#beingmathematical again

I got involved in another #beingmathematical chat on twitter last night. Here is the original prompt.

Problem: You have three discs, the first one has a 6 on the top, the second has a 7 and the third has an 8. There is a number written on the back of each one. When the discs are thrown the totals of the visible numbers are: 15, 16, 17, 18, 20, 21, 22 and 23 What numbers are on the back?

I had a quick go at the problem  during a break at work. Using some algebraic representation and mostly trial and improvement strategies I came up with 3 solutions. I knew I hadn't worked systematically so I was not at all convinced I had found all the solutions. 

During the chat I found it very difficult to work on the problem and contribute so I watched and listened a lot and asked (what I thought were) pertinent questions. I was very intrigued when people made assertions about possible solutions, especially when there wasn't space in twitter to fully explain their thoughts. I found that that gave me a chance to make my own meaning from their ideas. 

Someone stated that they knew the total of all the front and back numbers. It hadn't occurred to me initially that this might be useful and although I forgot instantly what the total was I decided that when I had another go at the problem I would try to use the knowledge that it was possible to calculate the total.  Also the fact that twitter isn't instant meant that I had time to reflect on statements before someone else contributed. 

Here are my follow up workings out, the post it notes are comments about how I might expect learners to approach the problem:

I wondered:

• would I use this with people who hadn't had much exposure to algebraic representations?
• would the need to label/describe/discuss the missing numbers encourage the use of symbols or letters to represent unknown values?
• could finding a way to represent all the possible outcomes be a valuable aim even if no solutions are found?
• might I get people to try their own examples?
• is 2 disks a useful simplification or does it over simplify the situation?
• could suggesting that people look for ranges for solutions be a useful prompt, even if they don't represent them in a traditional way?

PS: I used Graspable Math when I was having my first go at the problem. It was nice to be able to try lots of examples quickly though I wonder whether it stopped me being all that systematic. I was playing around a lot and I didn't limit my values much or record my findings at all. It did give me a bit of an idea about how the totals were changing though. 

Here are the 3 solutions I found initially: