The counting numbers fill only the right half. Negative numbers are how we finally name the other direction — the half to the left of zero.
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.
5.1.1 Quantities that come in opposite pairs
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 −.
Each amount comes in a matched pair. The number in the middle is the size; the + or − in front records which of the two opposite directions you mean.
Key idea
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.
🎮 Try itGive a real quantity its sign
Pick a scenario, set a size from 0 to 9, then flip the direction. Watch the signed number and its plain-English meaning change.
Scenario
Size3
Direction
5.1.2 Positive and negative numbers, written down
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.
Zero is the dividing line. +5 sits five steps above it, −5 sits five steps below it — equal distance, opposite directions.
"Negative," not "minus"
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.
🎮 Try itStep the thermometer above and below zero
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."
Temperature3
5.1.3 Zero is neither positive nor negative
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.
Zero is the fence between the two camps. To its left live the negatives; to its right live the positives; the fence post itself belongs to neither.
Key idea
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).
5.1.4 The three ingredients of a number line
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.
Three choices build the line: ① an origin at zero, ② a positive direction (the arrow), and ③ a unit length (the gap between ticks). Change any one and every label moves.
All three, or it falls apart
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.
🎮 Try itBuild the line yourself
Change the unit length and flip the positive direction. Watch every label slide to its new home — the same numbers, relocated by your choices.
Unit length
Positive points
5.1.5 Showing numbers on the number line
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.
Four numbers in their homes: −3 and −1 sit left of zero, +2 sits right, and +3.5 lands between the 3 and 4 ticks.
Worked example — plotting −2.5
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.
🎮 Try itWalk the marker to a number
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?
Marker0
★ The big ideas, in one breath
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.
Coming up next — 5.2
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.
✎ Exercises 5.1
Work each one out first, then open the answer to check your thinking.
A diver goes 30 feet below the surface. If "above the surface" is the positive direction, what signed number names the diver's depth?
Show answer
−30 feet. "Below" is the opposite of the chosen positive direction, so it takes a negative sign.
Write a signed number for each: (a) a deposit of $15, (b) a withdrawal of $15.
Show answer
(a) +15 (b) −15. A deposit adds to the account (positive); a withdrawal takes away (negative). Same size, opposite signs.
Read −7 out loud, the correct way. Why not "minus seven"?
Show answer
"Negative seven." We say "minus" only for the operation of subtracting; when the dash names a number, the word is "negative."
On a number line, which number sits at the origin — the dividing point between positives and negatives?
Show answer
0. Zero is the origin: everything to its right is positive, everything to its left is negative, and zero itself belongs to neither side.
Someone draws a line, marks 0 in the middle, and puts an arrow pointing right — but every tick is a different distance apart. Which of the three ingredients is missing?
Show answer
The unit length. The origin and the positive direction are set, but a number line needs one fixed step size, copied evenly. Unequal gaps mean the labels can't be trusted.
Plot these on a number line: −4, −1, +3. Which is farthest left?
Show answer
Going left from zero: −1 is one step left, −4 is four steps left; going right, +3 is three steps right. The farthest left is −4.
Where does −1.5 land? Name the two whole-number ticks it sits between.
Show answer
It sits halfway between −2 and −1. Start at zero, turn left, step one and a half units: that's past −1 but not all the way to −2.
True or false: "Zero is a positive number." Explain.
Show answer
False. Zero is the origin — the dividing line. It is neither positive nor negative. Positive means strictly to the right of zero; zero is not to the right of itself.
At 9 p.m. the temperature is 3°. Overnight it falls 7°. What is the morning temperature, as a signed number? Use the number line.
Show answer
−4°. Start at 3 and step 7 units left: 3 → 2 → 1 → 0 → −1 → −2 → −3 → −4. You cross zero on the way down and end up 4 below it. (This is the wall 3 − 7 that negative numbers tear down.)
Jordan says, "−5 isn't a real number — you can't have less than nothing." Give one real situation that proves −5 is a perfectly real number.
Show answer
Many work: a thermometer reading 5° below zero; owing $5 (a balance of −5 dollars); standing on floor 5 of an underground garage; a submarine 5 meters below sea level. "Less than zero" just means the opposite direction from zero — and that direction is everywhere in the real world.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
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.