## How Long? How Wide? How Tall? How Deep?

- Lesson

In this lesson, students use historical nonstandard units (digits, hand, cubit, yard, foot, pace, fathom) to estimate the lengths of common objects and then measure using modern standard units. They will discover the usefulness of standardized measurement units and tools.

Many students have not had enough experiences with nonstandard units and therefore have an incomplete understanding of measurement. This lesson provides more of these experiences as well as a bridge into familiar standard units of measuring length. Interested teachers could also connect this lesson to information about measurement in many ancient cultures.

To begin the lesson, read *How Big Is a Foot?*
to students. This amusing story tells of a king who wants to have a bed
made just the right size for his queen. He measures her width and
length with his king-size feet. The job of building the bed falls to a little apprentice who
carefully uses the king's dimensions, but uses his little feet as the
unit. Students enjoy explaining why the bed turns out to be too small
for the queen and posing solutions to the dilemma.

Explain to the students that, although this is a fictional story, it is based upon fact. Our standard unit of measure, the foot, actually did come from making a model of a king's foot; and the standardized tool became known as a "ruler." Show a ruler so students can imagine a king's foot.

Have each student trace around his or her shoe on construction paper and cut out about six of these paper feet. Tape them heel to toe. Let the students use this new "six-foot" measure to find and record the length of common objects around the room.

After about ten minutes, lead the class in a discussion, comparing their measurements. Chart the data to use as a visual reference. Ask questions that help students compare their findings, for example:

- Who measured the height of the desk? What did you find?
- Who found a different measurement for the height of the desk?
- Why do you think it was different from ____'s?
- Is the desk really taller for ____ than for ____?

Show the students a variety of rulers (wooden, plastic, metal). Ask, does anyone have an idea about why we use rulers instead of paper feet taped together? Enjoy the idea-sharing! Note levels of thinking, reasoning, and creativity.

Then, explain that *inches *began in medieval England and were
based upon the width of the human thumb. Thumbs were excellent
measuring tools because even the poorest individuals had them available
when they went to market.

Ask students to draw, along the edge of their construction paper, a line equal to the width of their thumbs. Cut the edge off the paper (about an inch wide), and accordion-fold the strip to show 12 student "inches."

Have students compare the length of their 12 inches to the tracing of their shoes. Share observations. (Note: 12 student inches should be about the same as 1 student foot.) Explain that body measurements were probably the most convenient references for length measurement long ago.

Distribute the Body Parts Activity Sheet. Define, model, and have students repeat each of the body measurements on the chart.

With partners, have students measure and record the lengths of their own digits, hands, cubits, yards, and fathoms.

After about ten minutes, call students together to discuss the term "cubit." The cubit was devised by the Egyptians about 3000 BC, and is generally regarded as the most important length standard in the ancient Mediterranean world. The Egyptians realized that a standardized cubit was necessary in order for measurements to be fair, so a master "royal cubit" was made of black granite. The present system of comparing units of measure with a standard physical tool (such as a ruler or yardstick) follows directly from this Egyptian custom.

Ask for a volunteer and attempt to measure his or her height using your forearm (cubit). Ask for solutions to the difficulty and awkwardness. [One solution should be to make a model that is the length of your own cubit.] Direct students to make a model of their cubits using either string, ribbon, adding machine tape, or interlocking cubes. Have partners check for accuracy.

Have students duplicate their cubit models and use them to estimate, measure, and record the height of several classmates. At the end of the activity (about ten minutes), have students share ideas of which models worked best for measuring height.

### Reference

Myller, Rolf. How Big is a Foot? Reprint ed. New York, NY: Yearling, 1991.

- String, ribbon, adding machine tape, interlocking cubes
- Tools for measuring length (rulers, yardsticks, retractable and folding measuring tapes, trundle wheels)
- Construction paper
*How Big Is a Foot?*by Rolf Myller- Body Parts Activity Sheet

**Assessment Options**

- Collect the Body Parts Activity Sheet. Note whether data was complete and reasonable.
- Have students record answers to key questions in math logs. Note whether students were able to explain their thinking or had insights about the mathematics.

**Questions for Students**

1. What did you learn, notice, or wonder about when measuring with nonstandard units (body parts)?

[Students may note that it was tricky using one unit over and over again, or that they got different answers each time they measured. They may even say using a ruler is better because it's not as embarrassing as a cubit!]

2. What were some interesting words (vocabulary) you used in this lesson?

[Possible answers: cubit, apprentice, standardized, and ruler (as another name for "King").]

3. Why is it important to estimate before actually measuring?

[To make your answer reasonable, to catch errors.]

4. Explain, in your own words, why standardized units and tools are important when measuring.

[So you get the same answer every time, other people will get the same answer as you, and so all projects turn out the same.]

5. Can you ever get an exact measurement of length? Why or why not?

[You can get closer and closer, but you'll never get an exact measurement. Tools and units can get very accurate, but things you're measuring might be floppy or squishy.]

**Teacher Reflection**

- How did the students demonstrate understanding of the materials presented?
- What were some of the ways that the students demonstrated that they were actively engaged in the learning process?
- What worked with classroom behavior management? How would you change what didn’t work?

### Learning Objectives

Students will:

- Become familiar with the language/vocabulary of measurement.
- Gain an understanding of measuring length by estimating, making comparisons, handling materials to be measured, and measuring with tools.
- Understand that all measurements are approximations.
- Understand the need for measuring with standard units.

### NCTM Standards and Expectations

- Understand such attributes as length, area, weight, volume, and size of angle and select the appropriate attribute.

- Understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems.

- Understand that measurements are approximations and how differences in units affect precision.

- Select and apply appropriate standard units and tools to measure length, area, volume, weight, time,

### Common Core State Standards – Mathematics

Grade 3, Measurement & Data

- CCSS.Math.Content.3.MD.B.4

Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units-- whole numbers, halves, or quarters.