Defining and Recognizing Constant Rate

Lesson 1 of 2: Definition and Language

In this lesson:

• Define constant rate of change
• Recognize constant rate language in verbal descriptions
• Distinguish constant rate from constant percent rate

What You Will Learn Today

1. Define "constant rate" as a fixed amount added per unit interval
2. Connect constant rate to slope and the equal-differences property
3. Identify constant rate from verbal descriptions using key phrases
4. Distinguish constant rate (linear) from constant percent rate (exponential)

Does the Faucet Add the Same Every Minute?

A faucet fills a tub at 4 gallons per minute.

After 1 minute: 4 gal. After 2 min: 8 gal. After 3 min: 12 gal.

Constant Rate: Same Amount Every Unit

Constant rate = the SAME AMOUNT is added (or subtracted) per unit of the independent variable.

• Minute 1 adds 4 gal. Minute 100 adds 4 gal.
• The rate does not depend on how much is already there.

Verbal form of equal differences over equal intervals.

The Faucet in Function Form

(gallons, = minutes)

Rate = 4 gal/min = slope

Constant Rate Equals Slope Equals Equal Differences

All three say: the output changes by the same fixed amount for every unit increase in input.

When the Rate Is NOT Constant

Runner: starts 8 mph, slows 0.5 mph per mile.

Speed is linear in miles. Distance over time is NOT.

Quick Check: Is the Rate Constant?

For each scenario, decide: constant rate or not? State the rate if yes.

1. A candle burns down 2 cm every hour.
2. A bank account earns 4% interest each year.
3. A train travels 80 miles every hour on flat track.

Think before the next slide.

Rate vs. Value: A Critical Distinction

• Value: what something IS now — "The tank has 200 gallons." → gives
• Rate: how fast it CHANGES — "Drains at 8 gal/min." → gives

"The temperature is 72°" is a VALUE, not a rate.

Key Phrases That Signal Constant Rate

Signal: per · each · every + fixed amount (not a percent)

• earns $15 per hour → linear
• loses 2 lb per week → linear
• costs $0.10 per text → linear
• adds 50 people per month → linear

Classifying Eight Constant Rate Phrases

Key: per/each/every + fixed amount → linear. Percent or factor → exponential.

Amount vs. Percent: The Key Distinction

"5% of what?" — bigger balance → bigger dollar increase.

Example: Percent vs. Amount Side by Side

• City A: adds 300 people per year → linear,
• City B: grows 3% each year → exponential,

City A: constant difference. City B: constant factor.

Quick Check: Classify Three Phrases

1. "A coffee shop earns $12 per customer."
2. "A bacteria colony doubles every 3 days."
3. "A hiker descends 150 meters each hour."

Classify each: constant rate (linear) or not? Identify the key phrase.

Pause before the next slide.

Practice: Classify Six Rate Descriptions

1. Factory: 200 units per day.
2. Investment: grows 8% annually.
3. Car: loses $1,500 each year.
4. River: rises 4 inches per hour.
5. Population: grows by factor 1.2 monthly.
6. Tank: fills at 6 liters per minute.

Constant rate or not? Identify the key phrase.

Answers to Six Classification Problems

1. 200 units/day → constant rate,
2. 8% annually → not constant rate (exponential)
3. −$1,500/yr → constant rate,
4. 4 in/hr → constant rate,
5. ×1.2/month → not constant rate (exponential)
6. 6 L/min → constant rate,

Key Takeaways from Lesson One

✓ Constant rate = same amount added per unit, not same percent
✓ Slope = constant rate; starting value = y-intercept
✓ Key phrases: per, each, every + a fixed amount

Values are not rates — look for "per/each/every"
Percent → exponential; fixed amount → linear
Even with "per," verify the amount is fixed

Coming Up in Lesson Two

Deck 2 covers:

• Writing linear functions from constant rate descriptions
• Identifying rate as slope and initial value as y-intercept
• Interpreting results with correct units
• Evaluating whether a constant rate model is reasonable