Topic 1 Whole Numbers.pdf

INTRODUCTION

Welcome to the first topic of Teaching of Elementary Mathematics Part IV. What is your expectation of this topic? Well, this topic has been designed to assist you in teaching whole numbers to primary school pupils in Years Five and Six. For hundreds of years, computational skills with paper-and-pencil algorithms have been viewed as an essential component of children’s mathematical achievement. However, calculators are now readily available to relieve the burden of computation, but the ability to use algorithms is still considered essential. In An Agenda for Action (NCTM, 2000, p. 6), the NCTM standards support the decreased emphasis on performing paper-and-pencil calculations with numbers more than two digits. Most of the operations in this topic will cover the content area of whole numbers to 1,000,000 in KBSR Mathematics.

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Whole Numbers

LEARNING OUTCOMES

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

1. Explain the importance of developing number sense for whole numbers to 1,000,000 in KBSR Mathematics;

2. List the major mathematical skills and basic pedagogical contentknowledge related to whole numbers to 1,000,000;

3. Show how to use the vocabulary related to addition, subtraction, multiplication and division of whole numbers correctly;

4. List the major mathematical skills and basic pedagogical contentknowledge related to addition, subtraction, multiplication and divisionof whole numbers in the range of 1,000,000; and

5. Plan basic teaching and learning activities for whole numbers,addition, subtraction, multiplication and division of whole numbers inthe range of 1,000,000.

TOPIC 1 WHOLE NUMBERS

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PEDAGOGICAL CONTENT KNOWLEDGE

Computation with whole numbers continues to be the focus of KBSR Mathematics. Thus, when you observe a classroom mathematics lesson, there is a high probability you will find a lesson related to computation being taught. The National Council of Teachers of Mathematics (NCTM) emphasises the importance of computational fluency, that is, “having efficient and accurate methods for computing” (NCTM, 2000, pg. 152). Computational fluency includes children being able to flexibly choose computational methods, understand these methods, explain these methods, and produce answers accurately and efficiently.

1.1.1 Whole Numbers Computation

A common but rather narrow view of whole numbers computation is that it is a sequence of steps to arrive at an answer. These sequence or step-by-step procedures are commonly referred to as algorithms. Tell your pupils, that there are three important points that need to be emphasised when teachers talk about whole numbers computation. (a) Computation is much broader than using just standard paper-and-pencil

algorithms. It should also include estimation, mental computation, and the use of a calculator. Estimation and mental computation often make better use of good number sense and place-value concepts.

(b) Children should be allowed ample time and opportunity to create and use

their own algorithms. The following shows a child’s procedure for subtracting (Cochran, Barson, & Davis, 1970):

64 - 28 - 4 +40 36

1.1

ACTIVITY 1.1

Talk to children in your classroom about the algorithms they use tosolve problems. Describe these algorithms.

TOPIC 1 WHOLE NUMBERS 3

What is the child doing? His thinking could be as follows: “4 minus 8 is -4, 60 minus 20 is 40. -4 plus 40 is 36”!

This child’s method might not make sense to all or most children, however,

it did make sense to that child, which makes it a powerful and effective method for him at that moment.

(c) There is no one correct algorithm. Computational procedures may be altered

depending on the situation. There are many algorithms that are efficient and meaningful. For this reason, teachers should be familiar with some of the more common alternative algorithms.

Alternative algorithms may help children develop flexible mathematical

thinking and may also serve as reinforcement, enrichment, and remedial objectives.

1.1.2 Estimation and Mental Computation

Estimation and mental computation skills should be developed along with paper-and-pencil computation because these help children to spot unreasonable answers. Teachers should also provide various sources for computational creativity for children. (a) Mental Computation Sometimes, we need to do mental computation to estimate the quantity or

volume. Mental computation involves finding an exact answer without the aid of paper and pencil, calculators, or any other device. Mental computation can enhance understanding of numeration, number properties, and operations and promote problem solving and flexible thinking (Reys, 1985; Reys and Reys, 1990).

When children compute mentally, they will develop their own strategies

and, in the process, develop good number sense. Good number sense helps pupils use strategies effectively. Teachers should explain to the children how to do mental computation. You should also encourage children to share and explain how they did a problem in their heads. Children often can learn new strategies by hearing their classmates’ explanations.

