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GCSE - Algebra

 

 

Foundation only

Foundation and Higher

Higher only

 

 

 

  

 

 

1. USE OF SYMBOLS

2. SEQUENCES
3. COORDINATES AND GRAPHS OF LINEAR FUNCTIONS
4. EQUATIONS
5. TRIAL AND IMPROVEMENT

6. REAL-LIFE GRAPHS

7. FORMULAE

8. QUADRATIC FUNCTIONS

9. INEQUALITIES AND REGIONS

10. SIMULTANEOUS EQUATIONS

11. OTHER FUNCTIONS

12. TRANSFORMING FUNCTIONS

13. ALGEBRAIC PROOFS

14. ALGEBRAIC MANIPULATION

15. NON-LINEAR GRAPHS AND GRAPHICAL SOLUTION OF EQUATIONS


USE OF SYMBOLS

Specification Learning objectives Grade Resources

distinguish in meaning between the words 'equation', 'formula', 'identity' and 'expression'.

manipulate algebraic expressions by collecting like terms, by multiplying a single term over a bracket, and by taking out common factors

Simplify expressions with one variable such as a + 2a + 3a F


W  simplifying blockbusters

W   factorising  

W   simplifying collect a joke

W  simplifying

F Expand over 1 bracket

Simplify expressions with more than one variable E
Multiply out expressions with brackets such as 3(x+2) D
Factorise expressions over one bracket D
Expand harder expressions involving indices or more than one bracket such as 3(x+2)-5(2x-1) C
expand the product of two linear expressions   Be able to expand expressions like (x+2)(x-4)

C

W  quadratics simplifying and expanding
F Expand 2 sets of brackets

 

SEQUENCES

Specification Learning objectives Grade Resources

generate terms of a sequence using term-to-term and position-to-term definitions of the sequence

.

Continue a sequence of numbers or diagrams G

P   W   number patterns

W   W  number sequences

 

W   Flash cards

Write the terms of a simple sequence G
Find a particular term in a sequence of positive numbers F
Write the term to term rule of a sequence involving positive numbers F

Find a particular term in a sequence of negative numbers or fractions

E

Write the term to term rule of a sequence involving negative numbers or fractions

E

generate common integer sequences (including sequences of odd or even integers, squared integers, powers of 2, powers of 10, triangular numbers)

Write the terms of a sequence or a series of diagrams given the nth term D

P   Sequence envelope

use linear expressions to describe the nth term of an arithmetic sequence, justifying its form by reference to the activity or context from which it was generated

Describe the nth term of a sequence or a series of diagrams C

W  W  sequences and formulae
W W  number patterns
W   nth term
W Sequences based on patterns

W  nth term

 

COORDINATES AND GRAPHS OF LINEAR FUNCTIONS

Specification Learning objectives Grade Resources
Understand that one coordinate identifies a point on a number line, two coordinates identify a point in a plane, and three coordinates identify a point in space, using the terms 1-D, 2-D, and 3-D.

use the conventions for coordinates in the plane

plot points in all four quadrants

and use axes and coordinates to specify points in all four quadrants

locate points with given coordinates

Use coordinates in the first quadrant G  

P  P  W 

W

  co-ordinate grids

W   co-ordinate fred  P (answers)

W   co-ordinate pictures

W   orc co-ordinates

 

W   W  coordinate bingo

W   3-D co-ordinates  

Use coordinates in all four quadrants F
Draw lines such as x=3 and y=x+2 E
Draw lines such as y=2x+3 D
Solve problems involving graphs, such as finding where the line y=x+2 crosses the line y=1 D
Use and understand coordinates in 3D C

find the coordinates of points identified by geometrical information

Solve geometrical problems using coordinates such as finding the 4th corner of a parallelogram given the other 3

D

 

find the coordinates of the midpoint of the line segment AB, given A and B.

Find the midpoint of a line segment

C  

recognise (when values are given for m and c) that equations of the form y = mx + c correspond to straight-line graphs in the coordinate plane

Recognise the equations of straight line graphs such as y=-4x+2 C

W   W  W  straight line graphs

 

 

find the gradient of lines given by equations of the form y = mx + c (when values are given for m and c)

understand that the form y = mx + c represents a straight line and that m is the gradient of the line, and c is the value of the y-intercept

Find the gradients of straight line graphs C
Find the gradient and y-intercept from the equation of a straight line B

investigate the gradients of parallel lines

Explore the gradients of parallel straight-line graphs B

explore the gradients of parallel lines and lines perpendicular to each other

Explore the gradients of perpendicular straight-line graphs

A

 

EQUATIONS

Specification Learning objectives Grade Resources

set up simple equations

 

solve simple equations by using inverse operations or by transforming both sides in the same way

 

solve linear equations, with integer  or fractional coefficients, in which the unknown appears on either side or on both sides of the equation

