Mathematics 10-3

The metric system was designed in France in the 1790s so that every unit steps by ten, and in 1960 it became the SI used around the world, Canada included since the 1970s.

Before it, every region had its own foot and its own pound, and a trader moving between towns had to convert everything.

A referent is a familiar length you can picture: a millimetre is about the thickness of a dime, a centimetre the width of a fingernail, a metre a long stride, and a kilometre about twelve minutes' walk; an inch is a thumb's width, a foot a 12-inch ruler, a yard a big step and a mile about twenty minutes' walk.

A referent turns a number into something you can see, so you can estimate without a tool and catch a wrong answer.

SI measures each quantity in its own unit: length in metres, mass in grams and kilograms, capacity in litres, temperature in degrees Celsius; area and volume are built from the metre, in square metres and cubic metres.

A unit belongs to one quantity and never crosses to another.

Estimating with a referent counts how many of it fit: a hallway eight of your 75 cm strides long is about 8 × 0.75 = 6 m.

Lay the referent along the length in your head or on the floor and count, then multiply by its size: eight strides of 75 cm is about 6 m; a counter four hand spans of 20 cm is about 80 cm; a lot 30 big steps wide is about 30 yd.

Six prefixes are powers of ten on a base unit: kilo 1000, hecto 100, deca 10, deci one tenth, centi one hundredth, milli one thousandth, so 1 km = 1000 m, 1 m = 100 cm and 1 cm = 10 mm.

Draw the ladder from big to small: km, hm, dam, m, dm, cm, mm, one power of ten per rung.

Angles are classed by size: acute under 90°, right exactly 90°, obtuse between 90° and 180°, straight exactly 180°, and reflex between 180° and 360°, so two arms always make two angles, such as 110° and 250°.

You met acute, right and obtuse angles before; straight and reflex complete the set.

A right triangle's hypotenuse is the side opposite the right angle and always the longest; the two legs meet at the right angle, and in a² + b² = c² the c is always the hypotenuse.

Find the square corner first; the side that does not touch it is the hypotenuse.

A strategy is a plan chosen from the puzzle's structure and shown to work every time: the Tower of Hanoi's three discs move to another peg in 7 moves by a plan, never by random sliding.

Spatial puzzles are about moving, turning, building and taking apart shapes: sliding tiles, rotating pieces, stacking discs, packing boxes.

Familiar angles are referents: a square corner is 90°, a flat edge 180°, a square folded corner to corner 45°, and on a clock each hour mark is 30°, so the hands at 1:00 make 30° and at 2:00 make 60°.

Carry a few angles in your head.

The theorem is about squares built on the sides: the square on the hypotenuse equals the two squares on the legs together, so on a 3-4-5 triangle 9 + 16 = 25.

Draw a right triangle with legs 3 and 4 on grid paper and a square on each side: 9, 16 and 25 unit squares.

A unit price is the price divided by the amount, in dollars for one unit: 750 g of cheese at $4.29 is 4.29 ÷ 7.5 = $0.572 for every 100 g.

Divide money by quantity so the answer is money for one unit, and choose a unit that gives a readable number: per 100 g, per litre, per item.

Income is earned in several ways: an hourly wage, a salary, commission on sales, piecework per item, a contract price per job, tips, bonuses and shift premiums, and many jobs combine two or more.

An hourly wage pays for every hour worked, often with overtime.

Two sizes compare by the same unit: 750 g at $4.29 is $0.572 per 100 g and 1.2 kg at $6.49 is $0.541 per 100 g, so the 1.2 kg pack is the better buy.

2 kg is 1200 g, twelve hundred-grams), divide each price by its amount, and compare: lower is better.

A salary is a yearly figure split by the number of cheques: $46 800 a year is $900 weekly (÷ 52), $1800 biweekly (÷ 26), $1950 semi-monthly (÷ 24) and $3900 monthly (÷ 12)

The divisor is the number of pay days in a year, not the number of months.

The bigger package is not always cheaper per unit, and the lower total price is not always the better value: 500 mL of shampoo at $2.19 is $0.438 per 100 mL while 2 L at $9.49 is $0.475.

Bulk sizes often cost less per unit, but a sale on the small size, a convenience 'family size' or a different brand can reverse it, so only the unit prices decide.

A formula is used by putting each known value in for its letter, with its unit: P = 2l + 2w with l = 3.2 m and w = 1.5 m gives 2(3.2) + 2(1.5) = 9.4 m.

Write the formula, write it again with numbers and units in place of the letters, then compute.

Each trade uses the formulas of its work: a flooring installer A = lw, a fence builder P = 2l + 2w, a framer c² = a² + b² for rafters and squaring, a surveyor the trigonometric ratios, and a payroll clerk gross pay = rate × hours.

Ask what the job buys or measures.

A job often needs several formulas in a row, each chosen for what that step measures: fencing and seeding a 12 by 8 m yard takes 40 m of fence by perimeter and seed for 96 m² by area, and a 4 by 3 by 0.1 m pad of concrete at $180 a cubic metre costs 1.2 × 180 = $216.

List what the question finally asks for, then the steps that lead there, and pick the formula for each step.

A good problem built around a formula gives a situation, the numbers it needs and one clear question: 'a dog run 6 m by 2.5 m needs fencing except for a 1 m gate; how many metres of fence?' is solved as P − 1 = 17 − 1 = 16 m.

To create a problem, choose a real situation, the formula it uses, numbers that make sense for it (a dog run is a few metres, not 300), and the question.

Rearranging keeps a formula true by doing the same thing to the whole of both sides: from A = ½bh, doubling both sides gives 2A = bh, and dividing both sides by b gives h = 2A ÷ b.

A formula is a balance, so each step is applied to each whole side.