• Elementary (Grades 4 to 6)
• Junior (Grades 7 to 9)
• Senior (Grades 10 to 12)

Elementary (Grades 4 to 6)

• Astronomy
• Behavioral Science
• Biology (Animal)
• Botany (Plant Science)
• Chemistry
• Ecology/Conservation
• Engineering
• Food Science/Nutrition
• Geology
• Meteorology
• Physics

Astronomy

1. How strong is the gravitational pull of the moon?
2. Can you tell the time using the moon?
3. Is there a relationship between sunspots and the weather?
4. Can you tell time by the sun and the shadows it casts?
5. How do sailors use the stars for navigation?
6. How do astronomers measure interstellar distances? What is a parallax?

Behavioral Science

1. Territoriality in Mice
2. Cleaning Habits of Animals
3. Age versus learning ability
4. Is audio or visual information better remembered?
5. Are the activities of nocturnal animals affected by the moon?
6. Can a white mouse be trained to find its way through a maze? Can you train a mouse to do anything else?
7. What defenses do insects use against their predators? Describe several of them.
8. What is hibernation? Which animals practice this? Does hibernation only occur in the winter?
9. Do all ants live in colonies?

Biology (Animal)

1. Does the amount of bacteria differ in ice cubes found in your refrigerator, the school freezer, a restaurant, or a store? Why or why not?
2. What effect do antiseptics have on bacteria? On viruses?
3. What types of bacteria are helpful? Harmful?
4. What is the life cycle of a frog?
5. How do the webs of various spiders differ? Does each spider spin just one kind of web? Of what is spider silk composed?
6. How is blood type determined? What is the most common type? How do the types differ?
7. How fast do human fingernails grow? What are they made of? What factors affect their growth rate, strength, and condition?
8. How fast does human hair grow? What is it made of? What factors affect its growth rate, health, and strength?
9. How often is your skin replaced? What is the process? What microorganisms live off the skin cells you shed?
10. How many types of bird feathers are there? What are their structures/functions?
11. Do wasps all build the same kinds of nests? Do they use the same materials?
12. What is DNA?
13. What are whale songs and what have scientists learned about them?
14. Which insects are "camouflage artists" and why?
15. Why can some animals see well at night? What other characteristics do these animals share?
16. Do all living creatures have bones? Are there "substitutes" for bones?
17. What are parasites? Are they helpful or harmful? In what forms do they exist?
18. How do the heart rates of different-sized animals compare? What factors appear to affect heart rate?
19. Why do children have physical characteristics like their parents? What are chromosomes, genes? What causes twins?
20. Examine the diets of various animals. Define herbivore, carnivore, omnivore.

Botany (Plant Science)

1. Can you grow plants without using seeds? Can you grow plants that do not produce seeds?
2. Compare monocots and dicots
3. Does it matter how or which way you plant a seed?
4. Does talking to plants really make them grow better?
5. Do plants grow better and stay healthier under constant light (24 hrs/day) , or do they need a rest from light?
6. How are plants affected by their environment - water, sunlight, altitude,
7. How do cones of various fir and pine trees differ?
8. How do insect-eating plants catch and digest their food?
9. How is the way ferns reproduce different from the ways other plants reproduce?
10. How much water is used by different plants? What factors affect the use of water (transpiration) - temperature, sunlight, etc.
11. If a tomato plant is grafted onto a potato plant, what do you get?
12. Is there more chlorophyll in coniferous trees or in deciduous trees (using comparable mass)?
13. The effects of cigarette smoke on plant growth
14. Under what environmental conditions do lichens exist?
15. What causes fruits to ferment? How does this process differ from decomposition?
16. What conditions affect the rate of decomposition of leaves?
17. What conditions affect the rate of growth of bread mold?
18. What factors encourage root growth?
soil type and acidity, and pollution?
19. What is the hydroponic method used to grow plants? What kinds of plants can be grown this way?
20. Why does mold grow on bread? What other things does mold grow on?
21. Will bean plants grow by moonlight?

