## Changing Twines: Exploring Area and Perimeter

### Jessica McDonald

#### Description

In this fun, hands-on activity, students actively engage to determine the relationships between area and perimeter measurements! Students are challenged to discover, prove, and write mathematical conjectures.

#### Objectives

The student solves real world or mathematical problems involving perimeter, area, circumference, surface area, and volume and how these are affected by changes in the dimensions of the figures.

#### Materials

-Twine pre-cut to the following whole centimeter lengths: 24 cm, 30 cm, 36 cm, and 40 cm. Cut enough of each for these different lengths to be evenly distributed throughout the class.
-Centimeter graph paper (three sheets per student)
-Activity log (see attached file)
-Evaluation checklist (see attached file)
-Overhead projector or dry-erase/chalk board and appropriate writing utensils
-Pencil and eraser for each student (provided by student)

#### Preparations

1. Cut twine to specified whole centimeter lengths (24 cm, 30 cm, 36 cm, and 40 cm). Cut one piece of twine per student. Cut equal amounts of the four different lengths to ensure that students have one of each length to compare in their cooperative groups.
2. Copy activity logs (one per student).
3. Copy evaluation checklist (one per student).
4. Secure overhead projector or dry-erase board and appropriate writing utensils.
5. Secure centimeter graph paper (three sheets per student).

#### Procedures

1. This lesson addresses only the effect of changing dimensions on area and perimeter of quadrilaterals. The activity requires prior knowledge of area and perimeter formulas for quadrilaterals. Write the formulas on the overhead, and briefly review using these formulas correctly.

2. Introduce this lesson by asking students to form a hypothesis about the relationship between area and perimeter of the same quadrilateral. Provide examples of several quadrilaterals on the overhead to aid the discussion and to elicit responses from students. Facilitate a brief discussion and list student hypotheses on the overhead. Conclude the discussion by introducing the activity as an investigation and review cooperative group procedures.

3. Hand out three sheets of centimeter graph paper to each student. Hand out the pre-cut twine randomly around the room. Be sure that neighboring students do not have the same length.

4. Students arrange the twine on the centimeter graph paper to create at least four non-congruent quadrilaterals. (Students may explore options other than squares and rectangles). Each quadrilateral’s perimeter must equal the entire length of the twine so that each quadrilateral has the same perimeter. Students remove the twine and record each quadrilateral on the graph paper.

5. Hand out activity logs. Students record the dimensions of each quadrilateral on the activity log. Then, students calculate and record the perimeter and area of each quadrilateral on the activity log. During this time, walk around and check the accuracy of calculations. Guide struggling students toward correct answers. After students have completed their calculations on the activity log, they trade logs with a partner and check each other’s logs for accuracy given the recorded dimensions. Students discuss corrections with partners and record accurate measurements.

6. Ask students to consider the results of the area calculations on their logs. Students identify the quadrilateral with the maximum area by writing “MAX” next to the area column. Students identify the quadrilateral with the minimum area by writing “MIN” next to the area column.

7. Assign students to form cooperative groups of four or less students. Each group has only one student with each string length. Remind students that the purpose of the activity is to investigate the hypotheses on the overhead. Read the two questions at the bottom of the activity log. Instruct students to review and discuss each other’s activity logs and graph paper as a group. Their discussions should compare and contrast the quadrilaterals and resulting perimeters and areas from each twine length.

8. In their groups, students decide the effect of changing dimensions on perimeter. Then, students decide the effect of changing dimensions on area. Students DO NOT record these decisions on the activity log.

9. Finally, students return to their desks to work individually. Students form conjectures by responding to the questions posed at the bottom of the activity log. Students submit the completed activity log, including conjectures, to be evaluated by the teacher.

#### Assessments

This lesson addresses only mathematical problems involving area, perimeter, and the effect of changing dimensions on quadrilaterals.

CRITERIA: Students find and record the dimensions of four quadrilaterals. Next, students calculate and record the area and perimeter of each quadrilateral. Then, students compare results of changing dimensions on area and perimeter. Finally, students write conjectures about the effect of changing dimensions on the area and perimeter.

EVIDENCE: Students complete the activity log. (See associated file.)

#### Extensions

This activity can be adapted to any polygon (triangles, pentagons, etc.). This may also require students to use the area and perimeter formulas provided on the FCAT Mathematics Reference Sheet.

#### Attached Files

This file contains the activity log used for recording student data.     File Extension:  pdf

This file contains the checklist used to evaluate the written conjectures.     File Extension:  pdf