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# Classroom Case Studies, K-2 Part C: Activities That Illustrate Measurement Reasoning (55 minutes)

## Session 10: K-2, Part C

In this part, you’ll look at several short activities that are appropriate for students in grades K-2. As you read through the activities, answer the following questions:

 Questions to Answer: What is the measurement content in the problem? What are the big ideas that you want students to consider and understand? What prior knowledge is required? What later content does it prepare students for? How does the content in this problem relate to the mathematical ideas in this course? What other questions might extend students’ thinking about the problem? What other instructional activities or problems might you use in conjunction with this one to further your content goals?

### Problem C1:

Exploring Capacity

Activity Summary
Students recognize situations that involve capacity and compare capacities of different containers.

Materials Needed:

• Containers of different sizes and shapes (including measuring spoons)
• Water, rice, or sand
• Pictures that illustrate capacity situations

Show students a variety of containers and ask them what types of things we might use to fill the containers. What might we measure using these containers? Next, show students pictures that illustrate capacity situations, such as a bottle of milk, a box, a sack of rice, a fish tank, and a swimming pool. For each picture, ask the students to describe what they could fill the object with. Students will often mention that they could fill the object with a liquid, but encourage them to also consider filling objects with solids, such as sugar or sand. Be sure to show the students objects or pictures of objects that cannot be filled — a square, a rock, a piece of string. You may want to start using the term capacity, which refers to the available space inside a container, in your discussion. But don’t expect your students to become comfortable with this term following just one lesson.

Work with a small group of students at a time, either at the sink or at the sand table. Provide them with a number of containers (use more containers with older students). Then ask them to predict which container holds the most and which holds the least, but do not expect students to be able to determine the greatest capacity merely by looking at the containers. Most students will need to pour materials from one container to another before they can make any sort of prediction.

Following experimentation with many containers, choose three or four containers that are different in height and diameter of base. For example, try to find three or four cans: a short, squat can; a tall, skinny can; and cans that are somewhere in between. Or use three rectangular prisms that differ in height. Have students predict which container holds the most and which holds the least, and then have them use filling (rice, sand, etc.) to put the containers in order from largest to smallest.

After all the groups have had an opportunity to work on the task, conduct a discussion about the results. Ask students to share what they discovered. Which can held the most, and which held the least? Ask the students how they arrived at their conclusions. Did tall cans or prisms always hold the most? What types of containers hold a lot of a particular filling, and what types hold very little? Continue to use the word capacity, and encourage students to talk about the capacity of the cans.

Problem C2

Comparing Distances
Activity Summary
Students indirectly compare distances traveled by toy cars. Older students use nonstandard and standard units to measure the distances.

Materials Needed:

Toy cars and ramp

• Paper tape and scissors
• Links, multilink cubes, or some other object to use as a nonstandard unit
• Inch rulers and yardsticks Note 3

Explain that today the students are going to compare the distances different toy vehicles travel. Have students work with a partner, and have each pair choose a small toy car to use. Each pair of students will release their toy vehicle from the starting line at the top of a ramp and then use a piece of paper tape to measure the distance the car traveled.

If you are working with younger students, you may wish to have them write their names on the end of the lengths. These lengths can then be taped to a bulletin board to make a bar graph. Conduct a discussion about the graph. In particular, ask students to compare the distance their cars traveled. Whose cars went the farthest? Whose cars went the shortest distance? How can we tell which cars traveled farther than Anita’s (pick a distance in the middle of the group) by looking at the graph? Can we tell how much farther one car went than another? Depending on the toy vehicles used, you may find that the heavier cars traveled the greatest distance.

Older students, or those who are ready to use numbers, can determine the distances the cars traveled by using nonstandard units, standard units, or both. Students write the number of units on the tape prior to making the bar graph. When discussing the graph, they can use either the lengths of the tapes and/or the number of units to determine which car went the farthest. Furthermore, if both nonstandard and standard units were used to measure the distances, this is a great time to discuss why the number of units is not the same (e.g., why the car went 65 inches but not 65 links) for both measures.

When students are measuring with units, notice how they approach the task. Do they place units end to end? Do they use iteration of one unit, or do they use rulers and yardsticks to measure? If using nonstandard units, it is easier to use units such as links or multilink cubes that can be connected together. During the measurement process is the perfect time to give students individualized instruction on how to measure accurately and precisely.

### Notes

Note 3

Multilink cubes are cubic units that can be connected together (cubic centimeters and 3/4 cubic inches). These can be purchased from:

Delta Education
80 Northwest Boulevard
P.O. Box 3000
Nashua, NH 03061-3000
Phone: 1-800-442-5444
http://www.delta-education.com

or

ETA/Cuisenaire
500 Greenview Court
Vernon Hills, IL 60061
Phone: 800-445-5985/800-816-5050 (Customer service)
Fax: 800-875-9643/847-816-5066
http://www.etacuisenaire.com

### Solutions

Problem C1

 Answers to Questions In this lesson, students are learning about volume or capacity, making estimates, gaining a physical “sense” of capacity, and getting the physical experience of measuring. A big idea is that the capacity of a three-dimensional object, such as a cylinder, is the measure of how much “stuff” it holds. This lesson is also likely to bring out the challenge students face in understanding important ideas such as conservation and transitivity. This lesson can build on similar experiences with area and prepare students for future work with two-dimensional and three-dimensional figures, such as understanding the difference between them, as well as looking at more concrete ways of working with and measuring volume (perhaps using unit cubes). In the course, volume was explored extensively to a higher degree of complexity than is typical for students at this level. Transitivity, conservation, and units of measurement were explored as well. If the containers have lids, you can turn them on their sides and have students consider the impact of repositioning on capacity. This is especially useful if students think “taller” cans hold the most. Building on this lesson and reconnecting with lessons where rectangles are covered in tiles to determine area, you could have students fill rectangular prisms (or boxes) with centimeter (or other appropriate-sized) cubes.

### Problem C2

 Answers to Questions: In this lesson, students are learning about the concept of length, comparing lengths indirectly, and learning how to measure length with standard and nonstandard units. They are also learning about different units and unit iteration (repetition of the same unit needed to measure) as well as transitivity (measuring by making a paper tape and understanding that it’s equal to the distance traveled). Finally, they are learning about bar graphs and getting some early exposure to data representation. This lesson builds on students’ previous experiences with ordering numbers and deepens their understanding of length or distance. For younger students, experiences such as these prepare them for measuring length with standard tools. This lesson also prepares students for considering dimensional attributes of shapes. Depending on students’ previous experiences, the lesson can either build on or prepare students for work in data display. Mathematics ideas from this course include unit iteration, comparison of measurements using different units, conservation, and transitivity related to measuring lengths. The course also looks at measurement as an approximation and at measurement error. Lastly, it focuses on the ability to distinguish measurable properties and the fact that length, unlike area and volume, is one-dimensional. If students are using numbers, you can ask them questions about how they arrived at their distance. Did they use any counting shortcuts or strategies? Students at this age need many opportunities to develop their understanding of the particular attributes they are comparing or measuring, and they need to practice measuring lengths with both standard and nonstandard units. Measuring distances that they create themselves can be very motivating to students. For example, they can measure the length of their stride or how far they can jump. Or they can measure how far they can walk in a set amount of time.