Odd numbers

Here you will learn about odd numbers, including examples of odd numbers, odd numbers on a number line and a number chart, and properties of odd numbers.

Students will first learn about odd numbers as part of operations and algebraic thinking in 2nd grade. They expand upon their knowledge of odd numbers in 3rd grade when they identify arithmetic patterns and properties of numbers.

What are odd numbers?

Odd numbers are whole numbers that are not divisible by 2 without remainders.

The last digit of an odd number is 1, 3, 5, 7, or 9.

Odd numbers are the opposite of even numbers, which are divisible by 2. An even number’s last digit is 0, 2, 4, 6, or 8.

For example

To identify an odd number, you can look at the last digit, use a number line, or use a number chart.

Last digit	The last digit of a number, or the digit in the ones place, will tell you if it is even or odd. The last digit of an odd number will always be \bf{1, 3, 5, 7,} or \bf{9} . 2,631,903,42\underline{3} In this number, the last digit is 3, so it is an odd number.
Number line	Start with a number whose last digit is 1, 3, 5, 7, or 9 and jump every other number, or count by twos.
Number chart	On a number chart each column that begins with 1, 3, 5, 7, or 9 includes odd numbers. This chart shows a list of odd numbers between 1 and 100.

Properties of even numbers and odd numbers

• Property of addition
  • If you add an even number to an even number, the sum will always be an even number. For example, 8 + 4 = 12 .
  • If you add an even number to an odd number, the sum will always be an odd number. For example, 8 + 3 = 11 .
  • If you add an odd number to an odd number, the sum will always be an even number. For example, 5 + 3 = 8 .

• Property of subtraction
  • If you subtract an even number from an even number, the difference will always be an even number. For example, 16-10 = 6 .
  • If you subtract an even number from an odd number or an odd number from an even number, the difference will always be an odd number. For example, 16-9 = 7; 17-10 = 7 .
  • If you subtract an odd number from an odd number, the difference will always be an even number. For example, 9-5 = 4 .

• Property of multiplication
  • If you multiply an even number by an even number, the product will always be an even number. For example, 4 \times 2 = 8.
  • If you multiply an even number by an odd number, the product will always be an even number. For example, 4 \times 5 = 20 .
  • If you multiply an odd number by an odd number, the product will always be an odd number. For example, 3 \times 5 = 15 .

Groups of objects

• To determine if a group of objects has an even number of objects or an odd number of objects, group the objects into pairs, or equal groups of 2. If each object can be grouped into a pair, there is an even number of objects. If there is one object left over after pairing, there is an odd number of objects.

For example,

Since all of the objects in this group can be grouped into pairs, there is an even number of objects.	After grouping the objects into pairs, there is one left over. Therefore, there is an odd number of objects in this group.

What are odd numbers?

Common Core State Standards

How does this relate to 2nd grade math?

• Grade 2 – Operations and Algebraic Thinking (2.OA.3)
  Determine whether a group of objects (up to 20 ) has an odd or even number of members, for example, by pairing objects or counting them by 2 s; write an equation to express an even number as a sum of two equal addends.

How to identify odd numbers

In order to identify odd numbers:

1. Look at the last digit, or the digit in the ones place. If the digit is \bf{1, 3, 5, 7,} or \bf{9,} the number is odd.
2. Use this strategy to answer the question.

In order to determine if the answer to an equation will be an odd number:

1. Recall the properties of addition, subtraction, or multiplication for odd numbers.
2. Apply the correct property.

In order to determine if there is an odd number of objects in a group:

1. Group the objects into pairs.
2. If all objects can be grouped into pairs, there is an even number of objects. If there is one left over, there is an odd number of objects.

Odd numbers examples

Example 1: identifying odd numbers

True or false: 823 is an odd number.

1. Look at the last digit, or the digit in the ones place. If the digit is \bf{1, 3, 5, 7,} or \bf{9,} the number is odd.

The last digit, or the digit in the ones place, is 3.

2Use this strategy to answer the question.

Since the last digit in 823 is 3, it is an odd number, so the statement is true.

Example 2: identifying odd numbers

List all consecutive odd numbers between 61 and 71.

Example 3: identifying odd numbers

Look at the list of numbers. Which numbers are odd?

17, 438, 282, 500, 901, 744, 355, 483, 809, 620

Look at the last digit, or the digit in the ones place. If the digit is \bf{1, 3, 5, 7,} or \bf{9,} the number is odd.

1\underline{7}, 43\underline{8}, 28\underline{2}, 50\underline{0}, 90\underline{1}, 74\underline{4}, 35\underline{5}, 48\underline{3}, 80\underline{9}, 62\underline{0}

Example 4: apply properties of odd numbers

Joe is subtracting an odd number from an odd number. He thinks the difference will be an odd number, but his friend Samira says it will be even. Who is correct?

Recall the properties of addition, subtraction, or multiplication for odd numbers.

Example 5: apply properties of odd numbers

Anna solves the following equation:

105,915 + 27,863 = 133,775

Her friend glances at the equation and tells her it’s incorrect. How does her friend know this is incorrect?

Recall the properties of addition, subtraction, or multiplication for even numbers.

Example 6: determine if a group of objects is odd

Look at the objects in the square. Is there an odd number of objects?

Group the objects into pairs.

Teaching tips for odd numbers

• Hang a number chart, or hundreds chart, in your classroom with odd numbers highlighted.

• Provide worksheets with practice questions that require students to find odd numbers between any given numbers. For example, instead of always starting at 0, give them a starting number of 63 and ask them to find the odd numbers through 100.

• Provide worksheets with practice questions that ask students to find the next odd numbers and find consecutive odd numbers to build on their understanding of odd numbers. Allow students to use a number line if needed.

Easy mistakes to make

• Thinking you can’t count by \bf{2} s using odd numbers
  Students may think that you can only count by 2 s using even numbers since odd numbers are not multiples of 2. However, if you start at an odd digit and count by 2 s, each number will be odd.

• Thinking that fractions and decimals can be odd numbers
  Only integers (whole numbers and their corresponding negative numbers) can be odd numbers. Students may think that a number such as 2.43 is an odd number because its last digit is a 3. However, this is not the case as fractions and decimals can not be even numbers or odd numbers.

Related types of numbers lessons

• Types of numbers
• Absolute value
• Prime numbers
• Rational numbers
• Natural numbers
• Even numbers
• Number sets
• Prime and composite numbers
• Integers
• Whole numbers

The next lessons are

• Rounding numbers
• Factors and multiples
• Fractions
• Number patterns

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