The second task that I presented to my students comes from the NRICH website. Here is the link to the video that I used to introduce the problem to my students:

http://nrich.maths.org/507

They really got a kick out of the British accents!

Once they watched the video, I had my students work in groups of 4 to answer the two questions the video posed (see below) and any other questions that came to mind as they watched the video.

*What is special about the numbers that can be written as two consecutive numbers or three consecutive numbers or 4 or 5, etc?*

*Can all numbers be written as the sum of consecutive numbers?*

Once again, I was so impressed with the level of thinking that my students showed. Most groups started with thinking about what types of numbers can written as the sum of 2 consecutive numbers. They quickly realized that all odd numbers can be written in this way. When I asked them why this was, they remembered that when you add one odd number and one even number, you always get an odd number. Some groups were also strategic in their approach and had some students looking at sums of three numbers, some looking at sums of 4 numbers, etc. The number of patterns that the students discovered were quite amazing. Some were even able to see the connection between this task and the last because, as an example, they realized that the sum of 5 consecutive numbers is always a multiple of 5 and the smallest one that will work is 15, which is the sum of the numbers from 1-5. When challenged by me to think about what numbers wouldn't work, they realized that all powers of 2 are not able to be written as sums of consecutive numbers. I also threw random numbers out at them (like 81 for example) and challenged them to think of all the ways that they could write this number as the sum of consecutive numbers.

For anyone interested in trying problem solving tasks in their classroom, I would highly recommend this problem, as it has a low entry point (all of my students realized that all odd numbers can be written as the sum of consecutive numbers) but a very high ceiling.