Place Value, Ordering Whole Numbers and Roman Numerals for Primary 4

Place Value, Ordering Whole Numbers and Roman Numerals

Performance Objectives

By the end of the lesson, students should be able to:

1. determine the place value of digits;
2. order whole numbers;
3. write in Roman numerals up to 1000;
4. solve problems on quantitative aptitude.

Understanding Place Value

Place value is the value of a digit based on its position in a number. Every digit in a number has a specific place, and this place tells its value. For example, in the number 23, the digit ‘2’ is in the tens place, and the digit ‘3’ is in the units place.

In Primary 4, numbers can have up to four digits. These places are Units, Tens, Hundreds, and Thousands.

1. Units (U): This is the first digit from the right. It represents single items.
2. Tens (T): This is the second digit from the right. It represents groups of ten.
3. Hundreds (H): This is the third digit from the right. It represents groups of one hundred.
4. Thousands (Th): This is the fourth digit from the right. It represents groups of one thousand.

Understanding place value helps in reading, writing, and working with larger numbers.

Determining the Place Value of Digits

To determine the place value of a digit, identify its position in the number. The value of the digit is the digit itself multiplied by the value of its place.

Consider the number 5,283:

1. The digit 3 is in the Units place. Its value is $3 \times 1 = 3$.
2. The digit 8 is in the Tens place. Its value is $8 \times 10 = 80$.
3. The digit 2 is in the Hundreds place. Its value is $2 \times 100 = 200$.
4. The digit 5 is in the Thousands place. Its value is $5 \times 1000 = 5000$.

Worked Examples: Place Value

Example 1: What is the place value of the digit 7 in the number 4,721?

Solution:

In the number 4,721:

1. 1 is in the Units place.
2. 2 is in the Tens place.
3. 7 is in the Hundreds place.
4. 4 is in the Thousands place.

The digit 7 is in the Hundreds place. Its value is $7 \times 100 = 700$.

Example 2: Write the place value of each digit in the number 9,065.

Solution:

1. Digit 5 is in the Units place. Its value is $5 \times 1 = 5$.
2. Digit 6 is in the Tens place. Its value is $6 \times 10 = 60$.
3. Digit 0 is in the Hundreds place. Its value is $0 \times 100 = 0$.
4. Digit 9 is in the Thousands place. Its value is $9 \times 1000 = 9000$.

Expanding Numbers

Expanding a number means writing it as the sum of the values of its digits. This shows how each digit contributes to the total value of the number.

How to Expand Numbers

To expand a number, follow these steps:

1. Identify the place value of each digit in the number.
2. Multiply each digit by its place value.
3. Add these values together.

Worked Examples: Expanding Numbers

Example 1: Expand the number 678.

Solution:

1. 6 is in the Hundreds place, so its value is $6 \times 100 = 600$.
2. 7 is in the Tens place, so its value is $7 \times 10 = 70$.
3. 8 is in the Units place, so its value is $8 \times 1 = 8$.

Expanded form: $600 + 70 + 8$.

Example 2: Expand the number 2,345.

Solution:

1. 2 is in the Thousands place, so its value is $2 \times 1000 = 2000$.
2. 3 is in the Hundreds place, so its value is $3 \times 100 = 300$.
3. 4 is in the Tens place, so its value is $4 \times 10 = 40$.
4. 5 is in the Units place, so its value is $5 \times 1 = 5$.

Expanded form: $2000 + 300 + 40 + 5$.

Example 3: Expand the number 8,009.

Solution:

1. 8 is in the Thousands place, so its value is $8 \times 1000 = 8000$.
2. 0 is in the Hundreds place, so its value is $0 \times 100 = 0$.
3. 0 is in the Tens place, so its value is $0 \times 10 = 0$.
4. 9 is in the Units place, so its value is $9 \times 1 = 9$.

Expanded form: $8000 + 0 + 0 + 9 = 8000 + 9$.

Ordering Whole Numbers

Ordering whole numbers means arranging them from smallest to largest (ascending order) or from largest to smallest (descending order). To do this, compare the numbers using their place values.

Comparing Whole Numbers

To compare two or more whole numbers:

1. Count the number of digits: The number with more digits is usually larger. For example, 543 is larger than 98 because 543 has three digits and 98 has two digits.
2. Compare digits from left to right: If numbers have the same number of digits, start comparing the digits from the leftmost place value (the largest place value).
   1. If the digits in the largest place value are different, the number with the larger digit in that place is the greater number.
   2. If the digits are the same, move to the next place value to the right and compare those digits. Continue this process until a difference is found.

