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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 $2;6;11;17;...$ 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

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

3. Use algebra to prove this conjecture.

First try some examples

$23+12=35148+31=17911+200=211$(1)

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 $2y-1$.

$Sum=2x+(2y-1)=2x+2y-1=(2x+2y)-1=2(x+y)-1$(2)

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 two-digit number by 11

Question

1. Consider the following examples of multiplying any two-digit number by 11. What pattern can you see?

$11×42=46211×71=78111×45=495$(3)

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

4. Use algebra to prove your conjecture.

Find the pattern

We notice in the answer that the middle digit is the sum of the two digits in the original two-digit 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

$11×67=73711×56=616$(4)

Therefore our conjecture only holds true if the sum of the two digits is less than 10.

Express algebraically

Any two-digit number can be written as $10a+b$. For example, $34=10(3)+4$. Any three-digit number can be written as $100a+10b+c$. For example, $582=100(5)+10(8)+2$.

$11×(10x+y)=110x+11y=(100x+10x)+10y+y=100x+(10x+10y)+y=100x+10(x+y)+y$(5)

From this equation we can see that the middle digit of the three-digit number is equal to the sum of the two digits x and y.