1029 science project ideas, from class 1 to college.

What it shows

A triangle cut from one end of a parallelogram and slid to the other end makes a rectangle of the same base and height, so the area of a parallelogram is base times height. Cutting the parallelogram along a diagonal gives two equal triangles, each with area half the base times the height.

You will need

• Two sheets of coloured card
• A ruler, set square and pencil
• Scissors
• A board with two card rails to slide the pieces
• Glue

Safety: An adult must help with the scissors for thick card.

Take it further: Cut a trapezium into two triangles and show that its area is half the sum of the parallel sides times the height.

What it shows

A cube of side 5 cm hinged along its edges opens flat into a net of six 5 cm squares, so its surface area is 6 times 25 = 150 square cm. A second box, 5 by 4 by 3 cm, opens to show three pairs of rectangles with total area 94 square cm.

You will need

• Thick chart paper or thin cardboard
• Cloth tape for hinges
• A ruler and pencil
• Centimetre grid paper glued on each face
• Scissors

Safety: An adult must help with cutting the cardboard.

Take it further: Make a cube with a lid to show that an open box has only five faces, 125 square cm for side 5 cm.

What it shows

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.

You will need

• A 30 cm square board
• Centimetre graph paper glued to the board
• A stiff card or aluminium strip 10 cm long as the arm
• A split pin or a bolt with a nut as the pivot
• A protractor fixed round the centre
• Thread with a bead as a plumb line

Safety: An adult must drill the pivot hole if a bolt is used.

Take it further: Add a vertical tangent line at x = 10 cm and extend the arm to meet it: the height where it meets is 10 times tan of the angle.

What it shows

A rope with 13 knots at equal spacing is pulled into a triangle with sides of 3, 4 and 5 spaces. The angle between the 3 and 4 sides is exactly a right angle, because 3 squared plus 4 squared equals 5 squared, a trick used to lay out buildings for thousands of years.

You will need

• About 4 metres of thin rope or thick string
• A measuring tape
• Three pegs or three friends
• A set square to check
• Chalk to mark the ground

Safety: Peg the rope outdoors on soft ground, or hold it with fingers; never pull it tight near anyone's face.

Take it further: Use the rope to check whether the corners of your classroom are true right angles and report the error in centimetres.

What it shows

At the same moment the Sun makes the shadow of a tree and the shadow of a metre stick in the same proportion. If a 1 metre stick casts a 1.5 metre shadow and the tree casts a 9 metre shadow, the tree is 9 divided by 1.5, which is 6 metres tall.

You will need

• A metre stick or a straight pole of known length
• A measuring tape at least 10 metres long
• Chalk or pebbles to mark shadow tips
• A notebook and calculator
• A sunny morning or afternoon when shadows are long

Safety: Never look directly at the Sun. Measure shadows of trees from the ground only; do not climb.

Take it further: Measure the Sun's angle from the stick and shadow using tan, and compare with the angle you get from a clinometer.

What it shows

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.

You will need

• A plywood board 30 cm by 45 cm
• 28 small nails in a triangle of seven staggered rows
• Thin card strips for eight collecting channels
• A clear plastic sheet from an old folder as the front
• About 200 mustard seeds or 5 mm beads
• A funnel at the top
• Tape and glue

Safety: An adult must hammer the nails and check none pokes through the back.

Take it further: Bend some nails so beads go right with probability more than a half, and show the peak shifting to one side like a skewed distribution.

What it shows

A spinner with four equal sectors lands on each colour about a quarter of the time. A spinner with sectors of 180, 90, 45 and 45 degrees lands on the big colour about half the time and on each small colour about one eighth, showing that probability is the share of the circle.

You will need

• Two card circles of radius 8 cm
• A protractor and colours
• Two split pins and two card arrows, or a pencil through the centre
• A tally sheet
• A calculator

Safety: An adult must push the split pins through the card.

Take it further: Spin both spinners together and record pairs to find how often both show red; compare with one quarter times one half.

What it shows

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.

You will need

• Permission from the school to collect day and month of birth only, no year or names
• A survey sheet for each section
• A calculator or a spreadsheet
• Chart paper for the results

Take it further: Work out how many people are needed for a 50 percent chance that someone shares your own birthday: it is 253, far more than 23.

What it shows

A survey of 30 classmates is shown as coloured bottle caps on a circle. If 12 chose cricket, cricket gets 12 out of 30 of the circle, which is 144 degrees, and the slices of the pie add up to the full 360 degrees of the class.

You will need

• About 30 bottle caps in four colours, or caps painted
• A card circle of radius 15 cm
• A protractor and ruler
• Glue
• A survey sheet and a tally
• A bar chart on graph paper for comparison

Take it further: Survey a second class and make a pie of the same size, then compare the slices for each game.

