A few years ago I saw a great investigation for year 8 students on TeachersTV about figuring out how many peas fit inside the classroom. I recently decided to slightly change the investigation to ask how many footballs fit inside. Inspired again by Dan Meyer’s 101questions, I made a video of myself putting balls two at a time into the classroom and then asked them if they had any questions which we could investigate.
Of course when you try to pack sphere-like objects into a cuboid-like object there will be space left over. The change from peas to footballs was predominantly down to the fact that the PE department at school have large cuboid boxes which fit about 50 footballs in, thus making it easier for the students to figure out the percentage of space left over from packing footballs into a cuboid. If a student was really struggling on how to find the space left over, I simply said: ‘There might be something down in PE that could help.’ Secondly, the diameter of a small pea is very difficult to accurately measure with a ruler which causes pretty big accuracy problems (I think engineers call a length such as the diameter of a sphere an “immeasurable” because it cannot be directly measured without tearing the thing apart). Thirdly, I wanted the students to have an intuitive feel for the problem and make a guess before they started doing the calculations. I’ve found in the past that students have really struggled to make a decent guess on how many peas fit inside the classroom, even when I ask them to give a lower bound and upper bound for the guess.
I usually ask the question and just let them take it away. The only piece of information I do give is the height of the classroom for safety purposes, but everything else they can do by themselves like for example searching for the volume of a sphere formula on the internet. I also like to make it a little more challenging by providing a sample of footballs of different sizes. In this way they might realise that finding the average radius of the footballs would be appropriate.
At the end of the investigation, they can compare their guess with the calculated answer. I usually wrap it up by asking them how many footballs they would buy if they had to fill the entire classroom. This promotes a good discussion on the assumptions made such as the balls being perfect spheres or the classroom being a perfect cuboid and also makes the students think about the accuracy of their measurements and calculations. Groups often add between 100-1000 balls to their calculated answer because they are so suspicious about the assumptions made or inaccuracies in the calculations.
Anyhow, a great investigation which always gets the students enthused whether you use peas, footballs or whatever!