Often the pupils are able to do the computation without understanding why it works. You as a teacher should be able to explain to the pupils why it works.

PEDAGOGICAL CONTENT KNOWLEDGE

One of the most important aspects of a number system is place value. You know, there are many number systems in existence; but without the place value you cannot do much with it. If you know Roman numerals, try to add two numbers in this system. School children often have difficulty understanding the concept of place value. Well, human civilisation took a long time to come out with the place system

1.1.1 Place Value

The idea of place value must be taught properly at an early age. This is to avoid misconceptions (or mislearning). They fail to make the connection between numbers and place value, so that when two numbers are added and the value exceeds 9; they have to understand that the number will move to the next ÂcolumnÊ. When the number moves to the left, it increases its value. Simple questions as follows should be emphasised (although they are supposed to have learnt it before). What is the place value of the 5 in 34856? Hence the pupil will know that the value of a digit in a numeral is dependent on its position. If the pupil has problem at this stage, they will bigger problem when it comes to addition, subtraction, multiplication and division.

1.1.2 Numbers to 10 000

In the previous year, the pupil has been exposed to numbers. From their everyday experience, they also deal with numbers. Almost always, these

1.1

TOPIC 1 WHOLE NUMBERS ! 3

numbers are not that big. However, the pupils look at these numbers as size or quantity of something. This is OK because the starting point for numeration is counting. In the Year 2 syllabus, your pupils have learnt numbers to 1000. However, counting does not stop with 1000. Seldom do they see the numbers as symbolic representation of the quantities they count. In this manner, they have difficulty of imagining higher numbers. Another difficulty they encounter is the word used to vocalise these big numbers. The English language can be strange at times especially to non-native English speaking pupil. At this stage try not to introduce words like ÂtenthsÊ, ÂhundredthsÊ or ÂthousandthsÊ. These words have different meanings to tens, hundreds and thousands. Students must be taught the vocabulary related to place value. Without it, they will continue to struggle, especially when symbols are brought into the picture. Example of Exercises 1. Let the pupil say numbers such 100 as one hundred. The number 1000 as one thousand. The number 10 000 as ten thousand. Continue with other numbers such as 6 000, 10 000 and so on and let

the pupils say it themselves. 2. You must give more complex examples, first writing the numbers, and

then say it in words. For instance, you write: 7191 The pupils will say this as „seven one nine one.‰ Once they are comfortable with this, you may then asked the pupil to

say „seven thousand one hundred and ninety one.‰ It is very important at this stage that your pupils really understand the concept of place value. It is true that they have learned it in previous year. They might be able to read and write a number, but we would like them to understand what they are doing. There are a number of activities at the end of this chapter to help them understand this concept.


1.1.3 Addition and subtraction

In teaching addition and subtraction of four-digit numbers, you should pay attention to the aspects of:

(a) Understanding of concepts and the process of regrouping; and

(b) Solving problems involving addition and subtraction in real life situations.

In planning and carrying out teaching and learning strategies, you should be attentive of the following activities.

(Source: Teacher Education Division, Malaysia. 1998. Primary School Mathematics Teaching and Learning of Whole Numbers Module. Kuala Lumpur. Page 36.)

Addition is related to counting on. Subtraction is related to counting back. Your pupils acquire the concrete experience informally. You use concrete objects to demonstrate before introducing formal mathematics symbols. You teach the regrouping process using teaching aids such as Multi-based blocks or abacus. The addition operation is related to the subtraction operation. Subtraction is the inverse of addition. Problem solving questions are based on real life situations. Your pupils have to solve non-routine problems so as to develop their critical and creative thinking skills.

1. We read numbers from left to right. A way of reading a four-digit number is to cover the last three digits, read the first digit as thousands, then read the three remaining digits in hundreds, tens and ones.

The pupil is given a big number. Ask them to break it up into ten-thousand, thousand, hundred, ten and one.

ACTIVITY 1.1


Before you continue, be sure to give a quiz as follows:

1. True or False: 546 is the same as 500 + 40 + 6

2. 501 + 158 = ?