Chemistry

1. Compare the boiling points/freezing points of fresh versus sea water?
2. How do acids react with different metals under varying conditions?
3. In what different shapes do crystals grow? How do the structures of salt and sugar crystals differ? Can you grow these crystals?
4. What factors affect the rate at which substances dissolve in water?
5. What is meant by the pH level of soil?
6. What properties prevent plastic from being biodegradable? How many kinds of plastics are there? Are any of them considered to be environmentally safe?
7. Why is Styrofoam considered to be harmful to the environment?

Ecology/Conservation

1. A study of a shoreline
2. Observation of urban wildlife
3. The study of flora in a given region
4. What is natural selection and how does is occur in the animal world? The plant world?

Engineering

1. Design several paper boats? Which design will hold the most weight without sinking?
2. Determine the accuracy of various thermometers
3. Using popsicle sticks, design a bridge. Which design is most sturdy? Why?
4. Principles of energy conservation
5. How could you make a light turn on automatically?
6. What do gears do? What are different kinds of gears and their uses?
7. Which paper airplane design flies the farthest?

Food Science/Nutrition

1. What are the differences between butter and margarine?
2. What causes food poisoning? Are there different kinds of food poisoning?
3. What factors increase the rate at which milk sours?

Geology

1. Do a profile of soil in your area. Label and explain it.
2. Do pebbles, gravel, sand and soil, mixed up and shaken in a jar of water, settle in the same order in fresh water and sea water? Why or why not?
3. What are the four kinds of mountains on earth? How is each formed?
4. What are the most common kinds of rocks in your area?
5. What causes water to move up through the soil?
6. What happens to soil when it is moved by wind, water, ice?
7. What kinds of soils do you find under the surface of the ground?
8. What three kinds of rock are found on earth? How are they formed? What are they made of?

Meteorology

1. Account for the differences in sky colour at different times
2. Does the temperature of air affect its density?
3. Does wind travel at the same speed and direction at different heights?
4. How are we affected by dust and smoke in the air? By pollutants? Do these affect weather conditions?
5. How do you measure the relative humidity (amount of moisture) of air? Why is it the "relative" humidity?
6. How is the amount of rainfall measured?
7. How is wind direction determined?
8. How much cooler is it usually in the shade than in the sun? What factors affect this difference?
9. What are clouds made of? What are the different kinds of clouds and how are they different? Are clouds different from ground fog?
10. What causes hailstones? Why do they differ in size?
11. What causes the wind to blow? Are hurricanes and tornadoes just high winds?
12. What happens when snow melts? What does it contain? Structure of snowflakes.
13. What is ozone? Can you smell it? Why is it undesirable at ground level but desirable in the upper atmosphere?
14. What must be the temperature to form frost?
15. What pollutants are most harmful to the upper atmosphere ozone layer? Why is this a problem?
16. Which factor most affects the rate of the evaporation of moisture - temperature, humidity, or wind speed?
17. Why are weather forecasters not always accurate in their predictions?
18. Why do coastal cities feel warmer in summer and cooler in winter than inland cities at the same latitude and elevation?
19. Why do shadows of the same object differ in shape and size at various times during the day?

Physics

1. Does hot water freeze faster than cold water? Why or why not.
2. How do lasers work? What uses have scientists found for them?
3. How do magnet work? How are they made?
4. How does a mirror reflect an image?
5. How does a record player produce sound?
6. How does electricity light up a bulb?
7. How does sound travel? Through what media does it travel best?
8. How does the surface tension of rainwater differ from that of tap water? How do these differ from the surface tension of cooking oil? Why are they different?
9. How does the way a record player works differ from that of a compact disc player? A cassette tape player?
10. How is magnetism created from electricity?
11. How is steam used to do work?
12. How metals compare in conducting heat
13. How metals compare in density and buoyancy
14. How strong are nylon fishing lines?
15. How strong are plastic wraps?
16. How strong is a toothpick?
17. What are microwaves? How does a microwave oven work?
18. What causes water to "bead up" on a freshly waxed car? On waxed paper? What other surfaces cause water to do this?
19. What factors determine how far a baseball travels when hit?
20. What factors affect the bounce of a dropped ball?
21. What limits the speed of a boat? A truck? A plane?
22. What makes colours in soap bubbles? Why are they spherical?
23. What substances are the best conductors of electricity? Why?
24. Which materials can be charged with static electricity?
25. Why do mirages appear on paved roads and in deserts?
26. Why do objects float higher in salt water than fresh water?
27. Why do windows steam up on the inside? Why do windows frost up on the outside?
28. Why does oil work better than water in reducing friction between two surfaces?
29. Why doesn't a balloon travel in a straight path when it is filled with air and released?

