Private: Learning Math: Measurement
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:
Students recognize situations that involve capacity and compare capacities of different containers.
- 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.
Students indirectly compare distances traveled by toy cars. Older students use nonstandard and standard units to measure the distances.
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.
Multilink cubes are cubic units that can be connected together (cubic centimeters and 3/4 cubic inches). These can be purchased from:
80 Northwest Boulevard
P.O. Box 3000
Nashua, NH 03061-3000
500 Greenview Court
Vernon Hills, IL 60061
Phone: 800-445-5985/800-816-5050 (Customer service)
|Answers to Questions
|Answers to Questions:
Session 1 What Does It Mean To Measure?
Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.
Session 2 Fundamentals of Measurement
Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.
Session 3 The Metric System
Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.
Session 4 Angle Measurement
Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.
Session 5 Indirect Measurement and Trigonometry
Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.
Session 6 Area
Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.
Session 7 Circles and Pi (π)
Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.
Session 8 Volume
Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.
Session 9 Measurement Relationships
Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.
Session 12 Classroom Case Studies, 6-8
Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.