What it shows

Equal weights hung from a numbered beam at the data values balance only when the support is at the mean. Weights at 2, 3 and 7 balance at 4, their mean, and moving one weight from 7 to 13 shifts the balance point to 6, showing how an extreme value pulls the mean.

You will need

• A straight wooden scale or a 60 cm batten marked 0 to 20
• A triangular block or a pencil fixed to a base as the pivot
• Six identical washers or coins with thread loops
• A nail or hook at every number mark
• A small level or a straight edge

Safety: An adult must fix the hooks and make sure the beam cannot fall on fingers.

Take it further: Hang two washers at one value to show frequency, and find the mean of a frequency table on the beam.

What it shows

Paper cups stacked in a triangle carry Pascal's numbers: each cup holds the sum of the two cups above it. Row sums double each time (1, 2, 4, 8, 16, 32), the second diagonal counts 1, 2, 3, 4, and the third diagonal holds the triangular numbers 1, 3, 6, 10, 15.

You will need

• 55 paper cups for ten rows
• A marker
• Coloured stickers for odd and even numbers
• A board or a wall space and tape
• A calculator to check the big numbers

Take it further: Read each row as a number (1, 11, 121, 1331, 14641) and show these are powers of 11 until carrying spoils it at row 5.

What it shows

Squares of side 1, 1, 2, 3, 5, 8 and 13 cm fit together into a rectangle of 13 cm by 21 cm, and a quarter circle in each square joins into a spiral. The ratio of the sides, 21 divided by 13, is about 1.615, close to the golden ratio 1.618, and the ratio gets closer as more squares are added.

You will need

• Chart paper and coloured paper
• A compass, ruler and set square
• Scissors and glue
• A calculator
• A board 30 cm by 40 cm

Safety: Keep the compass point capped when not in use.

Take it further: Measure the ratio of your height to the height of your navel and of several objects in the room, and see which come near 1.6.

What it shows

Rolling a round lid exactly one turn along a ruler marks out its circumference. For every lid, from a bottle cap to a steel plate, dividing the distance rolled by the diameter gives the same number, about 3.14, which is pi.

You will need

• Five round objects: a bottle cap, a tin lid, a bangle, a steel plate and a bucket lid
• A long ruler or measuring tape
• A marker to mark the start point on each rim
• A sheet of paper and a calculator
• A thread for a second check

Take it further: Measure the diameter of a bicycle wheel, count turns over 50 metres and check the distance with the formula.

What it shows

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.

You will need

• A square tray or a 30 cm square drawn on chart paper with a tape border
• A compass to draw the largest circle inside the square
• About 500 grains of rice or mustard seeds
• A tally sheet and a calculator
• A sieve or a sheet to scatter grains from a height

Take it further: Write a ten-line program that drops a million random points and compare its estimate with your rice result.

What it shows

A hanger with a cup on each side stays level only when both sides weigh the same. A sealed bag of marbles stands for x; with 3 loose marbles beside it balancing 7 on the other side, removing 3 from each side keeps the balance and reveals that the bag holds 4.

You will need

• A plastic clothes hanger
• Two identical paper cups hung on thread from the hanger ends
• About 30 identical marbles or large steel washers
• Three small opaque bags or film canisters as unknowns
• A hook or a string to hang the hanger from

Safety: Keep marbles away from small children and hang the hanger where it cannot swing into anyone.

Take it further: Solve 2x + 1 = x + 6 by placing bags on both sides and removing a bag and a marble from each side.

What it shows

Square tiles of side x, strips of x by 1 and unit squares are arranged into a rectangle. One x squared tile, five x strips and six unit tiles fit exactly into a rectangle with sides x + 2 and x + 3, so x squared + 5x + 6 = (x + 2)(x + 3).

You will need

• Thick cardboard or foam sheet in three colours
• A ruler and cutter
• A board with a right-angled corner frame
• Labels for x squared, x and 1
• A box to hold the tile set

Safety: An adult must cut the tiles with the cutter.

Take it further: Make red tiles for negative terms and factorise x squared take away x take away 6 as (x take away 3)(x + 2).

What it shows

Pins at grid points on a four-quadrant board let a string be stretched through any two points to show a straight line. The string through (0, 1) and (2, 5) rises 4 for every 2 across, slope 2, and crosses the y axis at 1, so its equation is y = 2x + 1.

You will need

• A cork sheet or soft board 40 cm square
• Centimetre graph paper with axes from minus 10 to 10
• A box of drawing pins
• Coloured thread or elastic
• Labels and a marker
• A set square for perpendicular lines

Safety: Drawing pins are sharp; store them in a closed tin and keep the board out of reach of small children.

