Challenging Math Problems for Students

2. It’s a problem! 1 From: FMSP

3. It’s a problem! 2 Ship’s Path A ship sails so that it is twice as far from a straight cliff edge as it is from a lighthouse. (A ratio of 2:1) What path does it take? Explore other ratios. From: The Core and More

4. It’s a problem! 3 Three fact triangles A triangle is a ‘3-fact triangle’ if any 3 of the following are true: • One side is 3cm • One side is 4cm • One angle is 90° • One angle is 30° How many 3-fact triangles are there? Put them in order from smallest perimeter to largest. Put them in order from smallest area to largest. From:RISPS

5. It’s a problem! 4a How many rectangles ? How many rectangles can you find in this diagram?

6. It’s a problem! 4b How many rectangles ? The diagram shown could be described as a 3 by 4 since it is divided into 3 rows and 4 columns. How many rectangles would there be: • In a 4 by 7 ? • Or a 2 by 6? • Or an m by n?

7. Pose your own problem! On the following slides are diagrams and other information (which your teacher will choose from). For each one write your own questions, adding labels or other information as necessary. Can you write a difficult question and an easier question for each one?

8. Pose your own problem 1

9. Pose your own problem 2

10. Pose your own problem 3

11. Pose your own problem 4

13. Teacher notes: Problem Solving and Assessment One of the issues with using problem solving in the classroom is assessing student work and giving feedback which will enable students to progress. Another issue is locating suitable problems for students to solve, and then how they are supported to develop their skills. In this resource there are a range of problems for students to solve and in these notes are suggestions for ways of assessing work and involving students in the assessment and feedback processes. A range of sources have been used to locate suitable problems and these are all listed on the final slide. One of the key considerations when ‘assessing’ problem solving is what will be of most help to the student? This could be quite different to ‘assessment’ that is helpful when learning subject matter.

14. Teacher notes: Problem Solving and Assessment Students assessing others work Students solve a problem and then assess 3 or 4 solutions written by others – or the teacher. Students can be asked to write feedback which identifies 1 or 2 things they like about the solution and 1 or 2 suggestions for improvement or development. They are not being asked to assess levels or give a mark. One of the advantages of this approach is that students become familiar with a range of approaches, which builds up their own ‘toolkit of strategies’. An example that would lend itself well to this is ‘How many rectangles?’ Alternatively,use some of the Bowland Assessment Tasks which have samples of student work. There are also assessment/ progression grids, but in this context they may not be appropriate to use.

15. Teacher notes: Problem Solving and Assessment Students Posing their own problems Give students a stimulus: diagram, chart, set of numbers, expression etc. and ask them to devise their own questions, adding labels or other information as they choose. Ask them to come up with several questions which have a range of difficulty. Students then come up with a model answer – or even a mark-scheme- for each of their questions. Alternatively, select several questions from the class for all to answer and/or devise model answers and mark-schemes for. The benefits of this approach are that the students have to think about what they know about a topic as well as think about what makes for a harder question and what makes for an easier question. If the information or diagram is similar to something that would be seen in an examination, then this approach can help with revision as well.

16. Teacher notes: Problem Solving and Assessment Assessing problem solving skills: thinking about progression When GCSE Mathematics had a coursework element, there was a set of generic criteria for assessing problem solving which Exam boards exemplified to guide teachers in their assessment of students’ work on particular problems. Although we no longer assess coursework, the generic statements related to: ‘Making and monitoring decisions to solve problems’, ‘Communicating mathematically’ and ‘Developing skills of mathematical reasoning’ and the breakdown of them could still prove helpful when thinking about progression and development in problem solving contexts.

17. Teacher notes: Problem Solving and Assessment Assessing problem solving skills: thinking about progression The Bowland Assessment tasks utilise a similar approach to this, setting up a grid and considering progression in fours strands: Representing, Analysing, Interpreting and evaluating, and Communicating. Some tasks have all fours strands, some have fewer and some have sub-strands. With each task the grid is bespoke. Trying a selection of these tasks and using the grids to help consider progression and development of problem solving skills could provide a basis for a department devising their own progression map or assessment grids for a range of other tasks.

18. Teacher notes: Problem Solving and Assessment Answers Annulus: Area= π(R2-r2)=25π where R is the radius of the large circle and r is the radius of the inner circle. Ship’s Path: An ellipse. Other ratios will give ellipses or parabola

19. Teacher notes: Problem Solving and Assessment Answers 3 fact triangles: 12 of them. How many rectangles: 60 m rows by n columns

20. Acknowledgements/ Sources of problems Bowland: http://www.bowlandmaths.org.uk/assessment/ FMSP: http://www.furthermaths.org.uk/gcse_problem_solving RISPs: http://www.risps.co.uk/ Williams, Tetlow, Tilley, Townson, Waddington. (1992) The Core and More: An activity based approach to teaching and learning mathematics at Advanced Level. The Mathematics Centre, WSIHE