 

solve linear equations that require prior simplification of brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution

Solve equations such as 3x=12 or x+5=9

F

W   forming equations

W   W  making equations

W   word equations

W   X 

 W

  solving equations

W    X (Answer grid)   Number grid

 

W W  X  X solving equations

W  re-arranging using given operations

W  balancing

X  linear equations

 

W  Collect a letter

W  simple equations

P  domino loops

W  solving equations

W  equation match

W   manipulating equations

W   solving equations

Solve equations suc as 3x-1=9 or x/2 = 7

E

Solve equations such as 3x-4=5 or 2(5x+1)=28

D

Solve equations such as 3x-12=2(x-5) and equations involving fractions or simple algebraic fractions C
Find a solution to a problem by forming an equation and solving it C
Solve equations with more than one algebraic fraction B

 

TRIAL AND IMPROVEMENT

Specification Learning objectives Grade Resources

use systematic trial and improvement to find approximate solutions of equations where there is no simple analytical method of solving them.

 Form and solve no-linear equations using trial and improvement methods

C

P Starter 

W   W  W  F trial and improvement

 
P P Examples

 

REAL-LIFE GRAPHS

Specification Learning objectives Grade Resources

interpret information presented in a range of linear and non-linear graphs

 

construct linear functions from real-life problems and plot their corresponding graphs

 

discuss and interpret graphs arising from real situations

Plot points of a conversion graph and read off positive values F

W  P  Conversion graphs without questions

W   conversion graphs questions

W  drawing conversion graphs

 

Read from a conversion graph for negative values E
Interpret distance-time graphs E

   P  travel graphs without questions

W   gradients and area under graph

 

P   P  W  animated travel graphs

 

Calculate simple average speeds from distance-time graphs D

Calculate complex average speeds from distance-time graphs

C
Interpret velocity-time graphs B
Discuss and interpret graphs modelling real situations B

 

FORMULAE

Specification Learning objectives Grade Resources

use formulae from mathematics and other subjects expressed initially in words and then using letters and symbols

Use a formula written in words G

X   number machines

 

Use a simple formula expressed in symbols F
derive/generate a formula Write an expression from a problem F

substitute numbers into a formula

 

substitute positive and negative numbers into expressions with indices.

Substitute positive numbers into a simple formula F

W  W  4 in a line

W W  W  substitution

W  substitution

Substitute negative numbers into a simple formula

E

Substitute negative numbers into a complicated formula

D

change the subject of a formula, including cases where the subject occurs twice, or where a power of the subject appears  

Rearrange linear formulae such as p=3q+5 C

W   W  W  changing the subject  

Rearrange formulae that include brackets, fractions and square roots B
Rearrange formulae where the variable appears twice A

 

QUADRATIC FUNCTIONS

Specification Learning objectives Grade Resources

factorise quadratic expressions including the difference of two squares and cancel common factors in rational expressions  

solve quadratic equations by factorisation, completing the square and using the quadratic formula  

Solve quadratic equations such as x-8x+15=0 by factorisation B

W  F F Factorise expressions
F F Factorise and solve


P P W F Factorisation Formula

 

 

F F Complete the square

Solve quadratic equations such as x-2x-1=0 by using the quadratic formula

Solve equations with algebraic fractions by converting them to a quadratic equation A*
Complete the square A*
Use completing the square to solve equations and find maximum and minimum values A*

 

INEQUALITIES AND

REGIONS

Specification Learning objectives Grade Resources

solve simple linear inequalities in one variable, and represent the solution set on a number line

Solve simple inequalities such as 3x<9 and represent solutions on a number line C

W   W  inequalities

Solve linear inequalities such as 4x-3<10 or 4x<2x+7 and represent solutions on a number line

C

Solve linear inequalities such as x+13>5x-3 and represent solutions on a number line

B

solve several linear inequalities in two variables and find the solution set

Solve a set of linear inequalities in two variables the solution as a region of a graph B


 

SIMULTANEOUS EQUATIONS

Specification Learning objectives Grade Resources

find the exact solution of two simultaneous equations in two unknowns by eliminating a variable, and interpret the equations as lines and their common solution as the point of intersection

 Solve a pair of simultaneous equations in two unknowns, such as 2x+y=5 and 3x-2y=4 B

P P P Starter Intructions Rules
W   W  W  W  P P simultaneous equations

W   spot the mistake

 

find the intersection points of the graphs of a linear and quadratic function, knowing that these are the approximate solutions of the corresponding simultaneous equations representing the linear and quadratic functions   Know that simultaneous equations can be represented by lines on a graph and that the point of intersection is the solution B

P  W  W  W  graphical solutions of equations  

W   Graphical method

solve exactly, by elimination of an unknown, two simultaneous equations in two unknowns, one of which is linear in each unknown, and the other is linear in one unknown and quadratic in the other

, or where the second is of the form x + y = r

Solve a pair of simultaneous equations where one is linear and one is non-linear such as y=3x-2 and y=x A

P Linear and Quadratic

find graphically the intersection points of a given straight line with this circle and know that this corresponds to solving the two simultaneous equations representing the line and the circle.

