GCSE A Level
Mathematics
Improving through Collaboration
Contents
Contents
Self-Assessment Tracking..........................................................................................................................3
1. Expanding brackets and simplifying expressions...................................................................................4
2. Surds and rationalising the denominator...............................................................................................6
3. Rules of indices...................................................................................................................................10
4. Factorising expressions.......................................................................................................................14
5. Completing the square........................................................................................................................17
6. Solving quadratic equations by factorisation.......................................................................................19
7. Solving quadratic equations by completing the square........................................................................21
8. Solving quadratic equations by using the formula...............................................................................23
9. Sketching quadratic graphs.................................................................................................................25
10. Solving linear simultaneous equations using the elimination method................................................27
11. Solving linear simultaneous equations using the substitution method...............................................29
12. Solving linear and quadratic simultaneous equations........................................................................31
13. Solving simultaneous equations graphically......................................................................................34
14. Linear inequalities.............................................................................................................................37
15. Quadratic inequalities.......................................................................................................................39
16. Sketching cubic and reciprocal graphs...............................................................................................41
17. Translating graphs.............................................................................................................................44
18. Stretching graphs..............................................................................................................................47
19. Straight line graphs...........................................................................................................................51
,20. Parallel and perpendicular lines........................................................................................................54
21. Pythagoras’ theorem.........................................................................................................................57
22. Proportion.........................................................................................................................................60
23. Circle theorems.................................................................................................................................64
24. Trigonometry in right-angled triangles..............................................................................................71
25. The cosine rule..................................................................................................................................75
26. The sine rule......................................................................................................................................78
27. Areas of triangles..............................................................................................................................81
28. Rearranging equations......................................................................................................................84
29. Volume and surface area of 3D shapes..............................................................................................87
30. Area under a graph............................................................................................................................91
Answers..................................................................................................................................................96
2
,Self-Assessment Tracking
Section Confidence Completed Confidence
from (1-5 (5 is
GCSE most
1-5 (5 is confident)
most
confident)
1. Expanding Brackets and Simplifying Equations
2. Surds and rationalising the denominator
3. Rules of Indices
4. Factoring Expressions
5. Completing the Square
6. Solving Quadratic Equations by factorisation19
7. Solving Quadratic Equations by factorisation
8. Solving Quadratic Equations by using the formula
9. Sketching quadratic graphs
10. Solving linear simultaneous equations using the
elimination method
11. Solving linear simultaneous equations using the
substitution method
12. Solving linear and quadratic simultaneous
13.Solving simultaneous Equations graphically
14. Linear Inequalities
15. Quadratic Inequalities
16. Sketching cubic and reciprocal graphs
17. Translating Graphs
18. Stretching Graphs
19. Straight Line graphs
20. Parallel and Perpendicular Lines
21. Pythagoras’ Theorem
22. Proportion
23. Circle Theorems
24. Trigonometry in Right-Angled Triangles
25. The Cosine Rule
26. The Sine Rule
27. Areas of Triangles
28. Rearranging Equations
29. Volume and Surface Area of 3D Shapes
30. Area under a Graph
3
,1. Expanding brackets and simplifying expressions
A LEVEL LINKS
Scheme of work: 1a. Algebraic expressions – basic algebraic manipulation, indices and surds
Key points
When you expand one set of brackets you must multiply everything inside the bracket by what is outside.
When you expand two linear expressions, each with two terms of the form ax + b, where a ≠ 0 and b ≠ 0, you
create four terms. Two of these can usually be simplified by collecting like terms.
Examples
Example 1 Expand 4(3x − 2)
4(3x − 2) = 12x − 8 Multiply everything inside the bracket
by the 4 outside the bracket
Example 2 Expand and simplify 3(x + 5) − 4(2x + 3)
3(x + 5) − 4(2x + 3) 1 Expand each set of brackets
= 3x + 15 − 8x – 12 separately by multiplying (x + 5) by
3 and (2x + 3) by −4
= 3 − 5x 2 Simplify by collecting like terms:
3x − 8x = −5x and 15 − 12 = 3
Example 3 Expand and simplify (x + 3)(x + 2)
(x + 3)(x + 2) 1 Expand the brackets by multiplying
= x(x + 2) + 3(x + 2) (x + 2) by x and (x + 2) by 3
= x2 + 2x + 3x + 6
= x2 + 5x + 6 2 Simplify by collecting like terms:
2x + 3x = 5x
Example 4 Expand and simplify (x − 5)(2x + 3)
(x − 5)(2x + 3) 1 Expand the brackets by multiplying
= x(2x + 3) − 5(2x + 3) (2x + 3) by x and (2x + 3) by −5
= 2x2 + 3x − 10x − 15
= 2x2 − 7x − 15 2 Simplify by collecting like terms:
3x − 10x = −7x
4
, Practice
Watch out!