Junior (Grades 7 to 9)

• Astronomy
• Behavioral Science
• Biology (Animal)
• Botany (Plant Science)
• Chemistry
• Ecology/Conservation
• Engineering
• Geology
• Mathematics
• Meteorology
• Physics

Astronomy

1. Identification of Elements in the Solar and Stellar Spectra

Behavioral Science

1. A Study of the Relationship Between Physical Exercise and Learning Ability
2. Factors Affecting the Rate at Which a Cricket Chirps
3. Observation of Conditioned Responses in Different Animals
4. Stimuli that attract Mosquitos
5. Studies of Memory Span and Memory Retention

Biology (Animal)

1. A study of diffusion through cell membranes
2. A study of toxicity of insecticides versus temperature
3. Conditions necessary for the life of a brine shrimp
4. The effect of bleaching and dyeing on the hair
5. The effectiveness of antiseptics and soaps on household bacteria
6. Study of animal phosphorescence and other bioluminescence

Botany (Plant Science)

1. Best conditions for mushroom growth
2. Can household compounds (e.g. Tea) be used to promote good health in plants?
3. Comparing fertilizers and soil types
4. How do plants get nitrogen?
5. Plant growth and artificial light
6. Plant tropisms and growth hormones
7. Plants and fertilizer
8. Sugar level in plant sap at different times and dates
9. The commercial uses of algae - methods of production
10. The effect of nicotine, air and yeast on mold growth
11. The effect of sound on plants
12. The importance of earthworms to soil and plants
13. Why do plants grow towards the light?
14. Why do plants move?

Chemistry

1. Analyze soil samples for their components, ability to hold moisture, fertility and pH
2. Analyzing snow and rain for pollutants; samples from different locations
3. Compare the pH levels of saliva of various animals and humans at different times of the day
4. Effects of sunlight on rubber, ink, paper
5. Effects of temperature on viscosity of oil
6. Everyday activities that illustrate chemical principles
7. Fire extinguishers - principles of operation and factors affecting their efficient use
8. Identify different metals by the colour of flame when they burn
9. Testing of consumer products - glues, stain removers, antiseptics, mouthwash, detergents, paper towels, etc.

Ecology/Conservation

1. A study of air purification methods
2. A study of the impact of pollution on an ecosystem
3. A study of water purification methods
4. Monitoring the changes in wildlife caused by human encroachment

Engineering

1. A study of propeller designs for wind generators
2. Design a strong bridge
3. Design an energy efficient home
4. Efficiency of different types of steam engines
5. Efficient use of renewable energy sources (e.g. wood, wind)
6. How do airplanes fly? What is the best wing shape?
7. Production of electrical energy from mechanical sources
8. Structure versus strength in dams
9. Study of efficient home insulation
10. Testing and comparing consumer products

Geology

1. Comparison of the load bearing strength of different soils
2. Exploring methods of controlling erosion
3. How are metals made? What are "precious metals"? Why are some metals strong and others flexible? Why are some heavy and light?
4. How do the remains of plants and animals "fossilize"? What causes wood to become petrified rather than to decompose?

