Place value
Place value is the value a digit has because of its position in a number.
What would you like to understand?
Search the published lessons, tools and activities. Upcoming topics are listed in each subject pathway.
Browse all published topics below, or search with JavaScript enabled.
Positive and negative numbers
Positive numbers are above zero; negative numbers are below zero. Zero is neither positive nor negative.
Order of operations
Order of operations is the shared set of rules for deciding which calculation to do first.
Fractions
A fraction describes equal parts of a whole. The denominator counts the equal parts; the numerator counts the parts selected.
Decimals
A decimal represents whole numbers and parts of a whole using places based on powers of ten.
Percentages
A percentage is a fraction out of one hundred. The symbol % means “per hundred”.
Ratios and proportions
A ratio compares quantities multiplicatively. A proportion states that two ratios are equal.
Estimation and rounding
Rounding replaces a number with a nearby value at a chosen level of precision. Estimation finds an approximate answer useful for a purpose.
Discounts
A discount reduces a price by a specified amount or a percentage of the original price.
Variables and expressions
A variable is a symbol that stands for a number. An expression combines numbers, variables and operations without asserting an equality.
Substitution
Substitution means replacing a variable with a known value while preserving the operations around it.
Solving linear equations
A linear equation has variables only to the first power. Solving it means finding values that make both sides equal.
Rearranging formulas
Rearranging a formula makes a different variable the subject while preserving the relationship.
Metric units and conversion
Metric prefixes describe powers of ten applied to a unit. Converting changes the numerical representation without changing the quantity.
Perimeter
Perimeter is the total distance around the outside boundary of a flat shape.
Area
Area measures the amount of surface inside a two-dimensional boundary, in square units.
Volume
Volume is the amount of three-dimensional space an object or substance occupies, measured in cubic units.
Pythagoras’ theorem
In a right-angled triangle, the square of the longest side equals the sum of the squares of the other two sides.
Coordinates and gradient
Coordinates locate a point using an ordered pair (x, y). Gradient measures how much y changes per unit change in x.
Mean, median and mode
The mean is the total divided by the number of values. The median is the middle of an ordered list. The mode is the most frequent value.
SI units
The International System of Units, or SI, provides a shared language for physical measurements.
Scalars and vectors
A scalar has magnitude only. A vector has magnitude and direction.
Speed and velocity
Average speed is total distance divided by elapsed time. Average velocity is displacement divided by elapsed time.
Acceleration
Acceleration is the rate of change of velocity with time.
Motion graphs
A motion graph shows how a quantity such as position or velocity changes with time.
Newton’s laws
Newton’s laws connect forces to changes in motion and describe how interacting objects exert forces on each other.
Mass and weight
Mass measures inertia and is measured in kilograms. Weight is the gravitational force on a mass, measured in newtons.
Friction and resultant force
Friction is a contact force that opposes relative sliding, or the tendency to slide, between surfaces.
Work and energy transfer
Work is energy transferred by a force acting through a displacement.
Kinetic energy
Kinetic energy is energy associated with motion. For ordinary speeds, translational kinetic energy is ½mv².
Gravitational potential energy
Near Earth’s surface, a change in gravitational potential energy is mass × gravitational field strength × change in height.
Current, voltage and Ohm’s law
Current is the rate of flow of charge. Voltage is energy transferred per unit charge. Resistance relates voltage to current for a component.
Electrical power and energy
Power is the rate of energy transfer. Electrical power in a steady direct-current circuit is voltage multiplied by current.
Frequency and wavelength
Frequency counts cycles per second. Wavelength is the distance between matching points on adjacent wave cycles.
Temperature and specific heat
Temperature describes thermal state. Specific heat capacity is the energy required to raise one kilogram of a substance by one kelvin.
Fraction visualiser
Split a whole into equal parts. Change the numerator and denominator to see how much is selected.
Percentage and discount calculator
Explore the saving and remaining price as you change an original price or discount.
Equation balance scale
Build an equation ax + b = c, then watch the same operation act on both sides.
Unit converter
Convert lengths, masses, volumes and speeds. Choose units of the same type.
Coordinate and gradient explorer
Move two points numerically and see the straight line, its rise, its run and its gradient.
Triangle and angle explorer
Change two angles to construct a triangle and find the third. The side lengths in the drawing are illustrative.
Motion graph simulator
Explore constant acceleration. The graph shows velocity against time; the signed area is displacement.
Force diagram builder
Set the forces on a block. Arrows show their directions; the resultant controls acceleration.
Circuit and Ohm’s law explorer
Change the voltage and resistance of a virtual circuit. Compare current and power.
Wave and sound visualiser
Explore how frequency changes wavelength at a fixed wave speed. The static trace is readable without animation.
Calculate supermarket discounts
You are comparing a shelf price with a sale offer. Enter the original price and the advertised percentage to work out what you save and pay.
Compare simple and compound interest
Imagine leaving a starting amount untouched. Compare adding interest only on the original amount with adding interest on the growing balance.
Estimate paint for a room
You are painting the four walls of a rectangular room. Subtract doors and windows, choose a coat count, and estimate the paint volume.
Understand an electricity bill
Separate energy usage from the daily supply charge. Enter illustrative charges to see how each contributes to the total.
Calculate vehicle fuel consumption
Record distance travelled and fuel used between comparable fill levels. Find litres per 100 kilometres and estimate the cost of that journey.
Explore speed and stopping distance
Explore why stopping distance grows quickly with speed. Separate the distance travelled during reaction from the distance covered while braking.
Explore speaker frequency and wavelength
Compare a low bass tone and a higher tone by entering their frequency. The wavelength gives a useful scale for thinking about a room.
Estimate file download time
A file size is measured in bytes, while connection speed is commonly advertised in bits per second. Convert carefully before estimating the wait.