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...
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"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!"
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Comment recorded on the 3 October 'Starter of the Day' page by S Mirza, Park High School, Colne:
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Comment recorded on the 19 June 'Starter of the Day' page by Nikki Jordan, Braunton School, Devon:
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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 2 April 'Starter of the Day' page by Mrs Wilshaw, Dunsten Collage,Essex:
"This website was brilliant. My class and I really enjoy doing the activites."
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 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."
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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!
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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.
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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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"A set of real life savers!!
Comment recorded on the 11 January 'Starter of the Day' page by S Johnson, The King John School:
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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."
Comment recorded on the 10 September 'Starter of the Day' page by Carol, Sheffield PArk Academy:
"3 NQTs in the department, I'm new subject leader in this new academy - Starters R Great!! Lovely resource for stimulating learning and getting eveyone off to a good start. Thank you!!"
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"I use Starter of the Day as a registration and warm-up activity for my Year 6 class. The range of questioning provided is excellent as are some of the images.
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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 21 October 'Starter of the Day' page by Mr Trainor And His P7 Class(All Girls), Mercy Primary School, Belfast:
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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.
Alternatively, for the more advanced student, there is an ever-growing collection of Exam-Style Questions with worked solutions on the topic of Calculus.
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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