3. 3782 1393 = ?

Relationship of counting and operations To build the concept of four-digit numbers, you may tell your pupils to count on by ones, tens, hundreds, and thousands. Odometer is a name given to the principle which describes the nature of all place-value positions to count like the ones position. For example:

(Adapted from: Tom Cooper, School of mathematics, Science and Technology

Education, QUT, 1999)

You have learnt number lines in the earlier modules. Again number lines can be used to show the position of the four-digit numbers. In the Figure 1.1 below, indicate somewhere that the first is the unit position and the second one the position of the one thousand.

1. Search through the last few years of the Arithmetic Teacher (available on the internet). Read an article on the teaching and learning of addition or subtraction of four-digit numbers using non-routine problems.

2. Discuss your article with your coursemates and tutor.

ACTIVITY 1.2


Figure 1.1: The unit position The material for teaching counting on is a calculator. For example, enter 5467 and then add 100 and keep pressing = . Similarly, you may tell your pupils to count back. For example:

Count on is similar to addition. For example,

On the other hand, count back is like subtraction. For example,

Using abacus Do you know what an abacus is? It can be used to show four-digit numbers and perform addition and subtraction operations subsequently. Piaget (1972) said that primary school pupils understand mathematics concept through concrete experience. Thus teaching aids play an important role in showing mathematics concept effectively (refer to Figure 1.2).


Figure 1.2: Parts of the abacus

Source: Chuah Lay Thiam, Institut Perguruan Temenggong Ibrahim, Johor Baru.

Movement of a lower bead indicates 1. Movement of upper bead indicates 5. The first rod on the right is the place value for ones. The next rods indicate tens, hundreds, and thousands. Refer to Figure 1.3 for the rods indication of lower and upper.

move up 1, move down 1

1) Thumb :

move up 1

2) Forefinger :

move down 1



1) Middle finger : 2) Middle finger:

move up 5 move down 5

2) Middle finger and

forefinger :

move down 9 at the

same time.


1) Thumb and middle

finger :

move up 9 at the

same time.

Figure 1.3: Rods indication of lower and upper

Traditional Algorithms In the modern approach in teaching mathematics, you introduce various strategies in written algorithms for performing addition and subtraction operations. To understand the process of addition and subtraction algorithm, consider the following example, which makes use of blocks. You may also use coloured cards.

1. Use an abacus to perform addition and subtraction of four-digit numbers. Demonstrate to your pupils the ways to do these operations by using abacus.

2. What are the benefits of using abacus to perform addtion and

subtraction? Discuss with your tutor and coursemates.

ACTIVITY 1.3


Figure 1.4: Model and language of the problem

Source: Adapted from Tom Cooper, School of Mathematics, Science and

Technology Education, QUT, 1999

Figure 1.5 shows two examples of algorithms for addition and Figure 1.6 illustrates two examples of algorithms for subtraction.

Figure 1.5: Examples of algorithms for addition

You may then ask the pupil what it all means. Take for example the sum 1467 + 1422. You start by breaking up the number into parts. Hence

PROBLEM: There are 1497 pupils in a boyÊs school and 1422 pupils in a girlsÊs school. How many pupils are there in the two schools? MODEL AND Show the 1497 Thousands ! Hundreds! Tens Ones LANGUAGE Ú. … Show the 1422 Thousands Hundreds Tens Ones . . Thousands Hundreds Tens Ones What do you want? [The total] ! ÚÚ What is the answer? Ú [2 thousands 9 hundreds 1 ten and 9 ones]


1437 = (1000) + (400) + (30) + 7 1422 = (1000) + (400) + (20) + 2 You then add up the various place values, to get (1000+1000) + (400+400) + (30+20) + (7+2) = 2000 + 800 + 50 + 9 = 2859 Now, make it slight more complicated: (1497 + 1422) That is: 1497 = (1000) + (400) + (90) + 7 1422 = (1000) + (400) + (20) + 2 Straight away you can see a problem with the addition (90 + 20). That is OK. Write it as 90 + 20 = (110) = (100) + (10) So the sum becomes: 1497 + 1422 = (1000+1000) + (400 + 400 + 100) + (10) + (7+2) The same principle can be used for subtraction. Whatever way you are showing, make sure there is a pattern. Once the pupil can see the pattern, they are on the way to abstract thinking.