Mental computation is often employed even when a calculator is used. For

example, when adding 1,350, 785, 448, and 1,150, a child with good number sense will mentally compute “1,350 plus 1,150” and key in 2,500 into the calculator before entering the other numbers (Sowder, 1990).


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(b) Estimation You should know that estimation involves finding an approximate answer.

Estimation may also employ mental computation, but the end result is only an approximate answer. Teachers should ensure that children are aware of the difference between Mental Computation and Estimation.

Reys (1986) describes four strategies for whole number computational

estimation. They are the front-end strategy, rounding strategy, clustering strategy, and compatible number strategy. The definition of each strategy is as follows:

(i) Front-end strategy The front-end strategy focuses on the left-most or highest place-value

digits. For example, for children using this strategy they would estimate the difference between 542 and 238 by subtracting the front-end digits, 5 and 2, and estimate the answer as 300.

(ii) Rounding strategy Children using this rounding strategy would round 542 to 500 and 238

to 200 and estimate the difference between the numbers as 300.

(iii) Clustering strategy The clustering strategy is used when a set of numbers is close to each

other in value. For example, to find the sum of 170 + 290 + 230, children would first add 170 and 230 to get 400, and then they can estimate the sum of 400 + 290, so it’s about 700.

(iv) Compatible number strategy When using the compatible number strategy, children adjust the

numbers so that they are easier to work with. For example, to estimate the answer for 332 , they would note that 333 is close to 332 and is divisible by 3, and that would give an estimated answer of 111.

1.1.3 Computational Procedure

When teachers engage their children in the four number operations of addition, subtraction, multiplication and division, it is important that they pay special attention to the following points:

(a) Use models for computation Concrete models, such as bundled sticks and base-ten blocks help children

to visualise the problem.

(b) Use estimation and mental computation These strategies help children to determine if their answers are reasonable.


(c) Develop bridging algorithms to connect problems, models, estimation and symbols

Bridging algorithms help children connect manipulative materials with symbols in order to make sense of the symbolic representation.

(d) Develop time-tested algorithms These algorithms can be developed meaningfully through the use of

mathematical language and models.

(e) The teacher poses story problems set in real-world contexts. Children are able to determine the reasonableness of their answers when

story problems are based in familiar and real-world contexts.

MAJOR MATHEMATICAL SKILLS FOR WHOLE NUMBERS

The introduction of the basics of whole number skills will help children to learn higher mathematical skills more effectively. Teachers should note that before children learn to name and write numbers they will already have developed considerable number sense. The major mathematical skills to be mastered by your pupil when studying the topic of whole numbers are as follows:

(a) Name and write numbers up to 1,000,000.

(b) Determine the place value of the digits in any whole number up to 1,000,000.

(c) Compare value of numbers up to 1,000,000.

(d) Round off numbers to the nearest tens, hundreds, thousands, ten thousands and hundred thousands.

(e) Add any two to four numbers to 1,000,000.

(f) Subtract one number from a bigger number less than 1,000,000.

(g) Subtract successively from a bigger number less than 1,000,000.

1.2

SELF-CHECK 1.1

1. Explain the three important points that need to be emphasisedwhen teaching whole number computations.

2. Explain Reys’ four strategies for whole number computationalestimation.


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(h) Solve addition and subtraction problems.

(i) Multiply up to five digit numbers with a one-digit number, a two-digit number, 10, 100 and 1,000.

(j) Divide numbers up to six digits by a one-digit number, a two-digit number, 10, 100 and 1,000.

(k) Solve problems involving multiplication and division.

(l) Calculate mixed operations of whole numbers involving multiplication and division.

(m) Solve problems involving mixed operations of division and multiplication.

TEACHING AND LEARNING ACTIVITIES

There are a few activities that can be carried out with pupils for better understanding about this topic.

1.3.1 Basic Operations of Whole Number

Now, let us look at a few activities to learn the basic operations of whole numbers in class.