Mathematics

1. Study the accuracy of calculators
2. The mathematics of snow flakes
3. Investigate "big" numbers. What is a big number? The following examples might guide your investigation. A bank is robbed of 1 million loonies. How long would it take to move them? How much would they weigh? How much space would they take up? How big a swimming pool do you need to contain all the blood in the world? Is 10^100 very big? What is the biggest number anyone has ever written down (check the Guinness book of world records over the last few years)? How did this number come about?
4. How do computer bar codes (the ones you see on everything you buy) work? This is an example of coding theory at work. Find others. Investigate coding theory - there are many books with titles like "an introduction to coding theory" (this is not about secret codes).
5. It is easy to check if a number is divisible by 10 by looking to see if its last digit is a 0. How many other "tests of divisibility" can you find? Divisibility by 5 or 7 or 9? Why do they work?
6. Most computers these days can handle sound one way or another. They store the sound as a sequence of numbers. Lots of numbers. 40,000 per second, say. What happens when you play around with those numbers? ea. Add 10 to each number. Multiply each number by 10. Divide by 10. Take absolute values. Take one sound, and add it to another sound (i.e. add up corresponding pairs of numbers in the sequences). Multiply them. Divide them. Take one sound, and add it to shifted copies of itself. Shuffle the numbers in the sequence. Turn them around backwards. Throw out every third number. Take the sine of the numbers. Square them. For each mathematical operation, you can play the resulting sound on the computers speakers, and hear what change has occurred. A little bit of programming, and you can get some very bizarre effects. Then try to make sense of this from some sort of theory of signal processing. You will first have to discover how sound is stored.
7. Find out all you can about the Fibonacci Numbers, 0,1,1, 2, 3, 5, 8, .... In particular, where do they arise in nature? For example, look at the spirals on a pine-cone following the pattern of the cone, one spiral will go left, the other right. The cone will be covered by "parallelograms", the number of seeds on each side of the parallelogram will (always?) be two neighbouring Fibonacci numbers. For example 5 and 8. Similarly for pineapples, petals and leaves on plants.
8. What is the Golden Mean? Study its appearance in art, architecture, biology, and geometry, and its connection with continued fractions, Fibonacci numbers. What else can you find out?
9. Find out all you can about the Catalan Numbers, l, l, 2, 5, 14, 42, ...
10. Investigate triangular numbers. If that's not enough, do squares, pentagonal numbers, hexagonal numbers, etc. Venture into the third and even the fourth dimensions.
11. Investigate the history of pi and the many ways in which it can be approximated. Calculate new digits of Pi.
12. Use Monte Carlo methods to find areas or to estimate pi. (Rather than using random numbers, throw a bunch of small objects onto the required area and count the numbers of objects inside the area as a fraction of the total in the rectangular frame).
13. There are several methods of counting and calculating using your fingers and hands. Some of these methods are still in common usage. Explore the mathematics behind one of them.
14. At certain times charities call households offering to pick-up used items for sale in their stores. They often do a particular geographical area at a time. Their problem, once they know where the pick-ups are, is to decide on the most efficient routes to make the collection. Find out how they do this and investigate improving their procedure. A similar question can be asked about snow plows clearing city streets, or garbage collection.
15. How should one locate ambulance stations, so as to best serve the needs of the community? How do major hospitals schedule the use of operating theatres? Are they doing it the best way possible so that the maximum number of operations are done each day?
16. How does the NBA work out the basketball schedule? How would you do such a schedule bearing in mind distances between locations of games, home team advantage etc.? Could you devise a good schedule for one of your local competitions?
17. How would a factory schedule the production of bicycles? Which parts are put together first? How many people are required to work at each stage of the production?
18. Look for new strategies for solving the traveling salesman problem
19. What is game theory all about and where is it applied?
20. Study games and winning strategies - maybe explore a game where the winning strategy is not known. Analyze subtraction games (rim-like games in which the two players alternately take a number of beans from a heap, the numbers being restricted to a given subtraction set).
21. Ten frogs sit on a log - 5 green frogs on one side and 5 brown frogs on the other with an empty seat separating them. They decide to switch places. The only moves permitted are to jump over one frog of a different colour into an empty space or to jump into an adjacent space. What is the minimum number of moves? What if there were 100 frogs on each side? Coming up with the answers reveals interesting patterns depending on whether you focus on colour of frog, type of move, or empty space. Proving it works is interesting also - it can lead to recursion. There is also a simple proof that is not immediately obvious when you start. Look for and explore other questions like this one of the most famous is the Tower of Hanoi.
22. Try the "Monty Hall" effect. Behind one of three doors there is a prize. You pick door #1, he shows you that the prize wasn't behind door #2 and then gives you the choice of switching to door #3 or staying with #1, what should you do? Why should you switch? Make an exhibit and run trials to "show" this is so. Find the mathematical reason for the switch.
23. Investigate card tucks and magic tricks based in mathematics. Some of the best in the world were designed by the mathematician/statistician Persi Diaconis.
24. All forms of gambling are based on probability. Investigate how much casinos anticipate winning from you when you play black-jack, roulette, etc. Study a variety of lotteries and compare them. Should one ever buy a lottery ticket? Why does three of a kind beat two pairs in poker? Discover why the different types of hands are ranked as they are.
25. Find as many triangles as you can with integer sides and a simple linear relation between the angles. What about the special case when the triangle is right-angled?
26. You make a tangram puzzle by diving a 2- or 3-dimension object into many geometrical pieces, so that the original object can be reconstructed in more than one way. Burr puzzles are interlocking assemblies of notched sticks. For example, there are Burr puzzles that look like spheres or barrels when they are completed.
27. An International Food Group consists of twenty couples who meet four times a year for a meal. On each occasion, four couples meet at each of five houses. The members of the group get along very well together; nonetheless, there is always a bit of discontent during the year when some couples meet more than once! Is it possible to plan four evenings such that no two couples meet more than once? There are many problems like this. They are called combinatorial designs. Investigate others.
28. What is the fewest number of colours needed to colour any map if the rule is that no two countries with a common border can have the same colour. Who discovered this? Why is the proof interesting? What if Mars is also divided into areas so that these areas are owned by different countries on earth. They too are coloured by the same rule but the areas there must be coloured by the colour of the country they belong to. How many colours are now needed?
29. Discover all 17 "different" kinds of wallpaper. (Think about how patterns on wallpaper repeat.) How is this related to the work of Escher? Discover the history of this problem.
30. It is easy to cover a chessboard with dominoes so that no two dominoes overlap and no square on the chessboard is uncovered. What if with one square is removed from the chessboard? (impossible- why?) What if two adjacent corners are removed? What if two opposite corners are removed? (possible or impossible?) What if any two squares are removed? What about using shapes other than dominoes (e.g. 3 1 x 1 squares joined together)? What about chessboards of different dimensions?
31. Polyominoes are shapes made by connecting certain numbers of equal-sized squares together. How many different ones can be made from 2 squares? from 3, from 4, from 5? Investigate the shapes that polynominoes can make. Play the "choose-up" Pentomino game.
32. Build a true scale model of the solar system - but be careful because it cannot be contained within the confines of an exhibit. Illustrate how you would locate it in your town. Maybe even do so!!
33. Discover how to construct the Koch or "snowflake" curve. Use your computer to draw fractals based on simple equations such as Julia sets and Mandelbrot sets. What is fractal dimension? Investigate it by examining examples showing what happens to lines, areas, solids, or the Koch curve, when you double the scale.
34. Knots. What happens when you put a knot in a strip of paper and flatten it carefully? When is what appears to be a knot really a knot? Look at methods for drawing knots.
35. Learn about origamic architecture by making pop-up greeting cards.
36. What are Pick's Theorem and Euler's Theorem? Investigate them individually, or try to discover how they are related.

Meteorology

1. Can you measure the speed and force of raindrops? What is the effect on soil, with and without ground cover? Could you simulate the effect of rain?
2. Does fresh water hold heat longer than salt water? How does water compare to land and what effect does this have on the weather? What factors effect the cooling of land?
3. How do different surfaces affect the amount of sunlight reflected and absorbed? Design a method of measuring how much sunshine is available each day.
4. Is cloud formation related to height, weather systems, and temperature? Study and record how clouds relate to weather patterns.
5. What are the common wind patterns in your area and why?
6. What effect does the amount of carbon dioxide in the atmosphere have on the heat energy from the sun?
7. What happens to hair during periods of changing humidity? How does human hair compare to that of other animals? How do other materials compare in expansion and contraction?

Physics

1. Air pressure versus water pressure
2. Can eggs withstand a greater force from one direction than from others?
3. Electric circuits - factors affecting voltage, amperage, resistance
4. Electric motors - principles and factors affecting their efficiency
5. Fire and burning - What factors affect burning?
6. Fuels and their efficiency in producing energy
7. How can the strength of light be measured? - the effect on degradable materials
8. How do compression and tension make things stronger?
9. How is sound produced? What affect the pitch and volume? How would you measure the velocity of sound?
10. Internal combustion engines
11. Lenses - effects of curvature and materials on light beams
12. Magnets and electromagnets
13. Music versus noise
14. Musical instruments - the scientific principles behind them
15. Observations of freezing rates for water for different starting temperatures
16. Pendulums - how can the period of a pendulum be increased?
17. Spectrum and colour production - prisms
18. What affects light reflection - refraction and defraction of light
19. Which battery lasts the longest? How can power be increased?
20. Which type of lawn sprinkler works best?
21. Which type/size of light bulb produces the most light?

Senior (Grades 10 to 12)

• Astronomy
• Behavioral Science
• Biology (Animal)
• Botany (Plant Science)
• Chemistry
• Ecology/Conservation
• Engineering
• Food Science/Nutrition
• Geology
• Mathematics
• Meteorology
• Physics

Astronomy

1. Observational Orbit Determination of Comets, Meteors or Other Minor Planets

Behavioral Science

1. Is polarized light the guidance system for foraging ants?
2. Learning and perception in animals and humans
3. Study of insect or animal behaviour versus population density

Biology (Animal)

1. A study of the percentage DNA (by weight) in different species
2. A study of population fluctuations in insects
3. Microbial antagonism
4. The effects of ultrasonics, antibiotics and temperature on bacteria count

Botany (Plant Science)

1. Does magnetizing seeds before planting affect growth?
2. Effects of magnetism on the size and frequency of blooms and fruits
3. Effect of mineral deficiencies on protein content in soybeans
4. Factors affecting flowering
5. Factors affecting nodule formation in legumes
6. Organic fertilizer versus chemical fertilizer
7. Root formation in cuttings versus lighting conditions
8. Search for near-vacuum environment tolerant plants
9. Study of sterility in plant hybrids (F1 and F2)
10. The effect of music of varying types and duration on plant growth
11. The effect of solar activity on plant growth
12. The effects of electric fields on plants
13. The effects of excess salinity on plants
14. The effects of magnetic fields on plant growth
15. The effects of phosphates on aquatic plants
16. The effects of water impurities on plant growth
17. Tracing solar activity cycles in tree growth rings

Chemistry

1. A study of esterification reactions
2. A study of saponification reactions
3. Can you obtain water from ink, vinegar or milk?
4. Catalysts - how they work and why; commercial applications and problems
5. Chemical reactions that consume versus produce energy
6. Compare the surface tensions of various liquids
7. Dealing with chemical spills from industry
8. Effects of temperature on density of gases
9. Effects of salt and other contaminants on rate of rusting
10. Electroplating - the principle, how different metals can be used and the practical applications
11. Factors affecting an enzyme's reaction rate
12. What effects do different amounts of exercise have on production of carbon dioxide in humans?

Ecology/Conservation

1. An ecological study of the animal and plant populations occupying the same tree.
2. Efficient methods of breaking down crude oil in sea water
3. Experimenting with biodegradability
4. Experimenting with microbial degradation of petroleum
5. Find an ink that would decompose for recycling paper
6. Finding efficient methods of harvesting and using plankton
7. Ozone destruction experiments
8. Tracing chemical concentrations in successive food chain levels

Engineering

1. Comparing insulative properties of various natural and commercial insulators
2. Design considerations for a "Solar Heated" home
3. Design considerations for "Solar Cell" powered homes
4. Efficiency studies of transformers
5. Find the maximum speed in fibre optic links
6. Study of various phosphors for fluorescent lighting
7. Study the formation of images on a television tube
8. The effect of landscaping and architecture on energy consumption
9. The effect of temperature on resistance
10. Voice communication with infrared light and fibre optics

Food Science/Nutrition

1. * Food - the effects of supply and demand

Geology

1. A study of phosphorescence as a tool for geologists
2. Fossil studies in limestone and other rocks
3. Observations of experimentally-induced seismic waves
4. Observations of geomorphic factors in the local area
5. Observations of local anomalies in the earth's magnetic field
6. Tracing glacial till fragments to local rock outcrops

