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Refresher · Signed numbers · about 12 min Pending teacher review

Opposites and absolute value

Goal: Find the opposite of any integer, evaluate expressions like $-(-3)$ and $|{-5}|$, and explain both as positions and distances on the number line.

Before this: Builds on Number-line meaning (distance from 0).

Step 1 · Launch (≈5 min)

Two players stand the same distance from the goal line, one on each side: at $4$ and at $-4$.

They are opposites — same distance from 0, opposite sides. The opposite of $4$ is $-4$; the opposite of $-4$ is $4$.

Absolute value is just the distance: $|4| = 4$ and $|{-4}| = 4$. A distance is never negative.

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.

Find $\;-(-5)$

Answer

Find $\;|{-12}|$

Answer

Find $\;-|{-3}|$

Answer

Which numbers have an absolute value of $9$?

AOnly $9$BOnly $-9$CBoth $9$ and $-9$

Show me how it works — worked explanation

Both ideas are about distance from 0. **Opposite** keeps the distance and flips the side. **Absolute value** keeps the distance and drops the side.

Opposite: $-(\,x\,)$ means "the opposite of $x$". The opposite of $-3$ is $3$, so $-(-3) = 3$.
Absolute value: $|x|$ is how far $x$ is from 0. $|{-5}| = 5$ and $|5| = 5$.
Nested expressions: work from the inside out. $-|{-4}|$: first $|{-4}| = 4$, then the opposite of 4 is $-4$.
Check with the number line: an opposite is a mirror image; an absolute value is a count of steps.

Step 3 · Wrap-up (≈5 min)

The minus sign in front of a number can mean "the opposite of". The opposite of a negative is positive: $-(-3) = 3$. Absolute value measures distance, so it is never negative: $|{-5}| = 5$.

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 read $-(-3)$ as "negative negative three" (confusing) or "the opposite of negative three" (helpful)?
Do they keep a minus sign inside absolute value bars?
On $-|{-4}|$, do they work inside-out?

What to ask

"What does the first minus sign mean here?"
"How far is that number from zero? Can a distance be negative?"
"Which part do you do first?"

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

Treats $-(-3)$ as $-3$
Looks like: Writes −(−3) = −3
Ask: "Where is −3 on the line? Where is its opposite?" Self-corrects → slip. Says "two minuses, still minus" → misconception; use the mirror picture.
Keeps the sign inside absolute value
Looks like: |−6| = −6
Ask: "How many steps from −6 to zero?"
Drops the outside minus after the bars
Looks like: −|−4| = 4
Ask: "You found |−4| = 4. What does the minus in front of the bars do to that 4?"

When to show an example

Show the worked explanation the first time $-(-a)$ goes wrong; the mirror picture usually settles it in one look.

Complete solutions (keep closed during the independent check)

Show practice solutions

Support

The opposite of $6$ is $-6$.
The opposite of $-2$ is $2$.
$|{-7}| = 7$ because $-7$ is 7 units from 0.

Core

$-(-5)$ means the opposite of $-5$, which is $5$.
$|{-12}| = 12$.
Inside first: $|{-3}| = 3$. Then the opposite: $-3$.
Both $9$ and $-9$ are 9 units from 0.

Extension

Inside: $-(-8)=8$; then the opposite of $8$ is $-8$.
$|{-6}| = 6$, $|2| = 2$, so $6 - 2 = 4$.

Repeat or move on?

Move on when the check is clean. This skill is short — repeating the core set once is normally enough.