Fence Optimisation: Find the length of a rectangle enclosing the largest possible area.
Maximum Product: Two numbers add up to 10. What's the largest possible product they could have?
Road Connections: Design roads to connect four houses that are on the corners of a square, side of length one mile, to minimise the total length of the roads.
Curriculum for Calculus:
Years 12 and 13
Pupils should be taught to understand and use the derivative of f (x) as the gradient of the tangent to the graph of y = f ( x) at a general point (x, y); the gradient of the tangent as a limit; interpretation as a rate of change, sketching the gradient function for a given curve, second derivatives, differentiation from first principles for small positive integer powers of x and for sin x and cos x. Understand and use the second derivative as the rate of change of gradient; connection to convex and concave sections of curves and points of inflection more...
Pupils should be taught to know and use the Fundamental Theorem of Calculus more...
Pupils should be taught to differentiate xn, for rational values of n, and related constant multiples, sums and differences. Differentiate ekx and akx, sin kx, cos kx, tan kx and related sums, differences and constant multiples. Understand and use the derivative of ln x more...
Pupils should be taught to integrate xn (excluding n = -1) and related sums, differences and constant multiples.
Pupils should be taught to apply differentiation to find gradients, tangents, normals, maxima, minima and points of inflection. Identify where functions are increasing or decreasing more...
Pupils should be taught to evaluate definite integrals; use a definite integral to find the area under a curve and the area between two curves more...
Pupils should be taught to differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions more...
Pupils should be taught to Understand and use integration as the limit of a sum. more...
Pupils should be taught to understand and use numerical integration of functions, including the use of the trapezium rule and estimating the approximate area under a curve and limits that it must lie between more...
Pupils should be taught to differentiate simple functions and relations defined implicitly or parametrically, for first derivative only more...
Pupils should be taught to carry out simple cases of integration by substitution and integration by parts; understand these methods as the inverse processes of the chain and product rules respectively (Integration by substitution includes finding a suitable substitution and is limited to cases where one substitution will lead to a function which can be integrated; integration by parts includes more than one application of the method but excludes reduction formulae) more...
Pupils should be taught to construct simple differential equations in pure mathematics and in context, (contexts may include kinematics, population growth and modelling the relationship between price and demand) more...
Pupils should be taught to integrate using partial fractions that are linear in the denominator more...
Pupils should be taught to evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions (Separation of variables may require factorisation involving a common factor.) more...
Pupils should be taught to interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution; includes links to kinematics more...
Comment recorded on the 26 March 'Starter of the Day' page by Julie Reakes, The English College, Dubai:
"It's great to have a starter that's timed and focuses the attention of everyone fully. I told them in advance I would do 10 then record their percentages."
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"Thank you for sharing such a great resource. I was about to try and get together a bank of starters but time is always required elsewhere, so thank you."
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"A set of real life savers!!
Comment recorded on the 25 June 'Starter of the Day' page by Inger.kisby@herts and essex.herts.sch.uk, :
"We all love your starters. It is so good to have such a collection. We use them for all age groups and abilities. Have particularly enjoyed KIM's game, as we have not used that for Mathematics before. Keep up the good work and thank you very much
Comment recorded on the 19 October 'Starter of the Day' page by E Pollard, Huddersfield:
"I used this with my bottom set in year 9. To engage them I used their name and favorite football team (or pop group) instead of the school name. For homework, I asked each student to find a definition for the key words they had been given (once they had fun trying to guess the answer) and they presented their findings to the rest of the class the following day. They felt really special because the key words came from their own personal information."
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"Love using the Starter of the Day activities to get the students into Maths mode at the beginning of a lesson. Lots of interesting discussions and questions have arisen out of the activities.
Comment recorded on the 14 September 'Starter of the Day' page by Trish Bailey, Kingstone School:
"This is a great memory aid which could be used for formulae or key facts etc - in any subject area. The PICTURE is such an aid to remembering where each number or group of numbers is - my pupils love it!
Comment recorded on the 2 May 'Starter of the Day' page by Angela Lowry, :
"I think these are great! So useful and handy, the children love them.
Comment recorded on the 18 September 'Starter of the Day' page by Mrs. Peacock, Downe House School and Kennet School:
"My year 8's absolutely loved the "Separated Twins" starter. I set it as an optional piece of work for my year 11's over a weekend and one girl came up with 3 independant solutions."
Comment recorded on the 9 October 'Starter of the Day' page by Mr Jones, Wales:
"I think that having a starter of the day helps improve maths in general. My pupils say they love them!!!"
Comment recorded on the s /Indice 'Starter of the Day' page by Busolla, Australia:
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Comment recorded on the 14 October 'Starter of the Day' page by Inger Kisby, Herts and Essex High School:
"Just a quick note to say that we use a lot of your starters. It is lovely to have so many different ideas to start a lesson with. Thank you very much and keep up the good work."
Comment recorded on the 3 October 'Starter of the Day' page by Fiona Bray, Cams Hill School:
"This is an excellent website. We all often use the starters as the pupils come in the door and get settled as we take the register."
