Low Threshold High Ceiling Tasks for Secondary Classrooms
Secondary classrooms
Take a look at Arithmagons. At the threshold students use trial and improvement to arrive at a solution. As they collect more evidence, they start to look for patterns which they can then generalise. They may use an algebraic notation to arrive at, or confirm, a conjecture about ways of solving the problem. And once the confident students have satisfied themselves that they can solve any triangle arithmagon, they can try out some different shaped ones which involve multi-step solutions.
Odds and Evens has been a favourite of ours for some time. At the threshold students work systematically to derive all possible results, recording their work in the most useful way. They use their results to justify which version of the game is fairest. Others can move on to working together to generate lots of data which they can sort and analyse. They may use it to form a conjecture about the balls needed for a fair game, and predict which other combinations would do so too, justifying their statements. The activity moves from experimental probability to theoretical probability (and a discussion of the difference) to sophisticated analysis.
Other secondary LTHC tasks we like
Opposite Vertices (and Vector Journeys as follow-up)
What's Possible (this links also to Plus Minus)
Number Pyramids (again with follow-ups)
Post-16 classrooms
Post-16 classrooms are often the ones where everyone does the same A level questions, perhaps at different speeds. It's good to be able to offer tasks which highly able students find a challenging alternative to racing through the exercises, but which less confident members of the group can still access.
Painting by Numbers is a practical activity which introduces students to the idea of topology, and hence the Four Colour Theorem. At the threshold students will be using trial and improvement to clarify their ideas about what it is that makes two apparently different diagrams have the same topology. More confident students will be able to use their new knowledge to engage with the theorem and some will even be able to understand it fully - which takes them up to university level pure mathematics.
The problem Whose Line Graph is it Anyway? has some inbuilt support and extension and is one of a new set of tasks we are developing which make links across and between the sciences and mathematics. Students have to match the statement, equation and graphical representation of various physical processes. At the threshold they will be consolidating existing knowledge as they match sets of which they are certain. More confident students will generalise what they know and apply it to new specific situations, looking for similarities between the physical processes. For the high flyers there are links to other activities which use similar content, and coming soon will be links to readings about the maths and science involved.
Other Post-16 LTHC tasks we like