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The place value of digits in whole numbers

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MATHEMATICS

Grade 4

WHOLE NUMBERS AND THEIR RELATIONSHIPS

Module 2

THE PLACE VALUE OF DIGITS IN WHOLE NUMBERS

Activity:

Recognise the place value of digits in whole numbers [LO 1.4]

Recognise and represent whole numbers in order to describe and compare them [LU 1.3]

OUR MODERN NUMBER SYSTEM: THE DECIMAL SYSTEM

  • Now that we have done oral counting exercises and mental calculations, we think about the meaning of our wonderful number system.
  • See what Johnny says about Susie. This sounds strange doesn’t it?
Figure 1
Figure 1 (Picture 1.png)
  • 1+ 1 is not eleven! But look at Roman numerals: I + I = II. Then it would be correct, because II is the way the Romans wrote 2. In Activity 5 we shall learn more about Roman numerals.

1. Now let’s look at a bigger number. Just what does the number 1 111 mean, and why? Try to write down what it means:

One might say this is what it means:

Figure 2
Figure 2 (Picture 2.png)

2. What number do you think this diagram represents?

Figure 3
Figure 3 (Picture 3.png)
  • Our decimal system works in groups of loose ones (units), tens, hundreds, thousands and ten thousands. We can have up to nine loose blocks. If we get one more, we say we have ten blocks/ 10 that is, one group of ten and nothing left loose. The “0” fills the empty place to say there is nothing left. With blocks, it would look like this:
Figure 4
Figure 4 (Picture 4.png)
  • Because we cannot always draw blocks, we use the POSITION of the digits to tell us the size of the group. So we have place value:
Table 1
THOUSANDS HUNDREDS TENS UNITS
1 000 100 10 1
10 x 10 x 10 10 x 10 10 1

Recap: Our Decimal Number System

In our number system we have nine symbols and “0”. We use these symbols, 1; 2; 3; 4; 5; 6; 7; 8; 9 and 0 to make any and all the numbers we need. We use the position of the digit in the number to indicate its value. So in the number 2 768 the 7 means 700 because of where it is in the number.

Figure 5
Figure 5 (Picture 5.png)

If there are no thousands (or digits in the other columns) we use 0 as a place holder.

Note: the 0 cannot be left out. If we left out the 0 the value of the whole number would change (e.g. 10 291 would become 1 291) so the 0 is very important.

  1. Now write each of the numbers below in EXPANDED NOTATION. The one at the top of the page looks like this: 2 768 = 2 000  700  60  8

Now complete the ones below:

Table 2
2 768 = 2 000  700  60  8
7 834 =
2 056 =
8 503 =
1 940 =
16 473 =
25 809 =

Note also:

When we write big numbers we leave a space between the thousands and the hundreds. This makes it easier to read the number. Key 10 403 into your calculator. Unfortunately the calculator does not leave this space. Do you see it is not so easy to read this number on the calculator when there is no space between the thousands and the hundreds? Remember to leave the space in the correct place when you are writing big numbers.

MAKING NUMBERS AND ARRANGING THEM IN ORDER

  • We have seen how each digit in a number has a value, for example:

3 967 = 3 000  900  60  7.

It can be written in columns like this:

Table 3
THOUSANDS1 000 HUNDREDS100 TENS10 UNITS1
3 9 6 7

Because there are:

3 × 1 000  9 × 100  6 × 10  7

4. Now create the largest and the smallest numbers with the digits: 2; 8; 4; 1. Write them and two other numbers, still using only the digits 2; 8; 4; 1 in columns:

Which of your numbers above is the largest number?

  • Now write your numbers. Begin with the largest, then the next largest, then the next largest, until you reach the smallest. This is called DESCENDING ORDER. (Moses DESCENDED from the mountain)

Descending order is when you start with the largest number and go Down

Example: 10; 9; 8; 7; 6; 5; 4; 3; 2; 1

4.2 Ascending order is when you start with the smallest and go up! Now write your numbers in ascending order. Remember to begin with the smallest:

4.3 Write these numbers from the smallest to the largest:

6 095; 9 065; 6 059; 9 506; 5 069

4.4 Write these numbers from the largest to the smallest:

8 315; 3 851; 5 318; 1 853; 8 513

5. EVEN AND ODD NUMBERS

5.1 Study the number line below:

Figure 6
Figure 6 (Picture 6.png)

All the numbers that have been written there are even numbers. They can be shared equally between two friends.

5.2 Between the even numbers are odd numbers. They cannot be kept whole and shared equally between two people. Fill in the names of the odd numbers on the number line below:

Figure 7
Figure 7 (Picture 7.png)

TEST YOUR KNOWLEDGE of odd and even numbers

a. List the even numbers between 2 800 and 2 812

b. Which odd number is just before 10 000?

c. What is the first even number after 2 998?

