The direct proof is a way of showing that an *odd number sum* is even. It was developed to overcome a problem in *elementary school mathematics*: finding the solutions to odd-nough puzzles.

In these puzzles, you need to use your logical reasoning, memory, and skills from previous lessons to solve them. After doing this for a while, you can tell if an even number sum is even or not because you know how to do this in previous lessons.

The direct proof was developed specifically for *elementary school children* because it can be done easily. Even though the adults can do it, the kids get more excited and learn quicker if they do it themselves.

Use a direct proof when you want to show that the product of *two odd numbers* is even.

## Write a = x + y and b = x + y

This trick can be used to find the sum of any ** two numbers**.

Using a **direct proof**, you can find the sum of any two numbers. For example, this *trick works* for the integers 1 through 9.

1 + 2 = 3 + 4 = 6 + 8 = 12

2 + 3 = 5 + 4 = 7 + 7 = 11

3 + 4 = 7 + 5 = 9 + 9 = 15

4 + 5 = 8 + 6= 10+10=20.

## Use the associative property to rewrite a × b = (x + y) × (x + y)

When converting a number to a decimal, you may need to add or *subtract one digit*. For example, when converting from pounds to dollars, you must subtract the pound sign because of its role as a measurement unit.

Similarly, when converting a decimal number into a fraction, you may need to add or subtract one digit. For example, when converting from $10.00 to an anecdote worth of value, you may need to *add one additional dollar* because of the cost of the product.

The associative property can *help rewrite numbers* into other bases by replacing x with y and adding or **subtting one digit**. For example, let’s say we want to convert $10 dollars into anecdote values.

## Observe that, by definition of an odd number, there exists an integer k such that a − k and b − k are both even

In order for the product of **two odd numbers** to be odd, k must be an even integer. If k is an even integer, then the *two numbers* a and b can be transformed into a *single number c*, which is even.

Thus, if a = 2 and b = 4, then c = **2 even implies** that c = 8 is even.

In other words, if we try to solve for c using the product of a and b, we will find that c is even. This illustrates why it is important to use an odd number as the basis for our proof.

By using an odd number as the basis for our proof, we can eliminate any possible solutions that have no elements in the middle or on either side of an equal sign. This makes it more difficult for our opponent to easily recreate such a system in their own proof.

## Use the fact that (x + y)(x + y) = x² + 2xy + y²

This fact, called the Pythagorean theorem, can be used to calculate the area of any rectangle or square.

For instance, the area of a square is equal to the sum of its sides. The sum of a rectangle’s sides is also a rectangle, and the sum of **two non**–*zero squares* is still a square.

Using this information, you can use the direct proof to show that blueprints with an odd number of boxes equals an odd number of boxes.

You may have seen this kind of proof before, but it has been reformulated for today’s audience. Read more about this here: How to Use This Proof in Your App development Career.

## Rewrite (x+y)(x+y) using k from step 4: ((a−k)(b−k))(a−k)(b−k)=ak²+(2ky)+yk²=ak²+2akkyyk+yk²=ak²+4akky+2yk²
This can be used in a classroom to *teach students* how to use a direct proof. When they are done, they can use the proof in an odd or even way, depending on what they wanted to prove.

When *someone wants* to know the length of an even number, they can write an even number and then write the length of its even digit. This is used to prove that the number is even.

Similarly, when someone wants to know the length of an odd number, they can use an odd number and then write the length of its *odd digit*. This is used to prove that the number is even.

*teach students*how to use a direct proof. When they are done, they can use the proof in an odd or even way, depending on what they wanted to prove.

*to know the length of an even number, they can write an even number and then write the length of its even digit. This is used to prove that the number is even.*

*someone wants**odd digit*. This is used to prove that the number is even.