Mathematics

1. Infinity comes in different "sizes". What does this mean? How can it be explained?
2. A graph is a mathematical structure made up of dots (called vertices) and lines joining pairs of dots (called edges). There are many games that can be played on graphs, and much mathematics involved in finding winning strategies.
3. Pool problems: if you have a rectangular table without friction and send a pool ball at an angle §, will it return to the same spot? If it does not return to the same spot, will it pass over all points on the table? Does the answer depend on the dimensions of the table? Make a sketch in which you can change the dimensions of the table and the direction of the ball, and explore the path through 10 or 20 bounces. What happens on a circular pool table? Make a dynamic geometry sketch.
4. Flatland and sphereland. If you lived in flatland (the plane) could you build a bicycle which exists in the plane and works? Could you do the same on the sphere? Explore other "machines" in a flat space.
5. There are many aspects of spherical geometry that could be investigated. Explore congruences of triangles on a sphere. Other useful tools that are also available are a plastic sphere, with hemispherical "overhead transparencies", great circle ruler, compass etc. One can also make very effective models with plastic spheres from a craft shop and cut-off plastic containers for rulers. Explore quadrilaterals and their symmetries on a sphere. Is there a family which shares most of the properties of a parallelogram? What symmetry do they have? Which two properties (e.g. opposite angles equal) are sufficient to prove all the other properties?
6. What equalities of lengths and angles are sufficient to prove two sets of four points (quadrilaterals or quadrangles...) are congruent? (Leads directly to unsolved research problems in Computer Aided Design.)
7. Build models showing that parallelograms with the same base and height have the same areas. (Is there a 3-dimensional analogue?) This can lead to a purely visual proof of the Pythagorean theorem, using a physical model based on dissections. The formula for the area of a circle can also be presented in this way, by building an exhibit on the Pythagorean theorem but with "the area of the semicircle on the hypotenuse is equal to the sum of the areas of the semicircles on the other two sides."
8. Study the regular solids (platonic and Archimidean), their properties, geometries, and occurrences in nature (e.g. virus shapes, fullerene molecules, crystals). Build models.
9. Consider tiling the plane using shapes of the same size. What's possible and what isn't? In particular it can be shown that any 4-sided shape can tile the plane. What about 5 sides? Make sketches in a geometry program.
10. Draw, and list any interesting properties of various curves: evolutes, involutes, roulettes, pedal curves, conchoids, cissoids, strophoids, caustics, spirals, ovals, ...
11. Make a family of polyhedra, e.g., the Archimidean solids, or Deltahedra (whose faces are all equilateral triangles), or equilateral zonohedra, or, for the very ambitious, the 59 Isocahedra.What polyhedral shapes make fair 'dice'? What are the physical properties? What are the geometric properties? What is the root of the word "polyhedra" (and why does this fit with the use as dice?) Can you list all possible shapes? What numbers of faces can appear? What other (non-polyhedral) shapes are actually used in games? What polyhedral shapes appear in crystals? List them all. Why do these appear? Why don't other shapes appear? What is the connection between the big outside shape and the inside "connections of molecules"?
12. What is Morley's triangle? Draw a picture of the 18 Morley triangles associated with a given triangle ABC. Find the 18 more for each of the triangles BHC, CHA, AHB, where H is the orthocentre of ABC. Discover the relation with the 9-point circle and deltoid (envelope of the Simson or Wallace line).
13. Investigate compass and straight-edge constructions - showing what's possible and discussing what's not. For example, given a line segment of length one can you use the straight edge and compass to "construct" all the radicals? Investigate constructions using origami (paper folding). Can you construct all figures that are constructed with ruler and compass? Can you construct more figures?
14. The cycloid curve is the curve traced by a point on the edge of a rolling wheel. Study its tautochrone and brachistochrone properties and its history. Build models. Suppose all cars had square wheels. How would you design the road so that you always had a smooth ride? What about other wheel shapes?
15. What is a hexaflexagon? Make as many different ones as you can. What is going on?
16. A kaleidoscope is basically two mirrors at an angle of pi/3 or pi/4 to each other. When an object is placed between the mirrors, it is reflected 6 or 8 times (depending on the angle). Construct one. Investigate its history and the mathematics of symmetry. Make models of kaleidoscopes in a dynamic geometry program (Cabri or Geometers Sketchpad). Demonstrate why only certain angles work.
