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# What Does It Mean To Measure? Part A: Comparing Rocks (15 minutes)

## Session 1, Part A

Measurement is used in all aspects of daily life, as well as in such fields as engineering, architecture, and medicine. We measure things every day. This morning you may have weighed yourself, poured two cups of water into the coffeemaker, checked the temperature outside to help you decide what to wear, cut enough gift wrap off the roll to wrap a present, decided on the size of a storage container for some leftover food, noted on your car’s odometer how far you’d driven, monitored both your car’s traveling speed and its gas gauge, and kept an eye on the time so that you wouldn’t be late.

All of the situations above are easily identifiable as measurement situations. Yet what is at the heart of all of these comparisons? In other words, in order to measure, what must we consider, and then what steps must we take? Note 2

To begin thinking about measurement, you will use, of all things, a rock.

### Problem A1

Make a list of attributes that could be used to describe the rock.

### Problem A2

Some of these attributes might be measurable, and some might not. How do we determine what we can measure?

If you are having difficulty sorting the attributes, consider which attributes can be quantified. For example, the texture of a rock (e.g., smooth, bumpy, rough) isn’t quantifiable using any of the standard units we know; in contrast, the weight of the rock is quantifiable and can be measured in ounces or grams.

Another suggestion is to see what happens when you combine your object with another, similar object. If the attribute is measurable, then it will increase when the objects are combined. For example, when you combine two rocks, the texture won’t increase or change in any way, but the weight certainly will.

### Problem A3

If you were to compare different rocks using each of the measurable attributes you listed in Problem A1, what units would you use?

### Problem A4

How could you measure these properties?

Think about what instruments, devices, or methods you might use.

### Notes

Note 2

Though we all use measurement daily, most adults have not considered the properties of an object that make it measurable in some way. This first activity is designed to focus attention on the many attributes that are used to compare objects (in this case, rocks) and to sort the attributes into two categories — those that are measurable and those that are not.

### Solutions

Problem A1

Answers will vary. Some answers might be the rock’s length, surface area, volume, weight, color, and texture.

Problem A2

A measurable property is a property that can be quantified using some kind of unit as a basis. For example, length is measurable, since there is a unit of length (an inch, a centimeter, etc.) and we are counting or measuring the number of units in our object. A non-measurable property is one without a standard unit. When we combine objects with a measurable property, the property must increase.

If we wanted to measure some of the properties not commonly measured, we would have to invent a method to do it. For example, to measure texture, we could look at the curvature over the small areas of the object; if the curvature doesn’t change much, we could say that the texture of the object was smooth. Some interesting modern research in mathematics focuses on such “nonstandard” measurements.

Problem A3

Answers will vary. For those listed in our solution to Problem A1, we can measure length in centimeters, surface area in square centimeters, volume in cubic centimeters, and weight in grams.

Problem A4

Answers will vary. We could measure the length of the rock using a ruler or tape measure. We could measure the weight using a scale, the volume using a beaker of water (for displacement), and the surface area using tinfoil.