1 Expand.
a 3(2x − 1) b −2(5pq + 4q2) When multiplying (or dividing)
c −(3xy − 2y2) positive and negative numbers, if
the signs are the same the answer is
‘+’; if the signs are different the
answer is ‘–’.
2 Expand and simplify.
a 7(3x + 5) + 6(2x – 8) b 8(5p – 2) – 3(4p + 9)
c 9(3s + 1) –5(6s – 10) d 2(4x – 3) – (3x + 5)
3 Expand.
a 3x(4x + 8) b 4k(5k2 – 12)
c –2h(6h2 + 11h – 5) d –3s(4s2 – 7s + 2)
4 Expand and simplify.
a 3(y2 – 8) – 4(y2 – 5) b 2x(x + 5) + 3x(x – 7)
c 4p(2p – 1) – 3p(5p – 2) d 3b(4b – 3) – b(6b – 9)
5 Expand (2y – 8)
6 Expand and simplify.
a 13 – 2(m + 7) b 5p(p2 + 6p) – 9p(2p – 3)
7 The diagram shows a rectangle.
Write down an expression, in terms of x, for the area of the rectangle.
Show that the area of the rectangle can be written as 21x2 – 35x
8 Expand and simplify.
a (x + 4)(x + 5) b (x + 7)(x + 3)
c (x + 7)(x – 2) d (x + 5)(x – 5)
e (2x + 3)(x – 1) f (3x – 2)(2x + 1)
g (5x – 3)(2x – 5) h (3x – 2)(7 + 4x)
i (3x + 4y)(5y + 6x) j (x + 5)2
k (2x − 7)2 l (4x − 3y)2
Extend
9 Expand and simplify (x + 3)² + (x − 4)²
10 Expand and simplify.
a b
5
Mathematics
Improving through Collaboration
Contents
Contents
Self-Assessment Tracking..........................................................................................................................3
1. Expanding brackets and simplifying expressions...................................................................................4
2. Surds and rationalising the denominator...............................................................................................6
3. Rules of indices...................................................................................................................................10
4. Factorising expressions.......................................................................................................................14
5. Completing the square........................................................................................................................17
6. Solving quadratic equations by factorisation.......................................................................................19
7. Solving quadratic equations by completing the square........................................................................21
8. Solving quadratic equations by using the formula...............................................................................23
9. Sketching quadratic graphs.................................................................................................................25
10. Solving linear simultaneous equations using the elimination method................................................27
11. Solving linear simultaneous equations using the substitution method...............................................29
12. Solving linear and quadratic simultaneous equations........................................................................31
13. Solving simultaneous equations graphically......................................................................................34
14. Linear inequalities.............................................................................................................................37
15. Quadratic inequalities.......................................................................................................................39
16. Sketching cubic and reciprocal graphs...............................................................................................41
17. Translating graphs.............................................................................................................................44
18. Stretching graphs..............................................................................................................................47
19. Straight line graphs...........................................................................................................................51
,20. Parallel and perpendicular lines........................................................................................................54
21. Pythagoras’ theorem.........................................................................................................................57
22. Proportion.........................................................................................................................................60
23. Circle theorems.................................................................................................................................64
24. Trigonometry in right-angled triangles..............................................................................................71
25. The cosine rule..................................................................................................................................75
26. The sine rule......................................................................................................................................78
27. Areas of triangles..............................................................................................................................81
28. Rearranging equations......................................................................................................................84
29. Volume and surface area of 3D shapes..............................................................................................87
30. Area under a graph............................................................................................................................91
Answers..................................................................................................................................................96
2
,Self-Assessment Tracking
Section Confidence Completed Confidence
from (1-5 (5 is
GCSE most
1-5 (5 is confident)
most
confident)
1. Expanding Brackets and Simplifying Equations
2. Surds and rationalising the denominator
3. Rules of Indices
4. Factoring Expressions
5. Completing the Square
6. Solving Quadratic Equations by factorisation19
7. Solving Quadratic Equations by factorisation
8. Solving Quadratic Equations by using the formula
9. Sketching quadratic graphs
10. Solving linear simultaneous equations using the
elimination method
11. Solving linear simultaneous equations using the
substitution method
12. Solving linear and quadratic simultaneous
13.Solving simultaneous Equations graphically
14. Linear Inequalities
15. Quadratic Inequalities
16. Sketching cubic and reciprocal graphs
17. Translating Graphs
18. Stretching Graphs
19. Straight Line graphs
20. Parallel and Perpendicular Lines
21. Pythagoras’ Theorem
22. Proportion
23. Circle Theorems
24. Trigonometry in Right-Angled Triangles
25. The Cosine Rule
26. The Sine Rule
27. Areas of Triangles
28. Rearranging Equations
29. Volume and Surface Area of 3D Shapes
30. Area under a Graph
3
,1. Expanding brackets and simplifying expressions
A LEVEL LINKS
Scheme of work: 1a. Algebraic expressions – basic algebraic manipulation, indices and surds
Key points
When you expand one set of brackets you must multiply everything inside the bracket by what is outside.
