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

use scale factors, scale diagrams and maps

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

use ratio notation, including reduction to simplest form

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)

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

understand and use proportion as equality of ratios

relate ratios to fractions and to linear functions

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

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

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

compare lengths, areas and volumes using ratio notation;

make links to similarity (including trigonometric ratios) and scale factors

understand that \(X\) is inversely proportional to \(Y\) is equivalent to \(X\) is proportional to \(\dfrac{1}{Y}\);

construct and interpret equations that describe direct and inverse proportion

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

recognise and interpret graphs that illustrate direct and inverse proportion

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

set up, solve and interpret the answers in growth and decay problems, including compound interest and work with general iterative processes