Measuring Objects Twice with Different Units - Second Grade Mathematics

One of the most fascinating discoveries in measurement is that the physical length of an object never changes, yet the number you write down to describe that length depends entirely on the size of the unit you choose! If you measure a wooden pencil using large paperclips, you might need 4 paperclips; but if you measure the exact same pencil using small paperclips, you will need 8 paperclips. In this chapter, you will explore the inverse relationship between unit size and unit count, discover why measuring twice with different units confirms physical constancy, and understand why smaller units always yield larger counts.

The Inverse Unit-Size Principle

The central law of measurement states: the smaller the unit of measurement, the more units are required to span the length of an object.

Measuring the Same Length Twice:

Object Length:  [==============================================]

Using INCHES (Larger Unit):
Units:          [  1 inch  ] [  1 inch  ] [  1 inch  ] [  1 inch  ]
Count:          4 INCHES

Using CENTIMETERS (Smaller Unit):
Units:          [1cm][1cm][1cm][1cm][1cm][1cm][1cm][1cm][1cm][1cm]
Count:          10 CENTIMETERS

The object did NOT stretch or shrink!
Larger Unit (Inches)  ======>  FEWER Units Needed (4)
Smaller Unit (Centimeters) =>  MORE Units Needed (10)

Why the Count Changes

Think of walking across a room: If an adult takes giant, long strides (large units), the adult might cross the room in only 8 steps. If a toddler takes tiny, short steps (small units), the toddler might need 24 steps to cross the exact same room! The room did not get bigger or smaller; because the toddler's steps are shorter, it takes more of them to cover the distance.

Comparing Inches and Centimeters on the Same Object

Let us measure common classroom objects twice using both standard customary inches and metric centimeters.

Double-Measurement Experiment:
Object:             Measured in Inches:      Measured in Centimeters:
Pencil              7 inches                 18 centimeters
Math Workbook       11 inches                28 centimeters
Standard Index Card 5 inches                 13 centimeters
Desktop Width       24 inches                61 centimeters

The Ratio Observation

Notice the mathematical pattern across all rows of the table: In every single case, the centimeter number is more than DOUBLE the inch number! Why? Because 1 inch is equal to approximately 2.5 centimeters. Since each centimeter is less than half the length of an inch, you need more than twice as many centimeters to cover the exact same physical length.

Measuring with Feet vs. Yards

The inverse relationship applies equally when comparing feet and yards.

Measuring a Classroom Rug:
Length: [=======================================================]
Measured in Feet:   9 Feet   (Foot is a medium unit)
Measured in Yards:  3 Yards  (Yard is a large unit: 1 yard = 3 feet)
Equation: 9 feet / 3 = 3 yards!

Predicting Counts Before Measuring

Before you pick up a measuring tool, you should always predict the direction of the count: If you switch from feet to inches: the unit gets smaller, so the count must get LARGER! If you switch from centimeters to meters: the unit gets larger, so the count must get SMALLER!

Practice Exercises

Exercise 1: An index card measures 6 inches long. When measured in centimeters, will the measurement number be greater than 6 or less than 6? Explain why.

Exercise 2: A student measures the length of a bulletin board as 12 feet. If the student measures the exact same bulletin board in yards, will the number of yards be more or fewer than 12? State the exact measurement in yards.

Exercise 3: A student measured a toy car with two different paperclips: giant paperclips and mini paperclips. The student got counts of 3 and 7. Which count goes with the giant paperclips, and which count goes with the mini paperclips?

Exercise 4: Explain why measuring an object twice with different units does not change the physical size of the object itself.

Solutions and Explanations

Solution to Exercise 1: The measurement in centimeters will be greater than 6 (it will be about 15 cm). Centimeters are smaller units than inches. Because each centimeter spans a shorter distance, it requires a greater number of centimeters to cover the exact same physical length.

Solution to Exercise 2: The number of yards will be fewer than 12. A yard is a larger unit than a foot (1 yard = 3 feet). Because the unit is larger, fewer units are needed. Calculating: 12 feet / 3 = 4 yards.

Solution to Exercise 3: The count of 3 goes with the giant paperclips, and the count of 7 goes with the mini paperclips. Giant paperclips are longer (larger units), so fewer of them are needed to span the car. Mini paperclips are shorter (smaller units), so more of them are required.

Solution to Exercise 4: An object's physical size is a fixed, permanent attribute that exists independently of human measuring systems. Units of measurement are simply arbitrary reference sticks created to describe that physical length. Changing from inches to centimeters changes the scale of the description, but leaves the physical object completely unchanged.