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Visit Stack Exchange I'm going to be starting teaching a course called algebra COE, which is for students who didn't pass the required state algebra exam to graduate and are now seniors, to do spaced-out exam-like extended problems after extensive support.I don't want to start the class out with "getting down to business" because I want the students to feel comfortable in the class, with me and with each other.
Some problems I have considered, yet I believe are too "math-feeling": (pp.
1335-1341) (pdf version) mentions a nice problem called "Dividing squares." Shown below are some pictures of a square divided into smaller squares, not necessarily of the same size.
(Image taken from the linked file.) A few possible questions you can ask: The frog jumping puzzle is nice, in that it is very simple but hides some surprising but manageable complexity. Have the students play with this, and then try to generalize to n male frogs and m female frogs.
For another favorite, see the map folding puzzle of Martin Gardner as problem 29 here.
‘Jack of all trades, master of none.’ This term is often applied to primary school teachers, as most have to work within a loaded curricula.
This often means that teachers aren’t able to teach all subjects as well as they would like, resulting in an adept ability to be masters of all subjects. ‘So many of the skills we took for granted—paragraphs, topic sentences, vocabulary choice, standardised spelling, punctuation, even grammar—are now passé.When you reach questions 3 and 4 that deal with how to work the problem out, ask students what problem solving approach they might take. Discuss with the students how they could use the process of elimination to assist in solving this problem.Use Newman’s prompts (PDF 41.18KB) as a whole class to model how to start thinking about solving the problem.This is a beautiful way to show the surprisingly delicate complexity arising in mathematics.I've broken all sorts of posting rules here, but perhaps someone will find better links. Continue with the same process as before, only this time introduce students to the strategy of looking for a pattern.Next Steps Provide lots of different examples where students can talk through the different problem solving strategies they could use.Students focus on the mathematics while the teacher focusses on observing and supporting the learners to make their own discoveries.On its own a rich task is not rich unless the classroom culture is one in which learners are not passive recipients of knowledge, but rather are curious, reflective constructors of their own understandings.At this stage, patterns and algebra problems often involve working backwards, using a process of elimination, trial and error and using inverse operations.These strategies need to be explicitly taught to students.