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Unit reasoning | Lesson

What is unit reasoning?

Suppose that we want to buy several pounds of snacks but the package only shows the mass in kilograms, or if we want to estimate the time our pizza delivery will arrive when we're told it will take

45

minutes. Unit reasoning helps us make sense of measurements by converting between measurement units and making reasonable estimates.

For the test, we need to know common measurement units for length, time, volume, and mass, as well as how common units are related in both metric and U.S. customary units.

What skills are tested?

Converting measurements from one unit to another
Using unit reasoning to interpret scale drawings
Recognizing appropriate units
Solving word problems involving unit conversions

How do we convert between measurement units?

We can convert measurements from one unit to another using

. To convert from a measurement in unit

A

to a measurement in unit

B

Recall the relationship between units $A$ ‍ and $B$ ‍ in terms of equivalent measurements.
Write the conversion factor, or the relationship between units $A$ ‍ and $B$ ‍, as a fraction. The unit we're converting to ( $B$ ‍) is in the numerator, and the unit we're converting from ( $A$ ‍) is in the denominator.
Multiply the measurement by the conversion factor. The units $A$ ‍ should cancel, leaving us with an equivalent measurement in unit $B$ ‍.

meas. in unit A \times \frac{unit B}{unit A} = meas. in unit B

How many seconds is equivalent to

2

minutes?

There are

60

seconds in

1

minute. Since we're converting from minutes to seconds, we need to write the conversion factor with seconds in the numerator and minutes in the denominator.

2 minutes \times \frac{60 seconds}{1 minute} = 120 seconds

$120$ ‍ seconds is equivalent to $2$ ‍ minutes.

How many feet is equivalent to

72

inches?

There are

12

inches in

1

foot. Since we're converting from inches to feet, we need to write the conversion factor with feet in the numerator and inches in the denominator.

72 inches \times \frac{1 foot}{12 inches} = 6 feet

$6$ ‍ feet is equivalent to $72$ ‍ inches.

Both metric and U.S. customary units appear on the test. The test will provide conversion information for less common unit relationships.

Type	Relationship
Length	$1 kilometer = 1,000 meters$ ‍
Length	$1 meter = 100 centimeters$ ‍
Length	$1 meter = 1,000 milliimeters$ ‍
Length	$1 mile = 5,280 feet$ ‍
Length	$1 yard = 3 feet$ ‍
Length	$1 foot = 12 inches$ ‍
Time	$1 day = 24 hours$ ‍
Time	$1 hour = 60 minutes$ ‍
Time	$1 minute = 60 seconds$ ‍
Volume	$1 liter = 1,000 milliliters$ ‍
Volume	$1 milliliter = 1 cubic centimeter$ ‍
Mass	$1 metric ton = 1,000 kilograms$ ‍
Mass	$1 kilogram = 1,000 grams$ ‍
Mass	$1 gram = 1,000 milligrams$ ‍
Mass	$1 pound = 16 ounces$ ‍

How do we use scale factors?

Architects use blueprints to construct large buildings, and engineers use schematics to build electronics with tiny components. In both cases, the plans are not the same size as the objects, but rather scaled representations of the objects. A scale factor is a relationship we use to translate between representation and reality. For example, each inch on a map represents a certain number of miles in the real world.

Unlike a conversion factor, which uses two equivalent measurements with different units, a scale factor can be chosen when connecting representation to reality. To use a scale factor to convert from unit

A

to unit

B

Determine the relationship between the representation and the real object or situation.
Write the scale factor, or the relationship between units $A$ ‍ and $B$ ‍, as a fraction. The unit we're converting to ( $B$ ‍) is in the numerator, and the unit we're converting from ( $A$ ‍) is in the denominator.
Multiply the quantity by the scale factor. The units $A$ ‍ should cancel, leaving us with a quantity in unit $B$ ‍.

qty. in unit A \times \frac{unit B}{unit A} = qty. in unit B

What are appropriate units?

We've all had to communicate measurements: the wait time before being seated at a restaurant, the distance between two locations, etc. To improve understanding, we use units appropriate for the task.

The most appropriate units are chosen so that:

The units match the types of measurement.
Typical measurements are numbers that are easy to read and understand. We tend to have an easier time reading and understanding large decimals and small integers compared to small decimals and large integers.

If we must also make a reasonable estimate of quantity, the quantity should fall within the typical ranges for what we're describing.

Things to remember

To convert from a measurement in unit

A