1.3

ACTIVITY 1.2

Learning Outcome:

To practise the algorithms of addition. Materials:

Clean writing papers; and

Task Sheet as below Procedures:

1. Divide the class into groups of four.

2. Give each pair some clean writing paper and a Task Sheet.


3. Each pupil in the group takes turn to fill in numerals from 0 to 9 randomly on the Task Sheet.

4. The teacher gives the instruction for addition by saying, Find the sum of any three three-digit numbers.

5. Each pupil identifies three three-digit numbers by reading the numerals from the square from left to right, right to left, top to bottom, bottom to top or even diagonally.

Each pupil in the group checks the calculation of their peers using the calculator.

Example: 841 + 859 + 768 = 2,469

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6

7

4

5

3

1

0

9

6. The winner for this round is the pupil with the highest sum and is

awarded 5 points.

7. Pupils in the group repeat steps (5) and (6) when the teacher gives the instruction for the next addition.

8. The teacher summarises the lesson on addition.


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In subsequent sections, some examples are provided for pupils to practise the algorithms of addition, subtraction multiplication and division. The next section discusses subtraction using the calculator and estimation of the product of two numbers. Let us look at Activity 1.3 first.

Learning Outcome:

To practise the algorithms of addition.

To increase the understanding of place value. Materials:

10 cards numbered 0 through 9

Task Sheet as below Procedures: 1. Divide the class into groups of four. 2. Give each pair some clean writing paper and a Task Sheet.

3. Each pupil in the group takes turns to draw a card and announces

the number on it. All players in the group write this number in oneof the addend boxes on the Task Sheet. Once a number has beenwritten on the Task Sheet, it cannot be moved or changed.

4. Replace the card and shuffle the cards.

5. Repeat steps (3) and (4) until all addend boxes are filled.

6. Pupils will compute their respective sum.

7. The winner is the pupil with the greatest sum and is awarded 5points.

8. Repeat steps (3) through (7) until the teacher stops the game.

ACTIVITY 1.3


Learning Outcomes:

To practise subtraction using the calculator.

To practise the algorithms of subtraction.

To increase the understanding of place value. Materials: Calculator Clean writing papers Procedures:

1. Pupils play this game in pairs.

2. Give each pair a calculator and some clean writing paper.

3. Throw a dice to decide who should start first.

4. Pupil A chooses three different single-digit numbers. For example: 1, 2, and 4.

5. Enter the selected digits into the calculator in order to create the largest number possible.

6. Enter “-“

7. Next, enter the same three selected digits to create the smallest number possible followed by the “=” sign.

Example: The largest number created from the three single-digit

numbers is 421. The smallest number created from the three single-digit numbers is 124.

421 - 124 297

ACTIVITY 1.4


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ACTIVITY 1

8. Repeat steps (5) through (7) with the digits 2, 7 and 9 (derived from the first subtraction) as shown below.

421 972 963 - 124 - 279 - 369 297 693 594

954 - 459 495

9. Pupil B will have to write out all the algorithms of the subtractions and Pupil A will check it.

10. If Pupil B had carried out all the subtractions correctly, the answer will eventually yield the magic number 495!

11. Pupil B repeats steps (4) through (8).

12. The game continues until the teacher instructs the the pupils to stop.

13. The teacher summarises the lesson on subtraction.


Learning Outcomes:

To estimate the product of two numbers.

To practise the algorithms of multiplication. Materials:

Calculator

Task Sheet as given Procedures:


2. Give each group some clean writing paper, a calculator and a Task Sheet.

3. Working in their group pupils will discuss the best strategy to fillin the missing numbers in the boxes.

4. Pupils will compute the algorithm of multiplication and fill in theblank boxes.

5. The winner is the group who obtained the correct answer in the shortest time.

6. Members of the winning group will explain to the class theirstrategy and also the algorithm of multiplication.

7. Teacher summarises the lesson on multiplication.

ACTIVITY 1.5


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ACTIVITY 1 TASK SHEET 1. Use only the numbers 4, 5, 6, 7, 8 and 9 to make

The largest possible product

X

The smallest possible product

X

2. Use your calculator to help you find the missing number.

X

8 6

2

1 9 2

5 9+


Learning Outcome:

Using calculators to develop number sense involving division. Materials:

Task Sheet

Four calculators Procedures:


2. Provide each group some clean writing papers, a Task Sheet andfour calculators.

3. Teacher explains the rules and starts the game.

4. Pupils will compete against members of their own group.

5. Pupils will use the calculator to determine a reasonable dividendand divisor.

6. The winner is the one in the group with the dividend and divisorthat results in a quotient closest to the target number.