6. LARGER AND SMALLER

LARGER THAN / SMALLER THAN

In Mathematics, this sign > means: LARGER THAN or GREATER THAN:

This sign < means: SMALLER THAN or LESS THAN

(Remember, the crocodile always opens his mouth towards the largest number because he is so hungry!)

Figure 8
Figure 8 (Picture 8.png)

500 > 400

or 500 is greater than 400

< 500

or 400 is less than 500

TEST YOUR UNDERSTANDING OF THESE SIGNS:

  • Fill in the correct sign from : <; =; >

a) 0  4 * 11 － 3

b) 13 － 6 * 0  7

c) 2  7 * 14 － 8

d) 13 － 5 * 7  4

6.2 Write down the missing number when you count in tens:

1 470 < …… ………….. < 1 490.

7. CALCULATOR GAME

Now you may play another calculator game. Try to puzzle out what the learners are doing this time. Then play the game with your friend.

Figure 9
Figure 9 (Picture 9.png)

Paul keyed in 187. He keyed in one operator and pressed = and the number 1 870 appeared on the screen. What operator did he key in? Yes, it was: X 10 because 187 X 10 = 1 870.

What operator did Reyhana key in? Yes, it was also X 10 because 1 870 X 10 = 18 700.

8. Now see if you can complete this table without a calculator. Then check your answers with a friend. (If you get stuck you may use a calculator.)

Table 4
Number Operator Answer
58 X 100  
145 X 10  
309   3 090
20  10  
1 000   10 000
520   52
1 690  10  
1 000  100  
10 000  10  

16 329

Hello! I am called Six Thousand. I am part of a very large number, which is: Sixteen thousand three hundred and twenty-nine.

9. Now write down the value of the digit that has been printed in bold type:

3 421

8 035

926

14 051

Now let us look at the number 2 848.

The 8 on the left means 800. The 8 on the right means 8.

What is the difference betweenthe values of the two 8s?

800 － 8 = 792

10. Now calculate the difference between the values of the numbers that have been made bold and underlined:

7 374

6 995

3 023

5 519

10 010

11. Now you may play a “place value” game with a friend and a calculator. This will strengthen your understanding of “place value”. It is important to play this game.

See if you can learn this game by reading what the two learners said:

Figure 10
Figure 10 (Picture 10.png)

The game continues until all the digits have been replaced by 0.

12. Now that you have played the “place value” game, try to do this exercise. Replace the digit that has been made bold (dark) with 0. (The first two have been done for you.)

Table 5
Number, bold digit to be replaced by 0 My suggestion: what to do. Calculator answer Π ρ What I should have done:
1 356 － 6 = 1 350 Π - -
2519 － 200 = 2 319   ρ － 2 000 =
6 723          
15 638          
13 642          
17 389          
590          
14 843          
7394          

1.3 Look at the first one again. Would it be correct to say  4? Yes, that is correct: 1 356  4 = 1 360 so we have replaced the 6 with a 0!

TEST YOUR SKILLS: PLACE VALUE and DESCRIBING AND COMPARING WHOLE NUMBERS

Now that you have learnt all about the importance of place value see if you can use this know ledge to complete the following exercises:

1. Write down the number that consists of:

6 000  0  20  9

2.1 Write the largest possible whole number with the digits:

6 ; 0; 9; 2; 7

2.2 Write down the odd number immediately before 4 521.

2.3 Write down the next even number:3 008.

3. What is the value of: the 6 in 16 708?

4. The number 17 538 is on the screen of my calculator. How can I change the 7 to 0 by keying in one instruction and = ?

5. Write down the whole number:

5.1 that is just before 10 000

5.2 that is just after 1 000

5.3 that is greater than 998 and less than 1 000

5.4 that is between 5 009 and 5 011

6. Write down the answers:

  • 347 － 47 =
  • 347 － 37 =
  • 254 － 54 =
  • 254 － 64 =
  • 254 － 44 =

TESTING YOUR PROGRESS SO FAR

18 408

1. Use the number in the frame to complete the following:

  • What Number System do we use?

  • What number symbols do we use to make all our numbers? Write them all down:

  • Write down the value of the underlined figure in the frame above.

1.4 Write down the value of the 8.

  1. a) on the left ………………… b) on the right.........…...............