Figure 1.6: Illustrates examples of algorithms for subtraction

Source: Adapted from Gan Teck Hock, et al, 2003, KPLI Mathematics Module,

Teacher Education Division, Ministry of Education Malaysia, Page 81


1.1.4 Estimation and Approximation

We teach Year 3 pupils to estimate quantities of objects up to 1000. This can be done by asking the pupils to count in tens, hundreds, and thousands.

Exercise Give the pupil several numbers. Asked them, to round the numbers to nearest 10s, 100s and 1000s. This re-enforce their understanding of place value. For example, 1234 = 1230 to nearest 10 This skill is very important. Sometimes the pupils make simple mistakes in additions (or subtraction). Most often they use a calculator and punch in the wrong number. You may ask them to check the calculation by having an approximate answer. For example, the pupil might make a mistake in the following addition: 1467 + 1422 = 22912 You may then ask them to „ball-park‰ the numbers. We then asked then to round-up to nearest 1000 first to see whether they are on the right track. 1467 + 1422 is (1000 + 1000) to nearest 1000, that is 2000. The pupil can straight away they see something is wrong.

Study the algorithms for addition and subtraction carefully. Which of these algorithms do you think is easier for your pupils to follow? Why?

Share your views with your coursemates.

ACTIVITY 1.4

When you buy a packet of ground nuts, can you estimate the number of ground nuts in the packet?

Can you suggest some ways to enhance the skill of estimation?

ACTIVITY 1.5


We also hope the pupil to have the ability to visually approximate things. We consider the following case. We take several glass beakers of the same size, put in water according to the amount shown in the figure above. We now asked to pupils to estimate to the amount of water in the first beaker. You must ask the pupils to think through the problem first. This is a higher level thinking and you need to ask them questions as to what is the proper procedure. When your pupils have acquired the skills of estimation and approximation, they wound be able to check answers in their calculation quickly. In planning the strategies for teaching and learning activities for the topic of estimation and approximation, you have to ensure that your pupils acquire skills in

(a) Estimation:

(i) Estimate to make wise decision; and

(ii) Count mentally.

(b) Approximation:

(i) They understand the concepts of place value of ones, tens and hundreds.

(ii) They know how to round off to an approximate value based on the place value.

In the activity below, pupils are asked to round up the price of some items. Although the pupils have not been formally introduced to money, by now they should have an intuitive idea of how to do some simple manipulation of money.


STAGES OF CONCEPTUAL DEVELOPMENT FOR WHOLE NUMBERS IN YEAR 3

1.2.1 Numbers to 10 000

In the teaching and learning of numbers to 10 000, you may follow the following sequence. Your pupils can learn more effectively as they follow a systematic approach. Major mathematics skills to be covered include:

(a) Say and use the number names in familiar contexts.

(b) Read and write numbers to 10 000.

1.2

Item Price Shampoo RM 9.90 Shower Gel RM 7.90 Tooth Paste RM 8.05 Tooth Brush RM 5.15 Soap RM 2.40

Try out the following activities. 1. Find the total cost of the items in the list. Here the pupils might

face some difficulties dealing with decimals. They need proper guidance in this instance:

(a) Find the estimation of the total cost of the items excluding the value of sen.

(b) Find the estimated total cost of the items with rounding off to the nearest ringgit.

2. Compare the answer in each case. Ask the pupil to draw some

simple conclusion?

ACTIVITY 1.6


(c) Know what each digit in a number represents.

(d) Understand and use the vocabulary of comparing and arranging numbers or quantities to 10 000.

(e) Understand and use the vocabulary of estimation and approximation.

