Patterns and conjecture
In mathematics, a conjecture is a mathematical statement which appears to be true, but has not been formally proven. A conjecture can be thought of as the mathematicians way of saying “I believe that this is true, but I have no proof yet”. A conjecture is a good guess or an idea about a pattern.
For example, make a conjecture about the next number in the pattern $2;\phantom{\rule{3.33333pt}{0ex}}6;\phantom{\rule{3.33333pt}{0ex}}11;\phantom{\rule{3.33333pt}{0ex}}17;...$ The terms increase by 4, then 5, and then 6. Conjecture: the next term will increase by 7, so it will be $17+7=24$.
Example 1: Adding even and odd numbers
Question

Investigate the type of number you get if you find the sum of an odd number and an even number.

Express your answer in words as a conjecture.

Use algebra to prove this conjecture.
Answer
First try some examples
Make a conjecture
The sum of any odd number and any even number is always odd.
Express algebraically
Express the even number as $2x$.
Express the odd number as $2y1$.
From this we can see that $2(x+y)$ is an even number. So then $2(x+y)1$ is an odd number. Therefore our conjecture is true.
Example 2: Multiplying a twodigit number by 11
Question

Consider the following examples of multiplying any twodigit number by 11. What pattern can you see?
$$\begin{array}{ccc}\hfill 11\times 42& =& 462\\ 11\times 71& =& 781\\ 11\times 45& =& 495\end{array}$$(3) 
Express your answer as a conjecture.

Can you find an example that disproves your initial conjecture? If so, reconsider your conjecture.

Use algebra to prove your conjecture.
Answer
Find the pattern
We notice in the answer that the middle digit is the sum of the two digits in the original twodigit number.
Make a conjecture
The middle digit of the product is the sum of the two digits of the original number that is multiplied by 11.
Reconsidering the conjecture
We notice that
Therefore our conjecture only holds true if the sum of the two digits is less than 10.
Express algebraically
Any twodigit number can be written as $10a+b$. For example, $34=10\left(3\right)+4$. Any threedigit number can be written as $100a+10b+c$. For example, $582=100\left(5\right)+10\left(8\right)+2$.
From this equation we can see that the middle digit of the threedigit number is equal to the sum of the two digits x and y.