Algebra Foundation Topics

**1. Use and interpret algebraic notation, including:**

- ab in place of a × b
- 3y in place of y + y + y and 3 × y
- a
^{2}in place of a × a - a
^{3}in place of a × a × a; a^{2}b 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) 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 binomials

(x –1)(x – 2) = x² – 3x + 2

(a + 2b)(a – b) = a² + ab – 2b² - factorising quadratic expressions:

x^{2}+ 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 = a^{5}; 3a^{3}÷ a = 3a³

y^{2}× y^{5}= y^{7}; y^{8}÷ y^{3}= y^{5}

(y^{2})^{4}= y×y × y×y × y×y × y×y = y^{8}

5. Understand and use standard mathematical formulae;

Circumference circle: 2πr = πd

Area circle: πr_{2}

Pythagoras’ theorem: a^{2} = b^{2} + c^{2}

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.

**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; x^{2} + x^{2} ≡ 2x^{2}

Argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments

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

**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;

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 = 2x^{2} + 1

y = x^{3} – 2x ; y=1/x+ 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**

12. Recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function y=1/x, with x ≠ 0;

e.g. y = 2 ; x = 1 ; y = 2x ; y=x^{2}

y = x^{3} – 2x ; y=1/x

**13. 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.

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

Find approximate solutions using a graph

**15. Solve quadratic equations like:**

x^{2} + bx + c, (NOT including those that require rearrangement) algebraically by factorising;

Find approximate solutions using a graph

Solve x^{2} – 5x + 6 = 0,

Find x for an x cm by (x + 1)cm rectangle of area 42cm^{2}

**16. Solve two simultaneous linear equations in two variables algebraically;**

e.g. Solve simultaneously

2x + 3y = 18 and

y = 3x – 5

Find approximate solutions using a graph

**17. 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

**18. Solve linear inequalities in one variable;**

e.g. Solve 3x – 1 = 5

Represent the solution set on a number line.

e.g 2x + 1 ≥ 7 and 1 < 3x – 5 ≤ 10

**19. 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

**20. Recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions ( r ^{n} where n is an integer, and r is a rational number > 0)**

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 x_{n} = x_{n –1} + x_{n–2}

21. Deduce expressions to calculate the n*th* term of linear sequence.

e.g. nth term = n^{2} + 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