It’s a problem! 1
From: FMSP
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
It’s a problem! 3
Three fact triangles
From:RISPS
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.
It’s a problem! 4a
How many rectangles ?
How many rectangles can you find in this
diagram?
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?
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?
Pose your own problem 1
Pose your own problem 2
Pose your own problem 3
Pose your own problem 4
45
40
35
30
25
20
15
10
5
0
0
1
2
3
4
5
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9
10
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.
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.
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-schemefor each of their questions. Alternatively, select several questions from
the class for all to answer and/or devise model answers and markschemes 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.
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.
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.
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
Teacher notes:
Problem Solving and Assessment
Answers
3 fact triangles: 12 of them.
How many rectangles: 60
m rows by n columns
m(m+1)
2
x
n(n+1)
2
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