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## Makeshift Measurements

• Lesson
3-5
1

There are many common objects that come in standard sizes. Students will verify measurements for some of these objects, use these objects to determine the size or spacing of other objects, and suggest additional objects that can be included in the class’ list of "makeshift measures."

Several days before the lesson, prepare the Measurement Task Cards. In addition to the situations described on the cards, you may wish to add some real life examples of your own. Copy the task cards onto heavy paper, cut them apart, and put them in a container.

The day before using this lesson in class, display a few common objects that come in standardized sizes. Possibilities include coins, dollar bills, credit cards, envelopes, index cards, DVDs, baking pans, and floor tiles. For the next day, ask students to bring in some common items that are always the same size. For instance, you might want to show them a standard letter envelope and say, "This is a standard envelope. It measures 4‑1/8 inches by 9‑3/8 inches. In fact, every standard envelope measures this same size." Explain that you want them to bring in other objects that have this same property.

You may want to send a note home to parents explaining the activity that students will be doing in class. Parents can help their students find a few items that could be used.

Begin the lesson with students describing the objects that they brought in, and have them share observations about the objects that other students brought to class. As students share their thoughts, emphasize the idea of size.

Ask students what they think today’s math lesson might be about, given the objects that they have been discussing. When a student suggests measurement, pass out rulers and direct students to find the dimensions (length and width) of their items. Students should record the name and measurements of each object on the Measurement Record Sheet. Make sure that all students have some items in common to measure. For instance, if some students brought in an index card, then distribute index cards to the other students, too. That way, they will have some items in common for later discussion.

After about 15 minutes, have students compare their data with other students. This can be whole‑group sharing with the teacher or students recording the information on a large chart, or it can be partners working together to check each others’ work.

Assemble the whole class and tell students that everyone’s measurements for the common items should be just about the same because these objects (and others) come in standard sizes. At this time, check to see if this is so. Have students explain and resolve any discrepancies.

Explain the term standard size, and ask students why they think these things come in a particular size. Students will probably think of reasons that apply to their lives. Be sure to suggest the following ideas if they don’t come up:

• For convenience, all objects should be the same size.
• Some items must work with other objects. For instance, a standard envelope is made to hold a sheet of folded paper, so it needs to be the correct size to do this.
• Manufacturing is more economical if machines make large quantities of a few sizes rather than lots of different sizes.
• Stores have limited display areas, and it’s easier to display things that are the same size. Also, it’s easier to store items (in a warehouse or store room, for example) that are the same size.

Ask students if they can think of any other common objects that come in standard sizes. Chart the objects and their measurements. Additional objects can be added as ideas arise.

Tell students that the mathematics people use most often are estimation and number sense. That is, people usually deal with reasonable numbers, and they don’t always need exact numbers. It also frequently happens that when someone needs to measure an item, the appropriate tool is not handy. Ask students to describe a situation when this might happen, and allow them to offer suggestions. Possibilities might include:

• Will a piece of furniture fit in the space available for it?
• Is a frame large enough for a painting or photograph?
• Will a gift fit into a gift box?
• Will a rug be too big or too small for the floor it needs to cover?

Ask students to explain how they might handle each of the situations they suggested. How could they get a good estimate if they didn’t have a ruler, tape measure, or other measuring device? Students should suggest that people could use items for which they know the size to make an estimate. For example, most adults carry credit cards, their driver’s license, and keys with them, and everyone always has their fingers and their feet with them. A person might estimate the dimensions of a floor by counting the number of floor tiles or by walking heel-to-toe and then converting the result to inches or feet.

Refer to the chart of common objects from earlier in the lesson. Take the rulers from students, and have them determine the length of some items in the classroom, such as a pencil or the top of a desk. On their own, students should realize that they can use the reference items from the chart to make good approximations. For instance, they might use a credit card to estimate the length of a pencil. Remind students that estimation is important to this lesson, so the computations can be approximate: "The pencil is roughly three credit cards long, or about ten inches."

After students have estimated the lengths of several items, discuss the results with the class. For each measurement, ask the class to indicate if the answer is reasonable (thumbs up) or not reasonable (thumbs down). Ask a few students to justify their answers.