When you expand two linear expressions, each with two terms of the form ax + b, where a ≠ 0 and b ≠ 0, you
create four terms. Two of these can usually be simplified by collecting like terms.
Examples
Example 1 Expand 4(3x − 2)
4(3x − 2) = 12x − 8 Multiply everything inside the bracket
by the 4 outside the bracket
Example 2 Expand and simplify 3(x + 5) − 4(2x + 3)
3(x + 5) − 4(2x + 3) 1 Expand each set of brackets
= 3x + 15 − 8x – 12 separately by multiplying (x + 5) by
3 and (2x + 3) by −4
= 3 − 5x 2 Simplify by collecting like terms:
3x − 8x = −5x and 15 − 12 = 3
Example 3 Expand and simplify (x + 3)(x + 2)
(x + 3)(x + 2) 1 Expand the brackets by multiplying
= x(x + 2) + 3(x + 2) (x + 2) by x and (x + 2) by 3
= x2 + 2x + 3x + 6
= x2 + 5x + 6 2 Simplify by collecting like terms:
2x + 3x = 5x
Example 4 Expand and simplify (x − 5)(2x + 3)
(x − 5)(2x + 3) 1 Expand the brackets by multiplying
= x(2x + 3) − 5(2x + 3) (2x + 3) by x and (2x + 3) by −5
= 2x2 + 3x − 10x − 15
= 2x2 − 7x − 15 2 Simplify by collecting like terms:
3x − 10x = −7x
4
, Practice
Watch out!
1 Expand.
a 3(2x − 1) b −2(5pq + 4q2) When multiplying (or dividing)
c −(3xy − 2y2) positive and negative numbers, if
the signs are the same the answer is
‘+’; if the signs are different the
answer is ‘–’.
2 Expand and simplify.
a 7(3x + 5) + 6(2x – 8) b 8(5p – 2) – 3(4p + 9)
c 9(3s + 1) –5(6s – 10) d 2(4x – 3) – (3x + 5)
3 Expand.
a 3x(4x + 8) b 4k(5k2 – 12)
c –2h(6h2 + 11h – 5) d –3s(4s2 – 7s + 2)
4 Expand and simplify.
a 3(y2 – 8) – 4(y2 – 5) b 2x(x + 5) + 3x(x – 7)
c 4p(2p – 1) – 3p(5p – 2) d 3b(4b – 3) – b(6b – 9)
5 Expand (2y – 8)
6 Expand and simplify.
a 13 – 2(m + 7) b 5p(p2 + 6p) – 9p(2p – 3)
7 The diagram shows a rectangle.
Write down an expression, in terms of x, for the area of the rectangle.
Show that the area of the rectangle can be written as 21x2 – 35x
8 Expand and simplify.
a (x + 4)(x + 5) b (x + 7)(x + 3)
c (x + 7)(x – 2) d (x + 5)(x – 5)
e (2x + 3)(x – 1) f (3x – 2)(2x + 1)
g (5x – 3)(2x – 5) h (3x – 2)(7 + 4x)
i (3x + 4y)(5y + 6x) j (x + 5)2
k (2x − 7)2 l (4x − 3y)2
Extend
9 Expand and simplify (x + 3)² + (x − 4)²
10 Expand and simplify.
a b
5