1.5 What number will you have if you leave out the “0”?

1.6 Add 4 to the number in the frame 18 408 + 4 =

1.7 Write the number in the frame in expanded notation:

1 × ………………. + 8 × ………………. + 4 × ……………..+ ………… × 10 + …………

1.8 You are counting in 2s. Begin with the number in the frame and write down the next 5 numbers:

2. Write down the missing numbers in this sequence:

18 408; 18 508; 18 608; 18 708; ……………..; …………………..; …………………

Assessment

Table 6
Learning outcomes(LOs)
 
LO 1
Numbers, Operations and RelationshipsThe learner will be able to recognise, describe and represent numbers and their relationships, and to count, estimate, calculate and check with competence and confidence in solving problems.
Assessment standards(ASs)
 
We know this when the learner:
1.1 count forwards and backwards in a variety of intervals (including 2s; 3s; 5s; 10s; 25s; 50s and 100s) between 0 and 10 000;
1.2 describes and illustrates various ways of counting in different cultures (including local) throughout history;
1.3 recognizes and represents the following numbers in order to describe and compare them:1.3.1 whole numbers to at least 4-digit numbers;
1.4 recognizes the place value of digits in whole numbers to at least 4-digit numbers;

Memorandum

ACTIVITY 1: PLACE VALUE

1. 1 000 + 100 + 10 + 1

2. 1 000 + 300 + 20 + 6 = 1 326

3. 7 834 = 7 000 + 800 + 30 + 4

2 056 = 2 000 + 0 + 50 + 6

8 503 = 8 000 + 500 + 0 + 3

1 940 = 1 000 + 900 + 400 + 0

16 473 = 10 000 + 6 000 + 400 + 70 + 3

25 809 = 20 000 + 5 000 + 800 + 0 + 9

4. Note: using 2; 8; 4; 1 there can be many numbers; room has been given for 4 numbers only, so all

Table 7
THOUSANDS1 000 HUNDREDS100 TENS10 UNITS1
2 8 4 1
2 4 8 1
2 1 8 4
2 8 1 4
2 4 1 8
2 1 4 8
8 4 2 1
8 4 1 2
8 2 4 1
8 2 1 4
8 1 2 4
8 1 4 2
4 8 2 1
4 8 1 2
4 2 8 1
4 2 1 8
4 1 8 2
4 1 2 8
1 8 2 4
1 8 4 2
1 4 8 2
1 4 2 8
1 2 8 4
1 2 4 8

Possibly not all learners will notice that there could be all these numbers, but some learners are sure to do so. This could start valuable discussion. The largest possible number is 8 421. The smallest is 1 248.

4.1 Again only 4 steps have been provided. Check that their answers are in descending order, e.g. 8 421; 4 821; 2 841; 1 248

Note: it is not necessary for the learners to put all the possible numbers into descending order; four numbers are sufficient to show that they understand the meaning of descending order.

4.2 Again, four numbers are sufficient to show that the learner understands the meaning of ascending order. Please check each learner’s answers.

E.g.: 1 248; 2 148; 4 218; 8 421

4.3 5 069; 6 059; 6 095; 9 065; 9 506

  • 8 513; 8 315; 5 318; 3 851; 1 853
  • 1; 3; 5; 7; 9; 11; 13; 15

TEST YOUR KNOWLEDGE: odd and even numbers

1. 2 802; 2 894; 2 806; 2 808; 2 810

2. 9 999

3. 3 000

6.LARGER AND SMALLER ; < ; =

6.1 (a) < (b) = c)  (d) <

  • 1 480

7.

Table 8
Number Operator Answer
58 X 100 5 800
145 X 10 1 450
309 X 10 3 090
20  10 2
1 000 X 10 10 000
520  10 52
1 690  10 169
1 000  100 10
10 000  10 1 000

9. 3 000; 8 000

20; 10 000

10.

Table 9
7 374 7 000 –70 = 6 930
6 995 9 00 – 90 = 810
3 023 3 000 – 3 = 2 997
5 519 5 000 – 500 = 4 500
2 454 400 –4 = 396
10 010 10 000 – 10 =9 990

11. Game

12. The first two are examples:

Table 10
Number, bold digit to be replaced by 0 What to do Calculator answer      
1 356 - 6or: + 4 1 3501 360      
2 519 - 200 2 319   × - 2 000 =
6 723 - 700or: + 300 6 0237 023      
15 638 - 30or: + 70 15 60815 708      
13 642 -10 000or: + 90 000 3 642103 642      
17 389 - 7 000or: + 3 000 10 38920 389      
590 - 500or: + 500 901 090      
14 843 - 3or: + 7 14 84014 850      
7 394 - 300or: + 700 7 0948 094      

13. Yes

TEST YOUR SKILLS: PLACE VALUE

  1. 6 029
  • 97 620 2.2 4 519 2.3 3 010

3. 6 000

  1. - 7 000 of + 3 000
  • 9 999 5.2 1 001
  • 999 5.4 5 010
  • 300
  • 310
  • 200
  • 190
  • 210

TESTING YOUR PROGRESS SO FAR

  • Decimal
  • 0; 1; 2; 3; 4; … 9
  • 400
  • (a) 8 000 (b) 8
  • 1 848
  • 18 412
  • 1 x 10 000 + 8 x 1 000 + 4 x 100 + 0 + 8
  • 18 410; 18 412; 18414; 18 416; 18 418

2. 18 808; 18 908; 19 008

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