Unit 16: The Grid

Two numbers find any spot

Draw two number lines crossing at right angles. Now every point on the page has an address: along first, then up.

The point (3, 2) is 3 along and 2 up. The point (2, 3) is somewhere else entirely — the order matters.

Where they cross is (0, 0), called the origin.

Four quarters

Go left of the origin and the first number is negative. Go below it and the second is.

So (-4, 3) is four to the left and three up, and (2, -5) is two right and five down.

The two lines cut the page into four quadrants, and the signs tell you which one you are in.

The midpoint is the average

To find the point exactly halfway between two others, average each number separately.

Halfway between (2, 3) and (8, 7) is ((2+8)/2, (3+7)/2), which is (5, 5).

That is all a midpoint is: the average of the two addresses.

Pythagoras on the grid

To find the distance between two points, draw the right-angled triangle underneath them.

The across side is the difference of the first numbers. The up side is the difference of the second. Then Pythagoras finishes it.

From (1, 2) to (4, 6): across 3, up 4, so the distance is sqrt(9 + 16) = 5.

A shape is just a list of points

Give four points and you have described a quadrilateral exactly — no drawing needed, no measuring, no argument about whether it looks right.

(0,0), (4,0), (4,3), (0,3) is a rectangle 4 across and 3 up. Anybody in the world would draw the same one.

That precision is the whole gift of coordinates.

Vertical and horizontal lengths are subtractions

The distance between (2, 1) and (2, 7) is simply 7 - 1 = 6. They share a first number, so the line is straight up.

Likewise (3, 5) to (11, 5) is 11 - 3 = 8.

Only when both numbers change do you need Pythagoras. Checking for a shared number first saves a lot of work.

A rule makes a table, a table makes a line

Take the rule y = 2x + 1. Feed in some values of x and see what comes out.

x = 0 gives 1. x = 1 gives 3. x = 2 gives 5. x = 3 gives 7.

Plot (0,1), (1,3), (2,5), (3,7) and they line up perfectly straight. Every rule with no powers in it does that.

Flat lines and upright ones

Two special cases are worth knowing before the general one.

y = 4 is a horizontal line: whatever x does, y stays at 4.

x = 4 is a vertical line: x never moves, and y can be anything.

They look similar written down and are at right angles on the page, so it is worth pausing over which is which.

Proving, not guessing

Is the triangle with corners (0,0), (6,0) and (3,4) isosceles?

Measure with Pythagoras. The two slanted sides are both sqrt(9 + 16) = 5. The base is 6.

Two sides equal, so yes — and you have proved it rather than squinting at a drawing.

The midpoint proves things too

To show a quadrilateral is a parallelogram, check whether its diagonals share a midpoint.

For corners (0,0), (5,1), (7,5) and (2,4): one diagonal runs (0,0) to (7,5), midpoint (3.5, 2.5). The other runs (5,1) to (2,4), midpoint (3.5, 2.5).

The same point. So the diagonals bisect each other, and by unit 12 it is a parallelogram.

Rise over run

How steep is a line? Count how far it climbs, and divide by how far it travels sideways.

From (1, 2) to (5, 10): it climbs 8 over a run of 4, so the gradient is 2. For every one step right, it goes two up.

That single number describes the whole line, because a straight line has the same steepness everywhere.

Flat is zero, upright has none at all

A horizontal line climbs nothing, so its gradient is 0.

A vertical line runs nowhere sideways, so the calculation would divide by zero. Its gradient is undefined — not infinity, not zero, simply not a number.

That is why y = mx + c cannot describe a vertical line. There is no m that would work.

Gradient: up divided by along

The gradient of a line is how much it climbs for every one step to the right.

From (1, 2) to (4, 8) it climbs 6 over a run of 3, so the gradient is 6/3 = 2.

A line going downhill has a negative gradient. A flat line has gradient 0.

The equation of a line

Every straight line can be written y = mx + c.

m is the gradient, and c is where it crosses the up-line — the intercept.

So y = 3x + 2 climbs 3 for every 1 across, and passes through (0, 2). Read it off, no working needed.

Parallel and perpendicular

Two lines are parallel when their gradients are equal — same steepness, never meeting.

They are perpendicular when the gradients multiply to -1. So a line of gradient 2 crosses at right angles to one of gradient -1/2.

Flip it over, change the sign. That is the whole rule.