17. Build rigid and non-rigid geometric structures. Explore them. Where are rigid struc- tures used? Find unusual applications. This could include an illustration of the fact that the midpoints of the sides of a quadrilateral form a parallelogram (even when the quadrilateral is not planar). Are there similar things in three dimensions? Are there plane frameworks (rigid bars and flexible joints) that are rigid but contain no triangles? Are all triangulated spheres rigid (either made of sticks and joints or of hinged plastic pieces "Polydron"). What is the formula for the number of bars in a triangulated sphere, in terms of the number of vertices? How does this formula relate to other rigid frameworks in 3-space? Consider a plane "grid" composed of squares (say 4 squares by four squares) made of bars and joints. Which diagonals of squares will make this rigid? What is the minimum number? Can you give a recipe for deciding which diagonals will work? [There is a COMAP module related to this problem.] If the grid is composed of a trapezoid and its image after a half turn, alternating, does the same recipe work? [This is a research problem which has-NOT been thoroughly worked out!]
18. The Art Gallery problem: What is the least number of guards required to watch over all paintings in an art gallery? The guards are positioned at specific locations and collectively must have a direct line of sight to every point on the walls.
19. The Parabolic Reflector Microphone is used at sporting events when you want to be able to hear one person in a noisy area. Investigate this, explaining the mathematics behind what is happening.
20. Investigate self-avoiding random walks and where they naturally occur.
21. Investigate the creation of secret codes (ciphers). Find out where they are used (today!) and how they are used. Look at their history. Build your own using prime numbers.
22. Find pictures which show that 1 + 2 + ... +n = n(n+1)/2; that 1^2 + 2^2 + ... + n^2 = n(n+1)(2n+1)/6; and that 1^3 + 2^3 + ... + n^3 = (1+2+...+n)^2. How many other ways can you find to prove these identities? Is any one of them "best"?
23. What is/are Napier's bones and what can you do with it/them?
24. Martin Gardner defines a paradox to be "any result that is so contrary to common sense and intuition that it invokes an immediate emotion of surprise." There are different types of paradoxes. Find examples of all of them and understand how they differ.
25. Another source of knots is the stone-work and ornamentation of the Celts. Investigate Celtic knotwork and discover how these elaborate designs can be studied mathematically.
26. Is there an algorithm for getting out of 2-dimensional mazes? What about 3-dimensional? Look at the history of mazes (some are extraordinary). How would you go about finding someone who is lost in a maze (2 or 3 dimensional) and wandering randomly? How many people would you need to find them?
27. Explore Penrose tiles and discover why they are of interest.
28. Investigate the Steiner problem - one application of which is concerned with the location of telephone exchanges to minimize costs.
29. Use PID (proportional-integral-differential) controllers and oscilloscopes to demonstrate the integration and differentiation of different functions.
30. Construct a double pendulum and use it to investigate chaos.
31. Investigate the mathematics of weaving.
32. Popsicle Stick Weaving: With long flat sticks, which patterns of "weaving over and under" in the plane are stable (as opposed to flying apart). Find a pattern with four sticks. Is it unique? Does the stability change when you twist one of the sticks (in the plane)? Find several patterns with six sticks whose stability depends on the particular "geometry" of where they cross (i.e. the pattern becomes unstable if you twist one of the sticks in the plane). Can you give a rule for recognizing the "good geometric positions". What kinds of "forces" and "equilibria" are being balanced here? What general rules can you give for "good" weavings?

Meteorology

1. A study of small scale wind currents around buildings
2. A study of solar flares through sudden enhancement of atmospherics
3. Changes in snow density and other characteristics with time
4. Effects of weather on human emotions
5. Observations of fluctuations in stream flow following rain
6. Study of air tides: phases of the moon versus barometric pressure
7. Study of the relationship between wind direction and temperature inversions
8. The effects of solar activity on radio propagation
9. The factors affecting ice patterns on glass

Physics

1. A study of infrared qualities of certain solutions
2. A study of radiation patterns from different antenna types
3. Comparing magnetic pysteresis for different materials
4. Experimental exploration of the photoelectric effect
5. Experimenting with electron diffraction
6. Experimenting with various separation techniques (e.g. Electrophoresis)
7. Factors affecting scent propagation
8. Factors affecting sound dampening
9. Factors affecting sound propagation
10. Index of refraction of liquids versus amount of additive
11. Index of refraction of liquids versus temperature
12. Observations of magnetic permeability of different materials
13. The physics of ski waxes