Maths
Maths projects for class 11 and 12: 25 ideas.
Each project below opens to its own page with what it shows, the science behind it, the materials, safety, step-by-step instructions, what you should see and viva questions. Many can be made from things at home; where a kit helps, the page links one.
Other subjects for class 11 and 12: Physics, Chemistry, Biology, Earth and space, Electronics and robotics, Environment, Computing and coding.
Fibonacci numbers in nature
In the sequence 1, 1, 2, 3, 5, 8, 13 and so on, each number is the sum of the two before it, and the ratio of neighbours gets closer to about 1.618, the golden ratio. The spirals on sunflower heads and pine cones often come in Fibonacci numbers.
Estimate pi by dropping needles (Buffon’s needle)
Drop a needle of length L onto lines spaced d apart, with L no longer than d. The chance that it crosses a line is 2L ÷ (πd), so counting crossings over many drops gives an estimate of π.
Unit circle trigonometry board with a swinging radius arm
A 10 cm arm pivots at the centre of a circle on a grid. At any angle, the height of its tip above the horizontal is 10 times the sine and the distance along the horizontal is 10 times the cosine, so at 30 degrees the tip is 5 cm up and 8.66 cm across.
Nail-board Galton machine that drops beads into a bell curve
Beads falling through seven rows of nails bounce left or right at random at each nail. With eight collecting slots at the bottom, the expected share of beads in each slot follows Pascal's triangle, 1, 7, 21, 35, 35, 21, 7, 1 out of 128, which draws a bell-shaped curve.
Birthday paradox survey across the sections of a school
In a group of just 23 people there is a better than even chance (about 50.7 percent) that two share a birthday, and with 30 people the chance is about 70.6 percent. Collecting birthdays from many class sections shows how often a shared birthday actually turns up.
Estimate pi by scattering rice over a square and its circle
Rice grains scattered evenly over a square land inside the circle drawn within it with probability equal to the circle's share of the area, which is pi divided by 4. Counting grains inside and outside and multiplying the inside fraction by 4 estimates pi, a Monte Carlo method used in real computer simulations.
Clay double cone cut to reveal all four conic sections
Slicing a cone with a flat cut gives a circle when the cut is parallel to the base, an ellipse when tilted, a parabola when parallel to the slanting side, and a hyperbola when steeper than the slanting side. Four clay cones, each cut one way, show the four curves side by side.
String art parabola made from straight lines only
Threads joined between two scales at right angles, the first point on one to the last on the other and so on, form no curve themselves, yet their edges outline a smooth parabola. The curve is the envelope of the lines, touched by every one of them.
Pin and string ellipse drawer with measured focus distances
A loop of string round two pins and a pencil draws an ellipse, because the pencil keeps the sum of its distances to the two pins constant. With pins 6 cm apart and a loop giving a major axis of 10 cm, the minor axis comes out as 8 cm and the eccentricity as 0.6, exactly as the formulas predict.
Seven bridges of Konigsberg board with removable bridges
Four land areas joined by seven bridges cannot be walked in one trip crossing every bridge once, because every land area has an odd number of bridges. Lifting any one bridge off the board leaves only two odd areas and the walk becomes possible, starting at one odd area and ending at the other.
Width of a pond or river from a baseline and the sine rule
You can find a distance you cannot measure directly by measuring a baseline and two angles. The sine rule in the triangle then gives the width.
Sine wave drawn by a turning wheel and a moving roll of paper
The height of a point on a turning wheel, measured above the centre, follows a sine curve in time. Tracing it on moving paper gives amplitude, period and a smooth wave.
Arm span against height: scatter plot, mean, spread and best-fit line
Two measurements on the same person can be linked. Collecting data for a class and drawing a scatter plot, a line of best fit and a correlation shows how statistics finds a pattern in data.
Area of a leaf by random dots: a Monte Carlo estimate with a check
The area of an odd shape can be estimated by throwing random points on a square that contains it. The share of points that land inside the shape estimates the share of the square it covers.
Monty Hall with three cups: should you switch? 100 trials and the proof
In the three-door puzzle, switching wins two times out of three, though many people feel that it is a fifty-fifty choice. Playing it many times and listing all the outcomes shows why.
How long is a game of snakes and ladders? 100 simulated games
The number of turns to finish a game is random but has a stable average. Playing or simulating many solo games gives the mean game length and shows how the snakes and ladders change it.
Balancing a cardboard triangle at its centroid, checked by coordinates
The point where the three medians of a triangle meet is its centroid, and a flat triangle balances there. Its coordinates are the average of the three vertex coordinates.
Taxicab geometry on a city grid: distances, midpoints and square circles
On a street grid you cannot travel diagonally, so the distance is the sum of the horizontal and vertical steps. With this rule a circle looks like a diamond, which shows that geometry depends on how distance is defined.
Bouncing ball: a geometric sequence of heights and its infinite sum
A ball that bounces back to a fixed fraction of its previous height gives a geometric sequence of heights. The total distance travelled until it stops is the sum of a geometric series.
Biggest open box from a square sheet: calculus against trial boxes
Cutting equal squares from the corners of a sheet and folding it gives a box. The volume rises and then falls as the cut grows, and calculus predicts exactly where it is greatest.
Area under a curve by rectangles and trapeziums before the integral
The area under a curve can be approximated by many thin strips. The more strips used, the closer the sum is to the definite integral.
Speed of a toy car from tangents to its distance-time graph
The slope of a distance-time graph at a point is the instantaneous speed, which is the derivative of distance with respect to time. A car on a ramp that speeds up gives a curve whose tangents get steeper.
Hanging chain or hanging bridge? Catenary and parabola compared
A chain hanging under its own weight forms a catenary, while a cable carrying a load spread evenly along the horizontal, as in a suspension bridge, forms a parabola. The two curves look alike but have different equations.
Two bags of beads: conditional probability and Bayes by experiment
When you know which colour was drawn, you can work backwards to the chance of which bag it came from. The experiment shows how new information updates a probability.
Range against launch angle: does 45 degrees really go farthest?
For a projectile launched at the same speed from ground level, the range is proportional to sin 2 theta, which is greatest at 45 degrees. Equal ranges occur for angles that add up to 90 degrees.