This year with my grade 8 students I am trying to spend one class every so often giving my students time to work collaboratively on mathematical tasks. The first one I did this year went really well. For this task, I had my students find the sum of the numbers from 1-10 and then from 1-20. For these, they just used their calculators and computed the answers. Then I asked them to find 1-50 and 1-100. Of course, some thought that using a calculator was still the best method, but there were others who knew that there had to be an "easier" way and worked to find it (especially once I asked them - what about from 1-1000?). This task led to some great thinking and some great ideas. We eventually got to the point where the students were able to come up with a formula to find the sums of consecutive numbers. I told them about Gauss and how when he was younger he did a similar exploration in his math class and came up with some amazing ideas in the area of number theory. For those that were interested, we extended this problem to think about what happens when you don't start at 1 (eg. find the sum of all the whole numbers from 10-90) or what happens if you are adding consecutive even or odd numbers.
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:
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.