Integers & the Four-Quadrant Plane

How to do it:

1. Name what 0 means in the story (sea level, empty account, freezing, no charge).

2. Assign positive to one direction and negative to the opposite.

3. Write a number that matches the situation.

Worked example: A trail crew starts at $0$ ft (trailhead). They climb to $250$ ft, then descend $400$ ft. Their elevation is $250 + (-400) = -150$ ft, which is 150 ft below the trailhead, not "negative hiking."

Check: an account of $-30$ dollars is a $30$ debt. The absolute value $|-30| = 30$ is the size of the debt. Replay fractions and signs in the Fraction converter if a mixed number appears.

Opposites sit the same distance from 0 on opposite sides. $-(-3) = 3\text{.}$ Zero is its own opposite.

How to do it:

1. Put 0 in the middle of your sketch.

2. Match equal steps left and right.

3. Compare with "to the right is greater" on a standard line.

Worked example: $-3 > -7$ because $-3$ sits to the right of $-7\text{.}$ In temperature, $-3°\text{C}$ is warmer than $-7°\text{C}\text{.}$

Check: a larger absolute value is not automatically a larger number. $|-10| = 10$ is bigger than $|-2| = 2\text{,}$ but $-10$ is less than $-2\text{.}$

Signs of the ordered pair $(x, y)$ name the quadrant: I $(+,+)\text{,}$ II $(-,+)\text{,}$ III $(-,-)\text{,}$ IV $(+,-)\text{.}$ Axes themselves are not in a quadrant.

How to do it:

1. Move $x$ first (right if positive, left if negative).

2. Then move $y$ (up if positive, down if negative).

3. Name the quadrant or say "on an axis" if one coordinate is 0.

Worked example: $(3, -2)$ is Quadrant IV. $(-4, 0)$ is on the negative $x$-axis, not in II or III.

Check: swapping $(3, -2)$ to $(-2, 3)$ jumps to Quadrant II. Order matters.

If two points share an $x$ or a $y\text{,}$ the distance is the absolute value of the other difference.

How to do it:

1. Check whether the $x$-coordinates match (vertical segment) or the $y$-coordinates match (horizontal segment).

2. Subtract, then take absolute value so distance is never negative.

3. Convert with a map scale if the grid units are not the real units.

Worked example: From $(-3, 2)$ to $(5, 2)\text{:}$ $|5 - (-3)| = 8$ units. If 1 grid unit $= 0.5$ mile, the walk is $4$ miles.

Check: $5 - (-3)$ is $8\text{,}$ not $2\text{.}$ The two minuses make addition. Review Polygons on the plane when you need area after you have the side lengths.