Divide students into pairs. Put the Measurement Task Cards in a bag or basket. Students should work together for the following Think-Pair-Share activity. Carefully explain the following process before students get started:

• Partners will find a good place in the classroom to work together.
• Pairs will pick one task card, and they will read the task out loud.
• The teacher will signal the beginning of Think, which is one minute for students to individually ponder the problem and consider possible strategies for solving it.
• The teacher will then signal the beginning of Pair, during which partners have a conversation about ways to estimate the length using common items. Students should understand that both partners should get a chance to share ideas during this five‑minute period.
• The teacher will signal the beginning of Share, which can either be done as an entire group, or one pair of students can share with another pair.

You may wish to use non‑verbal signals such as a bell or other sound, a change in lighting, or American Sign Language for think, pair, and share.

Note that there are plenty of task cards. The cards can be replaced in the container, and the activity can be repeated several times.

The lesson can end with students sharing their ideas from Think-Pair-Share, with all students completing the first Assessment Option below, or with students writing an answer to one of the Key Questions in their journals.

Assessment Options

1. Have the students reflect upon their learning by doing a self assessment.
• Three things I noticed during this lesson were…
• Two things I wondered about during this lesson were…
• One thing that could be added to our chart of referents is…
2. Ask students to consider other tasks that could be added to the set of Measurement Task Cards. By presenting some of the student ideas to the class, this could lead to an interesting discussion: "Here is another idea for a makeshift measurement card. Do you think it’s a good one? Why or why not?" During this discussion, do not mention student names. The discussion should focus on the mathematics. Allow the author of the task card to modify the task privately, if appropriate, after the discussion.
3. Students should select a classroom object and determine its approximate size by using one of the makeshift measures.The teacher should make note of each students’ skills. Does she arrive at a reasonable answer? Did he select an appropriate item to use? Does she use the reference object correctly? This assessment option can be done individually or in small groups while the rest of the class is engaged in another activity.

Extensions

1. Have students search for other objects that come in standard sizes and add them to the chart. Students could do an online search at an office store (such as Office Depot or Staples) to find common items. Many office supplies are even called by their sizes, such as 10" × 13" envelopes.
2. Have students pick a category of objects that come in standard sizes (baking pans or sticky notes, for example). Create a poster that could teach other students how to use them as makeshift measures. Display the posters in the hallway or other visible spot in the school.
3. Have students make wallet-sized reference card that list the dimensions of some of the common objects. Give these cards to friends and family.
4. Use common objects to estimate the area of objects, such as rugs, floor tiles, and game boards.
5. Students could research the story of the Smoot Bridge at MIT. A student whose last name was Smoot had some friends measure the length of a bridge by picking him up and laying him down across the entire span. The project got so much recognition that they renamed the bridge after him. Students could create a variation of this story that uses themselves and their friends to find the length of an object of their own choosing.

Questions for Students

1. Are your makeshift measures reasonable? How do you know?

[Students might justify their results using visualization (e.g., "I can imaginethat the pencil is about as long as three credit cards"), or comparison (they got a different answer than a friend.]

2. What other objects could be used to estimate measurements? Why would they be useful?

[Students could suggest many objects, such as a new pencil ("because there’s always one in my desk"), a bicycle ("because it stays the same size and is easy to move"), or paper clips ("because they are lightweight, of a standard size, and we have lots of them").]

3. Which is more useful as a referent, body parts or the objects we used today? Why?

[Students might suggest that either is more useful for different reasons. For instance, they may say that body parts are more useful, because you always have them with you, or they may say that objects are more useful because their size stays the same. Either answer could be correct, depending on the student’s justification. Students should realize that the answer depends on the item being measured and how accurate you have to be.

Teacher Reflection

• How did students demonstrate understanding of the materials presented?
• Was students’ level of enthusiasm/involvement high or low? Explain why.
• How did you lesson address auditory, tactile, and visual learning styles?
• What were some of the ways that the students illustrated that they were actively engaged in the learning process?

### Learning Objectives

Students will:

• Accurately measure dimensions of common objects collected by the teacher.
• Use standard size objects as references to determine sizes or spacing of other objects.
• Generate a list of additional objects that could be used as referents in measuring length.

### NCTM Standards and Expectations

• Select and use benchmarks to estimate measurements.