1. Windsor High School and Sixth Form
Windsor Sixth Form Summer Work Maths Year 12
Windsor High School and Sixth Form
Mathematics Faculty
Induction Pack

2. Getting Started…
Congratulations on completing your GCSE studies, you are now ready to begin the next stage of your education. It is important that you get some rest over the summer to re-energise, but it is also important that you spend time preparing for the A Level Mathematics course. This induction pack contains worked examples and practice exercises for you to complete prior to beginning your studies in September. All of the content contained within this induction pack is Higher GCSE content. It is essential that you have a thorough understanding of these topics and can confidently perform all of these skills with fluency. This will enable you to make the best possible start to the A Level Mathematics course. It is recommended that you spend at least one hour a week working through this induction pack over the summer. You will complete an assessment during your first week of lessons in order to assess your understanding of these vital skills. If you have taken the time to complete this induction pack to a good standard, you should feel confident in making a positive start to your studies. We look forward to welcoming you in September. Mr Wootton

3. Contents Factorising Indices Surds Solving Quadratics
Completing the Square
Simultaneous Equations
Equations of Straight Lines
Algebraic Fractions

4. Factorising 1 and -6 are the only two numbers that work here
Factorising is the reverse of expanding brackets. You will need to know how to factorise completely a wide range of expressions, but we will focus specifically on quadratic expressions here.
The same two numbers must multiply to give this value
1 and -6 are the only two numbers that work here
Two numbers add to give this value
This expands to give 6x2 – 11x – 10
as we require
The last terms in each bracket must multiply to give -10
The first terms in each bracket must multiply to give 6x2
Try out combinations that satisfy these two conditions,
such as 3x and 2x, -5 and 2, put them in double brackets, then expand to check your answer is correct

5. Factorising Exercise
Part A Part B

6. Indices
The laws of indices allow you to simplify powers of the same base. Here we will focus on exponents that are fractional and/or negative.
Using the second rule, x to the power of 6 over 3 is x2
The cube root of 125 is 5
Using the second rule, the square root of 25 is 5, which is then cubed to become 125
Using the third rule, a negative power means put 1 over the same base to the positive power

7. Indices Exercise
Part A Part B

8. Surds Expand single bracket as normal Expand top and bottom
Surds are irrational numbers which are written as the square root of a non-square integer. You need to recall how to manipulate these, including rationalising the denominator where necessary.
Expand single bracket as normal
Manipulate using the first rule
This is the second rule from the list below
Expand top and bottom
Simplify the denominator

9. Surds Exercise
Part A Part B

10. Solving Quadratics Equation must be made equal to zero first
Quadratic equations can be solved using a variety of techniques. Some can be factorised into two brackets first, but those which can’t be factorised may be solved by using the formula.
Equation must be made equal to zero first
Subtract 2 and 3x from both sides
The left factorises here
Solve each bracket equal to zero
Only one solution here as 4 is a repeated root

11. Solving Quadratics Exercise
Part A Part B

12. Completing the Square Apply the rule
This technique enables us to determine the turning point of a quadratic graph, in order to find its maximum or minimum value. It can also be used to solve quadratics, but this section will limit itself to the former case only.
Apply the rule
Simplify this part
Take the factor of 3 out first, then apply the rule to the bracket
This is a single bracket with 3 outside so expand it by multiplying (x + 1)2 by 3 and 12 by 3
Complete the square for x2 + 2x
p = 3, q = 1, r = -2

13. Completing the Square Exercise
Part A Part B

14. Simultaneous Equations
The solution to two simultaneous equations is found where their two graphs intersect. This can be performed algebraically for a range of functions, however we will restrict this to two linear, or a linear and a quadratic.
Rearrange the easier equation to make x or y the subject, then
substitute it into the other equation (here we will make y the subject)
We replace y in the second equation with 2x - 1
Substitute this x value into the
equation for y to find the value of y
Always substitute the linear equation
into the quadratic, rearrange if necessary
Here we replace all terms of x in the second equation with 3 – 2y
Factorise and solve to find all y values

15. Simultaneous Equations Exercise
Part A Part B

16. Equations of Straight Lines
The equation of a straight line graph in the Cartesian plane can be used for a variety of purposes, such as mathematical modelling where appropriate. Here we will focus on the calculation of the gradient and intercept.
HINT: draw a
sketch to help
Use the formula to calculate the gradient
Find the gradient first by using the formula
y = mx + c
Find c
Substitute one of the coordinates into this equation

17. Equations of Straight Lines Exercise
Part A Part B

18. Algebraic Fractions You must find a common denominator first
Fractions and their use are widespread throughout mathematics. Algebraic fractions and their manipulation are a requirement in many calculations, and you will be expected to be proficient in simplifying them.
Start by factorising the numerator and denominator where possible
Expand brackets in the numerator
Subtract the numerators
Simplify
You must find a common denominator first

19. Algebraic Fractions Exercise
Part A Part B