# odds & evens

How could you add the first four odd numbers?

1 + 3 + 5 + 7 = ?

Just add them, right? How difficult is that? You could even use your fingers or a calculator. It’s simple!

But how could you add the first 1000 odd numbers? Or the first 1,000,000 odd numbers?

Below are the odd numbers 1, 3, and 5 represented using blocks: • Notice how the pattern grows.
• Build similar models of odd numbers 7 and 9.
• Without changing the shape of each number, move them around and see if you can make them fit together to form a single shape.
• What did you notice? How can you find the sum of the first n odd numbers?

1 + 3 + 5 + 7 + … + (2n-1) = ? Can you find the answer using blocks?

See the animation below:  Below are the even numbers 2 and 4 represented using blocks: • Notice how the pattern grows.
• Build similar models of odd numbers 6, 8, and 10.
• Without changing the shape of each number, move them around and see if you can make them fit together to form a single shape.
• What did you notice? Can you find the sum of the first n even numbers in the same way you did with odd numbers?

2 + 4 + 6 + 8 + … + 2n = ? Can you find the answer using blocks?

The hint is in the animation below: #### The Natural Numbers

Aren’t odds and evens natural too?
Or are they unnatural?
The Natural numbers are the counting numbers.

1 , 2 , 3 , 4 , 5 #### Drawing Natural Numbers

Let’s build a growing pattern of the natural numbers. You can use building blocks or colour crayons and a grid (grid pdf file) as in the image below: • Notice how the pattern grows. Continue the pattern for numbers 4 and 5.
• How is this pattern different from the one you did for odds and evens?
• What happens if you make two copies of the natural numbers? Can you find the sum of the first n natural numbers?

1 + 2 + 3 + 4 + … + n = ? The answer is in the animation below: 