Problem Solving and STEP

A number sequence ...
Y12/13 Problem-Solving and STEP Activity Days:
- Our aim is to get sixth-form mathematics students to engage in
higher-level problem solving in a variety of ways,
- Our belief is that this will ultimately lead to increased proficiency in
solving STEP and/or AEA problems but this is not the main focus of
the initiative,
- We try to replicate the activities that ‘good’ mathematicians are
typically engaged in whilst solving problems: collaboration,
discussion, argument, thinking, accepting being stuck, trying
alternatives, cheating, banging head against wall and so on,
- We also try to ensure that students experience a number of different
problem-solving contexts within any given day and also across the
series of six days. We try to make the experience enjoyable.
Y12/13 Problem-Solving and STEP Activity Days:
In our current series there are six full-day events – 3 each in Y12 and Y13.
The Y12 events:
Autumn Term Year 12 – What is problem solving? Contexts for problemsolving.(Stand-up; Competition + Challenge)
Spring Term Year 12 – Longer problems; problems that extend students’
experience; practical or contextual problems.
(Master-class; London Eye)
Summer Term Year 12 – Mathematical Modelling.
(Beans problem; Thunder-looper task)
Different ways of creating problem-solving
Dragon Maths Competition
Autograph Challenge
Use what’s there …
Master-class – Quadratics and Cubics
Problems in context / practical problems…
Tin of Beans
Here is a standard tin of
3.7 cm
Is the amount of metal used
to make the tin minimal?
10 cm
Or is it possible to design a
cylindrical metal can which
holds the same quantity of
beans but uses less metal?
3.7 cm
Tins of Beans
10 cm
Tins of beans come in different sizes.
The two most common sizes are shown.
The smaller can has half the volume of
the larger can.
Assuming that these tins are sold in
equal volumes, what is the optimum
radius to use in order to minimise the
total amount of metal used to make the
3.7 cm
5 cm
What if, instead, the tins are sold in the
ratio 1 : a (= larger : smaller)?
Y12/13 Problem-Solving and STEP Activity Days:
The Y13 events:
Autumn Term Year 13 – Effective problem solving techniques; MAT preparation.
(Graph sketching; Support for teachers.)
Spring Term Year 13 – STEP and AEA – what is required to be successful.
(Selected problems that highlight useful techniques.)
Summer Term Year 13 – STEP and AEA preparation: choosing questions,
making progress on questions, etc.
Why use STEP?
The STEP is the University of Cambridge’s admissions test for
Mathematics. It serves three key purposes:
• It acts as a hurdle: success in the STEP is thought to be a good
indicator of potential to do well on a difficult course.
• It is useful preparation for the course: the sort of mathematics
found in the STEP questions is similar in style to undergraduate
• It tests motivation: the STEP requires dedicated preparation, by
working through past papers, for example. Those who are not
willing to make the effort are unlikely to thrive on a difficult
mathematics course.
Graph Sketching:
 STEP Examiners’ Report:
 Graph sketches were again weak. Students need much more practice
with sketching graphs of interesting functions (beyond the standard
A-level fare of quadratics, cubics, reciprocals of linears and x2, and
the basic trigonometrical functions). Sketching functions should
involve consideration of all of their main features: their axis-crossings,
their stationary points, their asymptotic behaviour and even such
basics as whether they are positive or negative in various regions.
Reciprocals of quadratics and cubics are good for learning this, as
well as more sophisticated functions.
 Graph Sketching is a good opportunity for student presentations
Sketch the graph y=xx
STEP I 1999, Question 4
 Sketch the following subsets of the x-y plane:
|x| + |y| ≤ 1;
|x − 1| + |y − 1| ≤ 1;
|x − 1| − |y + 1| ≤ 1;
|x| |y − 2| ≤ 1.
Sketch the graphs of:
a) y = sin(x)+0.1sin(10x)
b) y = sin(x)sin(10x)
Choosing Questions:
 Examiner’s Report 2009:
“This suggests that a skill lacking in many students
attempting STEP is the ability to pick questions effectively.
This is not required for A-levels, so must become an
important part of STEP preparation.”
 Examiner’s Report 2013:
“… but at least most candidates showed sufficient
skill to identify them and work on them as part of
their chosen selection of questions.”
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