Top Tip from Nrich!
Initially, it is easier to think of basic closed word problems than it is to devise open-problems, but luckily, a good source of open problems is actually these same basic closed problems.
Moses, Bjork and Goldenberg (1990) suggest analysing the problem in terms of what is known (information given), what is unknown (what needs to be found out) and what restrictions are placed on its solutions, which makes it easier to then open up the problem. For example: “How many 5p coins are needed to make 45p?”
|Kind of Knowledge||Details|
|Known||The final amount of money||45p|
|Unknown||The number of coins||?|
|Restrictions||All coins have the same value||5p|
Any of these variables can be altered to produce different problems. If the resulting problem has more than one solution, then it could be described as being more open. For example:
- Remove the restriction: How many coins does it take to make 45p?
- Remove known: I have a closed handful of 5p coins. How much do I have?
- Swap the known/unknown, change the restriction: I have 5 coins. Three are the same. How much money do I have?
- Swap the known/unknown, remove restriction: I have 5 coins. How much money could I have?
- Remove known and restriction, change unknown: I have some coins in my hand. How much money do I have?
- Change known, unknown, and restriction: What is the shortest/longest line that can be made with 5 coins?
Therefore, the invention of open problems is not dependent upon the creative mood of the teacher, but becomes a manageable procedure that can be applied to any problem. After a little practice, opening up a problem tends to become an automatic reaction to any basic word problem.
Funky Problems coming soon!