1.2.2 Addition and subtraction within the range of 10000

Major mathematical skills include:

(a) Understand addition as combining two groups of objects.

(b) Use and apply knowledge of addition in real life.

(c) Understand subtraction as „take away‰ or „difference‰ between two groups of objects.

(d) Recognise subtraction as the inverse of addition.

(e) Use and apply knowledge of subtraction in real life.

SAMPLES OF TEACHING AND LEARNING ACTIVITIES

The following are samples of teaching and learning activities for various mathematical skills in this topic. You can try out with your pupils.

1.3.1 Numbers to 10 000

Activity 1: Number representation

(a) Learning Outcomes (of pupil):

(i) Recognise numerals to 10 000

(ii) Count up to 10 000 objects by grouping them into thousands, hundreds and tens.

1.3

1. Name three examples of four-digit numbers that are used in real life.

2. What are the teaching aids suitable for teaching and learning of four-digit numbers and the addition and subtraction operations?

SELF-CHECK 1.1


(iii) Recognise the place value of numbers

(iv) Compare two numbers and say which is more or less. (b) Materials:

Four numeral cards, multi-based blocks, place value chart, pictures of cubes, big squares, rectangles, and small squares, worksheet.

(c) Procedures:

(i) Divide the class into groups of four.

(ii) Give each group a beg containing numeral cards, some multi-based blocks, some and some worksheets.

(iii) Appoint a pupil in the group to be the recorder.

(iv) Each pupil takes a numeral card. Pupils discuss to form two different four-digit numbers using the numeral cards.

(v) The recorder writes down the two numbers in the worksheet.

(vi) Other pupils in the group show the two numbers using multi-based blocks.

(vii) Group members cut the relevant pictures and paste them on the place value chart accordingly, then they compare the two numbers.

(viii) Allow group members to discuss their answers.

(ix) Repeat steps (4) to (8).


Activity 2: At the car park (Out-door)

(a) Learning Outcome: Position numbers in order on a number line. (b) Materials: Pencils, writing paper, erasers, cars, number lines with two positions,

four positions and eight positions. (c) Procedures:

(i) Divide the class in groups of four. Each group is given worksheet with number lines.

(ii) Bring pupils to the car park so that they can write down numbers by reading the car number plates. Each pupil jot down two numbers.

(iii) Pupils compare two numbers individually and write down the numbers on a number line of two positions.

(iv) Pupils compare four numbers in pairs. Allow pupils to discuss and write down the numbers on a number line of four positions.

For example:

(v) Pupils compare eight numbers in groups. After discussion, the recorder writes down the numbers on a number line with eight positions.

Activity 3: Tic - Tac - Toe

(a) Learning outcome: Round whole numbers less than 10 000 to the nearest 10. (b) Materials: Dice, answer paper with 9 squares, nine numbers less than 10 000. (c) Procedures:

(i) Pupils play this game in pairs.

(ii) Throw a die to decide who should start first.


(iii) Pupil A chooses a number and round off to the nearest 10. Match the answer on the answer sheet. Put a mark on the answer sheet with a cross or a nought.

(iv) Pupil B repeat step (3).

(v) The first person who gets a straight line on the answer sheet with right answers is the winner.

1.3.2 Addition and subtraction within the range of 10 000

Activity 4: Numbers and operations

(a) Learning Outcomes:

(i) Write numerals to 10 000.

(ii) Round whole numbers less than 10 000 to the nearest 10.

(iii) Add two four-digit numbers with the highest total of 10 000.

(iv) Subtract a three-digit number from a four-digit number. (b) Materials:

Cubes with numerals, a 16-hole tray, worksheet, abacus.


(c) Procedures:

(i) Divide the class into groups of four.

(ii) Give each group 16 dice, and some worksheets.

(iii) Each group member takes four cubes and put into the 16-hole tray.

(iv) Each pupil identifies a four-digit number and round off to the nearest hundreds. The numbers can be formed by reading the numerals on the dice from left to right, right to left, top to down, down to top or at any corner.

Question: Identify a four-digit number with the digit (6) as the numeral in the place value of tens. Round off your number to the nearest hundreds.

Answer: 4626 round off to 4600, 4253 round off to 4300, round off to 4200, 5612 round off to 5600.

(v) Each pupil writes a number sentence of addition or subtraction.

Question: Add two four-digit numbers. Round off your answer to the nearest tens.

Answer: 5612 + 2624 = 8240

Question: Subtract a three-digit number from a four-digit number.

Round off your answer to the nearest tens.

Answer: 4215 - 652 = 3560

(vi) Pupils are encouraged to use the abacus.

(vii) A pupil checks the calculation of his peer.

(viii) Repeat steps (3) to (7).

Source: Adapted from Ehsan, Kinta Teacher Training College, Ipoh


Activity 5: Problem solving

(a) Learning outcomes:

(i) Solve problems involving addition in real life.

(ii) Solve problems involving subtraction in real situations. (b) Materials:

A set of 6 problems on addition and subtraction, cubes with answers, colour pencils.

(c) Procedures:

(i) Pupils play this game in groups.

(ii) Each group is given a cube with answers randomly written on the faces, two colour pencils, a set of 6 questions. For example:

You want to buy a computer of RM2500. You only have RM1877 in your bank account. How much more money you need to save?

You have collected 578 stems. Your brother has collected 753 stems. How many stems all together?

You walk 1360 metres on the jogging track in the park. Your sister walks 2566 metres. How many more metres your sister walks?

A hawker sells 3450 rambutans. Another hawker sells 4670 rambutans. How many rambutans do they sell?

(iii) The pupils chose the questions to answer. After working out the answer, they match the answer on the cube and colour a face. Each pupil uses a different colour.

(iv) The first pupil who makes a path across the squares with his / her colour (top to bottom or right to left) wins.

1. Do you think abacus is suitable to be used in Activity 4?

Discuss with your coursemates.

ACTIVITY 1.7


Activity 6: Treasure Hunt

(a) Learning Outcome: Subtract numbers within the range of 10 000.

(b) Materials:

A worksheet with 8 questions on addition and subtraction. A map giving direction on the route for treasure hunt.

(c) Procedures:

(i) Each pupil is given a worksheet and a map on treasure hunt.

(ii) Pupils solve the questions on the worksheet.

(iii) Find the answers on the map.

(iv) Colour the space that contains the answer.

(v) Follow the track of coloured numbers to claim the treasure.

1640 63 =

2610 432 =

3577 888 =

4420 3752 =

2100 88 =

3060 399 =

5050 935 =

7765- 4592 =

766 6745 =


Figure 1.7: Treasure hunt

Source: Adapted from Enrichment and remedial mathematics programme, Inspire

project, University of Science, Malaysia, 1986

ACTIVITY 1.8

1. List six mathematical skills related to whole numbers.

2. Why do we use teaching aids in teaching whole numbers, and addition and subtraction operations? Discuss.

3. Plan a teaching and learning activity for addition with the highest total of 10 000.


Numerals are symbols used to represent numbers.

Concrete object such as based-ten blocks and abacus help your pupils to understand the idea of place value, and addition and subtraction algorithms.

The skills of estimation and approximation enable your pupils to check answers in calculation quickly.

There are various algorithms to perform addition and subtraction. There is no best algorithm to teach these two operations.

Teaching and learning activities should be engaging and interesting to your pupils.

Abacus

Addition

Estimation and approximation

Pedagogical content knowledge

Place value

Subtraction

Traditional algorithms

Whole numbers

Llewellyn, S. & Greer, A. (1996) Mathematics The basic skills. Britain: Stanley Thornes.

Reys, R. E. & Suydam, M. N. & Lindquist, M. M. (1989). Helping children

learn mathematics. New Jersey: Prentice Hall. Smith, Karl J. (2001). The nature of mathematics. US: Thomson Learning.

topic1 whole number

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