Measurement: Perimeter and Area in Grades 3 - 5
Session 3: Finding Areas and Perimeters of Complex Shapes
Objectives
 In this session, you will: Find the areas of complex shapes by cutting them into recognizable smaller units of area, such as units and half-units; Explore the relationships among shapes such as squares, rectangles, and triangles; Determine the perimeters of regular figures and of complex shapes; Determine the relationship between area and shape; Analyze and assess student thinking about the concepts of area and perimeter.

The following reading contains excerpts drawn from Issues in Mathematics Education: The Mathematical Education of Teachers, which discusses some current thinking about critical content, pedagogy, and policy related to the professional development of mathematics teachers. The reading focuses on geometry and measurement in elementary classrooms.

This assessment and sampling of student work from the Investigations in Number, Data, and Space curriculum unit "Flips, Turns, and Area" illustrates a range of student work dealing with the areas of complex shapes.

Activities
1. Do the "Going Around" activity with the pattern blocks virtual manipulatives to determine the perimeters of complex shapes.
2.

Do the "Exploring Area with Online Manipulatives" activity with the tangram virtual manipulatives to determine the areas of complex shapes.

3. View samples of student work from the "Exploring Area with Online Manipulatives" activity.
4. In your Learning Log, respond to the following:
• Review the assessment and the samples of student work from the Investigations in Number, Data, and Space curriculum unit "Flips, Turns, and Area."
• Explain how you would help Mark communicate more clearly about how he found the area of his shape.

• It is unclear whether Laurie Jo really understands the concept of how to use squares and triangles to measure the area of her shape. Write three questions you might pose to her to determine whether or not she has a clear understanding of the concept.
5. Use the Digital Drop Box to send your Learning Log to your facilitator. (Your Learning Log should include entries from Sessions 2 and 3). If you did not already do so, please print out the Digital Drop Box instructions and read the "Uploading Files to the Digital Drop Box" section. Alternatively, you can email your Learning Log, as an attachment, to your facilitator.
6.

In the following videos, students determined the areas of simple and complex figures. They did two activities that provided them with opportunities to use concrete manipulatives to compose and decompose shapes. Before doing these activities, students used virtual tangrams to construct shapes without determining their areas. They also explored the relationships among the tangram pieces.

• Watch the "Finding Areas of Tangram Pieces" video to observe students determining the area of each of the tangram pieces using the square tangram as the unit of measurement (one square unit).
• Click "Tangram Areas I" to see samples of student work from this activity.
7.
Watch the "A Common Misconception" video to observe one student's thinking as he tries to determine the area of the two-square-unit triangle. Notice how he confuses the idea of the fractional part of the whole with the concept of square units of area.
8.

Watch the "Finding Areas of Complex Figures" video to see students using tangram pieces to help them determine the areas of complex figures. Note the multiple routes that students took to determine their answers.

Discussion
Use the Discussion Board to address the following:
1.
Reflect on your observations of the students from the three videos. What misconceptions did you observe and what questions could you ask to clarify their misunderstandings or deepen their thinking?
2.
Read the responses in the Discussion Board and contribute to the conversation. If you like, you may review the expectations for contributing to the online discussion.