The first week of school has been a blast. This is my first year at a middle school, and I am really enjoying these seventh and eighth graders. They are an impressive bunch, and I am very excited for what is sure to be a great year.
Starting off the year, I got the idea from a friend of mine to present a really nice counting problem. We started looking at a soda can pyramid picture that I found. The bottom row of the pyramid has 11 cans with each successive row having one less can until the single can at the top.
The goal was to count the number of cans, and after students went through and added them all together, we tried to come up with a better way of counting that would be more efficient. A few students picked up on the trick commonly credited to Gauss, which was fun because I was able to show them a nice numberphile video about the counting problem after they worked it out.
The fun began when they were faced with a pyramid that has 50 cans at the bottom. They thought about it independently for a couple of minutes, then they were split into six groups to work out the problem on big white boards. Below are some of the great ideas they came up with and presented to the class.
This idea was about turning the pyramid into a square and subtracting off the area. The coolest part about this idea was that the group accounted for the single can at the top of the pyramid.
This idea was about adding up groups of tens, nines, eights, and so on. The group noticed that each successive group of numbers was reduced by five, which was very cool, and they were able to use that to quickly get to multiplying their sums.
This idea was very impressive to me. This group was trying to find a formula to use to find the number of cans. They went through a few different ideas, tried them out to see if they worked, and they finally landed on this beauty for any pyramid with base B.
The other three groups came up with similar solutions, and this served as a great problem for the start of the year. They loved working in groups, they really enjoyed using the white boards, and I enjoyed being able to go to each group and help them make sure that each member of the group was on board and asking good questions.
I look forward to posting more of these fun activities throughout the year. I am very excited about the start of the new year.