Example: Target Number = 6,438

Entered into the calculator: 32,195 5 Followed by = (within 5 sec.) : Display shows “6,439”

7. The winner will explain to the group members his strategy indetermining a reasonable answer.

8. The teacher summarises the lesson on division.

ACTIVITY 1.6


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TASK SHEET Target Numbers

446 815 845 490

6,438 654 8,523 6,658

29,881 31,455 44,467 51,118

Pick a target number and circle it.

Enter any number into your calculator.

Press the key.

Enter another number that you think will give you a product close to the target number.

Press the “=” key to determine your answer.

How close are you to the target number?


1.3.2 Estimation and Mental Computation

Below are the activities you can use to teach your pupils about estimation and mental computation.

Learning Outcomes:

To recognise patterns in whole number operations.

To practise estimation and computation of whole numbers. Materials: Calculator Procedures:


2. Ask each member of the group to choose a two-digit number.

3. Using the calculator ask them to multiply their numbers by 99.

4. Pupils in their group record and compare their results.

5. Ask them if they can see a pattern or relationship in their answers.

6. In their groups pupils will write a statement describing theirpattern.

7. Ask pupils to predict the results of multiplying 5 other numbers by99.

8. Repeat steps (2) through (7) but this time multiply the numbers by 999.

9. Ask pupils to compare results obtained from multiplication by 99and 999 and write statements describing the pattern

- The same as the one for two-digit numbers x 99.

- Different from the two-digit numbers x 999.

ACTIVITY 1.7


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Learning Outcome:

To practise estimation and computation of whole numbers. Materials:

Calculator

Task Sheet Procedures:


2. Give each group some clean writing paper, a calculator and a Task Sheet.

3. In their groups, ask pupils to discuss the best strategy to fill in themissing numbers.

4. Pupils will compute the algorithm of division and fill in the blankboxes.

5. The winner is the group that arrives at the correct answer in the shortest time.

6. Members of the winning group will explain to the class their strategy and also the algorithm of division.

7. Teacher summarises the lesson on division.

ACTIVITY 1.8


TASK SHEET 1. Use only the numbers 4, 5, 6, 7, 8 and 9 to make

The largest possible answer

)

The smallest possible answer

)

2. Use your calculator to help you find the missing number.

5 R 2

8 ) 6

0 7

8 ) 2 8


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In this topic, we have learned :

To explain the importance of developing number sense for whole numbers to 1,000,000 in KBSR Mathematics.

The major mathematical skills and basic pedagogical content knowledge related to whole numbers to 1,000,000.

How to use the vocabulary related to addition, subtraction, multiplication and division of whole numbers correctly.

The major mathematical skills and basic pedagogical content knowledge related to addition, subtraction, multiplication and division of whole numbers in the range of 1,000,000.

To plan basic teaching and learning activities for whole numbers, as well as the addition, subtraction, multiplication and division of whole numbers in the range of 1,000,000.

Addition

Division

Multiplication

Place value

Subtraction

Whole numbers

Hatfield, M. M., Edwards, N. T., & Bitter, G. G. (1993). Mathematics methods for

the elementary and middle school. Needham Heights, MA: Allyn & Bacon.

Kennedy, L. M., & Tipps, S. (2000). Guiding children’s learning of mathematics. US: Allyn &Wadsworth.

Rucker, W. E., & Dilley, C.A. (1981). Heath mathematics. Washington, DC: Heath and Company.


Tucker, B. F., & Weaver, T. L. (2006). Teaching mathematics to all children. Ohio: Merill Prentice Hall.

Vance, J. H., & Cathcart, W. G. (2006). Learning mathematics in elementary and middle schools. Ohio: Merrill Prentice Hall.

Documents

Topic 1 Whole Numbers.pdf