Generalising with odd and even numbers
🎯 In this topic you will
- Make and test general statements involving addition and subtraction of odd and even numbers.
🧠 Key Words
- counter-example
- even
- generalisation (general statement)
- odd
Show Definitions
- counter-example: An example that shows a general statement or rule is not always true.
- even: A whole number that can be divided exactly by 2, leaving no remainder.
- generalisation (general statement): A rule or statement that describes a pattern believed to be true for all cases.
- odd: A whole number that cannot be divided exactly by 2 and leaves a remainder of 1.
Odd numbers as L-shapes
Each L-shape shown is made from an odd number of dots. No matter how large the shape is, counting the dots always gives an odd total.
Combining two L-shapes
When two similar L-shapes are placed together, they form a rectangle. The total number of dots in each rectangle is even.
A pattern with odd numbers
For example, adding two odd numbers such as 3 and 3, 5 and 5, or 7 and 7 always results in an even number.
General statements in mathematics
A statement that uses the words odd and even to represent any odd or any even number is called a generalisation, or general statement. This kind of statement works for all examples.
What this section is about
In this section, you will add and subtract odd and even numbers and use patterns to explain what happens.
Key result: $\text{odd} + \text{odd} = \text{even}$
❓ EXERCISES
1. Find three examples that match these general statements.
a. The sum of two even numbers is even.
b. The sum of three odd numbers is odd.
👀 Show answer
a. Examples include $2 + 4 = 6$, $8 + 10 = 18$, and $12 + 6 = 18$.
b. Examples include $1 + 3 + 5 = 9$, $7 + 9 + 11 = 27$, and $3 + 5 + 7 = 15$.
2. Here are three cards.
Choose one card to complete this sentence.
When you add two odd numbers together the answer is ____ .
👀 Show answer
3. Here are six digit cards.
Use three cards to show the difference between two even numbers is even.
Think of two other even numbers and show the difference between them. Does this also show that the difference between two even numbers is even?
👀 Show answer
4. Hassan says, ‘Adding two odd numbers always gives an odd number answer’. Give a counter-example to show that Hassan is wrong.
👀 Show answer
5. Martha says, ‘I added three even numbers and my answer was $25$’. Explain why Martha cannot be correct. Discuss your answer with a partner.
👀 Show answer
6. Salem says, ‘When you add $5$ to any number the answer will be odd’. Is he correct? Explain how you know. Discuss with your partner.
👀 Show answer
7. Heidi says, ‘When you find the difference between two odd numbers the answer is odd’. Is she correct? Explain how you know. Discuss with your partner.
👀 Show answer
🧠 Think like a Mathematician
Odd lines
a. Place the numbers $1$ to $9$ inside the grid so that each row, column, and diagonal adds up to an odd number.
b. Extend this investigation by looking at the numbers $1$ to $16$ on a $4 \times 4$ grid.
You will show you are specialising when you find solutions to the problem.
You will show you are conjecturing if you make predictions about results on a $4 \times 4$ grid, based on those for a $3 \times 3$ grid.
Show Answers
- a. One possible solution is to place odd numbers so that each line contains an odd count of odd numbers. Since an odd total is needed, each row, column, and diagonal must contain an odd number of odd values.
- b. On a $4 \times 4$ grid, a prediction is that it may be impossible for all rows, columns, and diagonals to sum to odd numbers because each line contains an even number of entries. This conjecture is based on the patterns observed in the $3 \times 3$ grid.
💡 Quick Math Tip
Odd totals: A total is odd if it contains an odd number of odd numbers. When you add numbers in a row, column, or diagonal, check how many odd numbers are included to predict whether the sum will be odd or even.