You may also like

problem icon


Two circles of equal size intersect and the centre of each circle is on the circumference of the other. What is the area of the intersection? Now imagine that the diagram represents two spheres of equal volume with the centre of each sphere on the surface of the other. What is the volume of intersection?

problem icon

Power Up

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

problem icon

Fractional Calculus I

You can differentiate and integrate n times but what if n is not a whole number? This generalisation of calculus was introduced and discussed on askNRICH by some school students.

Interactive Workout - Mathmo

Stage: 5 Short Challenge Level: Challenge Level:1

You can use this workbook to practise your core mathematical skills. This will be very useful both whilst at school and when you move on to higher study, where a high degree of algebraic fluency is really useful.
Here are some hints and tips to improve your skills:
  1. Don't be tempted to look at the answer if you think that you can do the question unless you have actually done the question, on paper with a pen.
  2. Try not to use your calculator for the simple arithmetical parts of a question - this is a bad habit to get into.
  3. If you are stuck, don't give up immediately. Think about the problem. Perhaps look at one or two answers and ask: how does this answer work?
  4. Accuracy is important, but speed is also important. This workbook is good for speed training.
  5. If your answer differs from the answer given then think: is my answer wrong, or is it merely represented in a different form?
  6. Use your skills in some interesting rich mathematical problems from NRICH. You might want to look at the core mathematics curriculum document for suggestions or just browse the stage 5 content on the site.
  7. Keep a record of your best times. See how quickly you can complete one of each question type from each section.

What is a good time to aim for?

Fluency with mathematics means the ability to perform routine calculation BOTH quickly AND accurately. One without the other will hamper your progress, especially once you reach university.

To give you a feel for where you might be aiming, here are some average times for some of the questions as recorded by Judith, a second year mathematics undergraduate who worked with NRICH over the summer:

Algebra Curve sketching Differentiation
Quadratic equations 8s Modulus function for linear 20s Stationary points for quadratic 10s
Completing the square 8s Modulus function for quadratic 1m 10s Stationary points for cubic 50s
Inequalities for quadratics 10s Implicit differentiation 1m 10s
Inequalities for cubics 1m 10s
Partial fractions 1m 30s
Powers 1m
Logarithms 20s
Solving trig equations 40s
TOTAL 5m 6s

Can you match or beat Judith's times?

More times from other brave solvers will be posted periodically.