Thought for the day: When you use noticing and wondering prompts regularly, students begin to apply their new observing and questioning skills in other situations—often when you least expect it!

Concepts: sums of consecutive odd numbers; square numbers; patterns and formulas

Examples of noticing and wondering

I notice a square filled with L-shaped pieces.
I notice that each L-shaped piece has an odd number of squares.
I notice that you could build the image starting with the single red square and joining one L-shaped piece at a time.
I notice that each time you join a new L-shaped piece, it makes the next-larger square.
I notice the expression 1 + 3 + 5 + 7 + 9 + 11 + 13.

I wonder if I could use the image to find a shortcut for adding odd numbers.
I wonder if the sum of consecutive odd numbers is always a square number.
I wonder if I could find a formula for adding consecutive odd numbers.
I wonder if I could use rectangles to discover a formula for adding even numbers.
I wonder if I could use other shapes to discover other kinds of formulas.

Notes

This image can help students discover patterns in the sum of consecutive odd numbers beginning with 1.

1 = 1
1 + 3 =4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 11 = 25
etc.

The answers are the square numbers—and the image shows why this happens!