Solve a pair of simultaneous equations where one is linear and one is non-linear such as x+5y=13 and x+y=13

A*   F Quadratics

Solve a pair of simultaneous equations graphically, such as y=x-1 and x+y=16

A

Solve a pair of simultaneous equations graphically, such as y=2x-1 and x+y=27

A*

 

OTHER FUNCTIONS

Specification Learning objectives Grade Resources

plot graphs of functions in which y is given explicitly in terms of x, or implicitly; no table or axes given 

 

generate points and plot graphs of simple quadratic functions, then more general quadratic functions and find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function

Draw graphs of simple quadratic functions such as y=3x and y=x+4 D

W   understanding equations

 

W   quadratic graphs





W W W Graphical solutions

Draw graphs of harder quadratic functions such as y=x-2x+1

C
Find the points of intersection of quadratic graphs with lines C
Use graphs to find approximate solutions to quadratic equations C

plot graphs of simple cubic functions, the reciprocal function y=1/x (with x not zero), the exponential function y=kx for integer values of k and positive values of x.

recognise the characteristic shapes of all the above functions

Complete tables for, and draw graphs of cubic and reciprocal functions B

 

W  investigating graphs

Use graphs of cubic and reciprocal functions to solve equations B
Solve cubic equations by drawing appropriate lines on graphs A*
Plot and sketch graphs of exponential functions A*
Recognise the shapes of graphs of functions A*

 

TRANSFORMING FUNCTIONS

Specification Learning objectives Grade Resources
apply to the graph of y = f(x) the transformations y = f(x) + a, y = f(ax), y = f(x + a), y = af(x) for linear, quadratic, sin and cos functions f(x)   Transform the

graph of y = f(x), using the transformations

y = f(x) + a, y = f(ax), y = f(x + a), y = af(x) for linear, quadratic, sin and cos functions f(x)  

A*

P  transformations of graphs

W  functions  

 

 

ALGEBRAIC PROOFS

Specification Learning objectives Grade Resources
show step by step deduction in problem solving Decide with a reason whether a statement is true or false E-D  
recognise the importance of counter-examples Identify a counter example D
derive proofs using short chains of deductive reasoning

derive proofs using short chains of deductive reasoning

C
understand the difference between a practical demonstration and a proof

understand the difference between a practical demonstration and a proof

C

 

ALGEBRAIC MANIPULATION

Specification

Learning objectives

Grade

Resources

know the meaning of and use the words equation, formula, identity and expression

 

manipulate algebraic expressions by collecting like terms, by multiplying a single term over a bracket, and by taking out common factors

Be able to collect like terms

E

W   blockbusters

activity

multiply out simple expressions with brackets D  

Factorise simple expressions requiring one bracket.

D

 

factorise quadratic expressions including the difference of two squares and cancel common factors in rational expressions  

expand and simplify harder expressions

C

W  expanding & simplifying collect a joke

W  expanding and simplifying
F factorise where a=1

W  factorising including quadratics

F factorise where a is greater than 1 

Factorise quadratic expressions where the coefficient of x is 1

B

Factorise quadratic expressions where the coefficient of x is greater than 1

A

Simplify rational expressions

B

Simplify harder rational expressions

A*

expand the product of two linear expressions  

Expand and simplify quadratic expressions where the coefficient of x is 1

B

W  W  expanding and simplifying expressions 

including two brackets

Expand and simplify quadratic expressions where the coefficient of x is greater than 1

A

 

NON-LINEAR GRAPHS AND GRAPHICAL SOLUTION OF EQUATIONS

Specification

Learning objectives

Grade

Resources

generate points and plot graphs of simple quadratic functions, then more general quadratic functions and find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function  

Draw graphs of simple quadratic functions such as y=3x and y=x+4

D

P  explaining plotting curved graphs

W  quadratic graphs

 

 

Draw graphs of harder quadratic functions such as y=x-2x+1

C

find the intersection points of the graphs of a linear and quadratic function, knowing that these are the approximate solutions of the corresponding simultaneous equations representing the linear and quadratic functions

Find the points of intersection of quadratic graphs with lines

C

P  explaining intersection points for solving equations

W  W    graphical solutions of equations

 

W  using a graph plotter and deciding which graphs to plot.

Use graphs to find the approximate solutions of quadratic equations

C

Use points of intersection of a quadratic graph such as y=x-2x-4 with lines such as y=2x+1 to solve equations like x-2x-4=2x+1 and simplify this to x-4x-5

A

Solve equations using two graphs where the equation needs manipulating to match the graphs

A

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