Odd Numbers and Square Numbers

thingiverse

####Odd Numbers and Square Numbers Odd numbers **{ 2n – 1, where n is a natural number}** are not “odd” at all. They are in fact very interesting and geometrically beautiful. One activity appropriate for students at all levels is the sum of the first M odd numbers, which is M^2. For example, the sum of the first 5 odd numbers (1, 3, 5, 7, 9) is 5^2=25. Without 3D models, one can just use a square grid and have students color the odd numbers and talk about the progression of the sums. With 3D models, we could reach out to more students and allow them to see, feel, and play with the multiple connections between number concepts and their geometric implications and much more. There are certainly **many ways** to make a square or other shapes! Sure, it can be proved by induction. (1) When *M =1*, Sum=1^2, meaning the sum of the first odd number is just 1. (2) Assuming the sum of the first M odd numbers is M^2, the sum of the first ( M +1 ) odd numbers is then * M^2 + 2M+1*, which can be written as *(M + 1)^2*. (3) QED. This is nice but I still like the geometric aspect of the story. ####Among the Files 1. A set of odd numbers (1, 3, 5, 7, 9, 11) of the same height (20mm). 2. A set of odd numbers (1, 3, 5, 7, 9, 11) of varying height (step = 5mm) 3. The odd numbers can be printed all at once or one by one. ####Reference Conway, J. H. & Guy, R. (1995). The book of numbers. New York, NY: Copernicus.