### GCSE Maths (9-1) H: Algebra » Foundation Algebra

GCSE (9-1) Higher Mathematics is broken down into six main areas:

• Number (NH)
• Algebra (AH)
• Ratio, proportions, rates of change (RH)
• Geometry and Measures (GH)
• Statistics (SH)
• Probability(PH)

ALGEBRA:H

1. Use and interpret algebraic notation, including:

• ab in place of a × b
• 3y in place of y + y + y and 3 × y
• a2 in place of a × a
• a3 in place of a × a × a; a2b in place of a × a × b
• in place of a ÷ b
• coefficients written as fractions rather than as decimals
• brackets

2. Substitute numerical values into formulae and expressions, including scientific formulae

e.g. Given that v = u + at,
find v when t = 1, a = 2 and u = 7

v = √u² + 2as

with u = 2.1, s = 0.18, a = -9.8.

3. Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors

4. Simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by:

• collecting like terms
2a + 3a = 5a
• multiplying a single term over a bracket
2(a + 3b) = 2a + 6b
2(a + 3b) + 3(a – 2b) = 5a
• taking out common factors(factorise)
3a – 9b = 3(a – 3b)
2x + 3x² = x(2 + 3x)
• expanding products of two or more binomials
(x –1)(x – 2) = x² – 3x + 2
(a + 2b)(a – b) = a² + ab – 2b²
• factorising quadratic expressions:
ax2 + bx + c, and the difference of two squares
x² – x – 6 = (x – 3)(x + 2)
x² – 16 = (x – 4)(x + 4)
x² – 3 = (x – √3)(x + √3)
• simplifying expressions involving sums, products and powers, including the laws of indices
a × a × a = a³  ; 2a × 2b = 4ab
a³ × a² = a5  ; 3a3 ÷ a = 3a³
y2 × y5 = y7  ; y8 ÷ y3 = y5
(y2)4 = y×y × y×y × y×y × y×y = y8

5. Understand and use standard mathematical formulae;

Circumference circle: 2πr = πd
Area circle: πr2
Pythagoras' theorem: a2 = b2 + c2

Trigonometry formulae

Rearrange formulae to change the subject

1. where the subject appears once only.
Make d the subject of the formula
c = πd.
Make x the subject of the formula
y = 3x - 2.

ALGEBRA:H

6. Know the difference between an equation and an identity;

An Identity is an equation that is true for all values (use the ≡ identical symbol)
e.g. 3a + 2a ≡ 5a; x2 + x2 ≡ 2x2

Argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs
e.g. show that (x + 1)² + 2 = x² + 2x + 3

7. Where appropriate, interpret simple expressions as functions with inputs and outputs.

e.g. y = 2x + 3 as

interpret the reverse process as the 'inverse function' ; interpret the succession of two functions as a'composite function'

8. Work with coordinates in all four quadrants.

9. Plot graphs of equations that correspond to straight-line graphs in the coordinate plane;

Use the form y = mx + c to identify parallel lines and perpendicular lines

Find the equation of the line through two given points, or through one point with a given gradient

Use a table of values to plot graphs:
e.g. y = 2x + 3; y = 2x2 + 1

y = x3 – 2x ; + x

10. Identify and interpret gradients and intercepts of linear functions graphically and algebraically

y = 2x + 1 has gradient = 2, intercept = 1
y = 3 – x has gradient = –1, intercept = 3

11. Identify and interpret roots, intercepts, turning points (stationary points) of quadratic functions graphically;

Deduce roots algebraically and turning points by completing the square

12. Recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function , with x ≠ 0;

e.g. y = 2 ; x = 1 ; y = 2x ; y=x2
y = x3 – 2x ;

exponential functions y=kx for positive values of k, and the trigonometric functions (with arguments in degrees) y = sin x , y = cos x and y = tan x for angles of any size

13. Sketch translations and reflections of a given function

14. Plot and interpret graphs (including reciprocal graphs) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration.

i.e. distance-time, money conversion, temperature conversion

Straight line gradients = rates of change.
Gradient distance-time graph = velocity.

15. Calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts

16. Recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point

17. Solve linear equations in one unknown algebraically (including those with the unknown on both sides of the equation);

Find approximate solutions using a graph

18. Solve quadratic equations like:
x2 + bx + c, (including those that require rearrangement) algebraically by factorising; by completing the square and by using the quadratic formula;

Find approximate solutions using a graph

Solve x2 – 5x + 6 = 0,

Find x for an x cm by (x + 1)cm rectangle of area 42cm2

19. Solve two simultaneous linear equations in two variables linear/linear or linear/quadratic algebraically;

e.g. Solve simultaneously
2x + 3y = 18 and
y = 3x - 5

Find approximate solutions using a graph

20. Find approximate solutions to equations numerically using iteration

21. Translate simple situations or procedures into algebraic expressions or formulae;

e.g. Cost of car hire at £50 per day plus 10p per mile. The perimeter of a rectangle when the length is 2 cm more than the width.

Derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution

22. Solve linear inequalities in one or two variable; and quadratic inequalities in one variable.
e.g. Solve 3x - 1 = 5

Represent the solution set on a number line, using set notation and on a graph.
e.g 2x + 1 ≥ 7      and   1 < 3x - 5 ≤ 10

23. Generate terms of a sequence from either a term-to-term or a position-to-term rule

Continue the sequences
1, 4, 7, 10, ...
1, 4, 9, 16, ...
3, 4, 5, ... n + 2

24. Recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions ( rn where n is an integer, and r is a rational number > 0 or a surd) and other sequences

Triangular numbers: 3, 6, 10, 15,
Square numbers: 1, 4, 9, 16, 25,
Cube numbers: 1, 27, 81, 256,

Fibonacci: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34
The next number is found by adding up the two numbers before it.
The Rule is xn = xn –1 + xn–2

25. Deduce expressions to calculate the nth term of linear sequence.

e.g. nth term = n2 + 2n gives 3, 8, 15, ...

Find a formula for the nth term of an arithmetic sequence.

3, 7, 11, 14... 4n – 1
40, 37, 34, 31... 43 – 3n