Symbols used for comparison are:

1. $ > $ (greater than)
2. $ < $ (less than)
3. $ = $ (equal to)

Worked Examples: Comparing Numbers

Example 1: Compare 789 and 798.

Solution:

1. Both numbers have three digits.
2. Compare the Hundreds place: Both have 7.
3. Compare the Tens place: 8 in 789 and 9 in 798. Since 9 is greater than 8, 798 is greater than 789.

So, $789 < 798$ or $798 > 789$.

Example 2: Compare 1,234 and 1,230.

Solution:

1. Both numbers have four digits.
2. Thousands place: Both have 1.
3. Hundreds place: Both have 2.
4. Tens place: Both have 3.
5. Units place: 4 in 1,234 and 0 in 1,230. Since 4 is greater than 0, 1,234 is greater than 1,230.

So, $1,234 > 1,230$.

Ordering Numbers in Ascending Order

Ascending order means arranging numbers from the smallest to the largest.

Worked Examples: Ascending Order

Example 1: Arrange the following numbers in ascending order: 345, 123, 567, 210.

Solution:

1. Compare the numbers: 123 is the smallest.
2. Next is 210.
3. Then 345.
4. Finally, 567 is the largest.

Ascending order: 123, 210, 345, 567.

Example 2: Arrange the following numbers in ascending order: 4,567, 4,500, 4,600, 4,560.

Solution:

1. All numbers have 4 in the Thousands place.
2. Compare the Hundreds place: 4,500, 4,560, 4,567 all have 5 in the Hundreds place. 4,600 has 6. So 4,600 is the largest.
3. Now compare 4,500, 4,560, 4,567.
   1. Tens place: 4,500 has 0. 4,560 and 4,567 have 6. So 4,500 is the smallest among these three.
   2. Compare 4,560 and 4,567. Units place: 4,560 has 0. 4,567 has 7. So 4,560 is smaller than 4,567.

Ascending order: 4,500, 4,560, 4,567, 4,600.

Ordering Numbers in Descending Order

Descending order means arranging numbers from the largest to the smallest.

Worked Examples: Descending Order

Example 1: Arrange the following numbers in descending order: 987, 789, 897, 978.

Solution:

1. Compare the numbers: 987 is the largest.
2. Next is 978.
3. Then 897.
4. Finally, 789 is the smallest.

Descending order: 987, 978, 897, 789.

Example 2: Arrange the following numbers in descending order: 1,000, 999, 1,001, 990.

Solution:

1. Compare the numbers: 1,001 is the largest.
2. Next is 1,000.
3. Then 999.
4. Finally, 990 is the smallest.

Descending order: 1,001, 1,000, 999, 990.

Roman Numerals

Roman numerals are a system of numerical notation used by the ancient Romans. They use letters of the alphabet to represent numbers. The main symbols are:

1. I = 1
2. V = 5
3. X = 10
4. L = 50
5. C = 100
6. D = 500
7. M = 1000

Rules for Writing Roman Numerals

To write numbers using Roman numerals, follow these rules:

1. Repetition: A symbol can be repeated up to three times to add its value.
   1. II = $1 + 1 = 2$
   2. III = $1 + 1 + 1 = 3$
   3. XX = $10 + 10 = 20$
   4. CCC = $100 + 100 + 100 = 300$

   Symbols V, L, and D are never repeated.

2. Addition: If a smaller value symbol is placed after a larger value symbol, their values are added.
   1. VI = $5 + 1 = 6$
   2. XI = $10 + 1 = 11$
   3. LX = $50 + 10 = 60$
   4. CL = $100 + 50 = 150$
3. Subtraction: If a smaller value symbol is placed before a larger value symbol, the smaller value is subtracted from the larger value. This rule applies only to specific pairs:
   1. I can be subtracted from V and X (IV = $5 – 1 = 4$, IX = $10 – 1 = 9$).
   2. X can be subtracted from L and C (XL = $50 – 10 = 40$, XC = $100 – 10 = 90$).
   3. C can be subtracted from D and M (CD = $500 – 100 = 400$, CM = $1000 – 100 = 900$).

   Only one smaller symbol can be placed before a larger symbol for subtraction.

Writing Numbers in Roman Numerals (up to 1000)

To write a number in Roman numerals, break it down into its place values and apply the rules.

Worked Examples: Writing Roman Numerals

Example 1: Write 7 in Roman numerals.

Solution: $7 = 5 + 1 + 1 = \text{VII}$.

