Equations and Graphs of Proportional Relationships

Lesson 2 of 2

In this lesson:

• Write and use equations of the form
• Interpret and on graphs of proportional relationships
• Translate fluently among tables, graphs, equations, and verbal descriptions

What You Will Learn in Lesson 2

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

3. Write and explain what means in context
4. Interpret points on proportional graphs, especially and
5. Translate fluently among tables, graphs, equations, and descriptions

Writing Captures Everything

You know — write one equation to describe the entire relationship.

• alone: "$3.50 per gallon"
• : "total cost = $3.50 × gallons" for any number of gallons

Example 1: Factory Widget Production

Factory: 150 widgets/hour → , equation:

Forward: How many widgets in 8 hours?

Reverse: Hours to make 600 widgets?

Using in Both Directions

Widget examples: ·

Example 2: Vanilla Recipe ()

Vanilla: 1.5 tsp per batch → , equation:

Quick Check: Write the Equation from a Table

Find and write the equation using and .

Try it before the next slide.

Answer: Distance Equation

Verify: ✓ · ✓

Example 3: From Table to Equation

Currency exchange — find , write the equation.

→ equation:

Your Turn: Kilometers to Miles

1. Check all ratios — is it proportional?
2. Write the equation ( for miles, for km)
3. Find miles when

Equations and Graphs Are the Same Relationship

• is the slope — larger means steeper line
• The line always passes through
• Graph and equation are two views of the same relationship

Paint Cost Graph: Reading Every Point

The line tells a story. What does each point mean?

The Origin : Zero In, Zero Out

: zero gallons cost $0 — no base fee, no minimum charge.

• A fixed starting cost would break proportionality
• Required: the line must pass through

Reading the Unit Rate Point

: 1 gallon costs $28 — this is .

• At : — the unit rate is visible
• : same number as slope and constant of proportionality
• To find : locate , read off

Reading Any Point in Context

On the paint graph: what does mean?

• : 5 gallons purchased
• : total cost is $140
• Always read both coordinates and state what they mean together

Reading a point means telling the story of that purchase.

Non-Proportional Contrast: Line Misses Origin

Straight line — but misses . At : (fixed cost). Not proportional.

Quick Check: Which Graph Is Proportional?

Which one is proportional — and why?

1. Straight line through
2. Straight line with -intercept at
3. Curved line through the origin

Pick one and justify.

Answer: Only Graph 1 Is Proportional

Your Turn: Read This Proportional Graph

Line passes through and — speed in km per minute.

1. State and its meaning
2. Write the equation ( and )
3. What does mean?
4. Find when

Key Takeaways for Lessons 1 and 2

✓ works both ways: multiply for , divide for

✓ : zero in → zero out; required for proportionality

✓ : read at ; not

Proportional = straight line through origin only

Preview: Applying Proportional Relationships (7.RP.A.3)

Coming up: Using to solve real-world problems

• Percent problems: tax, tip, discount, percent change
• Multi-step ratio and rate problems
• Scale drawings and maps

All of 7.RP.A.3 builds directly on these two lessons.