Take it further: Pin the three vertices of a triangle and use the coordinates to find its area with the shoelace formula, then check by counting squares.

What it shows

Players hide paper ships on a grid running from minus 8 to 8 on both axes and call shots as ordered pairs like (minus 3, 5). Reading x before y and keeping the signs straight is the only way to win, so the game drills plotting points in every quadrant.

You will need

• Two shoe-box lids as game boards
• Two sheets of graph paper with axes from minus 8 to 8
• Paper ships of length 2, 3 and 4 squares
• Two colours of small stickers or pins for hits and misses
• A cardboard screen between the boards

Take it further: Allow diagonal ships and ask players to state the slope of each ship, which makes the game a review of gradients too.

What it shows

In a magic square every row, column and both diagonals add to the same total. The 3 by 3 square using 1 to 9 always totals 15 and has 5 in the centre; the 4 by 4 square 16, 2, 3, 13 / 5, 11, 10, 8 / 9, 7, 6, 12 / 4, 14, 15, 1 totals 34 in every line, in each corner 2 by 2 block and in the centre block too.

You will need

• Two card boards with 3 by 3 and 4 by 4 grids
• Number tiles from bottle caps or card squares
• A marker
• Velcro dots or a shallow lip so tiles stay in place
• A card with blank grids for visitors

Take it further: Build a 5 by 5 magic square with the Siamese method (start at the top middle, move up and right, drop down when blocked); its total is 65.

What it shows

Two players take 1, 2 or 3 sticks in turn from a pile of 21 and the player who takes the last stick wins. Whoever leaves a multiple of 4 after their move (20, 16, 12, 8, 4) can always win, so the second player wins a game of 21 by always making the two moves in a round add to 4.

You will need

• 21 matchsticks, straws or pencils
• A board marked with 21 slots
• A score sheet
• Two players and a referee

Safety: Use burnt matchsticks or straws so there is nothing to strike.

Take it further: Study the three-pile version of Nim and find the rule that uses binary addition without carrying.

What it shows

Discs of different sizes must be moved from one peg to another, one at a time, never putting a larger disc on a smaller one. The least number of moves is 1, 3, 7, 15, 31 and 63 for 1 to 6 discs, which is 2 to the power n minus 1, so 64 discs would take more than 18 quintillion moves.

You will need

• Six lids or caps of clearly different sizes, from a bottle cap to a jar lid
• Three pegs: pencils stuck upright in a plank or clay base
• A stopwatch
• A record sheet

Safety: An adult must fix the pegs; blunt the pencil tops.

Take it further: Write a short program or a flowchart for the recursive solution and let it print the moves for 4 discs.

What it shows

In any year the dates 4/4, 6/6, 8/8, 10/10, 12/12, 9/5, 5/9, 7/11, 11/7 and the last day of February all fall on the same weekday, called the doomsday. A wheel finds the doomsday for the year, and counting from the nearest of these dates gives the weekday of any date, such as 15 August 1947, a Friday.

You will need

• Two card circles, 20 cm and 14 cm, joined at the centre with a split pin
• A marker and ruler
• A list of the doomsday dates printed on the base
• A calculator to check
• An old calendar for testing

Safety: An adult must push the split pin through the card.

Take it further: Work out the weekday of your 50th birthday and of 26 January 1950, and show both with the wheel.

What it shows

The minute hand turns 6 degrees per minute and the hour hand half a degree per minute, so the angle between them at any time is the difference of 30 times the hours and 5.5 times the minutes. At 3:00 the angle is 90 degrees and at 3:30 it is 75 degrees, not 90, because the hour hand has moved on.

You will need

• A card clock face with a degree scale round the edge
• Two card hands on a split pin
• A protractor
• A table of times to test
• Optional: two toothed wheels from a broken toy with a 12 to 1 ratio

Safety: An adult must push the split pin through the card.

Take it further: Add a second hand and work out how often all three hands line up: only at 12:00.

What it shows

To square a number ending in 5, multiply the tens digit by the next number up and write 25 after it: 35 squared is 3 times 4 = 12 followed by 25, so 1,225. The flip board shows each step and the algebra behind it: (10a + 5) squared = 100a(a + 1) + 25.

You will need

• A cardboard base 40 cm by 20 cm
• Flip cards on rings or tape hinges for the tens digit and the answer
• A marker
• A calculator for checking
• A small whiteboard sheet for visitors to try

Take it further: Make a board for squaring numbers near 50: 50 + d squared is 2,500 + 100d + d squared, so 53 squared is 2,809.