For A Level Students age (16-18) this post on University Admissions Tests includes many resources to challenge their thinking. Whether or not they are taking any admissions test the resources will provide challenge for students aiming to achieve the highest grades.
Level 2 Further Maths (select for a collection) resources are also very useful for students aiming for the highest grades at GCSE.
I have used Underground Mathematics resources with my very able GCSE students; there are many Underground Mathematics Resource Types including Review Questions which in the words of the Underground Maths Team:
These are questions designed to test students’ understanding of one or more topics and to exercise their problem-solving skills. In many cases they can also be used as a classroom resource to help teach concepts and methods. They are mostly drawn from past examination questions and have been chosen as ones that are interesting in nature and require non-routine thinking. The hints and solutions are designed to explain the reasoning and highlight connections as well as giving the answer. In many cases, alternative methods or solutions are presented.
Checking the Review Question type in this category we see that O/AO-level questions are included.
I see several questions here that provide appropriate challenge for able GCSE students.
For example:
Can we fully factorise x4+4y4?
Starts with a Show that….
And then we factorise and will need to recall the difference of two squares.
We could get very sophisticated and look at those quadratic factors too; useful for those studying the Level 2 Further Mathematics Qualification.
Can we simplify these algebraic fractions?
Review algebraic fractions, simplification including the difference of two squares and quadratic equations. We could of course also talk about functions (including domain and range as these students are also studying AQA’s Level 2 Further Mathematics)
Can we simplify these simultaneous equations of degree 1 and 2?
To solve simultaneous equations, we’ll need simplification of algebraic fractions again and we can talk about the graphical solution of equations. We will also need to factorise a quadratic, 3y2−y−80 with a coefficient which is not 1 for the square term. We have all decided we are fans of the Box Method!
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