Use unit analysis to convert measurements and rates, choose appropriate units for a quantity, and report answers with a level of accuracy suited to the context.

What this topic is asking

The N-Q standards in the framework cover quantities: converting units, working with rates, choosing the right unit, and reporting an answer with appropriate precision. The Grade 10 MCAS embeds these inside contextual problems, so unit work is rarely a standalone question; it is the setup you must get right before the mathematics even begins. The reliable tool throughout is unit analysis (dimensional analysis), where you multiply by fractions equal to 1 so unwanted units cancel.

Unit analysis: cancel your way to the answer

The heart of conversion is multiplying by a fraction equal to 1. Because $12 \text{ in} = 1 \text{ ft}$, the fraction $\dfrac{12 \text{ in}}{1 \text{ ft}}$ equals 1, and multiplying by it changes the units without changing the quantity.

To convert 5 feet to inches: $5 \text{ ft} \times \dfrac{12 \text{ in}}{1 \text{ ft}} = 60 \text{ in}$. The feet cancel because one is on top and one is on the bottom. To go the other way, flip the fraction: $48 \text{ in} \times \dfrac{1 \text{ ft}}{12 \text{ in}} = 4 \text{ ft}$.

The rule that prevents errors: set up the fractions so the unit you are removing appears in the opposite position from where it starts, so it cancels. Then the unit you want is what remains.

Converting rates and compound units

A rate carries two units, and a conversion may need to change both. Chain the unit fractions in one line so each unwanted unit cancels.

When both units change, as in miles per hour to feet per second, stack three unit fractions: one for miles to feet, one for hours to seconds. The structure is the same; only the chain is longer.

Choosing appropriate units and precision

The MCAS sometimes asks which unit best reports a quantity, or whether an answer's precision is sensible. The principles:

• Match the unit to the scale. A room's area is reasonable in square feet, a country's in square miles. A reported speed for a person is sensible in miles per hour or feet per second, not feet per year.
• Round to the context. Money rounds to the cent. A count of objects that must be whole rounds down when asking how many fit or can be made, and may round up when asking how many are needed to cover a demand.
• Do not over-report accuracy. If measurements are given to the nearest tenth, an answer claiming five decimal places is false precision. The answer should not look more exact than the data allows.

For example, fitting 9-inch tiles along a 50-inch wall gives $\frac{50}{9} \approx 5.56$ tiles, but only 5 full tiles fit, so the contextual answer is 5. Conversely, if 5.56 buses are needed to carry a group, you need 6 buses, because a partial bus cannot carry anyone.

Try this

Q1. Convert 3 hours to seconds.

• Cue. $3 \times 60 \times 60 = 10{,}800$ seconds.

Q2. A wall is 140 inches long. How many 8-inch tiles fit across it without cutting?

• Cue. $\frac{140}{8} = 17.5$, so 17 full tiles fit.