Grade 10 Math

Two bags, two sizes, two prices — which one is the deal?

Meet the four symbols before they go to work: Q is a quantity of one ingredient — grams of flour, cups of milk — and S is the number of servings.

A map that says 1:50,000 is making you a promise: every 1 cm on the sheet stands for 50,000 cm of real world.

A 25% off sign is an instruction with a shortcut in it: S = L(1 - d) — read aloud: S equals L times one-minus-d.

The till has exactly one trick and it plays it on every purchase: T = P(1 + r) (read aloud: T equals P times one-plus-r). P is the price before tax, r is the tax rate as a decimal, and T is the total the register asks for — that last one…

Here old is the price before, new is the price after, and c is the change you are solving for — a fraction of the old price, times 100 to say it as a percent.

Meet the star of this chapter: slope, written m. It answers one friendly question — for every step you walk ACROSS, how much does the path climb?

Two points are all a line needs — and all its slope needs, too.

Every straight line can wear the same uniform: y = mx + b — read aloud, y equals m x plus b. x is whatever you put in — gigabytes used, weeks grown, kilometres driven — and y is what comes out, usually the total.

The same line owns three costumes.

Here it is, the chapter's best secret: on a distance–time graph, the slope IS the speed. Rise is kilometres gained, run is hours spent, so rise over run comes out in kilometres per hour.

Canadian pumps and window stickers speak in L/100km — read aloud, litres per hundred kilometres: the fuel a car drinks to cover 100 km.

Meet in the middle, coordinate by coordinate.

A rug is area; its fringe is perimeter. Keep those two apart and half this chapter's traps never touch you. Four area machines to meet, each with a read-aloud form you can own.

Exam papers love to hand you that slant. It always takes the scenic route, so it is always LONGER than the true height, and using it always overshoots.

Forgetting it is this lesson's headline mistake, and it always lands exactly double. A trapezoid has two parallel sides of different lengths, a and b.

Circles bring one number and one trap.

A sector is a pie slice; an arc is the crust on that slice. No new machinery — just a fair share.

So far the area was the answer. Now it's a GIVEN, and a side is hiding.

Nobody prints a formula for THIS floor — and nobody needs to. Every odd floor is friendly shapes holding hands: slice it into rectangles, triangles and half-circles, work each piece, and add.

So far you've measured flat shapes; this chapter measures what they can HOLD. Volume is counted in cubes — a box 4 m long, 3 m wide and 2 m tall holds 4 × 3 × 2 = 24 metre-cubes of space.

It's the circle's own number: every circle's distance around is π times its distance across, no exceptions, ever.

Here's the most satisfying demo in geometry. Take a cone and a cylinder with the SAME circle base and the SAME height. Fill the cone with water and pour it into the cylinder — it takes exactly three pours to fill it.

Notice who wears which power: the VOLUME cubes the radius, the SURFACE squares it. And cubes grow ferociously — double the radius and the volume jumps ×8; triple it and you're at ×27.

Two different questions live on every box: how much does it HOLD (volume, in cubes) and how much does it take to COVER it (surface area, in squares — paint, wrapping paper, cardboard).

Peel the label off a soup can and flatten it: a plain rectangle, as tall as the can and as long as the trip around the circle.

Good news: the litre is a cube in disguise.

So far the machines run forward: measurements in, volume out. Exams love reverse: here's the volume — find the edge, the radius, the height.

Every triangle in this chapter has one square corner — the right angle — and the side straight across from it is the hypotenuse (say it with confidence: hy-POT-en-use).

The two shorter sides (the legs, a and b) have squares that together exactly fill the square on the hypotenuse c.

The little letters are just first initials, and they are all side LENGTHS in the same unit: o is the opposite, a the adjacent, h the hypotenuse — the three names you dealt out last lesson.

A trig ratio with numbers in it is really an equation with one blank.

So far the angle was handed to you.

Surveyors gave us two tidy phrases. The angle of elevation opens UP from level ground to something high — a treetop, a tower’s tip.

Chapter 2 called it slope.

Money problems tell one little story, and it always has the same four characters. The principal P is the money you start with.

I = Prt is one fact wearing four outfits — solve it for I, P, r or t and it answers four different exam questions.

Simple interest pays rent on the principal only, forever.

Banks rarely compound once a year.

Two letters, and one of them has a twist: t is the wait in years, and R is the rate as its percent number — 6 for 6%, not the decimal 0.06 that every other formula in this chapter wants.

So far money has walked forwards in time, growing as it goes.

Same deposit, same rate, two paths. Simple interest adds the same flat rent every year — a straight line. Compound interest re-invests each year's rent — a curve that bends upward.