Choosing Direction

Lesson 3 of 3: Context Drives the Unit Rate

In this lesson:

• Decide which direction to compute from a given ratio
• Work with like-unit ratios (scale factors)
• Work with area and mixed-unit contexts

What You Will Learn in Lesson Three

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

1. Choose the direction for the unit rate
2. Set up the complex fraction
3. Compute and simplify
4. Explain what the rate means, with units
5. Apply this to lengths, areas, and like units

One Ratio Yields Two Unit Rates

Pipe: gal in min — two valid rates:

Read the Answer Unit First

The routine:

1. Read the question — identify what you want per what
2. Write the answer unit as a fraction: "miles per hour" → miles/hour
3. Put the A-quantity in the numerator, B-quantity in the denominator
4. Set up the complex fraction to match, then compute

Two Directions: The Printing Press

Same ratio — same context — two different unit rates.

Worked Example: Both Directions Computed

Press: pint per page — compute both rates:

Pints per page:

Pages per pint:

Quick Check: Choosing the Right Rate

A manager needs to order ink for a 500-page edition.

Which rate do you multiply by 500 to get pints of ink?

Think: what are the units of "500 pages × ___" that gives pints?

Answer: Pints per Page Is Correct

Multiply 500 pages × pints-per-page rate — the "pages" cancel:

Wrong choice: 500 × pages-per-pint gives "pages² per pint" — units that make no sense.

Like Units: When Units Cancel

Scale = actual feet per model foot ( ft model = ft actual):

Like Units: The Scale Factor Diagram

• Both quantities measured in the same unit → units cancel in the result
• The result is a scale factor — a pure multiplier, no unit label
• Applies to: maps, architectural plans, similar figures

Worked Example: Cost per Square Foot

A floor tile covers sq ft and costs dollar.

Cost per square foot:

Practice: Choose Direction Then Compute

Identify the answer unit before computing.

1. A hiker covers mile in hour. Speed in mph?

2. A map: inch represents mile. Scale in inches per mile?

3. Carpet: sq yd costs dollars. Cost per square yard?

Answers to Three Direction Problems

1. mph: mph

2. in/mi: inch per mile

3. $/sq yd: dollars per sq yd

Three Key Takeaways from Lesson Three

✓ One ratio → two unit rates (reciprocals of each other)
✓ Read the answer unit first — it sets the numerator
✓ Like units → dimensionless scale factor

Wrong direction gives the reciprocal — check magnitude

Always label units — "9/10 gal/min," not "9/10"

Series Complete: You Can Now Do Three Things

• Set up any fractional unit rate — fraction ÷ fraction
• Simplify complex fractions — reciprocal, cancel, multiply
• Choose direction — read the answer unit first

Coming up in 7.RP.A.2:
The unit rate is also the constant of proportionality in .