Adding on the Number Line

Skill 1.02 · Core CA CCSSM 7.NS.1b About 10 minutes

Every sum is a place to start and a jump to take — and the sign of the second number decides which way you jump.

You get in the elevator on level 2 of a parking garage. You press a button and the elevator moves five floors down. Where do you end up?

Level 3 of the basement. You did not need a rule for that. You knew where you started, you knew how far you were going, and you knew which direction.

That is all addition ever is. These notes make that idea precise so it keeps working when the numbers turn negative.

1. Every sum is a start and a jump

When you see p + q, read it as two separate instructions:

Start at p. That is your position on the number line before anything happens.
Jump q. The size of q tells you how far. The sign of q tells you which way.

Figure 1. Two instructions, never one. Where you start, and how far and which way you jump. The gold dot is the sum.

The rule for the whole page A positive second number means jump right. A negative second number means jump left. The size of the number is how far. Nothing else changes, ever.

2. Warm-up: when both numbers are positive

You have done this since second grade, but do it the new way, out loud: start at 2, jump 3 to the right.

Figure 2. 2 + 3 = 5. Start on 2, jump three to the right, land on 5.

Say it that way even when it feels too easy. The habit is what carries you through the hard cases.

3. Adding a negative number

Now: 2 + (−5). Same two instructions. Start at 2, jump 5 — and because the 5 is negative, jump left.

Figure 3. 2 + (−5) = −3. You started on the positive side, jumped past zero, and landed on the negative side.

Notice you crossed zero on the way. Nothing special happens there — zero is just another tick mark you jump over.

4. Starting in the negatives

The starting number can be negative too. That only changes where you stand, not how you jump. −4 + 6: start at −4, jump 6 to the right.

Figure 4. −4 + 6 = 2. A negative start plus a positive jump can land you anywhere — here it carried you past zero into the positives.

5. When both numbers are negative

This is the case that trips people. −3 + (−4). Do not reach for a rule. Read the two instructions: start at −3, jump 4 to the left.

Figure 5. −3 + (−4) = −7. You were already left of zero, and the jump took you further left. There is no way that lands you on the positive side.

The chips say the same thing. Three reds for the −3, four more reds for the −4. There are no yellows on the table, so nothing cancels. You just have seven reds.

Figure 6. No yellow chips means no zero pairs to sweep away. Seven reds stay on the table.

6. The mistake that costs the most points

Borrowing a multiplication rule and using it on addition

Somewhere you learned that two negatives make a positive. That is a true statement — about multiplication. A student who reaches for it here writes −3 + (−4) = 7, and the answer is not just wrong by a sign, it is on the wrong side of zero entirely.

Figure 7. The wrong version never starts at −3 at all. It throws away both minus signs, starts at zero, and jumps right twice.

Look at what the wrong walk actually does: it ignores where you were standing and turns both leftward jumps into rightward ones. Nothing in the problem said to do any of that.

The check: if both numbers are negative, you started left of zero and moved further left. Your answer must be negative, and it must be further from zero than either number you started with. If it is not, you used a multiplication rule on an addition problem.

7. When the jump lands exactly on zero

You met this in the last lesson. If the jump is the same size as your starting position but pointed the other way, you land on zero — a zero pair.

Figure 8. −4 + 4 = 0. The jump is exactly long enough to undo the start. This is the zero pair from Skill 1.01, seen as a walk.

8. A real situation

Worked example

At dawn the temperature in Big Bear was −6°C. A storm came through and it dropped another 5 degrees. What was the temperature then?

Figure 9. The number line does not have to lie flat. On a thermometer it stands up, and "jump left" becomes "jump down."

Start at −6. The word dropped means the jump is negative, so jump 5 down:

−6 + (−5) = −11

Eleven degrees below zero. Run the check from the last section: both numbers were negative, so the answer had to be negative and further from zero than either one. It is.

9. A look ahead: the jump does not have to be a whole number

Coming up next

Nothing on this page cared that the numbers were whole. You can start at −1.5 and jump 4 to the right exactly the same way — you just land between two tick marks.

Figure 10. −1.5 + 4 = 2.5. Same start, same jump, same direction — the only new thing is that you can stop between the ticks.

That is the next skill: the same walk with fractions and decimals. The thinking does not change at all, which is why it is worth getting this page exactly right now.

Words to know

sum: The result of an addition — where you land after the jump.
absolute value: The distance a number sits from zero, ignoring direction. Written |−4| = 4. It tells you how far to jump.
zero pair: A number and its opposite. They add to zero, because the jump exactly undoes the start.
rational number: Any number that can be written as a fraction of two integers. Every number on this page is one.

What to carry away

Read p + q as two instructions: start at p, jump q.
The size of the second number is how far. Its sign is which way.
Positive jumps go right. Negative jumps go left. That never changes.
Two negatives added together must give a negative — you started left and kept going left.
"Two negatives make a positive" is a multiplication rule. It has no business in an addition problem.

Check your understanding

You are ready to move on when you can answer these out loud, in your own words, without looking back:

What are the two instructions hiding inside p + q?
In −3 + (−4), which number tells you where to stand and which tells you how to move?
Why must the answer to −3 + (−4) be negative, before you compute anything?
In Figure 7, what exactly does the wrong walk ignore?
How is Figure 8 the same idea as the chips you swept off the table in the last lesson?