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LIAR!
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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)

Dragon Maths Competition

Autograph Challenge

Use what’s there …

3.7 cm
Here is a standard tin of beans.
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
3.7 cm
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.
5 cm
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 tins?
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 mathematics.
• 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.

 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 x 2 , 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

1) Sketch the graph y=x x
2) STEP I 1999, Question 4
 Sketch the following subsets of the x-y plane: i) |x | + |y| ≤ 1; ii) |x − 1| + |y − 1| ≤ 1; iii) |x − 1| − |y + 1| ≤ 1; iv) | x| |y − 2| ≤ 1.
3) Sketch the graphs of: a) y = sin(x)+0.1sin(10x) b) y = sin(x)sin(10x)

 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 .”