Example 2: Write 14 in Roman numerals.

Solution: $14 = 10 + 4 = \text{X} + \text{IV} = \text{XIV}$.

Example 3: Write 39 in Roman numerals.

Solution: $39 = 30 + 9 = \text{XXX} + \text{IX} = \text{XXXIX}$.

Example 4: Write 45 in Roman numerals.

Solution: $45 = 40 + 5 = \text{XL} + \text{V} = \text{XLV}$.

Example 5: Write 99 in Roman numerals.

Solution: $99 = 90 + 9 = \text{XC} + \text{IX} = \text{XCIX}$.

Example 6: Write 250 in Roman numerals.

Solution: $250 = 200 + 50 = \text{CC} + \text{L} = \text{CCL}$.

Example 7: Write 400 in Roman numerals.

Solution: $400 = \text{CD}$.

Example 8: Write 678 in Roman numerals.

Solution: $678 = 600 + 70 + 8 = \text{DC} + \text{LXX} + \text{VIII} = \text{DCLXXVIII}$.

Example 9: Write 999 in Roman numerals.

Solution: $999 = 900 + 90 + 9 = \text{CM} + \text{XC} + \text{IX} = \text{CMXCIX}$.

Example 10: Write 1000 in Roman numerals.

Solution: $1000 = \text{M}$.

Converting Roman Numerals to Standard Numbers

To convert Roman numerals to standard numbers, read the symbols from left to right and apply the addition and subtraction rules.

Worked Examples: Converting Roman Numerals

Example 1: Convert XIV to a standard number.

Solution: $\text{XIV} = 10 + (5 – 1) = 10 + 4 = 14$.

Example 2: Convert LIX to a standard number.

Solution: $\text{LIX} = 50 + (10 – 1) = 50 + 9 = 59$.

Example 3: Convert CCXL to a standard number.

Solution: $\text{CCXL} = 100 + 100 + (50 – 10) = 200 + 40 = 240$.

Example 4: Convert DCLXXV to a standard number.

Solution: $\text{DCLXXV} = 500 + 100 + 50 + 10 + 10 + 5 = 675$.

Example 5: Convert CMXCIX to a standard number.

Solution: $\text{CMXCIX} = (1000 – 100) + (100 – 10) + (10 – 1) = 900 + 90 + 9 = 999$.

Quantitative Reasoning

Quantitative reasoning involves solving problems that require understanding numbers, patterns, and relationships. These problems often combine different mathematical concepts like place value and Roman numerals.

Solving Quantitative Aptitude Problems on Place Values

These problems test the understanding of how digits contribute to the value of a number based on their position.

Worked Examples: Quantitative Reasoning (Place Value)

Example 1: What is the difference between the place value of 8 in 8,450 and the place value of 8 in 3,821?

Solution:

1. In 8,450, the digit 8 is in the Thousands place. Its place value is $8 \times 1000 = 8000$.
2. In 3,821, the digit 8 is in the Hundreds place. Its place value is $8 \times 100 = 800$.
3. Difference = $8000 – 800 = 7200$.

The difference is 7200.

Example 2: A number has 5 in the Thousands place, 0 in the Hundreds place, 7 in the Tens place, and 2 in the Units place. What is the number?

Solution:

1. Thousands place: 5
2. Hundreds place: 0
3. Tens place: 7
4. Units place: 2

The number is 5,072.

Example 3: If you swap the digits in the Hundreds and Units places of the number 6,123, what new number do you get?

Solution:

1. Original number: 6,123
2. Hundreds digit: 1
3. Units digit: 3
4. Swap them: The new Hundreds digit is 3, and the new Units digit is 1.

The new number is 6,321.

Solving Quantitative Aptitude Problems on Roman Numerals

These problems involve converting between standard numbers and Roman numerals, or performing simple operations with them.

Worked Examples: Quantitative Reasoning (Roman Numerals)

Example 1: What is the sum of the standard numbers represented by VI and IV?

Solution:

1. VI = $5 + 1 = 6$
2. IV = $5 – 1 = 4$
3. Sum = $6 + 4 = 10$.

The sum is 10.

Example 2: Which Roman numeral comes just before XL?

Solution:

1. XL = $50 – 10 = 40$.
2. The number before 40 is 39.
3. 39 in Roman numerals is XXXIX.

The Roman numeral is XXXIX.

Example 3: If a clock shows the time as IX, what is the time in standard numbers?

Solution:

1. IX = $10 – 1 = 9$.

The time is 9 o’clock.