Lesson · Signed numbers · about 20 min Pending teacher review

Subtracting integers: add the opposite

Goal: Subtract any two integers by rewriting $a - b$ as $a + (-b)$, and explain why subtracting a negative moves to the right.

Before this: Needs Adding integers (same and different signs) and Opposites (the opposite of $-b$ is $b$).

Step 1 · Launch (≈5 min)

You have $4$ positive chips. Someone takes away 7 positives. You do not have 7 to give — so first add 3 zero pairs (3 positives and 3 negatives), which changes nothing. Now remove 7 positives. What is left?

Now try it as a jump: subtracting $7$ is the same as adding $-7$ — a jump of 7 to the left from $4$.

Step 2 · Practice (≈10 min)

Pick the level that fits today. Hints are allowed — each one is recorded as assisted practice, which never changes an independent status.

Subtract: $\;-4 - 6$

Answer

Subtract: $\;2 - (-9)$

Answer

Subtract: $\;-7 - (-3)$

Answer

Which expression is equivalent to $\;-5 - 8$?

A$-5 + 8$B$-5 + (-8)$C$5 - 8$

Show me how it works — worked explanation

There is only one rule: **subtracting a number is the same as adding its opposite.** $a - b = a + (-b)$. After the rewrite, it is an addition problem you already know.

Rewrite: change "minus" to "plus the opposite". $4 - 7$ becomes $4 + (-7)$. $4 - (-7)$ becomes $4 + 7$.
Then add using jumps or chips: $4 + (-7)$ jumps 7 left from 4 → $-3$. $4 + 7$ jumps 7 right → $11$.
Why it works (chips): taking away a negative chip from a pile is the same as adding a positive chip — the pile moves up by one either way.
Check direction: subtracting a positive moves LEFT; subtracting a negative moves RIGHT.

Step 3 · Wrap-up (≈5 min)

Subtraction is adding the opposite: $a - b = a + (-b)$. That is why subtracting a negative moves you to the right — you are adding a positive. Once rewritten, every subtraction problem is an addition problem.

Step 4 · Independent check

4 fresh problems on this skill, done alone: no hints, no worked solutions, no feedback until the end. This is what moves the skill's status.

You can start the check any time, but the wrap-up is worth reading first.

Parent guide (for the adult)

What to notice

Does your learner rewrite before computing, or try to "just know" the answer?
On $4 - (-7)$, do they treat the two minus signs as one minus ("4 − 7")? That is the key misconception here.
On $-3 - 6$, do they subtract sizes and keep the sign ("−3")?

What to ask

"What is the opposite of the number you are subtracting?"
"Rewrite it as an addition problem first — what does it become?"
"Which direction does subtracting a negative move you? Why?"

Common errors — and a question that separates a slip from a misconception

Treats subtracting a negative as subtracting a positive
Looks like: 4 − (−7) = −3
Ask: "What is the opposite of −7? So what are we adding?" If the rewrite fixes it, it was a slip in reading; if they say "minus a negative is still minus", use the chip story: taking away negatives raises the pile.
Subtracts sizes and keeps the sign on −a − b
Looks like: −3 − 6 = −3
Ask: "Rewrite as −3 + (−6). Are both jumps going the same way?"
Flips the final sign
Looks like: 5 − 9 = 4
Ask: "Start at 5 and jump 9 left. Do you pass zero?"

When to show an example

Show the worked explanation the first time a double-negative problem goes wrong. Have your learner say "add the opposite" out loud for the next three problems.

Complete solutions (keep closed during the independent check)

Show practice solutions

Support

$3 - 5 = 3 + (-5) = -2$.
$-2 - 4 = -2 + (-4) = -6$.
$5 - (-1) = 5 + 1 = 6$.

Core

$-4 - 6 = -4 + (-6) = -10$.
$2 - (-9) = 2 + 9 = 11$.
$-7 - (-3) = -7 + 3 = -4$.
$a - b = a + (-b)$: $-5 - 8 = -5 + (-8)$.

Extension

$-6 - (-6) = -6 + 6 = 0$.
$-11 - (-2) = -11 + 2 = -9$: a drop of 9 degrees.

Repeat or move on?

Move on (this completes the published pathway) when the independent check is 4 of 4, or 3 of 4 with a self-corrected slip. If the double-negative error persists after one review, outside instruction with physical chips is the next step — do not loop retries.