Integers on the Line and the Plane
Below zero is a real place
Until now the number line started at zero and only grew. Integers let it run both ways: the counting numbers to the right, their opposites to the left, and zero in the middle as the meeting point. The negatives are not a trick — they name real situations. A temperature three below freezing, a diver sixty metres under the surface, eighty dollars owed on a card: each needs a number below zero to be honest. An integer is a whole number, its opposite, or zero, and zero here means a reference point, not nothing at all.
Locating an integer on the line
Every integer is one fixed place on the line, the same equal step apart as every other. To locate negative seven, you count seven steps left of zero; to locate four, four steps right. Because the spacing never changes, position quietly carries order: whatever sits further left is the smaller number. That makes negative seven less than negative two, even though seven feels like the bigger digit. Reading integers as places rather than as digits with a sign attached is the habit that stops the most common slip, treating a large negative as though it were large.
Every integer has an opposite
Fold the line at zero and each integer lands on its opposite: six meets negative six, and the two stand the same distance from zero on either side. That shared distance is the size of the number, and zero is the only integer that is its own opposite, sitting exactly on the fold. Opposites matter because they undo each other — a deposit cancels an equal debt, a step up cancels an equal step down. Seeing six and negative six as a mirrored pair, rather than two unrelated numbers, prepares the directed addition and subtraction that follow in later years.
When the balance falls below zero
Money is where integers do everyday work, and it is the financial context the curriculum names directly. A balance above zero is credit, money you hold; spend past it and the balance crosses into the negatives, recording what is now owed. The zero line is the divide between having and owing, and the same integers track both directions without needing two separate systems. A balance of negative twenty-five dollars is not an error — it is a precise statement that the account is twenty-five dollars short, the kind of reading that bank statements, temperatures and elevations all share.
Ordering runs left to right
To order a mixed set of integers, read the line from left to right: the most negative number comes first, then the rest climb toward the positives. The bars make the claim visible, hanging below the zero line for negatives and rising above it for positives, so a deep bar marks a small number, not a large one. This is the same ordering skill from earlier years, simply extended through zero. Children who picture the line rarely fall for the trap of ranking negative nine above three because nine looks bigger; on the line, negative nine is plainly the furthest left.
Two numbers fix one point
One integer fixes a place on a line; two integers fix a place on a plane. A coordinate pair reads across first, then up or down: the x value steps left or right of the origin, the y value steps above or below it, and together they pin exactly one point. The signs decide the quadrant — the point at negative four, three sits up and to the left, in the second quadrant, while four, negative one drops to the fourth. The origin, zero, zero, is where the two axes cross. This is the bridge from a single line of integers to the whole coordinate plane.
Integers, the line and the plane
Integers gather temperature, depth, money and elevation into one system: whole numbers, their opposites and zero, laid out as positions on a line and paired as coordinates on a plane. Read as places rather than digits with a sign, they order themselves, mirror into opposites, and locate points in all four quadrants. From here the same numbers learn to be added and subtracted with direction, and the coordinate plane becomes the stage for plotting patterns and, in time, the graphs of algebra.