3.3 Ratio, proportion and rates of change

R1

Basic foundation content

Additional foundation content

Higher content only

change freely between related standard units (eg time, length, area, volume/capacity, mass) and compound units (eg speed, rates of pay, prices) in numerical contexts

compound units (eg density, pressure)

in numerical and algebraic contexts

 

R2

Basic foundation content

Additional foundation content

Higher content only

use scale factors, scale diagrams and maps

   

Notes: including geometrical problems.

R3

Basic foundation content

Additional foundation content

Higher content only

express one quantity as a fraction of another, where the fraction is less than 1 or greater than 1

   

R4

Basic foundation content

Additional foundation content

Higher content only

use ratio notation, including reduction to simplest form

   

R5

Basic foundation content

Additional foundation content

Higher content only

divide a given quantity into two parts in a given part : part or part : whole ratio

express the division of a quantity into two parts as a ratio

apply ratio to real contexts and problems (such as those involving conversion, comparison, scaling, mixing, concentrations)

   

Notes: including better value or best-buy problems.

R6

Basic foundation content

Additional foundation content

Higher content only

express a multiplicative relationship between two quantities as a ratio or a fraction

   

R7

Basic foundation content

Additional foundation content

Higher content only

understand and use proportion as equality of ratios

   

R8

Basic foundation content

Additional foundation content

Higher content only

relate ratios to fractions and to linear functions

   

Notes: see also N11, R14

R9

Basic foundation content

Additional foundation content

Higher content only

define percentage as ‘number of parts per hundred’

interpret percentages and percentage changes as a fraction or a decimal, and interpret these multiplicatively

express one quantity as a percentage of another

compare two quantities using percentages

work with percentages greater than 100%

solve problems involving percentage change, including percentage increase/decrease and original value problems, and simple interest including in financial mathematics

   

Notes: see also N2, N12

R10

Basic foundation content

Additional foundation content

Higher content only

solve problems involving direct and inverse proportion, including graphical and algebraic representations

   

R11

Basic foundation content

Additional foundation content

Higher content only

use compound units such as speed, rates of pay, unit pricing

use compound units such as density and pressure

 

Notes: including making comparisons.

R12

Basic foundation content

Additional foundation content

Higher content only

compare lengths, areas and volumes using ratio notation

scale factors

make links to similarity (including trigonometric ratios)

 

Notes: see also G19, G20

R13

Basic foundation content

Additional foundation content

Higher content only

 

understand that is inversely proportional to is equivalent to is proportional to

 
 

interpret equations that describe direct and inverse proportion

construct and interpret equations that describe direct and inverse proportion

R14

Basic foundation content

Additional foundation content

Higher content only

 

interpret the gradient of a straight-line graph as a rate of change

recognise and interpret graphs that illustrate direct and inverse proportion

 

Notes: see also A15, R8

R15

Basic foundation content

Additional foundation content

Higher content only

   

interpret the gradient at a point on a curve as the instantaneous rate of change

apply the concepts of average and instantaneous rate of change (gradients of chords and tangents) in numerical, algebraic and graphical contexts

Notes: see also A15.

R16

Basic foundation content

Additional foundation content

Higher content only

 

set up, solve and interpret the answers in growth and decay problems, including compound interest

and work with general iterative processes