Mathematics.

Split two numbers into prime-factor trees, then test candidate common factors and multiples against both numbers to find the HCF and LCM, and use them to solve a next-departure problem.

Set two pumps and a fill time to make a drink, then fill bottles to an order: keep the recipe while making more or filling faster, and work out the flow rates that meet each deadline.

Slide a recurring decimal against a multiple of itself until the repeating tails cancel, then test whether your fraction divides back to the same digits.

Drag a true distance and time past the points where their rounded readings change, then work out how far a calculated speed can really vary.

Grow and shrink an amount year by year, find the one multiplier that does each year’s change, then run it backwards to recover the starting value.

Drag club members into the right parts of a Venn diagram, then choose a union, intersection or complement and send a message to see exactly who receives it.

Place one meeting on three local calendars in different time zones, convert times and dates between them, and find a start that keeps the whole meeting inside everyone’s day.

Join, cancel and regroup repeated factors to see why the index laws work, why a zero or negative index means what it does, and how roots undo powers.

Build the next stages of growing square patterns, check them, then use how they grow to predict a far stage and find the one that breaks the pattern.

Shape a journey on a graph and run it, then use gradients for speed and areas for distance to plan deliveries, draw the way home and design a journey of a set length.

Tilt and slide a line, measure two points on it, and turn their coordinates into an equation, then build lines from equations, through a point, and in the form ax + by = c.

Move two endpoints, work out the length, midpoint and gradient of the segment between them, then build the line that cuts it in half at right angles and find its equation.

Drag points round a circle, turn lines and move chords to find the angle and length facts that always hold, then use them to build a fresh construction.

Resize boxes, triangles, cubes and jars by one scale factor, and see how lengths, surface areas and volumes change by different amounts when you test your answers on the objects.

Stretch a prism-shaped case, see which faces change, then set its length for a target capacity, cover every face with material, and reshape it to hold the same with less.

Name the sides from an angle, fly an aircraft along a fixed angle to see which ratios stay the same, then use the right ratio to find a missing height or side.

Turn an arm round a circle to draw sine, cosine and tangent graphs, then pick every crossing of a target line to solve equations from 0° to 360°.

Swing a rod from a fixed angle to find when a triangle can be made zero, one or two ways, then use the sine rule, the cosine rule and the area formula and test each answer by building it.

Turn a floor line beneath a hanging cable, find the one line directly under it, then work out its length and the angle between the cable and the floor.

Slide mirrors, move centres and drag image points to make reflections, rotations, translations and enlargements, then reorder two transformations to see when the order matters.

Program a warehouse robot with vector commands: move, reverse and scale the arrows, put them in an order that avoids obstacles, then find how far the bay really is in a straight line.

Plot pairs on a scatter graph, move and tilt a line through the whole cloud, then test estimates inside the data and forecasts beyond it.

Move unequal-width histogram bars to see which holds more pupils, then build the cumulative frequency curve from the same data and read the median, quartiles, IQR and a percentile from it.

Spin a known fair spinner and a fairground booth, compare short and long runs, then estimate the booth’s rate and plan how many prizes to stock.

Draw a counter from a bag, keep it out or put it back, and see which second draws are still possible before you work out the conditional probability.