Interpreting and Modeling with Lines

Lesson 2 of 2: What m and b Mean

In this lesson:

• Interpret the rate of change and initial value in context
• Run the full modeling cycle: build, interpret, and predict

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Interpret the rate of change in terms of the situation, with units
2. Interpret the initial value as the starting amount at input zero
3. Run the full cycle: identify source, build, interpret, and predict

What Do the Numbers Mean?

Last lesson you built for a phone bill.

A friend asks: "So what does the 0.10 mean? What's the 25?"

Could you answer — in plain English?

Point at the Pieces of the Equation

Maya saves: $120 to start, $35 each week. So .

• Point at the 35: what does it do to her balance each week?
• Point at the 120: what is it before any week passes?

Interpreting the Rate of Change

The rate of change tells you the per-unit change. Say it as:

"For each additional ___, the ___ increases/decreases by ___."

Maya: "For each additional week, the balance increases by $35."

Interpreting the Initial Value in Words

The initial value is the output when the input is zero. Say it as:

"When ___ is zero, the ___ is ___."

Maya: "When zero weeks have passed, the balance is $120 — her opening deposit."

Interpret a Phone Plan Model

Bill is $35 at 100 min, $50 at 250 min. Build, then interpret.

• ; using : . So .
• Rate: "For each extra minute, the bill rises $0.10."
• Start: "At zero minutes, the bill is $25 — the fixed monthly fee."

Interpret a Hiker's Distance Table

Distance from the trailhead vs. hours hiked:

. "For each hour, distance grows 3 miles." But why start at 2?

Quick Check: Say the Rate

A candle model is (height in cm, time in hours).

On your own: write the full rate-of-change sentence, with units.

Interpret a Burning Candle Model

After 2 h it's 10 cm; after 5 h it's 4 cm.

. Burns out when : hours.

Your Turn: Build and Interpret

"A tank is draining. At 1 min it holds 38 L; at 4 min it holds 26 L."

Build , then interpret both numbers in full sentences (negative rate!).

Example: A Delivery Truck (b = 0)

Distance from the warehouse vs. time, through and :

, : . Zero is meaningful — it starts at the warehouse.

The Full Modeling Cycle, Step by Step

You now have every piece. Putting them together:

1. Identify the source (words, table, graph, points)
2. Build
3. Interpret both and in context
4. Predict by substituting or solving

Example: Maya, Start to Finish

models Maya's balance after weeks.

After 8 wks: . Reaches $750 when .

Your Turn: Drain the Pool

Pool water level (inches) over hours — no row:

Build, interpret both, then predict: when is the pool empty ()?

Interpretations That Actually Count as Real

• ✗ "The slope is 3." → ✓ "The savings increase by $3 each week."
• ✗ ", so there's no initial value." → ✓ "It starts at the origin — zero miles out."

A bare number isn't an interpretation. Name the unit and direction.

Quick Check: What Does b Mean?

For the pool, you found .

On your own: write one sentence saying what the 49 means in this situation.

Your Turn: The Whole Cycle

"A plant is 5 cm tall when you get it and grows 2 cm per week."

On your own: build, interpret both numbers, and predict its height after 6 weeks. No hints.

Key Takeaways From This Lesson

✓ Rate of change = a per-unit story, with units
✓ Initial value = the output at zero; means starts at the origin
✓ The cycle: identify → build → interpret → predict

Verify with a known point; read a negative as "decreases"

What Is Coming Up Next Lesson

You can now build a model, read its meaning, and predict with it.

Next: 8.F.B.5 drops the numbers entirely — you'll describe and sketch how any graph behaves, even when no equation exists.