Representation asks how a picture, table, graph, diagram or symbol changes what you can see. The same idea can become clearer when shown differently.
If one picture is confusing, try a different one. That is not a shortcut; it is mathematics.
Representation
How does changing the way we show something change what we understand?
Take the values 1, 2, 4, 8.
Show them as a list, bar chart and points on axes. What does each view make easier to notice?
Try noticing
Notice 1
What does this representation reveal?
Notice 2
What does it hide?
Notice 3
Would a table, graph, diagram or formula help more?
Notice 4
Can two different pictures describe the same structure?
Knowledge you may need
Tools and visualisations
ChangeRepresentationQEDReasoning
A Level Calculus Revision Lab
Revise tangent gradients, definite integrals and trapezium-rule estimates.
ToolChangeRepresentationOptimisation
Area Approximation and Integration Explorer
Compare rectangles, midpoint approximations, trapezia and exact integrals.
ToolPatternQEDReasoningRepresentation
Classical Ciphers and Frequency Analysis Lab
Encode, decode and attack simple ciphers using shifts, keys and letter frequencies.
ExplorationTransformationRepresentationRepetition
Complex Number and Argand Diagram Explorer
Explore modulus, argument, multiplication and roots of unity.
ToolChangeRepresentationRepetition
Differential Equation Slope Field Explorer
Explore slope fields, initial conditions and numerical solution curves.
ToolChangeRepresentation
Forces and Inclined Plane Explorer
Resolve forces, friction and acceleration on an inclined plane.
ToolTransformationRepresentationChange
Function Transformation Playground
Explore vertical and horizontal graph transformations across familiar functions.
ToolChanceRepresentation
GCSE Probability Tree Lab
Build two-stage tree diagrams and compare replacement with no replacement.
ToolOptimisationChangeRepresentation
Gradient Descent Visualiser
Watch optimisation move across a loss landscape.
Tool and explorationConnectionRepresentationQEDReasoning
Graph Theory Visualiser
Build a graph and step through breadth-first and depth-first search.
ExplorationChanceRepresentationOptimisation
Linear Regression and Overfitting Lab
Compare training error, test error and model complexity.
Tool and explorationTransformationRepresentation
Matrix Transformation Visualiser
See how 2 by 2 matrices move vectors, shapes, area and orientation.
ToolChanceRepresentationOptimisation
Monte Carlo Simulator
Estimate pi and area under a curve using random sampling.
Tool and explorationChanceRepresentationQEDReasoning
Normal Distribution and Hypothesis Testing Lab
See rejection regions, z statistics, p-values and conclusions.
ToolChanceRepresentation
Probability and Statistics Lab
Explore binomial distributions, sampling and sample means.
ToolChangeRepresentationOptimisation
Projectile Motion Lab
Adjust launch speed, angle and gravity to explore projectile motion.
ToolChangeOptimisationRepresentation
River Flow and Flood Risk Simulator
Explore rainfall, storage, release decisions and flood thresholds.
Tool and explorationWhere this appears in school maths
Functions, graphs and data
Coordinates, graphs, diagrams and tables make relationships visible.
Modelling
A model is a chosen representation of the world, useful because it leaves some things out.