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Prove the mid-point theorem and use it for lengths, apply the converse to identify a mid-point, use the equal intercept theorem on transversals, and show why joining the mid-points of any quadrilateral gives a parallelogram.
Read articleRank a triangle's angles from its sides and its sides from its angles, test whether three lengths can form a triangle at all, prove the perpendicular is the shortest route to a line, and order the parts of a figure.
Read articleProve that base angles of an isosceles triangle are equal, prove the converse to show a triangle is isosceles, calculate unknown angles with the angle sum and exterior angle, and handle riders that combine both with congruence.
Read articleChoose between SSS, SAS, AAS and RHS, write the correspondence in the right order, set out a congruence proof in logical steps, and use CPCTC to unlock unknown sides and angles.
Read articleDerive the product, quotient and power laws from the laws of indices, expand a messy logarithm into separate terms, collapse a sum of logs into one, and prove identities and solve equations safely.
Read articleRead the logarithm as the index a base needs, convert freely between exponential and logarithmic form, evaluate logs straight from the definition, and solve for a missing base or argument.
Read articleUse the three laws of indices, work out what zero, negative and fractional powers must mean, solve equations by equating indices, and simplify expressions whose bases look different.
Read articleApply the cross-multiplication formula to any pair of linear equations, then frame equations from word problems on ages, fractions, digits, speed and the sides of a rectangle.
Read articleSolve a pair of linear equations by substitution and by elimination, clear fractions and brackets before you start, and check a given solution or find a missing constant from it.
Read articleSplit the middle term of a quadratic whether the leading coefficient is one or not, factorise a sum or difference of cubes with the right signs, and take a composite expression apart completely.
Read articleTake out the highest common factor, split four or six terms into workable groups, spot a difference of two squares even when it is disguised, and manufacture one by completing the square.
Read articleExpand the cube of a binomial and use it to cube 21 and 99, climb from a plus one over a to its square and cube, substitute into an identity to get a value, and prove an identity outright.
Read articleExpand the square of a binomial and use it to square 103 and 97 mentally, square a three-term expression with the correct signs, expand a product of two binomials, and combine two expansions.
Read articleUse the amount formula for any number of years, halve the rate for half-yearly compounding, find whichever of P, r, n or A is missing, and handle population growth and machine depreciation.
Read articleWork compound interest out year by year on a growing principal, handle a rate that changes each year, find exactly why compound interest beats simple interest, and work backwards to the principal or rate.
Read articleSimplify any surd to its lowest form, add, subtract, multiply and divide surds, kill the root in a denominator with its conjugate, and order surds of different index.
Read articleSort numbers into rational and irrational, plot a square root on the number line by construction, prove root 2 irrational by contradiction, and turn any recurring decimal into a fraction.
Read articleLearn where the Red Cross came from and what it stands for, what it actually does in India and abroad, and how the World Health Organisation works through governments instead of volunteers.
Read articleLearn how a vaccine teaches the body to defend itself, why a serum works at once and briefly, where antiseptics end and disinfectants begin, and what the body's own first line of defence does.
Read articleLearn the symptoms and control of cholera, typhoid and tuberculosis, then AIDS, chicken pox and hepatitis, then malaria, amoebiasis and sleeping sickness, then ascariasis, taeniasis and filariasis.
Read articleSeparate communicable from non-communicable disease, learn what endemic, epidemic, pandemic and sporadic actually describe, follow the air-borne and water-borne routes, and control the three vectors.
Read articleLabel the epidermis and dermis and learn what each does, meet the five derivatives of the skin from sweat glands to mammary glands, and follow how vasodilation and vasoconstriction hold your temperature steady.
Read articleList what the skeleton does besides holding you up, split its 206 bones into axial and appendicular and check the count, then classify every joint and place the four movable kinds in the body.
Read articleLabel a tooth from enamel to pulp, write the dental formula and check the count, place every digestive organ and gland with its job, then separate digestion from absorption and assimilation.
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