Comment recorded on the i asp?ID_Top 'Starter of the Day' page by Ros, Belize:
"A really awesome website! Teachers and students are learning in such a fun way! Keep it up..."
Comment recorded on the 3 October 'Starter of the Day' page by S Mirza, Park High School, Colne:
"Very good starters, help pupils settle very well in maths classroom."
Comment recorded on the 16 March 'Starter of the Day' page by Mrs A Milton, Ysgol Ardudwy:
"I have used your starters for 3 years now and would not have a lesson without one! Fantastic way to engage the pupils at the start of a lesson."
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"Thank you so much for your wonderful site. I have so much material to use in class and inspire me to try something a little different more often. I am going to show my maths department your website and encourage them to use it too. How lovely that you have compiled such a great resource to help teachers and pupils.
Comment recorded on the 12 July 'Starter of the Day' page by Miss J Key, Farlingaye High School, Suffolk:
"Thanks very much for this one. We developed it into a whole lesson and I borrowed some hats from the drama department to add to the fun!"
Comment recorded on the 28 May 'Starter of the Day' page by L Smith, Colwyn Bay:
"An absolutely brilliant resource. Only recently been discovered but is used daily with all my classes. It is particularly useful when things can be saved for further use. Thank you!"
Comment recorded on the 10 April 'Starter of the Day' page by Mike Sendrove, Salt Grammar School, UK.:
"A really useful set of resources - thanks. Is the collection available on CD? Are solutions available?"
Comment recorded on the 11 January 'Starter of the Day' page by S Johnson, The King John School:
"We recently had an afternoon on accelerated learning.This linked really well and prompted a discussion about learning styles and short term memory."
Comment recorded on the 19 November 'Starter of the Day' page by Lesley Sewell, Ysgol Aberconwy, Wales:
"A Maths colleague introduced me to your web site and I love to use it. The questions are so varied I can use them with all of my classes, I even let year 13 have a go at some of them. I like being able to access Starters for the whole month so I can use favourites with classes I see at different times of the week. Thanks."
Comment recorded on the 3 October 'Starter of the Day' page by Mrs Johnstone, 7Je:
"I think this is a brilliant website as all the students enjoy doing the puzzles and it is a brilliant way to start a lesson."
Comment recorded on the s /Coordinate 'Starter of the Day' page by Greg, Wales:
"Excellent resource, I use it all of the time! The only problem is that there is too much good stuff here!!"
Comment recorded on the 17 June 'Starter of the Day' page by Mr Hall, Light Hall School, Solihull:
Comment recorded on the 19 June 'Starter of the Day' page by Nikki Jordan, Braunton School, Devon:
"Excellent. Thank you very much for a fabulous set of starters. I use the 'weekenders' if the daily ones are not quite what I want. Brilliant and much appreciated."
Comment recorded on the 1 May 'Starter of the Day' page by Phil Anthony, Head of Maths, Stourport High School:
"What a brilliant website. We have just started to use the 'starter-of-the-day' in our yr9 lessons to try them out before we change from a high school to a secondary school in September. This is one of the best resources on-line we have found. The kids and staff love it. Well done an thank you very much for making my maths lessons more interesting and fun."
Comment recorded on the 1 February 'Starter of the Day' page by Terry Shaw, Beaulieu Convent School:
"Really good site. Lots of good ideas for starters. Use it most of the time in KS3."
Comment recorded on the 6 May 'Starter of the Day' page by Natalie, London:
"I am thankful for providing such wonderful starters. They are of immence help and the students enjoy them very much. These starters have saved my time and have made my lessons enjoyable."
Comment recorded on the 9 May 'Starter of the Day' page by Liz, Kuwait:
"I would like to thank you for the excellent resources which I used every day. My students would often turn up early to tackle the starter of the day as there were stamps for the first 5 finishers. We also had a lot of fun with the fun maths. All in all your resources provoked discussion and the students had a lot of fun."
It is said that the word calculus comes from the Latin word for the small pebble used for counting and calculations. The two major branches, differentiation and integration, are studied by pupils only towards the end of their school days but does then form a major part of their studies. A course in calculus is a prerequisite for other, more advanced courses in mathematical analysis.
Differentiation: Practise the technique of differentiating polynomials with this self marking exercise.
Integration: Exercises on indefinite and definite integration of basic algebraic and trigonometric functions.
Trapezium Rule: Practise using the trapezium rule to find an approximate value for the area under a curve.
Alternatively, for the more advanced student, there is an ever-growing collection of Exam-Style Questions with worked solutions on the topic of Calculus.
Differentiation Video: A reminder of how to differentiate different types of functions and how to find the equations of tangents and normals.
Introduction to Calculus: This video will give you a brief introduction to calculus. It does this by explaining that calculus is the mathematics of change. A couple of examples are presented, and then limits, derivatives, and integrals are introduced.
The Birth Of Calculus: A fascinating BBC Two television programme from 1986. A documentary on Leibniz and the calculus.
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