An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

In Fill Me Up we invited you to sketch graphs as vessels are filled with water. Can you work out the equations of the graphs?

Can you sketch graphs to show how the height of water changes in different containers as they are filled?

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

Can you draw the height-time chart as this complicated vessel fills with water?

Examine these estimates. Do they sound about right?

Where should runners start the 200m race so that they have all run the same distance by the finish?

If I don't have the size of cake tin specified in my recipe, will the size I do have be OK?

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

Use trigonometry to determine whether solar eclipses on earth can be perfect.

Analyse these beautiful biological images and attempt to rank them in size order.

This problem explores the biology behind Rudolph's glowing red nose.

Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?

Simple models which help us to investigate how epidemics grow and die out.

What shape would fit your pens and pencils best? How can you make it?

How would you design the tiering of seats in a stadium so that all spectators have a good view?

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

Formulate and investigate a simple mathematical model for the design of a table mat.

Get some practice using big and small numbers in chemistry.

Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.

In which Olympic event does a human travel fastest? Decide which events to include in your Alternative Record Book.

Can you work out which processes are represented by the graphs?

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .

How do you write a computer program that creates the illusion of stretching elastic bands between pegs of a Geoboard? The answer contains some surprising mathematics.

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

Work with numbers big and small to estimate and calculate various quantities in biological contexts.

Two trains set off at the same time from each end of a single straight railway line. A very fast bee starts off in front of the first train and flies continuously back and forth between the. . . .

Can Jo make a gym bag for her trainers from the piece of fabric she has?

Which dilutions can you make using only 10ml pipettes?

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

Explore the relationship between resistance and temperature

Is it cheaper to cook a meal from scratch or to buy a ready meal? What difference does the number of people you're cooking for make?

Make an accurate diagram of the solar system and explore the concept of a grand conjunction.

To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...

Invent a scoring system for a 'guess the weight' competition.

Use your skill and knowledge to place various scientific lengths in order of size. Can you judge the length of objects with sizes ranging from 1 Angstrom to 1 million km with no wrong attempts?

Can you deduce which Olympic athletics events are represented by the graphs?

Various solids are lowered into a beaker of water. How does the water level rise in each case?

Which countries have the most naturally athletic populations?

The triathlon is a physically gruelling challenge. Can you work out which athlete burnt the most calories?

How would you go about estimating populations of dolphins?

Which units would you choose best to fit these situations?

Work with numbers big and small to estimate and calculate various quantities in physical contexts.