A B C

Solutions for Session 10, Part C

See solutions for Problems: C1 | C2

Problem C1

a.

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.

b.

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).

c.

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.

d.

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.

e.

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

a.

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.

b.

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.

c.

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.

d.

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?

e.

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.