Why the numbers you know run out — and the new ones that pick up where they stop.
Point 1 of 5 in this lesson: 5.1.1 Quantities that come in opposite pairs
You can do 7 − 3 with no trouble — you have seven things, you take three away, four are left. But what is 3 − 7? You have three things and you are asked to take away seven. With the counting numbers alone there is no answer; you simply run out of stuff. Yet the real world keeps going right past that wall. On a winter morning the temperature drops below zero. You can owe money you don't have. A parking garage has floors below the ground floor. Reality has directions the counting numbers can't name — and negative numbers are how we name "the other direction."
By the end of this lesson you'll be able to attach a sign to opposite-direction quantities, read and write numbers like +5 and −5, explain why 0 is neither, name the three ingredients every number line needs, and plot any signed number in its right place. We keep one steady habit of color throughout: positive numbers are teal, negative numbers are red, zero is slate, and a distance from zero is amber.
Stop and notice how many everyday amounts arrive in matched opposite pairs — the same size, but pointing opposite ways. Walk 3 steps east or 3 steps west. Deposit $20 into an account or withdraw $20. A morning that is 5° warmer than yesterday or 5° colder. Ride 2 floors up in an elevator or 2 floors down.
Here is the snag: the size alone — the "3," the "20," the "5," the "2" — can't tell the two apart. "Three steps" doesn't say east or west. So we attach a small mark, a sign, that records the direction. One direction we call positive and mark with +; the opposite direction we call negative and mark with −.
A sign is not part of the size — it is a direction tag. Pick one direction of an opposite pair to be positive (+); the other is automatically negative (−). The size says how much; the sign says which way.
Pick a scenario, set a size from 0 to 9, then flip the direction. Watch the signed number and its plain-English meaning change.
Now we turn the idea into symbols. Think of a thermometer. The mark for "five degrees above zero" we write as +5, read aloud as "positive five." The mark for "five degrees below zero" we write as −5, read aloud as "negative five." The little dash in −5 doesn't mean "subtract" here — it means the opposite direction, the mirror image of +5 across zero.
A small convention saves ink: positive numbers may be written with or without the plus. The temperature +5 and the plain number 5 are exactly the same thing, so we usually drop the + and just write 5. But a negative number always keeps its sign — −5 can never be shortened, or you'd lose the very thing that makes it negative.
When the dash names a number, say "negative five," not "minus five." Save the word "minus" for the operation of subtracting (as in "seven minus three"). Same dash on the page, two different jobs — and getting the words right keeps them straight in your head.
Tap − and + to move the reading from −10 to +10. Notice how the sign flips the moment you cross zero, and how zero alone has no "above" or "below."
If +5 is "above" and −5 is "below," what is 0? It is the dividing line itself — the place that is neither above nor below. Picture sea level: a mountain rises above it, a submarine dives below it, but sea level itself is the reference both are measured from. It has no direction of its own.
On the number line we call this special spot the origin. Everything to its right is positive; everything to its left is negative; and 0 sits exactly on the boundary, carrying no sign. So zero is not a "small positive" or a "big negative" — it is the one number that belongs to neither side.
0 is the origin and the dividing line. It is not positive and not negative. Every other number is either to the right of it (positive) or to the left of it (negative).
A bare ruled line is just a line — it can't tell you where any number lives. To turn it into a true number line you must choose three things. Miss any one and the numbers have no home.
① The origin. Pick the point where 0 sits. Everything is measured from here. ② The positive direction. Decide which way counts as positive — by custom we point the arrow to the right, so the other way is automatically negative. ③ The unit length. Choose how far one whole step is. That single chosen length, copied over and over, places 1, 2, 3 on one side and −1, −2, −3 on the other.
Forget the origin and you don't know where zero is. Forget the direction and you can't tell positive from negative. Forget the unit and you can't tell 2 from 200. A real number line needs all three at once.
Change the unit length and flip the positive direction. Watch every label slide to its new home — the same numbers, relocated by your choices.
Once the line is built, plotting any number is a three-word recipe: start, turn, step. Start at the origin. Turn right if the number is positive, left if it is negative. Step that many units. Where you stop is the number's point.
To plot +2: start at 0, face right, take 2 steps. To plot −3: start at 0, face left, take 3 steps. And numbers needn't land on a tick — a half lands between ticks. −2.5 sits exactly halfway between −3 and −2; 3.5 sits halfway between 3 and 4.
Plot −2.5. Start at the origin. The sign is negative, so turn left. Step two and a half units left. That lands you halfway between the −2 tick and the −3 tick — a little farther from zero than −2, a little closer than −3.
Step the marker left and right between −6 and 6. The readout names exactly where you are. Then try the challenge: can you land on −4?
Many real quantities come in opposite pairs of the same size, so we tag each with a sign: one direction is positive (+), the opposite is negative (−). We write five above as +5 (or just 5) and five below as −5, read "negative five." Zero is the origin — the dividing line that is neither positive nor negative. A true number line needs three ingredients: an origin, a positive direction, and a unit length. And to plot any number you start at zero, turn right for positive or left for negative, and step that many units — halves landing between the ticks.
You've met +5 and −5 as mirror images across zero. Next lesson gives that mirror a name — opposites — and asks a sharper question: how far is a number from zero, no matter which side it's on? That distance is its absolute value, and it's the amber idea we've been hinting at all along.
Work each one out first, then open the answer to check your thinking.
Six questions to lock it in. Tap the answer you think is right.
This lesson opens Stage 5 and is aligned to the U.S. Common Core grade-6 number-system standards. It uses positive and negative numbers to describe quantities with opposite directions and meanings — temperature above/below zero, deposits/withdrawals, floors up/down — with 0 as the reference point (6.NS.C.5). It establishes that the negative sign indicates the opposite of a number and that 0 is its own opposite (6.NS.C.6a), and it has students locate signed numbers, including a half value, on the number line (6.NS.C.6c). The number line is treated as a built object with an origin, a positive direction, and a unit length, so plotting is a deliberate "start, turn, step" procedure rather than guesswork. The single most common misconception is "you can't have less than zero" — the belief that negatives aren't real numbers. The antidote, used throughout, is concrete opposite-direction contexts: debt, below-zero temperatures, and below-ground floors make "less than zero" tangible long before any formal arithmetic with signs.