Adding Digits Alternately Tests a Number for Eleven
Learn the digit-sum tests for 3 and 9, the alternating-sum test for 11, the algebra that proves why all three work, and how to crack cryptarithm puzzles in addition and multiplication.
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Free study notes for CBSE & ICSE, Classes 6–12.
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One Counterexample Destroys a Rule That Looks Obvious
Learn how divisibility passes to multiples and factors, why a divisor of two numbers divides their sum and difference, when two divisors combine into a bigger one, and how to disprove a claim with a single example.
Read articleA Square Is Secretly Both a Rectangle and a Rhombus
Learn the properties of a parallelogram and a rhombus, recognise a kite and a trapezium from given measures, and place any quadrilateral correctly in the family hierarchy.
Read articleOne Diagonal Proves Every Quadrilateral Adds to 360
Learn why the angles of any quadrilateral total 360 degrees, use the properties of a rectangle and its equal diagonals, apply the extra properties of a square, and calculate unknown sides and angles.
Read articleWithout Zero You Could Not Tell 105 From 15
Learn to convert numbers between base 10 and bases 2, 5, 7, 8 and 20, read off place values in any base, see why zero completes a place value system, and compare how efficiently two bases write the same number.
Read articleIn Roman Numerals the Same Symbol Always Means the Same Thing
Learn to read and write tally, Egyptian, Roman, Mesopotamian and Mayan numerals, identify the landmark numbers each system uses, and see what separates a positional system from a non-positional one.
Read articleEvery Number Can Be Written as One Digit and a Power of Ten
Learn why anything to the power zero is 1, turn negative exponents into positive ones, write huge and tiny numbers in scientific notation, and compare quantities by their powers of ten.
Read articleDoubling Beats Adding Even When It Starts Behind
Learn to write repeated multiplication in exponential form, use the three same-base laws, apply the two same-index laws, and compare multiplicative growth with additive growth over many steps.
Read articlePrime Factors in Threes Reveal a Perfect Cube
Learn to cube integers and recognise perfect cubes, test a number by grouping its prime factors in threes, find cube roots, and work out the smallest multiplier that makes a number a perfect cube.
Read articleThe Last Digit Alone Can Rule Out a Perfect Square
Learn to square numbers and spot perfect squares, use the units-digit and trailing-zero tests to rule numbers out, find square roots by prime factorisation, and build squares from consecutive odd numbers.
Read articleCutting One Forest Empties More Than One Habitat
Learn the three great threats to life on Earth, how burning fossil fuels warms the planet, how habitat destruction spreads loss through a food chain, and which actions genuinely help.
Read articleOne Parent Makes a Copy, Two Parents Make Something New
Learn how air, water and sunlight sustain life, how soil and rocks are formed and renewed, how living things depend on one another, and how asexual and sexual reproduction differ.
Read articleMove the Earth a Little Closer or Further and Life Ends
Learn what makes the Earth unique among the planets, compare the planets on size, distance, temperature and atmosphere, see why the habitable zone allows liquid water, and how gravity and the ozone layer protect us.
Read articleNature Does Work for Us That No Money Could Buy
Learn how interactions keep an ecosystem in balance, list the services healthy ecosystems provide, compare a natural ecosystem with a farm or aquarium, and see how pollinators and soil life support farming.
Read articleEvery Meal on Earth Traces Back to Sunlight
Learn to build a food chain and label producers, consumers and decomposers, see how chains link into a web, follow energy up the trophic levels, and predict what one change does to the whole web.
Read articleRemove One Kind of Animal and the Whole Place Changes
Learn to separate the living and non-living parts of a surrounding, tell habitat from population and community, see why every organism matters, and sort relationships into competition, predation, parasitism and mutualism.
Read articleA Purely Lunar Calendar Slides Its Seasons Out of Place
Learn how the day, month and year come from real motions, measure a solar day by shadows, compare lunar, solar and luni-solar calendars, see why leap years exist, and link festivals to the sky.
Read articleThe Moon Lines Up Every Month yet Eclipses Are Rare
Learn the phases of the Moon in order, tell waxing from waning, predict where and when a given phase appears, understand why phases happen at all, and see why eclipses skip most months.
Read articleA Lens Can Set Paper Alight With Nothing but Sunshine
Learn to tell a convex lens from a concave one by shape and by what it does to light, concentrate sunlight safely to a bright spot, compare the images each lens forms, and pick the right lens for a job.
Read articleOne Mirror Makes You Bigger and the Other Shrinks the Whole Room
Learn to tell a concave mirror from a convex one, compare the images all three mirror types form, apply the laws of reflection to find an angle, and explain lateral inversion.
Read articleHeating Air Makes It Lighter Without Removing Any of It
Learn why an object floats by comparing it with the water it displaces, calculate density from mass and volume, predict floating or sinking from density alone, and see how heat and pressure change density.
Read articleWarm Water Dissolves More Sugar but Less Gas
Learn to name the solute and solvent in any solution, tell a saturated solution from an unsaturated one, see how temperature changes the solubility of solids, and why gases behave the opposite way.
Read articleMixing Two Metals Makes Something Better Than Either
Learn to tell elements from compounds, sort elements into metals and non-metals, see why alloys beat pure metals for real uses, and follow a metal from its ore to the finished article.
Read articleAir Looks Like One Gas but Is a Mixture of Many
Learn to pick out the components of everyday mixtures, tell uniform from non-uniform mixtures, sort mixtures by the states of their components, and separate a pure substance from a mixture.
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