Translating Contexts to Signed Numbers and Opposites

Lesson 2 of 2

In this lesson:

• Translate verbal contexts to signed numbers and back
• Distinguish a sign from a subtraction operation
• Recognize opposites — same distance from zero, flipped sign

What You Will Be Able to Do

By the end, you will:

1. Translate verbal descriptions to signed numbers
2. Interpret a signed number back into context
3. Identify the opposite of any signed number

Mia Opens a Bank Account

Mia opens a savings account with $50.

In the next week, she:

• Deposits her allowance ($30)
• Buys a textbook ($40)
• Pays her phone bill ($60)

How does each transaction change the balance?

Recap: Signing Quantities From Yesterday

You already wrote signed numbers from cue words:

• "Basement ft below ground level" → ft
• "Balloon rises ft above launch" → ft
• "Ocean floor m below sea level" → m

The cue word ('above', 'below') tells you the sign.

Verbal to Signed: Read the Cue Word

The procedure:

1. Identify the context (what is being measured?)
2. Identify the zero (where is the reference?)
3. Read the directional cue word ('above', 'below', 'owe', 'gain')
4. Write the magnitude with the sign that matches

Three Worked Translations From Words to Signs

Each cue word selects the sign.

Signed to Verbal: Name the Zero First

The reverse procedure:

1. Recognize the context from units or framing
2. Name what zero means there
3. State the magnitude with the direction in words

This direction is harder — it tests interpretation, not just reading.

Three Reverse Translations Into Words

• ft elevation → "25 ft below sea level"
• A balance of dollars → "you owe $30" (or "overdrawn by $30")
• °C → "12 degrees above freezing"

Plain English, with direction expressed.

A Sign Is Not a Subtraction

Read as 'negative five', not 'minus five'.

Predict: Speech Says "Ten Below Zero"

A weather forecaster says "ten below zero."

What do you write down?

• A.
• B.
• C.

Pick before advancing.

Speech Drops the Sign; Writing Keeps It

The answer is B: .

• Spoken English often drops the sign — "ten below zero", "down by five"
• The directional word ("below", "down") carries the sign verbally
• When writing, you must restore the sign explicitly

Mia's Running Balance Crosses Zero

Start at , then , then , then (overdraft, all amounts in dollars).

Quick Check: A Diver Descends

Translate to a signed number:

"From the surface, a scuba diver descends ft."

Try, then advance.

Bridge: Deposit and Withdrawal — Opposites?

A deposit of dollars and a withdrawal of dollars are clearly opposite actions.

What about the numbers and ?

Are they opposites too? In what sense?

Every Number Has an Opposite

The opposite of a number is the number with:

• The same magnitude (same distance from zero)
• The opposite sign

The opposite of is . The opposite of is .

Zero is its own opposite — there is no other number at distance 0.

Contextual Opposites Match Sign Opposites

Reverse the action → flip the sign.

Game Score: Signs Beat Two Lists

A team's points: .

Net: .

Compared to listing gains and losses separately, then — same answer, less work.

Your Turn: Track a Scuba Sequence

A diver starts at the surface (0 ft):

• Descends ft
• Ascends ft
• Descends ft
• Ascends ft

Sign each change. Track the running depth. What is the final depth?

Mixed Practice — Both Directions, Plus Opposites

Translate:

1. "Earned $200" → ?
2. °C → ?
3. Find the opposite of
4. Find two opposite pairs in:

Pause and try all four.

Answers and the Common Errors to Watch

1. "Earned $200" → dollars
2. °C → " degrees below freezing"
3. Opposite of is
4. Pairs: and ; (none for or here)

Watch out: speech-dropped sign; reading "−" as subtraction; opposite means same magnitude.

Key Takeaways: Signing the World

✓ Translate both ways: verbal signed
✓ Sign on a single number is not subtraction
✓ Opposites have same magnitude, flipped sign

Watch out:

• Don't drop the sign because speech does
• Don't read "−5" as "minus 5"
• Opposites must have the same magnitude

Coming Up Next: The Number Line

In 6.NS.C.6, you will:

• Place rational numbers on a horizontal number line
• Extend to the coordinate plane (all four quadrants)
• Use "opposite of the opposite" to recover any number

Today's "zero is somewhere; opposites mirror it" makes that work feel inevitable.